Middle School Math 7: Proportions, Equations & Probability

Four degrees below zero, then forty-five above

On the morning of 22 January 1943, the recording thermometer at the Montana-Dakota Utilities office in Spearfish, South Dakota read -4°F, four degrees below zero. At about half past seven a warm wind called a Chinook rolled down off the Black Hills. Two minutes later the same thermometer read 45°F. Later that morning the reading reached 54°F, and then the cold air came back so fast that in 27 minutes the temperature plunged from 54°F all the way to -4°F again. Newspapers across the country ran the story, and it turned up in a Ripley's Believe It or Not cartoon.

Every reading in that story is the answer to an addition. Start at -4 and rise 49 degrees: where do you land? Start at 54 and fall 58 degrees: where do you land? You already know negative numbers from sixth grade, where you placed them on a number line and compared them. Now you will add with them, and the same method works whether the numbers are temperatures, heights above and below sea level, money in an account, or yards gained and lost on a football field.

This lesson covers the Common Core standards 7.NS.A.1 parts a, b and d (adding rational numbers on a number line, opposites that make zero, and using the properties of addition) and 7.NS.A.3 (real problems with rational numbers). A rational number is any number you can write as a fraction of two integers, so it includes -4, 45, 3/4, -2.5 and every other number in this lesson.

A sum is a place to start and a move to make

Read p + q as an instruction in two parts. The first number, p, tells you where to stand on the number line. The second number, q, tells you which way to move and how far. If q is positive you move up, or to the right on a line drawn sideways. If q is negative you move down, or to the left. How far you move is the distance of q from zero, which is its absolute value, written |q|. So |49| = 49 and |-58| = 58: the absolute value throws away the direction and keeps only the size.

A thermometer is a number line standing on its end, so here is the Spearfish morning drawn that way. Each arrow is one move.

The first arrow is the sum -4 + 49. The second is 45 + 9. The third is 54 + (-58). The brackets around -58 are there so the plus sign of the addition and the minus sign of the number do not run into each other; read it as "54 plus negative 58."

Key idea: In p + q, the first number is where you stand and the second is the move. A positive move goes up or right, a negative move goes down or left, and the length of the move is the absolute value of the second number.

Two moves from the Spearfish morning, worked

Worked example 1: -4 + 49. The temperature was -4°F and rose 49 degrees.

Step 1. Stand at -4, four steps below zero.

Step 2. The move is +49, so you go up 49.

Step 3. The first 4 degrees of the rise only bring you back to zero. That uses up 4 of the 49, leaving 49 - 4 = 45 degrees still to climb.

Step 4. Climb the remaining 45 from zero. You land on 45.

So -4 + 49 = 45. Check it against the story: the thermometer read 45°F two minutes after the Chinook arrived. It matches.

Worked example 2: 54 + (-58). The temperature was 54°F and fell 58 degrees.

Step 1. Stand at 54.

Step 2. The move is -58, so you go down 58.

Step 3. The first 54 degrees of the fall bring you down to zero. That uses up 54 of the 58, leaving 58 - 54 = 4 degrees still to fall.

Step 4. Fall the remaining 4 below zero. You land on -4.

So 54 + (-58) = -4, which is exactly where the thermometer ended. Notice that in both examples the real work was a subtraction of distances, 49 - 4 in the first and 58 - 54 in the second, because part of each move was spent getting back to zero.

The rule the number line gives you

You do not have to draw a picture every time. The picture gives a rule, and there are only two cases: the numbers have the same sign, or they have different signs.

When the signs are the same, both moves go the same way, so the distances pile up: add them and keep the shared sign. When the signs are different, the moves pull against each other, so the distances partly cancel: subtract the smaller distance from the larger, and the answer takes the sign of whichever number is farther from zero, because that one wins the tug of war.

Build one habit before any arithmetic: predict the sign first. Look at -38 + 25. The signs differ, and -38 is farther from zero than 25, so the answer must be negative. Then do the distances: 38 - 25 = 13. The answer is -13. If your calculation ever gives a sign that disagrees with your prediction, one of them is wrong, and you have caught the mistake before anyone else did.

Remember: Same signs, add the distances and keep the sign. Different signs, subtract the distances and take the sign of the number farther from zero.

Opposites cancel: why the thermometer ended where it began

Add up the whole Spearfish morning as moves. It rose 49, rose 9 more, and then fell 58.

49 + 9 + (-58) = 58 + (-58) = 0.

The total change was zero, which is why the thermometer finished at -4°F, the same reading it started from. A rise of 58 and a fall of 58 are opposites: the same distance from zero on different sides. A number and its opposite always add to zero, and for that reason they are also called additive inverses. So 58 + (-58) = 0, -2.5 + 2.5 = 0, and 3/4 + (-3/4) = 0.

The Common Core standard for this lesson gives an example from science: a hydrogen atom has a total charge of 0 because it is made of one proton with a charge of +1 and one electron with a charge of -1, and +1 + (-1) = 0. You can find the same thing in everyday life. Earn $15 mowing a lawn and spend $15 at the cinema, and your money is back where it started. Gain 6 yards on one play and lose 6 on the next, and the ball is back on the same line.

Opposites also explain the step you kept using in the worked examples. To find -4 + 49, you split the 49 into 4 + 45:

-4 + 49 = -4 + 4 + 45 = 0 + 45 = 45.

The -4 and the 4 are opposites, so they cancel, and what is left over is the answer. Getting back to zero first is not a trick; it is the opposites rule at work.

In short: A number and its opposite add to zero. When you add numbers with different signs, the part that cancels disappears, and the leftover is the answer.

Fractions and decimals move the same way

Nothing in the rule cares whether the numbers are whole. A move of 4.75 is a move of 4.75, and a move of 3/4 is a move of 3/4. Only the arithmetic of the distances changes.

Worked example 3: a diver, with decimals. A scuba diver is 12.5 metres below the surface, which is -12.5 m. She rises 4.75 m. Where is she now?

Step 1. Write the sum: -12.5 + 4.75.

Step 2. Predict the sign. The signs differ, and -12.5 is farther from zero than 4.75, so the answer is negative. She is still under water, which makes sense: she has not risen far enough to reach the surface.

Step 3. Subtract the distances, lining up the decimal points: 12.50 - 4.75 = 7.75.

Step 4. Attach the predicted sign: -12.5 + 4.75 = -7.75. She is 7.75 m below the surface.

Check by undoing the move. From -7.75, sink 4.75 m again: -7.75 + (-4.75). Same signs, so add the distances, 7.75 + 4.75 = 12.5, and keep the negative sign: -12.5. That is where she started, so the answer is right.

Worked example 4: a tide mark, with fractions. At low tide the water at a harbour wall is 3/4 of a foot below a painted mark. Over the next hour it rises 1 1/2 feet. Where is the water now, compared with the mark?

Step 1. Write the sum: -3/4 + 1 1/2.

Step 2. Give both fractions the same denominator. 1 1/2 = 3/2 = 6/4. The sum is -3/4 + 6/4.

Step 3. Predict the sign. The signs differ, and 6/4 is farther from zero than 3/4, so the answer is positive.

Step 4. Subtract the distances: 6/4 - 3/4 = 3/4.

So -3/4 + 1 1/2 = 3/4. The water is 3/4 of a foot above the mark.

Check on a number line marked in quarters. Start at -3/4 and take six quarter-steps up: -2/4, -1/4, 0, 1/4, 2/4, 3/4. You land on 3/4.

Adding a long list: reorder, then regroup

Addition has two properties that make long sums easy. The commutative property says you can add in any order: 3 + (-8) = (-8) + 3. The associative property says you can group the numbers any way you like: (3 + 5) + (-8) = 3 + (5 + (-8)). Together they let you rearrange a long sum into the friendliest order before you start.

Worked example 5: a football drive. A team starts on its own 25-yard line. On six plays it gains 7 yards, loses 3, loses 5, gains 12, loses 2 and gains 8. What is the net change, and where does the drive end?

Step 1. Write every play as a signed number: 7 + (-3) + (-5) + 12 + (-2) + 8.

Step 2. Gather the positives: 7 + 12 + 8 = 27.

Step 3. Gather the negatives: (-3) + (-5) + (-2) = -10. Same signs, so the distances add, 3 + 5 + 2 = 10, and the sign stays negative.

Step 4. Combine the two totals: 27 + (-10). Different signs, 27 - 10 = 17, and 27 is farther from zero, so the answer is positive: 17.

Step 5. The team gained 17 yards overall, so the drive ends on its own 25 + 17 = 42-yard line.

Check by adding left to right instead, keeping a running total after each play: 7, then 7 + (-3) = 4, then 4 + (-5) = -1, then -1 + 12 = 11, then 11 + (-2) = 9, then 9 + 8 = 17. Both methods give 17. The running total went negative after the third play, which is exactly the kind of place where adding left to right invites a slip, and it is why gathering positives and negatives first is usually safer.

Worked example 6: a lunch account. Many school cafeterias let a student's lunch account dip below zero. Yours shows -$3.75. You deposit $20.00, then buy two lunches at $4.25 each. What is the balance now?

Step 1. Write the sum: -3.75 + 20.00 + (-4.25) + (-4.25).

Step 2. Look for friendly pairs. -3.75 and -4.25 have the same sign, and their distances make a whole number: 3.75 + 4.25 = 8.00. So -3.75 + (-4.25) = -8.00.

Step 3. Add the other lunch: -8.00 + (-4.25) = -12.25.

Step 4. Add the deposit: 20.00 + (-12.25). Different signs, 20.00 - 12.25 = 7.75, positive because 20.00 is farther from zero.

The balance is $7.75. Check: you started $3.75 in the hole and spent $8.50 on lunches, so you needed $3.75 + $8.50 = $12.25 of your deposit, and $20.00 - $12.25 = $7.75 is left. Same answer.

Worth holding on to: In a long sum you may reorder and regroup freely. Gather the positives, gather the negatives, look for opposites or friendly pairs, and combine at the end.

Common misconceptions

"Two negatives make a positive, so -5 + (-3) = 8." That rule belongs to multiplication, which is two lessons away, and even there it needs explaining. In addition, two negative moves both go down: start at -5 and go down 3 more, and you reach -8. If you owe a friend $5 and then borrow $3 more, you owe $8, not own $8.

"The answer takes the sign of the first number." The Spearfish sum -4 + 49 starts with a negative and ends at a positive 45. The sign comes from whichever number is farther from zero, wherever it sits in the sum.

"When the signs are different, I add the distances." This gives -4 + 49 = 53 or -53. But the first 4 degrees of the rise only undid the 4 degrees below zero. When signs differ, part of each number cancels, so you subtract the distances.

"Adding always makes a number bigger." Adding a positive number moves you up. Adding a negative number moves you down, so 54 + (-58) is less than 54. Addition means combining, not growing.

Where this leaves us

• Read p + q as a start and a move: stand at p, then move |q| up for a positive q or down for a negative q.
• Same signs: add the distances and keep the sign. Different signs: subtract the distances and take the sign of the number farther from zero.
• Predict the sign before you calculate, so a wrong sign cannot slip through.
• A number and its opposite are additive inverses and add to zero, which is why a rise of 58 and a fall of 58 left Spearfish exactly where it began.
• Fractions and decimals follow the same rule; only the arithmetic of the distances changes.
• In long sums, reorder and regroup: positives together, negatives together, opposites cancelled.

Subtraction comes next, and it turns out not to need a new rule at all: every subtraction can be rewritten as one of the additions you just practised.

Sources

1. National Weather Service, Rapid City Forecast Office. (n.d.). The Black Hills remarkable temperature change of January 22, 1943. National Oceanic and Atmospheric Administration. weather.gov
2. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, The Number System (7.NS.A.1, 7.NS.A.3). thecorestandards.org
3. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 3.2, Add integers. Prealgebra 2e. OpenStax. openstax.org
4. Wikipedia contributors. (n.d.). Additive inverse. Wikipedia. en.wikipedia.org

A wrong answer from Death Valley

Badwater Basin, a salt flat in Death Valley National Park, California, lies 282 feet below sea level, the lowest point in North America. West of it, in the Sierra Nevada, the summit of Mount Whitney stands 14,505 feet above sea level, according to the National Park Service. How much higher is the summit than the salt flat?

Here is an answer a lot of careful students give:

14,505 - 282 = 14,223 feet.

It looks reasonable. It uses both numbers, the subtraction is done correctly, and 14,223 is in the right general area. It is still wrong, and wrong by more than five hundred feet. This lesson starts by finding exactly where the reasoning slips, because the fix is the rule that makes every subtraction of signed numbers work.

The lesson covers Common Core standards 7.NS.A.1 parts c and d (subtraction as adding the additive inverse, distance as the absolute value of a difference, and using the properties of operations) and 7.NS.A.3 (real problems with rational numbers).

Tracing the mistake on a number line

Stand an elevation number line on its end, with sea level at 0. Badwater Basin is below the zero mark at -282. The summit of Whitney is far above it at 14,505.

Now climb from the salt flat to the summit in two stages.

Stage 1. From -282 up to sea level at 0 is a climb of 282 feet.

Stage 2. From sea level up to 14,505 is a climb of 14,505 feet.

The whole climb is 282 + 14,505 = 14,787 feet.

So the answer 14,223 is 564 feet short, and 564 is exactly 282 twice. The student's subtraction, 14,505 - 282, measures from a point 282 feet above sea level up to the summit. It treats Badwater as if it were on a hill. That throws away the 282 feet below sea level that you have to climb just to reach zero, and then it subtracts another 282 on top. The mistake is not in the arithmetic. It is in the sign of the starting number, which is -282, not 282.

The correct subtraction keeps the sign: summit minus salt flat is 14,505 - (-282). The number line says that equals 14,787, which is 14,505 + 282. Subtracting -282 gave the same result as adding 282. That is not a coincidence, and the rest of this lesson shows why it always happens.

The point: A subtraction between two readings measures the move from one to the other. When one reading is below zero, the move has to cross zero, and the part below zero adds to the distance instead of taking away from it.

What subtraction really asks

You met subtraction years ago as taking away: 9 - 4 is what is left when 4 is removed from 9. That picture stops working as soon as the numbers can be negative, because you cannot remove -282 apples from a basket. There is a second meaning of subtraction that keeps working everywhere.

The difference p - q answers the question: what do I add to q to get p? Check it on something familiar. 9 - 4 = 5 because 4 + 5 = 9.

Now ask the Death Valley question that way. 14,505 - (-282) asks: what do I add to -282 to get 14,505? Starting at -282 and landing on 14,505 is an upward move of 282 to reach zero and then 14,505 more, which is 14,787. So 14,505 - (-282) = 14,787, and you can confirm it with an addition from the last lesson: -282 + 14,787 = 14,505.

That same question gives a second habit: a change is always end minus start. Whatever you add to the start to reach the end is the change. If the answer is positive, the reading went up. If it is negative, the reading went down.

The rule: subtracting is adding the opposite

Watch what happens to 5 - q as q goes down by one each row.

Each time you subtract one less, one more is left over, so the answers climb by one per row. The first four rows are ordinary arithmetic. The pattern then forces the last three: 5 - (-1) has to be 6, 5 - (-2) has to be 7, and 5 - (-3) has to be 8. And in every row, the subtraction gives the same answer as adding the opposite of the second number.

That is the rule, and the Common Core standard for this lesson writes it exactly this way:

p - q = p + (-q).

To subtract a number, add its opposite. The first number stays as it is. The subtraction becomes an addition. The second number changes sign. Some teachers call this keep, change, change. The name is fine as long as you remember which things change: the operation and the second number, never the first number.

Money gives the same rule a meaning you can feel. Suppose your bank account shows a balance that includes a -$6.50 charge for a fee the bank should not have taken. When the bank removes that charge, it is subtracting -6.50 from your balance, and your balance goes up by $6.50. Taking away a debt is the same as being given money.

Why this matters: Once every subtraction is rewritten as an addition, you only need the addition rules from the last lesson. There is nothing new to memorise.

Worked examples: a record cold snap, a bank fee and a lake

Worked example 1: the Death Valley climb, with the rule.

Step 1. Summit minus salt flat: 14,505 - (-282).

Step 2. Add the opposite: 14,505 + 282.

Step 3. Same signs, so add: 14,787 feet.

Check: start at -282 and add 14,787. Signs differ, 14,787 - 282 = 14,505, and the larger distance is positive. You land on the summit.

Worked example 2: a change that goes down. On 23 January 1916 the temperature in Browning, Montana was 44°F. By the next day it was -56°F, a drop that is still quoted as a world record for 24 hours. What was the change?

Step 1. Change is end minus start: -56 - 44.

Step 2. Add the opposite: -56 + (-44).

Step 3. Same signs, so add the distances, 56 + 44 = 100, and keep the negative sign: -100.

The temperature changed by -100 degrees, a fall of 100 degrees. Check: 44 + (-100) = -56, the reading at the end. A common wrong answer here is -12, which comes from treating -56 - 44 as -56 + 44. The subtraction sign and the sign of 44 have to be dealt with together, which is what rewriting as -56 + (-44) does for you.

Worked example 3: removing a bank fee. An account balance is -$18.25. That balance includes a $6.50 overdraft fee, recorded as -6.50, which the bank agrees to remove. What is the new balance?

Step 1. Removing the fee subtracts it: -18.25 - (-6.50).

Step 2. Add the opposite: -18.25 + 6.50.

Step 3. Signs differ. Subtract the distances, 18.25 - 6.50 = 11.75, and -18.25 is farther from zero, so the answer is negative: -11.75.

The new balance is -$11.75. The account is still overdrawn, but by $6.50 less. Check: if the fee were charged again, -11.75 + (-6.50) = -18.25, which is where you started.

Worked example 4: a lake level, with fractions. After a dry spring, a reservoir sits 2 1/4 feet below its normal level. During June it drops another 1 1/2 feet. Where is it now, compared with normal?

Step 1. Start at -2 1/4 and take away 1 1/2: -2 1/4 - 1 1/2.

Step 2. Add the opposite: -2 1/4 + (-1 1/2).

Step 3. Write both with quarters: -2 1/4 + (-1 2/4).

Step 4. Same signs, so add the distances, 2 1/4 + 1 2/4 = 3 3/4, and keep the negative sign: -3 3/4.

The reservoir is 3 3/4 feet below normal. The answer is more negative than the start, which is what a drop should do. In decimals, -2.25 - 1.5 = -3.75, the same number.

Distance: the absolute value of a difference

Sometimes you do not care which way the change went, only how far apart two numbers are. The distance between two numbers on the number line is the absolute value of their difference, and it does not matter which one you subtract from which.

Worked example 5: the distance between -3.5 and 2.

Step 1. Subtract in either order. 2 - (-3.5) = 2 + 3.5 = 5.5. The other order is -3.5 - 2 = -3.5 + (-2) = -5.5.

Step 2. Take the absolute value: |5.5| = 5.5 and |-5.5| = 5.5.

The two numbers are 5.5 units apart, whichever way you subtract. Count it on the diagram: from -3.5 to 0 is 3.5 units, and from 0 to 2 is 2 more, for 5.5 in all.

Worked example 6: how far apart are two elevations? A hiker's watch reads 1,250 m at the top of a pass and 380 m in the valley below. How far apart are the two points vertically, and what was the change for someone walking down?

Step 1. The change walking down is end minus start: 380 - 1,250 = 380 + (-1,250) = -870. The walker's elevation changed by -870 m.

Step 2. The vertical distance is the absolute value: |-870| = 870 m.

Both answers are right. They answer different questions.

Change or distance? Order decides

Subtraction does not let you swap the order: 3 - 8 = -5 but 8 - 3 = 5. For a change, the order carries real information, so always put the end first. For a distance, the absolute value removes the sign, so the order stops mattering.

Bottom line: Change is end minus start and keeps its sign. Distance is the absolute value of the difference and is never negative.

Common misconceptions

"Subtracting always makes a number smaller." Only when you subtract a positive number. 5 - (-3) = 8 is larger than 5, just as removing a debt leaves you better off.

"-56 - 44 is -12." This comes from reading the subtraction sign as if it belonged to nothing and then using the different-signs rule on -56 and 44. Rewrite first: -56 - 44 = -56 + (-44), which has the same signs, so the answer is -100.

"Keep, change, change means change the signs of everything." The first number never changes. In -5 - 3, keep the -5, change subtraction to addition, change 3 to -3: -5 + (-3) = -8. Writing 5 + 3 = 8 changes the first number too, which gives a different problem.

"It doesn't matter which number I subtract from which." It matters for a change, because 3 - 8 and 8 - 3 have opposite signs. It stops mattering only when you take the absolute value to find a distance.

What to carry forward

• The difference p - q is the number you add to q to get p, so a change is always end minus start.
• Every subtraction can be rewritten as addition of the opposite: p - q = p + (-q). After that, the addition rules do all the work.
• Subtracting a negative number gives a bigger result: 14,505 - (-282) = 14,787.
• The distance between two numbers is the absolute value of their difference, and it is the same whichever order you subtract in.
• When an answer seems off, sketch the two numbers on a number line and count across zero in two stages.

With addition and subtraction settled, the next lesson takes on the rule that causes the most arguments in seventh grade: why a negative number times a negative number comes out positive.

Sources

1. National Park Service. (n.d.). Seeing and climbing Mt. Whitney. Sequoia and Kings Canyon National Parks. nps.gov
2. Wikipedia contributors. (n.d.). Badwater Basin. Wikipedia. en.wikipedia.org
3. Wikipedia contributors. (n.d.). Browning, Montana: Climate. Wikipedia. en.wikipedia.org
4. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, The Number System (7.NS.A.1c). thecorestandards.org
5. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 3.3, Subtract integers. Prealgebra 2e. OpenStax. openstax.org

Where was the water four minutes ago?

A rain barrel under a downspout has sprung a leak, and it is losing 3 litres of water every minute. Right now the gauge on its side reads 60 litres. Here is the question this lesson is built on: what did the gauge read four minutes ago?

Think about it before you calculate. The barrel is losing water, so four minutes ago it had more water than it has now. The answer has to be bigger than 60. Each minute costs 3 litres, and four minutes cost 12, so four minutes ago the barrel held 60 + 12 = 72 litres.

You did not need any rules to answer that. But now write the same reasoning with signed numbers, the way a scientist would. A loss of 3 litres per minute is a rate of -3 litres per minute. Four minutes ago is a time of -4 minutes, since the past is before now, the way temperatures below zero are before zero on a thermometer. The change in the water level is rate times time:

change = (-3) × (-4).

You already know the change had to be +12, because the barrel was fuller in the past. So (-3) × (-4) = 12. A negative times a negative came out positive, and not because someone announced a rule. The situation demanded it.

This lesson covers Common Core standards 7.NS.A.2 parts a and c (multiplying rational numbers, with the rules for signs coming from the properties of operations, especially the distributive property) and 7.NS.A.3 (real problems with rational numbers).

First, a positive times a negative

Start with the easier case. Multiplying by a positive whole number means repeated addition, and that still works when the thing being repeated is negative.

Suppose $3 is taken from your account four times. That is 4 × (-3) = (-3) + (-3) + (-3) + (-3) = -12. Four withdrawals of $3 change your balance by -$12. On a number line, it is four jumps of 3 to the left, starting from zero.

What about (-3) × 4, with the negative number first? Multiplication can be done in either order, so (-3) × 4 = 4 × (-3) = -12 as well. In the barrel's language: losing 3 litres a minute, four minutes from now, the level will have changed by (-3) × 4 = -12 litres, so the gauge will read 48.

What matters here: A positive times a negative, in either order, is negative. Repeating a loss gives a bigger loss.

Second way to see it: follow the pattern

Here is (-3) multiplied by a number that goes down by one each row. Every row above the line uses only the rule you just saw.

Each time the second number drops by 1, you are taking away one fewer copy of -3, so the answer rises by 3. The pattern reaches 0 at (-3) × 0 and has nowhere to go but up. For the pattern to continue, (-3) × (-1) must be 3, and (-3) × (-4) must be 12, the same answer the rain barrel gave.

A pattern is persuasive, but it is not yet a reason. Someone could say the pattern just happens to break at zero. The third way closes that door.

Third way, the real reason: the distributive property

The distributive property is the rule you use when you work out 7 × 12 as 7 × 10 + 7 × 2 = 70 + 14 = 84. In general, a × (b + c) = a × b + a × c. You have trusted it for years, and every multiplication method you know depends on it. The Common Core standard says the sign rules come from insisting that it keeps working when the numbers are negative. Here is how.

Step 1. Anything times zero is zero, so (-3) × 0 = 0.

Step 2. Zero can be written as 4 + (-4), since opposites add to zero. So (-3) × (4 + (-4)) = 0.

Step 3. Use the distributive property on the left side: (-3) × 4 + (-3) × (-4) = 0.

Step 4. You already know (-3) × 4 = -12. So -12 + (-3) × (-4) = 0.

Step 5. The only number that adds to -12 to make 0 is its opposite, 12. So (-3) × (-4) = 12.

There was no choice at any step. If a negative times a negative were anything other than positive, the distributive property would fail, and so would the ordinary arithmetic built on it. The same argument with 1 in place of 3 and 4 gives the famous fact (-1) × (-1) = 1.

The upshot: A negative times a negative is positive because that is the only answer that keeps the distributive property true. The barrel story and the pattern agree with it, which is why they were worth trusting.

The sign rules, and products of many numbers

The size of the product is always the product of the sizes, so the only question is the sign. When there are more than two factors, multiply two at a time, and each pair of negatives turns positive. That gives a shortcut: count the negative factors. An even number of negatives gives a positive product, and an odd number gives a negative product. A factor of zero anywhere makes the product zero, whatever the signs.

Fractions and decimals follow exactly the same sign rules. Multiply the sizes the way you always have, top times top and bottom times bottom for fractions, then attach the sign.

Worked examples

Worked example 1: the rain barrel, checked. Level now: 60 litres. Rate: -3 litres per minute. Time: -4 minutes.

Step 1. Change = rate × time = (-3) × (-4) = 12.

Step 2. Level four minutes ago = 60 + 12 = 72 litres.

Check by running time forwards from then to now: 72 + 4 × (-3) = 72 + (-12) = 60. The gauge reads 60 now, as it should.

Worked example 2: a phone plan, forwards and backwards. A phone plan takes $15 from an account every week.

Step 1. Six weeks from now the balance will have changed by 6 × (-15) = -90, that is, $90 less.

Step 2. Six weeks ago is a time of -6 weeks, so the change from now back to then is (-6) × (-15) = 90. Six weeks ago the balance was $90 more than it is today.

That second answer makes sense: the plan has been draining the account, so it had more money in it before.

Worked example 3: a falling temperature, with a decimal. All evening the temperature has been falling 2.5°F per hour. It is 41°F now. What was it three hours ago?

Step 1. Rate: -2.5 degrees per hour. Time: -3 hours.

Step 2. Change = (-2.5) × (-3). Two negatives, so the product is positive. Sizes: 2.5 × 3 = 7.5. So the change is 7.5.

Step 3. Temperature three hours ago = 41 + 7.5 = 48.5°F.

Check: from 48.5, fall 2.5 degrees for 3 hours: 48.5 + 3 × (-2.5) = 48.5 + (-7.5) = 41.

Worked example 4: fractions. A weather balloon's height is changing by -3/4 metre per second as it sinks. What is its change in height over the next 2/5 of a second?

Step 1. Change = (-3/4) × (2/5). One negative factor, so the product is negative.

Step 2. Sizes: (3 × 2)/(4 × 5) = 6/20 = 3/10.

Step 3. The change is -3/10 of a metre. The balloon drops 0.3 m in that time.

Worked example 5: many factors. Find (-2) × (-5) × (-3) × 4.

Step 1. Count the negatives: -2, -5 and -3. That is three, an odd number, so the product is negative.

Step 2. Multiply the sizes: 2 × 5 = 10, then 10 × 3 = 30, then 30 × 4 = 120.

Step 3. The product is -120.

Check by pairs: (-2) × (-5) = 10, then 10 × (-3) = -30, then -30 × 4 = -120.

Worked example 6: let the properties choose the order. Find (-25) × 7 × (-4).

Step 1. Multiplication can be reordered and regrouped, so pair the numbers that make a friendly product: (-25) × (-4) = 100.

Step 2. Then 100 × 7 = 700.

Two negatives, so the answer is positive, and the whole calculation fits in your head.

So what?: The sign of a product depends only on how many factors are negative. Settle the sign first, then multiply the sizes in whatever order is easiest.

Common misconceptions

"Two negatives make a positive, so -3 + (-4) = 7." The saying is only about multiplication and division. In addition, two negatives make a bigger negative: -3 + (-4) = -7. If you owe $3 and borrow $4 more, you do not suddenly have $7.

"The sign of the product is the sign of the bigger number." That idea belongs to adding numbers with different signs. For multiplication, size has nothing to do with sign: (-2) × 8 = -16 and 2 × (-8) = -16.

"-3² is 9." The exponent is applied before the negative sign, so -3² means -(3 × 3) = -9. To square negative three you need brackets: (-3)² = (-3) × (-3) = 9.

"Multiplying always makes a number bigger." Multiplying by a negative flips the number to the other side of zero, and multiplying by a fraction between 0 and 1 shrinks it: 5 × (-2) = -10 and 8 × 1/2 = 4.

Summing up

• A positive times a negative is negative, because repeating a loss makes a bigger loss.
• A negative times a negative is positive. A situation run backwards in time shows it, the pattern of products shows it, and the distributive property proves it: any other answer would break a × (b + c) = a × b + a × c.
• For many factors, count the negatives: even gives a positive product, odd gives a negative one, and any zero factor gives zero.
• Fractions and decimals follow the same sign rules as integers.
• Reorder and regroup factors to make the arithmetic easy.

Division is the other half of the story. Because every division can be undone by a multiplication, the same sign rules will carry straight over, and the next lesson also shows why some fractions become decimals that stop while others repeat forever.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, The Number System (7.NS.A.2a). thecorestandards.org
2. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 3.4, Multiply and divide integers. Prealgebra 2e. OpenStax. openstax.org
3. Wikipedia contributors. (n.d.). Negative number: Multiplication. Wikipedia. en.wikipedia.org
4. Wikipedia contributors. (n.d.). Distributive property. Wikipedia. en.wikipedia.org

Ten dollars, three friends

Three friends order a $10.00 pizza and split the bill evenly. Type 10 ÷ 3 into a calculator and it shows 3.3333333, as many threes as the screen can hold. Nobody can pay a third of a cent, so two friends pay $3.33, one pays $3.34, and the bill is covered. Now suppose eight friends split a $10.00 order of garlic bread instead. The calculator shows 1.25, and it stops. Each pays exactly $1.25.

Why does one division stop and the other run on forever? And what happens to division when the numbers are negative, as they are when a temperature falls or a debt is shared out? This lesson answers both questions with one tool you already own, long division, and one fact from the last lesson: division undoes multiplication.

It covers Common Core standards 7.NS.A.2 parts b, c and d (dividing rational numbers, the positions of the negative sign, and converting a rational number to a decimal by long division) and 7.NS.A.3 (real problems with rational numbers).

Division undoes multiplication, so the signs come along

A division is a multiplication with a missing factor. 12 ÷ 3 = 4 because 3 × 4 = 12. The quotient is the number that the divisor has to be multiplied by to give the number you started with. Keep that meaning, and the sign rules for division come straight from the ones you learned for multiplication.

Read down the last column. When the two numbers have the same sign, the quotient is positive. When they have different signs, the quotient is negative. Those are exactly the multiplication rules, and they have to be, because each division is a multiplication read backwards.

Key idea: Same signs give a positive quotient, different signs give a negative one. To check any quotient, multiply it by the divisor and see whether you get back the number you divided.

Where the negative sign can sit, and the one division you cannot do

A fraction bar is a division sign, so a negative fraction can be written three ways that all mean the same number:

-(3/4) = (-3)/4 = 3/(-4) = -0.75.

Check the last two with the division rule: (-3) ÷ 4 has different signs, so it is negative, and so is 3 ÷ (-4). Each equals -0.75. The Common Core standard states this in general for integers p and q as -(p/q) = (-p)/q = p/(-q). What you cannot do is put a negative sign on both top and bottom and call it the same: (-3)/(-4) has the same sign on top and bottom, so it is positive 3/4.

There is one division with no answer at all. What is 12 ÷ 0? It would have to be a number that gives 12 when multiplied by 0. But every number times 0 is 0, so no such number exists. Division by zero is undefined, which is why the standard insists on a divisor that is not zero. The other way round is fine: 0 ÷ 12 = 0, because 12 × 0 = 0.

Worked examples: quotients that mean something

Worked example 1: a steady drop. Over 6 hours one winter night, the temperature fell steadily from 14°F to -4°F. What was the change per hour?

Step 1. Total change is end minus start: -4 - 14 = -4 + (-14) = -18 degrees.

Step 2. Change per hour = -18 ÷ 6. Different signs, so negative. 18 ÷ 6 = 3. The quotient is -3.

Step 3. The temperature changed by -3 degrees per hour, a fall of 3 degrees every hour.

Check: 6 × (-3) = -18, the total change. The negative sign in the answer carries meaning: it says falling, not rising.

Worked example 2: two negative decimals. Find (-4.5) ÷ (-0.9).

Step 1. Same signs, so the quotient is positive.

Step 2. Make the divisor a whole number by multiplying both numbers by 10: 4.5 ÷ 0.9 = 45 ÷ 9 = 5.

Step 3. The quotient is 5. Check: (-0.9) × 5 = -4.5.

A real meaning for it: a debt of $4.50 shared out as -$0.90 per person covers 5 people.

Worked example 3: a fraction divided by a fraction. A pond's water level falls 3/4 of an inch in 1/2 hour. What is the change per hour?

Step 1. Change per hour = (-3/4) ÷ (1/2).

Step 2. Dividing by a fraction is multiplying by its reciprocal, the fraction flipped: (-3/4) × (2/1).

Step 3. One negative factor, so negative. Sizes: 6/4 = 3/2.

Step 4. The level changes by -3/2 = -1 1/2 inches per hour. That makes sense: if it drops 3/4 inch every half hour, it drops twice that in a full hour.

Long division, end to end: 3/8

Every fraction is a division, top divided by bottom, and long division turns it into a decimal. Here is 3/8 worked in full.

Step 1. 8 does not go into 3, so write 0 and a decimal point, and treat 3 as 3.000 with as many zeros as you need.

Step 2. Bring down a zero to make 30. 8 goes into 30 three times, since 8 × 3 = 24, with remainder 30 - 24 = 6. The first decimal digit is 3.

Step 3. Bring down a zero to make 60. 8 goes into 60 seven times, 8 × 7 = 56, remainder 4. The next digit is 7.

Step 4. Bring down a zero to make 40. 8 goes into 40 five times exactly, remainder 0. The next digit is 5.

Step 5. The remainder is 0, so every later step would be 0 ÷ 8 = 0. The division is finished: 3/8 = 0.375.

Check: 0.375 × 8 = 3.000. A decimal like this, which stops, is called a terminating decimal. The garlic bread was the same kind: 10 ÷ 8 = 1.25.

Change one number: 3/11 never stops

Now keep the top the same and change the bottom from 8 to 11.

Step 1. Write 0 and a decimal point. Bring down a zero to make 30. 11 goes into 30 twice, 11 × 2 = 22, remainder 8. First digit: 2.

Step 2. Bring down a zero to make 80. 11 goes into 80 seven times, 11 × 7 = 77, remainder 3. Next digit: 7.

Step 3. Bring down a zero to make 30. But that is exactly where Step 1 began.

Once a remainder comes back, every step after it is a replay. The digits 2 and 7 will alternate forever: 3/11 = 0.272727... A decimal like this is a repeating decimal. The block that repeats is written once with a bar over it in most textbooks; here it is shown with dots, 0.2727..., meaning the pattern never ends. The pizza bill was the same kind: 10 ÷ 3 = 3.333..., because the remainder was 1 at every step.

A repeating decimal is an exact value only when it goes on forever. Any stopped version is an approximation. 0.27 is close to 3/11 but not equal to it, and $3.33 is close to a third of $10 but falls short by a third of a cent, which is why one friend had to pay the extra cent.

Remember: Long division stops when a remainder of 0 appears, giving a terminating decimal. When a remainder repeats instead, the digits repeat from that point on forever.

Why every fraction has to stop or repeat

Here is the reason, and it is short enough to hold in your head. When you divide by 7, every remainder is a whole number smaller than 7, so the only possible remainders are 0, 1, 2, 3, 4, 5 and 6. If a 0 ever appears, the decimal stops. If not, there are only six other remainders available, so by the seventh step at the latest one of them must turn up for a second time, and from then on the digits repeat.

Worked example 4: 2/7, the longest repeat a seventh can have.

All six non-zero remainders appear, and then remainder 6 returns at step 7, so the block 285714 repeats: 2/7 = 0.285714285714... It took six digits, the most a denominator of 7 allows.

Worked example 5: a repeat that starts late. 1/6. Step 1: 10 ÷ 6 = 1, remainder 4. Step 2: 40 ÷ 6 = 6, remainder 4. Step 3: 40 ÷ 6 = 6, remainder 4 again. So 1/6 = 0.1666..., where the 1 appears once and only the 6 repeats. A decimal can run for a while before its repeating block begins.

Can you tell in advance which kind you will get? Yes. Write the fraction in lowest terms and look at the bottom number. If its only prime factors are 2s and 5s, the decimal terminates, because a bottom made of 2s and 5s divides evenly into some power of 10. If the bottom has any other prime factor, such as 3, 7 or 11, the decimal repeats. So 3/8 terminates because 8 = 2 × 2 × 2, while 3/11 and 1/6 repeat, because 11 is prime and 6 = 2 × 3 contains a 3.

Negative fractions follow the same pattern with a sign in front: -7/4 = -1.75 terminates, and -2/3 = -0.666... repeats. Either way, the decimal of a fraction is never a random string of digits that goes on without a pattern. That is what the standard means when it says the decimal form of a rational number terminates in 0s or eventually repeats.

In short: A fraction's decimal must end or repeat, because long division has only a limited set of remainders to choose from. Lowest terms with a bottom made only of 2s and 5s means it ends.

Common misconceptions

"A negative divided by a negative is negative." Check with multiplication: if (-12) ÷ (-3) were -4, then (-3) × (-4) would have to be -12, but it is +12. The quotient is 4.

"Anything divided by zero is zero." That is true the other way round: 0 ÷ 5 = 0. But 5 ÷ 0 would need a number that gives 5 when multiplied by 0, and there is none. It is undefined, not zero.

"0.33 is the same as 1/3." It is close, but 3 × 0.33 = 0.99, not 1. Only the never-ending 0.333... equals 1/3 exactly.

"6/24 repeats, because 24 has a 3 in it." Reduce first. 6/24 = 1/4, and 4 = 2 × 2, so the decimal is 0.25 and it terminates. The 2s-and-5s test only works on a fraction in lowest terms.

The short version

• Division undoes multiplication, so the sign rules match: same signs give a positive quotient, different signs a negative one. Check a quotient by multiplying back.
• -(p/q) = (-p)/q = p/(-q), but (-p)/(-q) is positive.
• Division by zero is undefined; zero divided by a nonzero number is zero.
• A quotient in a real problem is usually a rate, and its sign tells you the direction: -3 degrees per hour means falling.
• Long division turns any fraction into a decimal. A remainder of 0 means it terminates; a repeated remainder means it repeats forever.
• In lowest terms, a denominator with no prime factors except 2 and 5 gives a terminating decimal; any other prime factor gives a repeating one.

Next comes practice at choosing the form of a number, fraction, decimal or percent, that makes a real multi-step problem easiest, and at estimating to catch a wrong answer before it leaves your page.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, The Number System (7.NS.A.2b, 7.NS.A.2d). thecorestandards.org
2. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 5.3, Decimals and fractions. Prealgebra 2e. OpenStax. openstax.org
3. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 3.4, Multiply and divide integers. Prealgebra 2e. OpenStax. openstax.org
4. Wikipedia contributors. (n.d.). Repeating decimal. Wikipedia. en.wikipedia.org

A towel bar on a bathroom door

You are fixing a towel bar to the back of a bathroom door. The door is 27 1/2 inches wide and the bar is 9 3/4 inches long, and you want the bar exactly in the middle. How far in from each edge of the door should the first screw hole go?

This exact example appears in the Common Core standards for seventh grade, with a remark attached: you will need to place the bar about 9 inches from each edge, and this estimate can be used as a check on the exact computation. That remark is the heart of this lesson. A real problem usually takes several steps, the numbers arrive in whatever form the world uses, and the answer is only useful if it is right. So you need two skills at once: choosing a form of the numbers that makes the steps easy, and making a rough estimate that tells you whether your careful answer can be trusted.

The lesson covers Common Core standards 7.EE.B.3 (multi-step problems with rational numbers in any form, converting between forms, and checking reasonableness with mental computation and estimation) and 7.NS.A.3 (real problems with the four operations on rational numbers).

The towel bar three ways

Worked example 1: the towel bar. First think about the shape of the problem. The door's width is made of three pieces: a gap on the left, the bar, and a gap on the right. The two gaps are equal. So take the bar's length away from the door's width, and split what is left in half.

gap = (door width - bar length) ÷ 2.

Now compare three ways of carrying that out.

Read the table across. The fraction and decimal routes give the same number, 8 7/8 = 8.875, because 7/8 = 0.875. The estimate, 9, sits just above both, which is exactly what you would expect: rounding 27 1/2 up to 28 and 9 3/4 up to 10 barely changes their difference, so the estimate should be close. If your exact answer had come out as 17 3/4 (you forgot to halve) or 18 5/8 (you added instead of subtracting and then halved), the estimate of 9 would have flagged it instantly.

Which exact route is better? For a tape measure marked in eighths and sixteenths, the fraction answer, 8 7/8 inches, is the one you can actually find on the tape. For a calculator, the decimal route is quicker. Neither is more correct. The skill is noticing which one the situation wants.

Key idea: Before calculating, sketch the problem to find its steps. Then pick the form of the numbers that suits the steps and the answer you need, and make an estimate you can compare against at the end.

One number, three forms

Changing form is easier when a few conversions are automatic. These come up constantly.

The last two rows carry a warning from the last lesson. Thirds are repeating decimals, so any decimal version you write down is rounded. If a problem involves thirds, stay in fractions until the very end. Finding 2/3 of 27 is easy in fractions, since 27 ÷ 3 = 9 and 9 × 2 = 18, but 0.67 × 27 = 18.09 carries a small error that has crept in from the rounding.

Worth holding on to: Fractions are exact and easy when the bottom divides the other number. Decimals are easy on a calculator and with money. Percents are easy to compare. Convert whenever a different form makes the next step simpler.

Estimating is calculating with friendly numbers

An estimate is not a guess. It is a real calculation done with numbers you have deliberately made easy. Three moves do most of the work.

• Round to friendly numbers. 19.7 becomes 20, 3 1/4 stays 3 1/4 or becomes 3. Then 20 × 3 1/4 = 65 is quick.
• Use benchmarks. 0.25 is a quarter, 0.48 is about a half, 1.9 is nearly 2, and 0.09 is about a tenth.
• Keep track of which way you rounded. If you rounded every number in a product up, the estimate is too high, and the exact answer should come in a little under it.

The estimate's job is to catch big mistakes, such as a decimal point in the wrong place, an added number that should have been subtracted, a step forgotten, or a sign that flipped. It cannot tell you whether the last digit is right. For that, you still check exactly.

Four more problems, each estimated and then solved

Worked example 2: a raise. The standard gives this one too. A woman earning $25 an hour gets a 10% raise. What is her new hourly pay?

Step 1. Pick a form. 10% is 1/10, and a tenth of anything is easy: move the decimal point one place left. 1/10 of $25 is $2.50.

Step 2. Add the raise to the old pay: $25.00 + $2.50 = $27.50 an hour.

Estimate check: a 10% raise on about $25 should add a couple of dollars, not twenty and not twenty cents. $27.50 fits.

Worked example 3: a descent, with a negative rate. A hiker starts at an elevation of 1,240 m and walks downhill for 3 1/2 hours. Her elevation changes by -180 m every hour. Where does she finish?

Estimate first: about -200 m an hour for about 3 1/2 hours is about -700 m, and 1,240 - 700 is about 540 m. Because 200 is more than 180, the true drop is smaller, so she should finish a bit higher than 540 m.

Step 1. Total change = 3 1/2 × (-180). Convert 3 1/2 to 3.5, or split it: 3 × (-180) = -540 and 1/2 × (-180) = -90, so the total is -540 + (-90) = -630 m.

Step 2. Final elevation = 1,240 + (-630) = 610 m.

610 m is a bit higher than the estimate of 540 m, just as predicted. The answer is reasonable.

Worked example 4: scaling a recipe. A recipe uses 2 1/4 cups of flour to make 12 muffins. How much flour do you need for 30 muffins?

Estimate first: a little over 2 cups per dozen, and 30 muffins is 2 1/2 dozen, so a little over 5 cups.

Step 1. Flour per muffin is 2 1/4 ÷ 12. In fractions, 2 1/4 = 9/4, and 9/4 ÷ 12 = 9/48 = 3/16 of a cup.

Step 2. For 30 muffins: 30 × 3/16 = 90/16 = 45/8 = 5 5/8 cups.

A little over 5 cups, as the estimate said. Here fractions were the right form, because measuring cups are marked in fractions, and 5 5/8 cups is something you can measure; the decimal 5.625 is harder to scoop.

Worked example 5: a fundraiser with costs. A class sells 48 raffle tickets at $1.25 each. It pays $22.40 for the prizes and a $5.75 printing fee. What is the profit?

Estimate first: about 50 tickets at $1.25 is about $62; take away about $22 and about $6 and you have about $34. Since 48 is a little less than 50, the true profit should be a little less than $34.

Step 1. Money in: 48 × $1.25. A quarter of 48 is 12, so 48 × 0.25 = 12, and 48 × 1.25 = 48 + 12 = $60.00.

Step 2. Money out, written as negatives: -22.40 + (-5.75) = -28.15.

Step 3. Profit = 60.00 + (-28.15) = $31.85.

$31.85 is a little under $34, as predicted.

Worked example 6: catching a misplaced decimal point. A student works out 4.8 × 0.25 and writes 12. Is that believable?

Estimate: 0.25 is a quarter, and a quarter of about 5 is about 1.25. An answer of 12 is ten times too big, so a decimal point has slipped.

Exact: 4.8 × 0.25 = 4.8 ÷ 4 = 1.2. The estimate caught the mistake without any long multiplication at all.

The core of it: Estimate before you calculate, then compare. If the two disagree by a lot, one of them is wrong, and it is almost always worth finding out which.

Common misconceptions

"Estimating is just guessing." A guess has no working behind it. An estimate is a real calculation with rounded numbers, and you can explain every step of it, as in the fundraiser: 50 × 1.25 - 22 - 6.

"Always change everything into decimals first." Thirds and sixths become never-ending decimals, and rounding them early builds in error. For 2/3 of 27, fractions give exactly 18 in two easy steps.

"If my answer is close to my estimate, it must be exactly right." An estimate only rules out big errors. 8 7/8 and 8 5/8 are both close to 9, but only one is the right gap for the towel bar. Check the exact answer too, for example by adding the two gaps and the bar back together: 8 7/8 + 9 3/4 + 8 7/8 = 27 1/2.

"Round every number up and the estimate will be safe." Rounding everything the same way pushes the estimate off in one direction. That is fine as long as you remember which direction and expect the exact answer on the other side, as the hiker example did.

Pulling it together

• Sketch or describe the problem first to find its steps: for the towel bar, subtract the bar from the door, then halve.
• Use whichever form of number makes each step easy: fractions for thirds and tape measures, decimals for money and calculators, percents for comparing.
• Know the common conversions: 1/4 = 0.25 = 25%, 1/8 = 0.125 = 12.5%, 1/3 = 0.333... = 33 1/3%.
• Estimate with friendly numbers before calculating, note which way you rounded, and compare at the end.
• An estimate catches big mistakes; an exact check, such as putting the pieces back together, confirms the final digits.

The next module takes one kind of multi-step problem and studies it closely: problems where two quantities grow together at a fixed rate, which is what the word proportional means.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Expressions and Equations (7.EE.B.3, including the raise and towel bar examples). thecorestandards.org
2. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, The Number System (7.NS.A.3). thecorestandards.org
3. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 9.1, Use a problem solving strategy. Prealgebra 2e. OpenStax. openstax.org
4. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 5.2, Decimal operations. Prealgebra 2e. OpenStax. openstax.org

Two-fifths of a fence in three-quarters of an hour

You have been paid to paint a neighbour's fence. You start at 9:00, and at 9:45 you step back and see that you have finished 2/5 of it. Two questions come to mind at once. At this pace, how much fence do you paint in one hour? And how long will the whole fence take?

Both answers are unit rates, and both need a division in which the top and the bottom are fractions: 2/5 of a fence in 3/4 of an hour. Try to guess the second answer before reading on. You have done less than half the fence in less than an hour, so the whole fence will take somewhere between one and two hours. Keep that guess; it will check the calculation.

This lesson covers Common Core standard 7.RP.A.1: computing unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

What a unit rate is, and why there are always two

A rate compares two quantities measured in different units: miles and hours, dollars and pounds, fence and hours. A unit rate is a rate written per one of the second quantity: 2 miles per 1 hour, $6 per 1 pound. You met unit rates in sixth grade, where the numbers were whole, such as 120 miles in 3 hours giving 120 ÷ 3 = 40 miles per hour.

Every rate gives two unit rates, depending on which quantity you put first:

• 120 miles in 3 hours is 120 ÷ 3 = 40 miles per hour.
• The same trip is 3 ÷ 120 = 1/40 of an hour per mile, which is 1.5 minutes per mile.

Both are correct. They answer different questions. Miles per hour answers how far you go in a given time. Hours per mile answers how long a given distance takes. The rule for either one is the same: divide the quantity you want by the quantity you want it per. For fence per hour, divide fence by hours. For hours per fence, divide hours by fence.

Dividing a fraction by a fraction, two ways

When both numbers are fractions, the division is written as a fraction whose top and bottom are themselves fractions. That is called a complex fraction, and it looks harder than it is. The standard's own example is a person who walks 1/2 mile in each 1/4 hour. The unit rate is

(1/2) / (1/4) miles per hour.

There are two good ways to simplify it.

Way 1: multiply by the reciprocal. Dividing by a fraction is the same as multiplying by that fraction flipped. (1/2) ÷ (1/4) = (1/2) × (4/1) = 4/2 = 2.

Way 2: scale the top and bottom. A fraction keeps its value when you multiply its top and bottom by the same number. Choose a number that clears both small denominators; here 4 works. Top: 1/2 × 4 = 2. Bottom: 1/4 × 4 = 1. So the complex fraction equals 2/1 = 2.

Way 2 has a real-world meaning worth noticing. Multiplying both parts by 4 is the same as asking about four quarter-hours instead of one: in 4 × 1/4 = 1 hour the walker goes 4 × 1/2 = 2 miles. That is exactly what a unit rate is.

The point: A unit rate from fractions is still just one quantity divided by the other. Multiply by the reciprocal, or scale the top and bottom until the bottom is 1.

Solving the fence problem

Worked example 1: fence per hour, and hours per fence. You painted 2/5 of the fence in 3/4 of an hour.

Step 1. Fence per hour means fence divided by hours: (2/5) ÷ (3/4).

Step 2. Multiply by the reciprocal: (2/5) × (4/3) = 8/15.

Step 3. You paint 8/15 of the fence per hour. That is a little more than half, which fits: in 3/4 of an hour you did 2/5, so in a full hour you should do a bit more.

Step 4. Hours per fence means hours divided by fence: (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8.

Step 5. 15/8 hours = 1 7/8 hours. Seven-eighths of an hour is 7/8 × 60 = 52.5 minutes, so the whole fence takes 1 hour 52 1/2 minutes, finishing at about 10:52.

That lands between one and two hours, as the guess said. Notice also that the two unit rates, 8/15 and 15/8, are reciprocals of each other. That is always true, because one is the other divided the opposite way round, and it gives you a free check.

Worked examples: a walk, a recipe, a price and a paint tin

Worked example 2: the walk to school. Your friend walks 1/2 mile in each 1/4 hour. How fast is that in miles per hour, and how many minutes does each mile take?

Step 1. Miles per hour: (1/2) ÷ (1/4) = 2 miles per hour, as above.

Step 2. Minutes per mile: 1/4 hour is 15 minutes, so the walk is 15 minutes per 1/2 mile, and (15) ÷ (1/2) = 15 × 2 = 30 minutes per mile.

Check: at 2 miles per hour, one mile takes half an hour, which is 30 minutes.

Worked example 3: lemonade with the same taste. A lemonade recipe uses 3/4 cup of lemon juice for every 2 1/2 cups of water. How much juice goes with each cup of water, and how much juice do you need for 8 cups of water?

Step 1. Juice per cup of water: (3/4) ÷ (2 1/2). Write the mixed number as a fraction: 2 1/2 = 5/2.

Step 2. (3/4) × (2/5) = 6/20 = 3/10 cup of juice per cup of water.

Step 3. For 8 cups of water: 8 × 3/10 = 24/10 = 2 2/5 cups of juice, which is 2.4 cups.

Check by scaling the original: 8 cups of water is 8 ÷ 2 1/2 = 3.2 batches, and 3.2 × 3/4 = 2.4 cups of juice. The two routes agree.

Worked example 4: which cheese is the better buy? A deli sells a 3/4 pound block of cheddar for $4.50 and a 1 1/4 pound block of the same cheddar for $7.25.

Step 1. Dollars per pound for the small block: 4.50 ÷ (3/4) = 4.50 × (4/3) = 18.00 ÷ 3 = $6.00 per pound.

Step 2. Dollars per pound for the large block: 7.25 ÷ (1 1/4) = 7.25 ÷ 1.25 = 725 ÷ 125 = $5.80 per pound.

Step 3. The large block is cheaper per pound by 20 cents, so it is the better buy if you will eat it before it goes off.

Worked example 5: a ratio of an area to a volume. The standard mentions ratios of areas and quantities in different units. A painter covers 2 1/2 square metres of wall with 1/3 of a litre of paint. How many square metres does one litre cover?

Step 1. Square metres per litre: (2 1/2) ÷ (1/3) = (5/2) × 3 = 15/2.

Step 2. One litre covers 15/2 = 7 1/2 square metres.

Step 3. The reciprocal rate is litres per square metre: 2/15 of a litre. For a wall of 30 square metres, you need 30 × 2/15 = 4 litres. Check with the first rate: 4 litres × 7 1/2 = 30 square metres.

Why this matters: Every question in these examples, how far, how long, how much, which is cheaper, was answered by one unit rate followed by one multiplication. Getting the unit rate right is most of the work.

Comparing two rates

Worked example 6: who walks faster? Maya walks 1/2 mile in 1/4 hour. Sam walks 3/4 mile in 1/3 hour. Who is faster?

You cannot compare the two trips as they stand, because they last different lengths of time. Convert both to the same unit rate.

Sam is faster, by 1/4 mile per hour. Maya covered her distance in less time, but she also covered less distance, so the time alone tells you nothing. Only the unit rate makes the two trips comparable.

Common misconceptions

"Always divide the bigger number by the smaller one." The order is set by the question, not by the sizes. Fence per hour is fence ÷ hours even though 2/5 is smaller than 3/4. Dividing the other way gives hours per fence, a different rate.

"(1/2) ÷ (1/4) is 1/8." That is (1/2) × (1/4). Dividing by 1/4 asks how many quarters fit into a half, and two quarters do, so the answer is 2. When you divide by a number less than 1, the answer is bigger than what you started with.

"Flip the first fraction." Only the divisor, the second fraction, gets flipped. (2/5) ÷ (3/4) becomes (2/5) × (4/3), not (5/2) × (3/4).

"A unit rate has to be a whole number." 8/15 of a fence per hour and 3/10 cup per cup are perfectly good unit rates. The word unit refers to the 1 on the bottom, per one hour or per one cup, not to the answer.

What you now know

• A unit rate is a rate per one of the second quantity. Divide the quantity you want by the quantity you want it per.
• Every rate has two unit rates, which are reciprocals: 8/15 of a fence per hour and 15/8 hours per fence.
• A complex fraction such as (1/2)/(1/4) simplifies by multiplying by the reciprocal of the bottom, or by scaling the top and bottom until the bottom is 1.
• Unit rates answer how far, how long and how much with one more multiplication, and they make different trips or prices comparable.
• Dividing by a fraction less than 1 gives a larger number, so a quick size check catches the most common error.

A unit rate that stays the same all the way through a situation is the signature of a proportional relationship. The next lesson shows how to spot one in a table and on a graph.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Ratios and Proportional Relationships (7.RP.A.1). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 4, Lesson 2: Ratios and rates with fractions. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 4.3, Multiply and divide mixed numbers and complex fractions. Prealgebra 2e. OpenStax. openstax.org
4. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 5.6, Ratios and rate. Prealgebra 2e. OpenStax. openstax.org

Six dollars a ticket, five dollars a game

On Main Street a cinema charges $6 a ticket. Next door, a bowling alley charges $5 a game, plus $4, paid once, to rent shoes. You and your friends have $24 between you. That buys 4 cinema tickets. It also buys 4 games of bowling: 4 × $5 = $20 for the games and $4 for the shoes.

So far the two places look alike. Now ask a different question: if you buy twice as much, do you pay twice as much? One cinema ticket costs $6 and two cost $12, exactly double. One game of bowling costs $9 with the shoes, and two games cost $14, which is not double. That small difference, whether doubling one quantity doubles the other, is the whole subject of this lesson.

This lesson covers Common Core standard 7.RP.A.2a: deciding whether two quantities are in a proportional relationship, by testing for equivalent ratios in a table or by graphing on a coordinate plane and seeing whether the graph is a straight line through the origin.

What proportional means

Two quantities are in a proportional relationship when one is always the same number times the other. Cinema tickets pass: the cost is always 6 times the number of tickets. So if you divide the cost by the number of tickets, you get 6 every time, however many tickets you buy:

• $12 ÷ 2 tickets = 6
• $18 ÷ 3 tickets = 6
• $24 ÷ 4 tickets = 6

The pairs 2 tickets for $12, 3 for $18 and 4 for $24 are equivalent ratios: each one simplifies to 6 : 1. The number they share, here 6 dollars per ticket, is called the constant of proportionality. It is the unit rate from the last lesson. The next lesson is all about that number; today you only need to decide whether one exists.

Two more facts follow straight from the definition, and both are useful checks:

• Zero goes with zero. Zero tickets cost 0 × $6 = $0.
• Scaling one quantity scales the other by the same factor. Three times as many tickets cost three times as much.

Key idea: A relationship is proportional when y ÷ x is the same number for every pair. Every other test in this lesson grows out of that one.

Four relationships, side by side

Here are four relationships you could meet on one Saturday. The first two are the cinema and the bowling alley. The third comes from a table that the U.S. National Institute of Standards and Technology (NIST) publishes to help people read both thermometer scales: freezing is 0 degrees Celsius or 32 degrees Fahrenheit, a cold day is 10 °C or 50 °F, a nice day is 20 °C or 68 °F, and a hot day is 30 °C or 86 °F. The fourth is square floor tiles, where the side length decides the area.

Only one column of quotients stays the same all the way along. Before reading on, say which relationship it is, and what goes wrong in each of the other three.

Reading the table, one relationship at a time

Worked example 1: the cinema.

Step 1. Divide cost by tickets in each row: 6 ÷ 1 = 6, 12 ÷ 2 = 6, 18 ÷ 3 = 6, 24 ÷ 4 = 6.

Step 2. Every quotient is 6, so the ratios are equivalent and the table fits a proportional relationship at 6 dollars per ticket.

Step 3. Check by scaling: 4 tickets are twice as many as 2, and $24 is twice $12.

Step 4. Here you also know the pricing rule, $6 for every ticket, so the relationship is proportional for any number of tickets, not only the four in the table.

Worked example 2: the bowling alley.

Step 1. Divide cost by games: 9 ÷ 1 = 9, 14 ÷ 2 = 7, 19 ÷ 3 = 6.33 (rounded), 24 ÷ 4 = 6.

Step 2. The quotients change, so the ratios are not equivalent. The relationship is not proportional.

Step 3. Find the reason in the situation. Each game adds $5, but the $4 for shoes is paid once however many games you play. That fixed $4 is shared out over more games as you play more, which is why the cost per game keeps falling: $9 for one game, but only $6 a game over four.

Step 4. Scaling check: if bowling were proportional, 2 games would cost twice as much as 1 game, $18. They cost $14.

Notice what does not decide it. The bowling cost goes up by the same $5 for each extra game, just as the cinema cost goes up by $6 for each extra ticket. Steady steps happen in both tables. Only the quotients tell them apart.

Worked example 3: the thermometer.

Step 1. The first row settles it on its own: 0 °C is 32 °F. In a proportional relationship 0 must go with 0, because 0 times any number is 0.

Step 2. The other rows agree: 50 ÷ 10 = 5, 68 ÷ 20 = 3.4, 86 ÷ 30 = 2.87 (rounded). The quotients are all different.

Step 3. The reason: NIST gives the exact rule as degrees Fahrenheit = degrees Celsius × 1.8 + 32. The + 32 plays the same part as the shoe rental. It is added whatever the Celsius reading is.

Step 4. A consequence you can check: doubling a Celsius temperature does not double the Fahrenheit reading. 20 °C is 68 °F, not 2 × 50 = 100 °F.

Worked example 4: the tiles.

Step 1. Divide area by side: 1 ÷ 1 = 1, 4 ÷ 2 = 2, 9 ÷ 3 = 3, 16 ÷ 4 = 4.

Step 2. The quotient grows with the side, so it is not constant. The relationship is not proportional.

Step 3. The reason: area is side times side, so the multiplier is the side itself, and the side changes. Doubling the side from 2 feet to 4 feet makes the area four times as big, 16 square feet instead of 4.

Step 4. Notice that a side of 0 feet would give an area of 0 square feet. This relationship does pair 0 with 0, and it is still not proportional.

Now put the four results next to each other:

Read the table across. Only the first row matches the last row in every column. Equal steps do not decide it, since three of the four relationships have them. Zero with zero does not decide it on its own, since the tiles have it and still fail. The constant quotient is the test.

Remember: Rising together is not enough, and neither are equal steps. Divide y by x in every row; a single mismatch rules proportionality out.

The same relationships on a graph

Plot the cinema and bowling tables on one grid, with the number of tickets or games across the bottom and the cost up the side.

The cinema points lie on a straight line that runs back to the corner of the grid, the origin (0, 0). The bowling points lie on a straight line too, but when you run that line back to 0 games it meets the cost axis at $4, the shoe rental. The two lines cross at 4 and $24, the point you found at the start. Sharing one point does not make two relationships the same.

That gives the graph test for proportionality. It has two parts, and both must hold:

1. The points lie on a straight line.
2. The line passes through the origin (0, 0).

Each of the other two relationships fails one part. The thermometer's points lie on a straight line, but it meets the vertical axis at 32. The tiles start at (0, 0), but their points (1, 1), (2, 4), (3, 9) and (4, 16) climb more steeply at every step, so they bend into a curve instead of a line.

The graph test and the table test always agree, because underneath they are the same test. If y ÷ x is always the same number k, then every point is (x, k × x). Each step of 1 to the right goes up by k, which makes a straight line, and x = 0 gives y = 0, which puts the line through the origin.

Filling in a proportional table

Once you know a relationship is proportional, the constant quotient fills in any missing value.

Worked example 5: fuel and distance. A car's distance is proportional to the fuel it uses, and on 3 gallons it goes 84 miles. Fill in the table.

Step 1. Find the constant: 84 ÷ 3 = 28 miles per gallon.

Step 2. For 5 gallons, multiply by the constant: 5 × 28 = 140 miles.

Step 3. For 196 miles, divide by the constant: 196 ÷ 28 = 7 gallons.

Step 4. For 10 1/2 gallons: 10.5 × 28 = 294 miles.

Check each new row with the quotient test: 140 ÷ 5 = 28, 196 ÷ 7 = 28 and 294 ÷ 10.5 = 28.

Worked example 6: pancakes, with fractions. A recipe card lists 3/4 cup of flour for 6 pancakes, 1 1/2 cups for 12, 2 1/4 cups for 18 and 3 cups for 24. Is it proportional, and how much flour do 30 pancakes need?

Step 1. Pancakes per cup in each row: 6 ÷ 3/4 = 6 × 4/3 = 8; 12 ÷ 1 1/2 = 12 × 2/3 = 8; 18 ÷ 2 1/4 = 18 × 4/9 = 8; 24 ÷ 3 = 8.

Step 2. Every quotient is 8, so the card fits a proportional relationship at 8 pancakes per cup of flour.

Step 3. For 30 pancakes you need 30 ÷ 8 = 3 6/8 = 3 3/4 cups of flour.

Check: 3 3/4 cups is 5 times 3/4 cup, and 30 pancakes is 5 times 6 pancakes. Both quantities were scaled by 5.

You may divide either way round, as long as you do it the same way in every row. Flour per pancake is 3/4 ÷ 6 = 1/8 cup in every row, and a constant 1/8 tells you exactly what a constant 8 told you.

A table shows only some of the pairs

Worked example 7: a copy shop. A copy shop's price list reads: 20 pages $2.00, 50 pages $5.00, 100 pages $10.00, 500 pages $40.00.

Step 1. Dollars per page: 2.00 ÷ 20 = 0.10, 5.00 ÷ 50 = 0.10, 10.00 ÷ 100 = 0.10, 40.00 ÷ 500 = 0.08.

Step 2. Three rows agree and the last does not, because big orders get a bulk price. One mismatch is enough: the relationship is not proportional.

Now imagine the price list had stopped at 100 pages. Every quotient would have been 0.10, and the table would have looked proportional. That is a real limit of the table test. A constant quotient in the rows you can see shows that the relationship could be proportional, and it says nothing about rows you cannot see. You can be sure only when you know the rule itself, such as "10 cents a page, whatever the size of the order". A different quotient in even one row, though, settles the question for good.

Common misconceptions

"Both numbers go up, so it is proportional." When you were 4, your cousin was 1. Now you are 12 and she is 9. Both ages went up together, but her age ÷ your age went from 1/4 to 9/12 = 3/4. What stays fixed is the difference, 3 years, and a fixed difference is not a fixed ratio.

"Each row goes up by the same amount, so it is proportional." The bowling costs go up by $5 a game, and they are not proportional. Equal steps make a straight line; only a straight line that also passes through (0, 0) is proportional.

"Any straight-line graph is proportional." The thermometer line is perfectly straight and meets the vertical axis at 32. The line has to pass through the origin.

"The graph goes through (0, 0), so it is proportional." The tile graph starts at the origin and bends upward. Both parts of the graph test must hold.

"I checked one pair and it scaled, so it is proportional." Test every row. In the copy shop's list, 100 pages cost exactly five times what 20 pages cost, and the relationship is still not proportional.

Recap

• Two quantities are proportional when y ÷ x is the same number for every pair. That number is the constant of proportionality.
• In a table, divide y by x in every row. One different quotient rules proportionality out. A constant quotient in the rows shown means it could be proportional, and knowing the rule settles it.
• On a graph, a proportional relationship is a straight line through the origin. A straight line that misses the origin, like the bowling costs or the thermometer, is not proportional, and neither is a curve through the origin, like the tile areas.
• Both quantities rising, equal steps, and one pair that scales are not tests of proportionality.
• Once a relationship is known to be proportional, its constant fills in any missing value with one multiplication or one division.

The next lesson gives the constant a letter, k, writes every proportional relationship as y = kx, and reads k straight off the graph.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Ratios and Proportional Relationships (7.RP.A.2a). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 2, Lesson 7: Comparing relationships with tables. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 2, Lesson 10: Introducing graphs of proportional relationships. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. National Institute of Standards and Technology. (2010, updated 2026). SI units: Temperature. U.S. Department of Commerce. nist.gov

Five grams, every time

A United States nickel weighs 5.000 grams. Not roughly five: the coin is made to that weight, and every nickel in your family's coin jar should match it. Suppose the jar is full of nickels and you want to know what they are worth. Counting them one at a time would take most of an evening. A kitchen scale can do it in a minute, and following how it does so is a good way to meet the most useful equation in this unit.

This lesson covers Common Core standards 7.RP.A.2b, 2c and 2d: identifying the constant of proportionality in tables, graphs, equations, diagrams and verbal descriptions; representing proportional relationships by equations; and explaining what a point (x, y) on the graph of a proportional relationship means, with special attention to (0, 0) and (1, r), where r is the unit rate.

First weighing: finding the constant

You do not have to trust the 5-gram figure. Test it. Count out 20 nickels, put them on the scale, and write down the reading. Then add coins until there are 50, and later 80.

Worked example 1: the constant from a table.

Step 1. Divide weight by number in every row, as in the last lesson. Every quotient is 5.

Step 2. The weight is proportional to the number of nickels, and the number every count is multiplied by is 5 grams per nickel.

That number has a name. In a proportional relationship, the fixed number that you multiply x by to get y is the constant of proportionality, usually written k. It is the same thing as the unit rate: 5 grams per 1 nickel. For the coin jar, k = 5.

Writing it as an equation

If every nickel adds 5 grams, then n nickels weigh 5 × n grams. Written as an equation:

w = 5n

Every proportional relationship can be written this way, as y = kx: the second quantity equals the constant times the first. The Common Core standard gives the shopping version. If each item costs p dollars, the total cost t of n items is t = pn, and the price p is the constant.

Worked example 2: using the equation. You tip the whole pile of nickels onto the scale, and it reads 1,340 grams. How many nickels are there, and what are they worth?

Step 1. Put the reading into the equation: 1,340 = 5n.

Step 2. Ask what number times 5 makes 1,340. Divide: n = 1,340 ÷ 5 = 268 nickels.

Step 3. Each nickel is worth $0.05, so the value is 268 × $0.05 = $13.40.

Check: 268 × 5 = 1,340, so 268 nickels do weigh 1,340 grams.

In short: A proportional relationship is y = kx. Find k once, from any pair, and the equation answers every other question with one multiplication or one division.

The jar that got in the way

Here the story takes a wrong turn that is worth following. The first time you weigh the pile, you leave the nickels in their glass jar, and the scale reads 1,690 grams. Divide by 5 and you get 338 nickels, worth $16.90. That is 70 nickels too many.

The empty jar weighs 350 grams, and 350 ÷ 5 = 70: the jar has been counted as 70 imaginary nickels. With the jar on the scale, the reading is w = 5n + 350, and that is not a proportional relationship at all. Zero nickels would still read 350 grams, just as zero games of bowling still meant paying $4 for shoes.

A kitchen scale has a button for exactly this, called tare. Put the empty jar on, press it, and the scale reads 0. From then on it shows only what you add, so the reading goes back to w = 5n. Taring puts the relationship back through (0, 0).

What the graph says, point by point

Here is w = 5n as a graph for up to 8 nickels. Every point on it is a pair (number of nickels, weight in grams) that fits the equation.

Worked example 3: reading points. Say in words what each marked point means.

Step 1. The point (6, 30) has x = 6 and y = 30: 6 nickels weigh 30 grams. Any point on the line reads the same way, as a number of nickels and the weight that goes with it.

Step 2. The point (0, 0) says that 0 nickels weigh 0 grams. On a tared scale that is exactly right, and it is why the graph starts at the origin.

Step 3. The point (1, 5) says that 1 nickel weighs 5 grams. Its y-value is the unit rate, which is the constant k. That is true of every proportional graph: go across to x = 1, and the height of the line there is k.

Step 4. The constant also shows in the steepness. Each step of 1 nickel to the right climbs 5 grams. A coin with a bigger k would give a steeper line.

Two equations for one relationship

You can turn the question round. Instead of weight from nickels, you might want nickels from weight, which is what you wanted at the start.

Worked example 4: the reverse equation.

Step 1. Each nickel weighs 5 grams, so each gram is 1/5 of a nickel. The number of nickels is n = (1/5)w, which is n = 0.2w.

Step 2. Test it on the pile: 0.2 × 1,340 = 268 nickels, the same answer as before.

Step 3. Notice the two constants, 5 grams per nickel and 0.2 of a nickel per gram. They are reciprocals, just like the paired unit rates in the fractions lesson: 5 × 0.2 = 1.

The jar holds a third relationship too, between the weight and the money. A nickel is 5 cents and weighs 5 grams, so every gram of nickels is worth exactly 1 cent. The value in dollars is v = 0.01w. Check it on the pile: 0.01 × 1,340 = $13.40, the same value as before. Three equations, w = 5n, n = 0.2w and v = 0.01w, all describe the same jar.

The upshot: Each proportional relationship has two equations, one for each direction, and their constants are reciprocals. Write the one whose left side is the quantity you are trying to find.

Which coin made this line?

Other coins are made to their own fixed weights. A quarter weighs 5.67 grams. A penny made since 1982, copper-plated zinc, weighs 2.5 grams. So each coin gives its own proportional relationship, with its own k.

Worked example 5: identify the coin from its graph. A friend graphs weight against number for a bag of one kind of coin. The line passes through the origin and the point (8, 20). Which coin is it?

Step 1. For a proportional graph, k = y ÷ x at any point other than the origin: k = 20 ÷ 8 = 2.5 grams per coin.

Step 2. The line therefore also passes through (1, 2.5): one coin weighs 2.5 grams.

Step 3. That matches a penny made since 1982. A quarter line would pass through (1, 5.67) and would be steeper than the nickel line through (1, 5).

Worked example 6: from a verbal description. A school printer prints 3 pages every 2 seconds. Write the equation, find k, and say what (1, k) means.

Step 1. Pages per second: 3 ÷ 2 = 1.5. So k = 1.5 and the equation is p = 1.5t, where p is pages and t is seconds.

Step 2. The point (1, 1.5) says that in 1 second the printer prints 1.5 pages. A point can mean half a page finished; it does not have to be a whole-number situation.

Step 3. Use it: in a minute, 60 seconds, the printer prints 1.5 × 60 = 90 pages. The reverse equation is t = (2/3)p, since each page takes 2/3 of a second.

Worked example 7: when the model breaks. Your grandmother's penny jar holds coins from many years. Pennies from before 1982 are 95 percent copper and weigh 3.11 grams; newer ones weigh 2.5 grams. The pennies in the jar weigh 600 grams in total. How many are there?

Step 1. If every penny were new, the count would be 600 ÷ 2.5 = 240.

Step 2. If every penny were old, it would be 600 ÷ 3.11 = 192.9, so about 193.

Step 3. With a mixture, the true count is somewhere between about 193 and 240, and the scale cannot say where. The equation y = kx needs every item to have the same weight. When the items differ, weight and number are no longer proportional, and the trick stops working.

Common misconceptions

"k is where the line starts on the vertical axis." A proportional line always starts at (0, 0). The constant is the height of the line at x = 1, not at x = 0.

"Divide whichever way gives the bigger number." In y = kx, k is y ÷ x, with the y quantity on top. For weight from nickels that is 5; dividing the other way gives 0.2, which is the constant of the other equation, n = 0.2w. Both are correct for their own equation, and mixing them up is the commonest error in this topic.

"The point (1, k) has to be in the table." The nickel table never listed 1 nickel. You can always compute k from any row, and the point (1, k) is on the line whether anyone measured it or not.

"A steeper line means more coins." Steepness shows k, the weight per coin, not how many coins there are. The quarter line is steeper than the nickel line because each quarter is heavier.

What to remember

• The constant of proportionality k is the number that x is multiplied by to get y. It is the unit rate, and it equals y ÷ x for any pair other than (0, 0).
• Every proportional relationship can be written y = kx, such as w = 5n for nickels or t = pn for a total cost at a price of p each.
• On the graph, (0, 0) means none of the first quantity goes with none of the second, and (1, k) shows the unit rate. Any other point (x, y) is a matching pair, such as 6 nickels and 30 grams.
• Each relationship has a reverse equation whose constant is the reciprocal: w = 5n and n = 0.2w.
• The model needs a real reason to be proportional. An added fixed amount, like a jar on the scale, or items that differ, like old and new pennies, breaks it.

The next module puts proportional reasoning to work on the thing it is used for most often in daily life: percentages on real prices.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Ratios and Proportional Relationships (7.RP.A.2b-d). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 2, Lesson 5: Two equations for each relationship. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 2, Lesson 11: Interpreting graphs of proportional relationships. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Wikipedia contributors. (n.d.). Nickel (United States coin). In Wikipedia. Retrieved September 24, 2026, for the 5.000 gram mass. en.wikipedia.org
5. Wikipedia contributors. (n.d.). Penny (United States coin). In Wikipedia. Retrieved September 24, 2026, for the 2.5 gram and 3.11 gram masses. en.wikipedia.org

A sign that says save 33%

A video game that cost $60 last week has a new sticker on it: $45. Above the shelf, a hand-written sign says SAVE 33%! The price has certainly dropped. The question is whether the sign has the right number, and the way to find out is to trace exactly how somebody got 33.

This lesson covers Common Core standard 7.RP.A.3, using proportional relationships to solve multistep percent problems, including percent increase and decrease and percent error, and the checking habits of 7.EE.B.3.

Tracing the 33

Step 1. The saving in dollars is $60 - $45 = $15. Nobody disputes that part.

Step 2. To turn $15 into a percent, it has to be divided by something. Try the two prices: 15 ÷ 45 = 1/3, which is 33.3 percent, and 15 ÷ 60 = 1/4, which is 25 percent. The sign's author divided by $45, the new price.

Step 3. Test the sign's claim directly. If you really saved 33 percent of the old price, you would save 0.33 × $60 = $19.80 and pay $40.20. You pay $45, so the sign overstates the saving.

Step 4. Test 25 percent the same way: 0.25 × $60 = $15 off, leaving $60 - $15 = $45. That matches the sticker. The price fell by 25 percent.

The mistake is not in the arithmetic. 15 ÷ 45 really is 33.3 percent. The mistake is in the choice of what to divide by.

The rule, and why the original is the base

"Save 25 percent" means that 25 percent of the price you started with has been taken away. A percent change always describes the change as a part of the starting amount, so the rule is

percent change = (change in amount) ÷ (original amount) × 100%.

The original amount is the base. When the new amount is bigger, the change is an increase; when it is smaller, a decrease. OpenStax's Prealgebra gives the same two steps: first find the amount of change, then find what percent that change is of the original amount.

Here is the twist that explains the sign. Suppose next week the game goes back up from $45 to $60. The change is again $15, but now the original is $45, so the increase is 15 ÷ 45 = 33 1/3 percent. The same $15 gap is a 25 percent decrease going down and a 33 1/3 percent increase going up. The sign's author used the number that belongs to the trip back up.

So what?: Every percent change question has a hidden word, "of". It is a percent of the original amount, and you find that amount by asking which number came first in time.

Worked examples: up, down, and a whole country

Worked example 1: a concert ticket goes up. A ticket cost $40 last year and costs $52 this year.

Step 1. Change: $52 - $40 = $12, an increase.

Step 2. Divide by the original: 12 ÷ 40 = 0.30.

Step 3. Percent: 0.30 × 100% = 30 percent increase.

Check: a 30 percent increase multiplies the price by 1.30, and $40 × 1.30 = $52.

Worked example 2: a skateboard goes down. A skateboard's price falls from $80 to $68.

Step 1. Change: $80 - $68 = $12, a decrease.

Step 2. Divide by the original: 12 ÷ 80 = 0.15, so 15 percent decrease.

Check: a 15 percent decrease leaves 85 percent, and $80 × 0.85 = $68.

Notice the checks. A percent increase of 30 percent is the same as multiplying by 1 + 0.30 = 1.30. A percent decrease of 15 percent is the same as multiplying by 1 - 0.15 = 0.85. That single multiplier is often the fastest way to work forwards, and it is the key to working backwards, below.

Worked example 3: the population of the United States. The census counted 308,745,538 people living in the United States in 2010 and 331,449,281 on April 1, 2020. By what percent did the population grow?

Step 1. Change: 331,449,281 - 308,745,538 = 22,703,743 people.

Step 2. The original is the 2010 count: 22,703,743 ÷ 308,745,538 = 0.0735, rounded to four decimal places.

Step 3. Percent: about 7.4 percent increase.

Check against the source: the Census Bureau's own announcement gives exactly these three numbers, an increase of 22,703,743, or 7.4 percent, from 308,745,538. If you had divided by the 2020 count instead, you would have got 22,703,743 ÷ 331,449,281 = 0.068, or 6.8 percent, and your answer would disagree with the Census Bureau's.

Three more wrong answers, traced

Wrong answer 2: "Up 50 percent, then down 50 percent, and you are back where you started." A trading card is worth $20. Its value rises 50 percent, then falls 50 percent.

Step 1. Up 50 percent: $20 × 1.5 = $30.

Step 2. Down 50 percent: the base is now $30, the value before this change. $30 × 0.5 = $15.

Step 3. The card ends at $15, not $20. The two 50 percents were percents of different amounts, $20 and then $30, so they do not cancel.

Wrong answer 3: "To undo a 20 percent increase, take 20 percent off." After a 20 percent price rise, a game costs $60. What did it cost before?

Step 1. The tempting route: 20 percent of $60 is $12, and $60 - $12 = $48. Check it: $48 × 1.20 = $57.60, not $60. Wrong.

Step 2. The rise was 20 percent of the old price, not of $60. So $60 is 120 percent of the old price: old price × 1.20 = $60.

Step 3. Undo the multiplication by dividing: $60 ÷ 1.20 = $50.

Check: $50 × 1.20 = $60. The old price was $50.

Wrong answer 4: a guessing jar. At a school fair you guess that a jar holds 400 jelly beans. It actually holds 500. A friend says your percent error is 100 ÷ 400 = 25 percent. That divides by the wrong number, for the same reason the shop sign did, and the next section shows the right one.

Percent error: how far off, compared with the truth

Percent error describes how far an estimate or a measurement is from the correct value, as a percent of the correct value:

percent error = |estimate - correct value| ÷ correct value × 100%.

The bars are absolute value, from the first module: the error is a distance, so it is never negative. Illustrative Mathematics defines it the same way, as the error expressed as a percentage of the actual amount.

Worked example 4: the jelly beans, fixed.

Step 1. Size of the error: |400 - 500| = 100 jelly beans.

Step 2. Divide by the correct value, 500: 100 ÷ 500 = 0.20.

Step 3. The percent error is 20 percent, not 25.

Worked example 5: a bag of chips. A bag is labelled 8 ounces, and on a kitchen scale the chips weigh 7.6 ounces.

Step 1. Size of the error: |7.6 - 8| = 0.4 ounce.

Step 2. The correct value is what the bag should hold, 8 ounces: 0.4 ÷ 8 = 0.05.

Step 3. The percent error is 5 percent. Dividing by 7.6 instead would give 5.3 percent.

Worked example 6: pacing out a classroom. You pace the width of your classroom and estimate 9 metres. A tape measure says 7.5 metres.

Step 1. Size of the error: |9 - 7.5| = 1.5 metres.

Step 2. Divide by the measured, correct value: 1.5 ÷ 7.5 = 0.2.

Step 3. The percent error is 20 percent. Your estimate was too big, but the percent error is still written as a positive 20 percent; say "20 percent too high" if you want to keep the direction.

Worth holding on to: Percent increase and decrease divide by the original amount. Percent error divides by the correct value. In both, the base is the number the change or the error is measured from, never simply the bigger or the newer number.

Common misconceptions

"Divide by whichever number is newer." That is how the sign got 33 percent. Percent change is measured from the original amount, the one that came first.

"Equal percent ups and downs cancel." They are percents of different amounts. The $20 card went up to $30 and down to $15.

"Take the same percent off to undo an increase." Divide by the multiplier instead. $60 after a 20 percent rise came from $60 ÷ 1.20 = $50, not from $48.

"A percent can never be more than 100." A decrease cannot be more than 100 percent, because you cannot take away more than all of something. An increase can be: a price that goes from $20 to $50 has risen by 30 ÷ 20 = 150 percent.

Putting it together

• Percent change = change ÷ original amount × 100%. An increase uses the multiplier 1 + rate, and a decrease uses 1 - rate.
• The same gap in dollars is a smaller percent of a bigger base: $60 to $45 is a 25 percent decrease, and $45 to $60 is a 33 1/3 percent increase.
• To work back to an original amount, divide by the multiplier: $60 ÷ 1.20 = $50.
• Successive percent changes act on different bases, so a 50 percent rise followed by a 50 percent fall leaves you lower than you began.
• Percent error = |estimate - correct value| ÷ correct value × 100%, always written as a positive percent.
• Check any percent answer by applying it to the starting amount and seeing whether you land on the other number.

The next lesson uses these multipliers on one real receipt: a discount, a sales tax and a tip, in order.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Ratios and Proportional Relationships (7.RP.A.3). thecorestandards.org
2. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 6.2, Solve general applications of percent. Prealgebra 2e. OpenStax. openstax.org
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 4, Lesson 14: Percent error. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. U.S. Census Bureau. (2021, April 26). 2020 Census apportionment results delivered to the President (Press Release CB21-CN.30). census.gov

$48 on the tag, $38.52 at the till

A pair of trainers has a tag that says $48. A sign above the rack says 25 percent off, and the shop is in Indiana, where the state charges a sales tax of seven percent on the sale. At the till you pay $38.52. By the end of this lesson you will be able to produce that number in two different ways, and to explain every line of the receipt that prints it.

This lesson covers Common Core standard 7.RP.A.3, which names tax, markups and markdowns, and gratuities (tips) among the multistep percent problems you should be able to solve, together with 7.EE.A.2, rewriting an expression to see a problem differently, and the estimation checks of 7.EE.B.3.

The receipt, one line at a time

Worked example 1: the trainers. Follow the order in which a till works: first the discount comes off the tag price, then the tax is added to what is left.

Step 1. The discount is 25 percent of the tag price: 0.25 × $48 = $12.

Step 2. The sale price is $48 - $12 = $36.

Step 3. The tax is 7 percent of the sale price, not of the tag price, because the tax is charged on what you actually pay for the goods: 0.07 × $36 = $2.52.

Step 4. The total is $36 + $2.52 = $38.52.

Estimate as a check: a quarter off $48 leaves about $36, and 7 percent of $36 is a little over $2.50, so the total should be about $38.50. It is.

The same receipt in one line

Taking 25 percent off leaves 75 percent, so the sale price is 48 × 0.75. Adding 7 percent tax to an amount a gives a + 0.07a, which is 1.07a. The Common Core standard uses the same idea as its example of rewriting an expression: a + 0.05a = 1.05a means that "increase by 5 percent" is the same as "multiply by 1.05".

Worked example 2: the trainers with multipliers.

Step 1. Discount multiplier: 1 - 0.25 = 0.75. Tax multiplier: 1 + 0.07 = 1.07.

Step 2. Total = 48 × 0.75 × 1.07.

Step 3. 48 × 0.75 = 36, and 36 × 1.07 = 38.52. The total is $38.52, as before.

Now try the multipliers the other way round: 48 × 1.07 = 51.36, and 51.36 × 0.75 = 38.52. The same total. Multiplication can be done in any order, so it makes no difference to a percent discount and a percent tax which one the till applies first.

The core of it: Each percent step on a receipt is one multiplier: 1 minus the rate for a discount, 1 plus the rate for a tax, a markup or a tip. Chain the multipliers and you have the whole receipt in one line.

Before the tag: the shop's markup

The $48 on the tag did not come from nowhere. A shop buys goods at a wholesale price and adds a markup, a percent of the wholesale price, to reach the list price on the tag. OpenStax's Prealgebra sets the problem out the same way.

Worked example 3: the shop's side of the trainers. The shop paid $30 a pair and marked them up 60 percent.

Step 1. Markup: 0.60 × $30 = $18.

Step 2. List price: $30 + $18 = $48. With a multiplier: 30 × 1.60 = 48.

Step 3. During the sale the shop takes 25 percent off, so it receives $36 for the pair. That is still $36 - $30 = $6 more than it paid.

Step 4. As a percent of the wholesale price, the shop's gain is 6 ÷ 30 = 0.20, or 20 percent.

The multipliers show it at once: 1.60 × 0.75 = 1.20. A 60 percent markup followed by a 25 percent discount is a 20 percent markup overall. Sixty minus twenty-five would say 35 percent, which is wrong for the reason you met in the last lesson: the 25 percent is taken of $48, not of $30.

A dinner bill: tax and tip

A tip, or gratuity, is money you choose to add for the people who served you, usually as a percent of the bill. Many people work out the tip on the food before tax, and this example does that.

Worked example 4: dinner for a family. The food comes to $64.00. The sales tax is 7 percent and the family tips 18 percent of the food bill.

Step 1. Tax: 0.07 × 64 = $4.48.

Step 2. Tip: 0.18 × 64 = $11.52.

Step 3. Total: 64 + 4.48 + 11.52 = $80.00. Both percents were taken of $64, so a shortcut works too: 64 × (1 + 0.07 + 0.18) = 64 × 1.25 = $80.

If the family tipped 18 percent of the bill with tax instead, the tip would be 0.18 × 68.48 = 12.3264, which rounds to $12.33, and the total would be $80.81. The difference is 81 cents. Either choice is fine; what matters is knowing which amount the percent is of.

A word on cents. Money is rounded to the nearest cent at each line of a receipt. Tax of 7 percent on a $19.99 game is 0.07 × 19.99 = 1.3993, which rounds to $1.40, so the game costs $21.39. Look at the thousandths digit: 5 or more rounds the cents up, and 4 or less leaves them.

Change one number and the better deal flips

You have two coupons and may use only one: $10 off, or 25 percent off. Which is better? The answer depends on the price, and seeing why is the point of this example.

Worked example 5: the coupons on two different prices.

Step 1. On the $48 pair, 25 percent is $12, more than $10. The percent coupon wins.

Step 2. On the $32 pair, 25 percent is only $8. The dollar coupon wins.

Step 3. Find where they tie: 25 percent of the price equals $10 when the price is 10 ÷ 0.25 = $40. Above $40 the percent coupon saves more; below $40 the $10 coupon does.

The percent coupon grows with the price and the dollar coupon does not, which is why one number changing, the price, flips the answer.

Stacked discounts, and working backwards

Worked example 6: 40 percent off, then an extra 20 percent off. A $50 hoodie is on a 40 percent off rack, and the register takes an extra 20 percent off the sale price.

Step 1. First discount: 50 × 0.60 = $30.

Step 2. Extra discount, of the sale price: 30 × 0.80 = $24.

Step 3. You pay $24, so the total saving is $26, and 26 ÷ 50 = 0.52: 52 percent off, not 60. With multipliers, 0.60 × 0.80 = 0.48, so you pay 48 percent of the price.

Worked example 7: the price before tax. A receipt shows a total of $53.50, including 7 percent tax. What was the price before tax?

Step 1. The total is 107 percent of the price: price × 1.07 = 53.50.

Step 2. Undo the multiplication: 53.50 ÷ 1.07 = $50.00.

Step 3. Check: 50 × 1.07 = 53.50. Taking 7 percent off the total instead gives 53.50 × 0.93 = 49.755, about $49.76, which is wrong, because the tax was 7 percent of $50, not of $53.50.

Bottom line: Before you multiply, say out loud what the percent is a percent of: the tag price, the sale price, the food bill, or the unknown price before tax. Nearly every receipt error is a percent taken of the wrong amount.

Common misconceptions

"25 percent off and 7 percent tax is 18 percent off." That gives 48 × 0.82 = $39.36, not $38.52. The tax is 7 percent of the smaller sale price, so the two percents cannot simply be subtracted. Multiply the multipliers: 0.75 × 1.07 = 0.8025.

"40 percent off plus an extra 20 percent off is 60 percent off." The extra 20 percent is taken of the sale price, so the total is 52 percent off.

"A markup and a sale of the same size cancel." A 25 percent markup on $40 gives $50, and 25 percent off $50 gives $37.50, not $40. Each percent is of a different amount.

"To remove the tax, take 7 percent off the total." Divide by 1.07 instead. The price before tax of a $53.50 total is $50.00, not $49.76.

The takeaway

• A till works in order: discount off the tag price, then tax on the sale price, each rounded to the nearest cent.
• Each step is one multiplier: 0.75 for 25 percent off, 1.07 for 7 percent tax, 1.60 for a 60 percent markup. The order of the multipliers does not change the product.
• A markup is a percent of the wholesale price; a tip is a percent of the bill you choose, often the food before tax.
• A percent-off coupon grows with the price and a dollars-off coupon does not, so the better coupon depends on the price. They tie where the percent of the price equals the dollars off.
• Stacked percents multiply, they do not add, and working backwards from a total means dividing by the multiplier.

The next lesson turns to the other side of the till: the percent a salesperson earns as commission, and the percent a bank pays as interest.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Ratios and Proportional Relationships (7.RP.A.3) and Expressions and Equations (7.EE.A.2). thecorestandards.org
2. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 6.3, Solve sales tax, commission, and discount applications. Prealgebra 2e. OpenStax. openstax.org
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 4, Lesson 10: Tax and tip. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Indiana Department of Revenue. (n.d.). Sales tax. State of Indiana. Retrieved September 24, 2026, for the seven percent state rate. in.gov

Three ways to be paid for selling bikes

A bike shop offers your cousin a summer job, and lets her choose how she is paid. Plan A is $400 a week, whatever she sells. Plan B is $150 a week plus 10 percent of her sales. Plan C is 20 percent of her sales and nothing else. She thinks she can sell about $2,000 of bikes a week. Which plan should she take?

No single calculation answers that, because the answer depends on the sales. What answers it is a table, read carefully. The same is true of the second half of this lesson, where the choice is between ways of saving and borrowing money.

This lesson covers Common Core standard 7.RP.A.3, which lists commissions, fees and simple interest among the multistep percent problems you should be able to solve, and uses the equation thinking of 7.EE.B.4.

Commission is a percent of sales

A commission is pay worked out as a percent of the value of what someone sells. At 10 percent commission, selling a $300 bike earns 0.10 × 300 = $30. Commission is how many salespeople, estate agents and some shop workers are paid, either on its own or on top of a fixed base wage.

Worked example 1: one week at $2,000 of sales.

Step 1. Plan A pays a flat $400.

Step 2. Plan B pays the base plus 10 percent of sales: $150 + 0.10 × $2,000 = $150 + $200 = $350.

Step 3. Plan C pays 20 percent of sales: 0.20 × $2,000 = $400.

At $2,000 of sales, Plans A and C tie at $400 and Plan B pays least. But one week's sales could easily be higher or lower, so look at more than one.

The three pay plans side by side

Read the table down each column, then across each row.

• Plan A never changes. It is the safe choice for a slow week.
• Plan C starts lowest and grows fastest, $100 more for every extra $500 of sales. It is proportional: pay = 0.20 × sales, a straight line through the origin, because zero sales earn $0.
• Plan B grows too, but only half as fast as Plan C, and its $150 base means it is not proportional: zero sales still pay $150.
• Across the rows, Plan A wins below $2,000 of sales, Plan C wins above $2,000, and they tie at exactly $2,000.

One more thing the table shows, if you look for it: Plan B is never the best plan. It beats Plan C only when sales are under $1,500, where Plan A beats both, and it beats Plan A only above $2,500, where Plan C is already ahead. A plan that sounds like a sensible middle choice turns out to lose to one of the others at every level of sales.

What matters here: A fixed amount wins when sales are low and a percent wins when they are high. To choose, find where the plans tie and ask which side of that point you expect to be on.

Worked example 2: where Plans A and C tie.

Step 1. Plan C pays 0.20 × sales, and Plan A pays $400. They tie when 0.20 × sales = 400.

Step 2. Ask what number times 0.20 is 400: sales = 400 ÷ 0.20 = $2,000.

Check: 20 percent of $2,000 is $400, matching the table.

Worked example 3: working backwards from a paycheck. One week on Plan B, your cousin is paid $620. What were her sales?

Step 1. Take off the base: $620 - $150 = $470. That is the commission part.

Step 2. The commission is 10 percent of sales, so sales = 470 ÷ 0.10 = $4,700.

Check: $150 + 0.10 × $4,700 = $150 + $470 = $620.

Fees: a percent added on

Some charges work like a commission paid by the customer. A ticket website might add a service fee of a percent of the ticket price.

Worked example 4: concert tickets with a fee. Four tickets cost $45 each, and the website adds a 12 percent service fee.

Step 1. Ticket total: 4 × $45 = $180.

Step 2. Fee: 0.12 × $180 = $21.60.

Step 3. Total: $180 + $21.60 = $201.60. As a multiplier: 180 × 1.12 = 201.60.

Simple interest: I = Prt

When you put money in a savings account, the bank pays you for the use of it, and when you borrow money you pay the lender. That payment is interest. With simple interest, the interest each year is the same percent of the original amount, called the principal, so the total interest is

I = P × r × t, usually written I = Prt,

where P is the principal in dollars, r is the yearly interest rate written as a decimal, and t is the time in years. It is a proportional relationship in t: each extra year adds the same amount of interest.

Worked example 5: a savings deposit. You deposit $500 at 2 percent simple interest a year for 3 years.

Step 1. Write the rate as a decimal: 2 percent = 0.02.

Step 2. I = 500 × 0.02 × 3 = 10 × 3 = $30. That is $10 each year for 3 years.

Step 3. The balance at the end is the principal plus the interest: $500 + $30 = $530.

A real example of interest paid this way: the U.S. Treasury sells notes that pay a fixed rate of interest every six months until they mature. If a $1,000 note paid 4 percent a year, each payment would be half a year's interest, 1,000 × 0.04 × 0.5 = $20, and over 2 years the four payments would add up to 1,000 × 0.04 × 2 = $80.

Three savings plans side by side

Read the table. Plan 2 has the lowest rate and still earns the most interest, because its principal is the largest and its time is the longest. Plan 3 has the highest rate and earns the least, because the money stays in for only 2 years. Plans 1 and 3 start with the same $600: the higher rate in Plan 3 does not make up for having half the time. Three numbers multiply to make the interest, and a big value of one can be outweighed by small values of the others.

Key idea: Simple interest is principal times rate times time. Doubling any one of the three doubles the interest, which is why a table of all three, not the rate alone, decides which plan earns more.

Worked example 6: borrowing for 18 months. You borrow $300 at 6 percent simple interest a year and repay it after 18 months.

Step 1. Time must be in years: 18 months = 18 ÷ 12 = 1.5 years.

Step 2. I = 300 × 0.06 × 1.5 = 18 × 1.5 = $27.

Step 3. You repay $300 + $27 = $327.

Worked example 7: finding the rate. A $400 deposit earns $36 of simple interest in 2 years. What was the rate?

Step 1. Put what you know into I = Prt: 36 = 400 × r × 2 = 800 × r.

Step 2. Divide: r = 36 ÷ 800 = 0.045.

Step 3. As a percent, the rate is 4.5 percent a year. Check: 400 × 0.045 × 2 = 36.

Common misconceptions

"Commission is a fixed amount per sale." Commission is a percent of the value sold, so a $600 sale earns twice the commission of a $300 sale. A fixed amount per sale, like $3 a sale, is a different kind of pay.

"The plan with the biggest percent always pays the most." The 20 percent plan paid least of all at $1,000 of sales. Percents are compared at particular amounts, which is what the table is for.

"5 percent interest for 3 years means 5 percent in total." The rate is per year. Simple interest at 5 percent for 3 years is 15 percent of the principal.

"Put the months straight into the formula." In I = Prt the time is in years. Six months is t = 0.5, not t = 6; using 6 makes the interest twelve times too big.

"Use the rate as 4, not 0.04." A rate of 4 percent is 4/100 = 0.04. Using 4 makes the interest a hundred times too big: $600 at 3 percent for 4 years would come out as $7,200 instead of $72.

Looking back

• Commission is a percent of sales. A commission-only plan is proportional to sales; a base wage plus commission is not, because zero sales still earn the base.
• To compare pay plans, make a table over a range of sales, find where the plans tie, and read which plan wins on each side of that point.
• A percent fee is added to a price the same way a tax is, with a multiplier such as 1.12.
• Simple interest is I = Prt, with r as a decimal and t in years. The final balance is P + I.
• Working backwards means undoing the steps in reverse order: take off a base, then divide by the rate; or divide the interest by P × t to find r.

The next module steps back from percents to the expressions underneath them, such as 150 + 0.10s, and the ways of rewriting them.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Ratios and Proportional Relationships (7.RP.A.3). thecorestandards.org
2. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 6.4, Solve simple interest applications. Prealgebra 2e. OpenStax. openstax.org
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 4, Lesson 11: Percentage contexts. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. U.S. Department of the Treasury, Bureau of the Fiscal Service. (n.d.). Treasury notes. TreasuryDirect. Retrieved September 24, 2026. treasurydirect.gov

Twelve kits and one coupon

The art club is ordering supplies for its 12 members. Each member gets a kit of 3 brushes, at b dollars a brush, and 2 tubes of paint at $4.50 a tube. The club also has one $15 coupon for the whole order. The club's treasurer writes the cost of the order as 12(3b + 9) - 15. The art shop's website shows the same order as 36b + 93. Is the website right? And how could you know, without simply trusting one of them?

Try it yourself before reading on. Where did the 9 come from, and where could 36 and 93 have come from?

This lesson covers Common Core standard 7.EE.A.1: applying properties of operations as strategies to add, subtract, factor and expand linear expressions with rational coefficients.

Working the order

Worked example 1: two ways of writing the art order.

Step 1. One kit costs 3 brushes at b dollars, which is 3b, plus 2 tubes at $4.50, which is 2 × 4.50 = $9. So a kit costs 3b + 9 dollars. That is the treasurer's 9.

Step 2. Twelve kits cost 12(3b + 9). The brackets matter: all 12 members get the brushes and the paint.

Step 3. Multiply each part of the kit by 12: 12 × 3b = 36b, and 12 × 9 = 108. So 12(3b + 9) = 36b + 108. That is the order before the coupon, written as 36 brushes and $108 of paint.

Step 4. Take off the coupon: 36b + 108 - 15 = 36b + 93.

The website is right. Now check it without algebra. Suppose brushes cost $2. The treasurer's version gives 12(3 × 2 + 9) - 15 = 12 × 15 - 15 = 180 - 15 = $165. The website's version gives 36 × 2 + 93 = 72 + 93 = $165. Try a second price, $1.25: 12(3.75 + 9) - 15 = 12 × 12.75 - 15 = 153 - 15 = 138, and 36 × 1.25 + 93 = 45 + 93 = 138. Two expressions that give the same value for every b are called equivalent expressions, and two matching checks with ordinary numbers make a good test.

Why not check with b = 0? Because a common wrong answer, 3b + 93, where someone multiplied only the 9 by 12, also gives 93 when b = 0. Zero wipes out every term with a letter in it, so it cannot catch a mistake in those terms. Check with numbers such as 2 or 1.25 instead.

The point: The two expressions describe the same order in two ways: 12(3b + 9) - 15 says "twelve kits, less a coupon", and 36b + 93 says "36 brushes and $93 of everything else". Rewriting changes the description, never the cost.

The words for the parts of an expression

In 36b + 93, the parts joined by + or - are called terms: 36b is one term and 93 is another. The number multiplying a letter is its coefficient, so the coefficient of b is 36. A term with no letter, like 93, is a constant. Like terms have the same letter part: 36b and -4b are like terms, and so are 93 and 15, but 36b and 93 are not.

The rule that did the work in Step 3 is the distributive property: a(b + c) = ab + ac. Multiplying a sum by a number multiplies every term in the sum by that number. Going from 12(3b + 9) to 36b + 108 is called expanding.

Combining like terms

Like terms can be added because they count the same kind of thing. If Maya's basket has 3 apples and 2 bananas and Leo's has 5 apples and 1 banana, together they have 8 apples and 3 bananas. With a for the price of an apple and n for the price of a banana, (3a + 2n) + (5a + n) = 8a + 3n. You cannot turn 8 apples and 3 bananas into 11 of anything, and for the same reason 8a + 3n does not simplify any further.

Behind the scenes this is the distributive property again: 3a + 5a = (3 + 5)a = 8a.

Worked example 2: like terms with fractions. Simplify (3/4)x + 2 - (1/2)x + 5/2.

Step 1. Group the like terms, keeping each sign attached to the term after it: (3/4)x - (1/2)x, and 2 + 5/2.

Step 2. Combine the x terms: 3/4 - 1/2 = 3/4 - 2/4 = 1/4, giving (1/4)x.

Step 3. Combine the constants: 2 + 5/2 = 4/2 + 5/2 = 9/2.

Step 4. The result is (1/4)x + 9/2.

Check with x = 4: the original gives 3 + 2 - 2 + 2.5 = 5.5, and the result gives 1 + 4.5 = 5.5.

Subtracting an expression, and negative coefficients

A minus sign in front of brackets means "subtract everything inside". Since subtracting is adding the opposite, -(3x - 4) is -1 × (3x - 4) = -3x + 4. The sign of every term inside changes.

Worked example 3: subtract one expression from another. Simplify (5x + 2) - (3x - 4).

Step 1. Distribute the minus: -(3x - 4) = -3x + 4.

Step 2. Rewrite: 5x + 2 - 3x + 4.

Step 3. Combine: 5x - 3x = 2x, and 2 + 4 = 6. The result is 2x + 6.

Check with x = 1: (5 + 2) - (3 - 4) = 7 - (-1) = 8, and 2 + 6 = 8. The common wrong answer 2x - 2 comes from changing only the first sign; at x = 1 it gives 0, and the check catches it.

Worked example 4: a negative fraction outside the brackets. Expand -(1/2)(6x - 8).

Step 1. Multiply the first term: -(1/2) × 6x = -3x.

Step 2. Multiply the second term: -(1/2) × (-8) = +4. A negative times a negative is positive, as in the second module.

Step 3. The result is -3x + 4. Check with x = 2: -(1/2)(12 - 8) = -(1/2)(4) = -2, and -6 + 4 = -2.

Factoring: the distributive property run backwards

Factoring undoes expanding. You look for a factor that every term shares, take it outside the brackets, and write what is left inside. The art order runs backwards too: 36b + 108 = 12(3b + 9), which says the order is 12 identical kits.

Worked example 5: factor 18x + 24.

Step 1. Find the greatest common factor of 18 and 24. Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The greatest shared one is 6.

Step 2. Divide each term by 6: 18x ÷ 6 = 3x and 24 ÷ 6 = 4.

Step 3. The result is 6(3x + 4). Check by expanding: 6 × 3x + 6 × 4 = 18x + 24.

2(9x + 12) is also equivalent, but it is not factored completely, because 9x + 12 still shares a factor of 3.

Worked example 6: factor out a fraction. Write (2/3)x - 4 with 2/3 outside the brackets.

Step 1. The first term: (2/3)x ÷ (2/3) = x.

Step 2. The second term: 4 ÷ (2/3) = 4 × 3/2 = 6, so -4 becomes -6 inside.

Step 3. The result is (2/3)(x - 6). Check by expanding: (2/3)x - (2/3) × 6 = (2/3)x - 4.

Remember: Expanding multiplies every term inside by the factor outside; factoring divides every term by the factor you take out. Either way, expand your answer or substitute two numbers to check it.

A perimeter with fractions

Worked example 7: a garden bed. A rectangular garden bed is 2x + 3/2 feet long and x - 1/4 feet wide. Write its perimeter as simply as possible.

Step 1. Perimeter is two lengths plus two widths: 2(2x + 3/2) + 2(x - 1/4).

Step 2. Expand each part: 4x + 3 and 2x - 1/2.

Step 3. Combine like terms: 4x + 2x = 6x, and 3 - 1/2 = 5/2. The perimeter is 6x + 5/2 feet.

Check with x = 1: the bed is 3.5 feet by 0.75 feet, so the perimeter is 2 × 3.5 + 2 × 0.75 = 7 + 1.5 = 8.5 feet, and 6 + 2.5 = 8.5.

Common misconceptions

"3p + 2n = 5pn." 3p and 2n are unlike terms: three pencils and two notebooks are not five pencil-notebooks. The expression 3p + 2n is already as simple as it gets.

"2(x + 5) = 2x + 5." The factor multiplies every term inside the brackets: 2(x + 5) = 2x + 10. Check with x = 1: 2 × 6 = 12, but 2 + 5 = 7.

"-(3x - 4) = -3x - 4." Subtracting the whole bracket changes the sign of every term, so -(3x - 4) = -3x + 4.

"x + x = x squared." Adding two x's gives 2x. Multiplying x by x gives x squared, a different expression: at x = 3, 2x = 6 but x squared = 9.

"7 - 2(x - 3) = 5(x - 3)." Multiplication comes before subtraction, so the 2 belongs to the brackets and the 7 waits. The right expansion is 7 - 2x + 6 = 13 - 2x.

Where this leaves us

• Terms are the parts joined by + and -, a coefficient multiplies a letter, and like terms have the same letter part.
• The distributive property, a(b + c) = ab + ac, expands brackets by multiplying every term inside; it works the same with fractions, decimals and negatives.
• Combining like terms adds their coefficients, (3/4)x - (1/2)x = (1/4)x, and unlike terms stay apart.
• A minus sign in front of brackets changes the sign of every term inside.
• Factoring takes out a common factor, whole or fractional, by dividing every term by it; expand the result to check.
• Two expressions are equivalent if they agree for every value of the letter. Checking two ordinary numbers, not 0, catches almost every slip.

The next lesson uses these moves for a different purpose: rewriting an expression so that it shows something about a situation that the first form hid.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Expressions and Equations (7.EE.A.1). thecorestandards.org
2. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 7.3, Distributive property. Prealgebra 2e. OpenStax. openstax.org
3. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 2.2, Evaluate, simplify, and translate expressions. Prealgebra 2e. OpenStax. openstax.org
4. Illustrative Mathematics. (n.d.). Grade 7, Unit 6, Lesson 19: Expanding and factoring. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
5. Illustrative Mathematics. (n.d.). Grade 7, Unit 6, Lesson 22: Combining like terms (Part 3). IM 6-8 Math. Kendall Hunt. im.kendallhunt.com

Twenty-four tiles around a square

Mr. Okafor's seventh-grade class is building a square garden bed in the school courtyard, and they want a border of 1-foot square paving tiles all the way around it. They have not decided how big the bed will be, so he asks each student to write an expression for the number of border tiles when the garden is s feet on each side. To help, he draws the case s = 5 on the board: a 5-by-5 garden with a ring of 24 tiles around it.

Four students hand in four different expressions. All four of them are right. Following why they are all right, and what each one notices that the others do not, is the story of this lesson.

This lesson covers Common Core standard 7.EE.A.2: understanding that rewriting an expression in different forms in a problem context can show how the quantities in it are related, as in the standard's own example, where a + 0.05a = 1.05a means that "increase by 5 percent" is the same as "multiply by 1.05".

Ana, Ben, Chloe and Dev

Worked example 1: Ana's expression, 4s + 4. Ana counted the sides and the corners separately. Each of the 4 sides has s tiles running along the garden, which makes 4s, and then the 4 corner squares are left over.

Step 1. Sides: 4 × s = 4s. Corners: 4.

Step 2. Total: 4s + 4. For the drawing, s = 5: 4 × 5 + 4 = 24. The drawing has 24 tiles.

Worked example 2: Ben's expression, 4(s + 1). Ben walked round the border. He took one side of s tiles together with the corner at its end, which is s + 1 tiles, and saw that the border is four such pieces, turning at each corner.

Step 1. One piece: s + 1. Four pieces: 4(s + 1).

Step 2. Expand to compare with Ana: 4(s + 1) = 4s + 4. The two expressions are equivalent.

Worked example 3: Chloe's expression, 2s + 2(s + 2). Chloe used the top and bottom rows as long rows that include the corners: each is s + 2 tiles. The left and right sides, between those rows, are s tiles each.

Step 1. Two long rows: 2(s + 2). Two short sides: 2s.

Step 2. Expand and combine: 2s + 2s + 4 = 4s + 4. Equivalent again.

Worked example 4: Dev's expression, 4(s + 2) - 4. Dev pretended every side was a long row of s + 2 tiles. That counts each corner twice, once for each side it touches, so he took the 4 extra corners back off.

Step 1. Four long rows: 4(s + 2) = 4s + 8.

Step 2. Remove the double-counted corners: 4s + 8 - 4 = 4s + 4. Equivalent.

A fifth student wrote 4(s + 2), with no correction. For s = 5 that gives 28, four too many, and when you expand it you get 4s + 8, which is not equivalent to 4s + 4. Rewriting catches the mistake that the drawing alone might not.

Why this matters: Equivalent expressions give the same number every time, but they are different descriptions. Each form records a way of seeing, and a form that matches the question you are asking makes that question easier.

The form that answers the question

The class then learns that the school has 60 spare tiles. How big can the garden be if the border uses all 60?

Worked example 5: working backwards with Ben's form.

Step 1. Ben's form says the border is 4 equal pieces, so each piece has 60 ÷ 4 = 15 tiles.

Step 2. Each piece is s + 1 tiles, so s + 1 = 15 and s = 14.

Step 3. Check with Ana's form: 4 × 14 + 4 = 56 + 4 = 60. The garden can be 14 feet on each side.

With Ana's form you could still get there, 4s + 4 = 60, but Ben's grouping lets you do it in your head: split the tiles into four, then take one off. Which form is best depends on the question, not on which one is shortest.

The plant sale: percents as single multiplications

To pay for the plants, the class runs a seedling sale. They buy seedlings for c dollars each and sell them at a 40 percent markup, as in the percent module.

Worked example 6: two forms of one price.

Step 1. Selling price = cost plus 40 percent of the cost: c + 0.4c.

Step 2. Combine like terms: c + 0.4c = 1c + 0.4c = 1.4c.

Step 3. The two forms say the same thing in two ways. c + 0.4c says "the cost plus the markup". 1.4c says "140 percent of the cost", which is quicker on a calculator. For a seedling costing $2.50: 2.50 + 0.4 × 2.50 = 2.50 + 1.00 = $3.50, and 1.4 × 2.50 = $3.50.

Then the principal asks for a teachers' discount of 10 percent off the selling price. The teacher price is 1.4c - 0.1(1.4c). Factoring out 1.4c gives (1 - 0.1)(1.4c) = 0.9 × 1.4c = 1.26c. That form tells the class something they did not see before: on a sale to a teacher, their markup is 26 percent, not 40. For the $2.50 seedling, 1.26 × 2.50 = $3.15, which is $3.50 less 35 cents.

In short: Adding a percent of an amount is the same as multiplying by 1 plus the rate, and taking a percent away is the same as multiplying by 1 minus the rate. Rewriting in that form turns a chain of percents into one number you can read.

Sharing the cost

Worked example 7: one group's share. The garden is planted by 6 groups. The soil for the whole bed costs $48, shared equally, and each group also buys its own plants for x dollars. What does one group pay?

Step 1. Total cost: 48 + 6x.

Step 2. One group's share is the total divided by 6: (48 + 6x) ÷ 6. Dividing by 6 is multiplying by 1/6, so the distributive property applies: (1/6)(48) + (1/6)(6x) = 8 + x.

Step 3. The rewritten form explains itself: each group pays $8 for its share of the soil plus the cost of its own plants. If a group's plants cost $5.25, it pays 8 + 5.25 = $13.25. Check with the first form: (48 + 6 × 5.25) ÷ 6 = (48 + 31.50) ÷ 6 = 79.50 ÷ 6 = $13.25.

Common misconceptions

"Only one of the four border expressions can be the right one." All four are right because they are equivalent: each rewrites to 4s + 4. A situation usually has many correct expressions, and a teacher who marks only one form correct is testing memory, not mathematics.

"The shortest expression is the best." The best form is the one that answers your question. 4(s + 1) was longer to write than 4s + 4, and it solved the 60-tile question in two mental steps.

"Increase by 5 percent means multiply by 0.05." Multiplying by 0.05 gives only the increase. The new amount is p + 0.05p = 1.05p.

"Two increases of 20 percent make an increase of 40 percent." Written as expressions, 1.2 × 1.2t = 1.44t, a 44 percent increase, because the second 20 percent is taken of the bigger amount.

What to carry forward

• One situation can be described by many equivalent expressions, such as 4s + 4, 4(s + 1), 2s + 2(s + 2) and 4(s + 2) - 4.
• Expanding and combining like terms show that two forms are equivalent; substituting numbers is a quick check, and expanding catches forms that are not equivalent, such as 4(s + 2).
• Each form carries a way of seeing the situation, and the form that matches the question makes it easiest: 4(s + 1) = 60 gives s + 1 = 15 at once.
• Percent changes rewrite as single multiplications: c + 0.4c = 1.4c, and 0.9 × 1.4c = 1.26c.
• Dividing an expression by a number shares out every term: (48 + 6x) ÷ 6 = 8 + x.

The next module turns expressions into equations, such as 4(s + 1) = 60, and solves them step by step.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Expressions and Equations (7.EE.A.2). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 6, Lesson 23: Applications of expressions. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 6, Lesson 20: Combining like terms (Part 1). IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 7.3, Distributive property. Prealgebra 2e. OpenStax. openstax.org

$21.50 for a taxi ride

A taxi in your town charges $3.50 the moment you get in, and then $2.25 for every mile. Your ride home from the train station costs $21.50. How far is it from the station to your house?

You could guess and check: 5 miles would cost 3.50 + 5 × 2.25 = $14.75, too little, and 10 miles would cost $26.00, too much. Guessing works, but slowly. This lesson gives you a method that goes straight to the answer and works just as well when the numbers are fractions or negative.

This lesson covers Common Core standard 7.EE.B.4a: solving word problems that lead to equations of the form px + q = r, solving such equations fluently, and comparing an algebraic solution with an arithmetic one by identifying the sequence of operations used in each.

Writing the equation

Let m be the number of miles. The miles cost 2.25 × m, which is 2.25m, and the $3.50 is added once. The fare is $21.50, so

2.25m + 3.50 = 21.50.

That is the form px + q = r, with p = 2.25, q = 3.50 and r = 21.50: a number times the unknown, plus a number, equals a number. An equation says two expressions are equal, and a solution is a value of the letter that makes the equation true.

A balance that must stay level

Think of an equation as a balance scale that is level. Whatever is on the left weighs exactly as much as whatever is on the right. You may change the scale however you like, as long as you do the same thing to both sides, because then it stays level.

Here is a smaller equation on the scale: three boxes of unknown weight x and two 1-unit weights on the left, eleven 1-unit weights on the right. Take 2 units off each side and it stays level: 3x = 9. Now the three boxes balance nine units, so split both sides into three equal groups: one box balances 3 units, and x = 3. Those two moves are the whole method.

• You may add or subtract the same number on both sides.
• You may multiply or divide both sides by the same number, as long as it is not zero.

The taxi, end to end

Worked example 1: solve 2.25m + 3.50 = 21.50. The plan is to get m on its own. The m has been multiplied by 2.25 and then had 3.50 added, so undo those in reverse order: take away the 3.50 first, then divide by 2.25.

Step 1. Subtract 3.50 from both sides: 21.50 - 3.50 = 18.00, so 2.25m = 18.

Step 2. Divide both sides by 2.25: 18 ÷ 2.25 = 1800 ÷ 225 = 8, so m = 8.

Step 3. Check in the original equation: 2.25 × 8 + 3.50 = 18 + 3.50 = 21.50. It is true, so the ride is 8 miles.

Step 4. Check against the guesses: 8 lies between 5 miles, which was too cheap, and 10 miles, which was too dear. The answer is reasonable.

The arithmetic route uses the same steps

You could also solve the taxi problem without an equation. Take off the $3.50 start charge, leaving $18.00 for the miles; at $2.25 a mile that is 18 ÷ 2.25 = 8 miles. The Common Core standard asks you to notice that this is the same pair of operations, in the same order, as the balance method.

Worked example 2: the standard's own rectangle. The perimeter of a rectangle is 54 cm and its length is 6 cm. What is its width?

Check: a 6 cm by 21 cm rectangle has perimeter 6 + 21 + 6 + 21 = 54 cm. Both routes subtract 12 and then divide by 2. The arithmetic route keeps the story in your head; the algebra route writes each state down, which is what lets it handle harder numbers without losing track.

The upshot: Solving px + q = r means undoing what was done to x, in reverse order: remove the q first, then divide by p. The balance rule, same to both sides, is what makes each step safe.

Fractions, negatives and decimals

Worked example 3: a growing plant. A seedling is 5 cm tall and grows 2/3 cm a day. After how many days will it be 17 cm tall?

Step 1. Equation, with d days: (2/3)d + 5 = 17.

Step 2. Subtract 5 from both sides: (2/3)d = 12.

Step 3. Divide both sides by 2/3, which is the same as multiplying by 3/2: d = 12 × 3/2 = 18.

Step 4. Check: (2/3) × 18 + 5 = 12 + 5 = 17. It takes 18 days.

Worked example 4: a falling temperature. At 6 p.m. the temperature is 12 °C, and it falls 1.5 °C every hour. After how many hours will it be -3 °C?

Step 1. Falling 1.5 degrees an hour is a change of -1.5 each hour, so after h hours the temperature is 12 - 1.5h. The equation is 12 - 1.5h = -3.

Step 2. Subtract 12 from both sides: -1.5h = -3 - 12 = -15.

Step 3. Divide both sides by -1.5: h = -15 ÷ -1.5 = 10. A negative divided by a negative is positive.

Step 4. Check: 12 - 1.5 × 10 = 12 - 15 = -3. It happens 10 hours after 6 p.m., at 4 a.m.

Worked example 5: books and a bookmark. You spend $38.50 on 3 paperbacks of the same price and a $4.75 bookmark. What does each paperback cost?

Step 1. Equation, with b for the price of a book: 3b + 4.75 = 38.50.

Step 2. Subtract 4.75: 3b = 33.75.

Step 3. Divide by 3: b = 11.25.

Step 4. Check: 3 × 11.25 + 4.75 = 33.75 + 4.75 = 38.50. Each book costs $11.25.

Change one number, and the answer stops making sense

Worked example 6: a fare that cannot happen. A friend says her ride in the same taxi cost $2.60. Solve 2.25m + 3.50 = 2.60 exactly as before.

Step 1. Subtract 3.50: 2.25m = 2.60 - 3.50 = -0.90.

Step 2. Divide by 2.25: m = -0.90 ÷ 2.25 = -0.4.

Step 3. The algebra has worked perfectly, and the check passes: 2.25 × (-0.4) + 3.50 = -0.90 + 3.50 = 2.60. But a ride of -0.4 miles is not a ride. The situation says the fare is at least $3.50, since you pay that before the taxi moves, so no ride costs $2.60. Your friend has misremembered, or took a different taxi.

Changing one input, the fare, from $21.50 to $2.60 turned a sensible answer into a solution the situation rejects. An equation always obeys its own rules; only you can ask whether its answer fits the world it came from.

So what?: Every solution gets two checks. Put it back into the equation to test the algebra, then read it back into the story to test the sense: a negative number of miles, a fraction of a person or a time before the start should make you stop and think.

Common misconceptions

"Divide first." Dividing 2.25m + 3.50 = 21.50 by 2.25 has to divide every term, the 3.50 as well, which gives messy numbers. Students who divide only the 2.25m and the 21.50 get m + 3.50 = 9.56 and a wrong answer of about 6.06 miles. Removing the added number first keeps it simple.

"Do it to one side only." Subtracting 3.50 from the left and not from the right tips the balance. Every step happens to both sides, or the new equation is not equivalent to the old one.

"-1.5h = -15, so h = -10." A negative divided by a negative is positive, so h = 10. The check catches this: 12 - 1.5 × (-10) = 27, not -3.

"If the check works, the answer must make sense." The check tests the algebra only. The -0.4 miles passed the check and still made no sense as a ride.

Summing up

• A problem of the form "a number times the unknown, plus a number, equals a total" gives an equation px + q = r. Define the letter before you write it.
• Keep the balance: add, subtract, multiply or divide both sides by the same number, never zero, and the equation stays true.
• Undo in reverse order: remove q, then divide by p. With a fraction p, dividing is multiplying by its reciprocal; with a negative p, watch the sign.
• The arithmetic route uses the same operations in the same order, subtract then divide; the algebra route writes each step down.
• Check every solution twice: in the equation, and in the situation.

The next lesson solves equations with brackets, p(x + q) = r, in two different orders, and traces a mistake that the brackets invite.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Expressions and Equations (7.EE.B.4a). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 6, Lesson 7: Reasoning about solving equations (Part 1). IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 6, Lesson 11: Using equations to solve problems. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 8.3, Solve equations with variables and constants on both sides. Prealgebra 2e. OpenStax. openstax.org

Jordan's answer: $11.67 a ticket

Three friends go to the cinema. Each of them buys a ticket and a $4 popcorn, and together they spend $39. Jordan wants to know what one ticket costs, and he writes:

A ticket for $11.67 and a popcorn for $4 would be $15.67 each, and three of those would be $47, not $39. Something has gone wrong. The job in this lesson is to find the exact line, fix it, and learn the two reliable ways to solve equations like this one.

This lesson covers Common Core standard 7.EE.B.4a for equations of the form p(x + q) = r, where p, q and r are rational numbers, including comparing an algebraic solution with an arithmetic one.

Checking first, then tracing

Put Jordan's answer back into his first line, which is the equation the story gave: 3(11.67 + 4) = 3 × 15.67 = 47.01. The left side should be 39. So the answer fails the check, and the mistake is somewhere between line 1 and line 4.

Now test each line against the one before it.

• Line 1 is right. Each friend spends t + 4 dollars, and there are three friends, so 3(t + 4) = 39.
• Line 2 is where it breaks. The 3 outside the brackets multiplies everything inside: three tickets and three popcorns. So 3(t + 4) = 3t + 12, not 3t + 4. Jordan multiplied only the t.
• Lines 3 and 4 are correct steps from line 2, which is why they look fine. A wrong line produces a sensible-looking chain of steps after it.

Worth holding on to: When a check fails, test each line against the one before it, not just the last line. The error is usually one line, and everything after it is correct algebra on a wrong equation.

Two correct routes to the same ticket

Worked example 1: solve 3(t + 4) = 39 by dividing first.

Step 1. The left side is 3 groups of (t + 4). Divide both sides by 3: t + 4 = 39 ÷ 3 = 13.

Step 2. Subtract 4 from both sides: t = 9.

Worked example 2: solve 3(t + 4) = 39 by distributing first.

Step 1. Expand: 3t + 12 = 39. This time every term inside is multiplied by 3.

Step 2. Subtract 12 from both sides: 3t = 27.

Step 3. Divide both sides by 3: t = 9.

Check either way: 3(9 + 4) = 3 × 13 = 39. A ticket costs $9.

The arithmetic route matches the first method exactly. Share the $39 among three friends, which is 39 ÷ 3 = $13 each, then take off each friend's $4 popcorn, which leaves $9. Divide, then subtract: the same operations in the same order as dividing first.

Brackets or no brackets: read the story

Change the story slightly. The three friends buy one large popcorn for $4 to share, and they spend $37 altogether. Now the popcorn is added once, not three times, and the equation is 3t + 4 = 37, the form px + q = r from the last lesson: 3t = 33, so t = $11. Check: 3 × 11 + 4 = 37.

So the question to ask is whether the extra amount belongs to each of the equal groups or is added once to the whole. If each group gets it, the story is p(x + q) = r. If it is added once, the story is px + q = r. Jordan's line 2 is exactly the shared-popcorn equation, which is why his mistake is so easy to make: it turns one story into the other.

Three more wrong answers, traced

Worked example 3: a negative outside the brackets. Solve -2(x - 5) = 18. A student writes -2x - 10 = 18, then -2x = 28, then x = -14.

Step 1. Check: -2(-14 - 5) = -2 × (-19) = 38, not 18. The answer fails.

Step 2. Trace: -2 times -5 is +10, because a negative times a negative is positive. The first line should be -2x + 10 = 18.

Step 3. Fix, distributing first: -2x + 10 = 18, so -2x = 8 and x = -4. Or divide first: x - 5 = 18 ÷ (-2) = -9, so x = -4.

Step 4. Check: -2(-4 - 5) = -2 × (-9) = 18.

Worked example 4: a fraction outside the brackets. You and a friend split the price of a game and its $6 delivery charge equally, and your half comes to $11. A student writes (1/2)g + 6 = 11 and gets g = 10.

Step 1. Check in the story: a $10 game plus $6 delivery is $16, and half of that is $8, not $11.

Step 2. Trace: you pay half of the delivery as well as half of the game, so the equation is (1/2)(g + 6) = 11. Writing (1/2)g + 6 halves only the game.

Step 3. Fix, dividing first: dividing by 1/2 is multiplying by 2, so g + 6 = 22 and g = 16.

Step 4. Check: (1/2)(16 + 6) = (1/2)(22) = 11. The game costs $16.

Worked example 5: decimals. Solve 2.5(y - 1.2) = 7. A student divides 7 by 2.5 correctly, gets 2.8, and then writes y = 2.8 - 1.2 = 1.6.

Step 1. Check: 2.5(1.6 - 1.2) = 2.5 × 0.4 = 1, not 7.

Step 2. Trace: after dividing, the equation is y - 1.2 = 2.8. To undo subtracting 1.2 you add 1.2, so y = 2.8 + 1.2 = 4. The student used the same operation instead of the inverse one.

Step 3. Check: 2.5(4 - 1.2) = 2.5 × 2.8 = 7.

Choosing the easier route

Both routes always give the same answer, so choose the one that keeps the numbers friendly. Illustrative Mathematics suggests looking at the numbers first, and two examples show why.

Worked example 6: distribute first. A square flower bed has sides of x + 1/4 metres, and its border is 9 metres long, so 4(x + 1/4) = 9.

Step 1. Distributing clears the fraction at once: 4x + 1 = 9.

Step 2. Subtract 1: 4x = 8. Divide by 4: x = 2.

Step 3. Check: each side is 2 1/4 metres, and 4 × 2 1/4 = 9. Dividing first also works, x + 1/4 = 9/4, but it keeps the quarters longer.

Worked example 7: divide first. Solve (3/4)(x + 10) = 12.

Step 1. Distributing would give (3/4)x + 7.5, a decimal. Dividing by 3/4 instead means multiplying both sides by 4/3: x + 10 = 12 × 4/3 = 16.

Step 2. Subtract 10: x = 6.

Step 3. Check: (3/4)(6 + 10) = (3/4)(16) = 12.

The core of it: For p(x + q) = r, divide first when r divides neatly by p, and distribute first when multiplying p through the brackets clears a fraction. Either way, the number outside the brackets belongs to every term inside.

Common misconceptions

"3(t + 4) = 3t + 4." The factor outside the brackets multiplies every term inside: 3(t + 4) = 3t + 12. Three friends buy three popcorns.

"-2(x - 5) = -2x - 10." A negative times a negative is positive, so -2 × (-5) = +10 and -2(x - 5) = -2x + 10.

"Subtract q first, before dealing with p." In p(x + q) = r, the q is inside the brackets, so it cannot come off first. Either divide by p to free the brackets or distribute to open them.

"Undo subtracting by subtracting." From y - 1.2 = 2.8, add 1.2 to both sides; the inverse of subtracting is adding.

The short version

• In p(x + q) = r, the p multiplies every term in the brackets. Solve by dividing both sides by p first, or by distributing first; both give the same answer.
• Dividing first matches the arithmetic route: share out the total, then undo the q.
• Pick the route that keeps the numbers friendly: divide first when r divides neatly by p, distribute first when it clears a fraction.
• Read the story: an amount added to each of the equal groups gives p(x + q) = r; an amount added once gives px + q = r.
• When a check fails, test each line against the line before it to find the one that broke.

The next lesson changes the equals sign into "at least" and "at most", and meets the one rule that makes inequalities different from equations.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Expressions and Equations (7.EE.B.4a). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 6, Lesson 6: Distinguishing between two types of situations. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 6, Lesson 10: Different options for solving one equation. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 8.3, Solve equations with variables and constants on both sides (general strategy). Prealgebra 2e. OpenStax. openstax.org

At least $100 by Friday

The seventh-grade trip club has $50 in its account. It makes $3 of profit on every box of cookies it sells, and a $100 deposit for the trip is due on Friday. How many boxes does the club need to sell?

The question is not "exactly how many". Selling more than enough is fine. The club needs its money to be at least $100, and that small phrase turns an equation into an inequality. The Common Core standard's own example is nearly the same story: a salesperson paid $50 a week plus $3 a sale who wants to earn at least $100.

Worked example 1: the cookie boxes.

Step 1. Let b be the number of boxes sold. The money will be 50 + 3b, and it must be at least 100: 50 + 3b ≥ 100.

Step 2. Subtract 50 from both sides, exactly as with an equation: 3b ≥ 50.

Step 3. Divide both sides by 3: b ≥ 16 2/3.

Step 4. Read it back into the story. The club cannot sell two-thirds of a box, so the numbers that work are the whole numbers 17, 18, 19 and so on.

Step 5. Check on both sides of the boundary. 16 boxes give 50 + 48 = $98, which is not enough. 17 boxes give 50 + 51 = $101, which is enough. The club must sell at least 17 boxes.

This lesson covers Common Core standard 7.EE.B.4b: solving word problems leading to inequalities of the form px + q > r or px + q < r, graphing the solution set, and interpreting it in the context of the problem.

What an inequality says

An inequality compares two expressions that need not be equal. There are four signs, and each matches everyday words:

An equation usually has one solution. An inequality usually has infinitely many, and all of them together are its solution set. On a number line you draw the set with a circle at the boundary and a ray showing the direction. The circle is closed, filled in, when the boundary number is itself a solution, as with ≥ and ≤, and open when it is not, as with > and <.

The one rule that is different: the flip

Almost everything you know about solving equations works for inequalities. You can add or subtract the same number on both sides, and you can multiply or divide both sides by the same positive number. Watch what happens with a true statement, 3 < 7, when you try each move:

Multiplying by a negative number swaps the order. On the number line, 3 is left of 7, but -6 is right of -14: multiplying by a negative reflects every number to the other side of zero, and a reflection turns left and right round. So the rule is:

When you multiply or divide both sides of an inequality by a negative number, reverse the inequality sign. Adding, subtracting, and multiplying or dividing by a positive number never reverse it.

Bottom line: Solve an inequality exactly as you would an equation, with one extra rule: dividing or multiplying by a negative flips < to >, ≤ to ≥, and the other way round.

A phone battery: the flip in a real problem

Worked example 2: when does the battery drop below 20 percent? Your phone is at 80 percent, and streaming video uses 6 percentage points of battery an hour. The phone warns you when it falls below 20 percent. When does that happen?

Step 1. After h hours the battery is 80 - 6h percent, and you want it below 20: 80 - 6h < 20.

Step 2. Subtract 80 from both sides: -6h < -60. No flip yet: subtracting never flips.

Step 3. Divide both sides by -6, and reverse the sign because -6 is negative: h > 10.

Step 4. Check with test values. At h = 11, the battery is 80 - 66 = 14 percent, below 20: a solution, as h > 10 says. At h = 9, it is 80 - 54 = 26 percent, not below 20: not a solution. At exactly h = 10 it is 20 percent, which is not below 20, so 10 is not included and the circle is open.

More inequalities, worked

Worked example 3: at most, at the arcade. You have $40. The arcade charges $8 to get in and $1.25 a game. How many games can you play?

Step 1. Spending must be at most $40: 8 + 1.25g ≤ 40.

Step 2. Subtract 8: 1.25g ≤ 32.

Step 3. Divide by 1.25, a positive number, so no flip: g ≤ 25.6.

Step 4. In the story, g is a whole number of games, from 0 up to 25. Check: 25 games cost 8 + 31.25 = $39.25, which you can afford, and 26 games would cost 8 + 32.50 = $40.50, which you cannot. You can play at most 25 games.

Worked example 4: a fraction coefficient. Solve (1/2)x - 3 > 4.

Step 1. Add 3: (1/2)x > 7.

Step 2. Multiply both sides by 2, a positive number: x > 14.

Step 3. Check: x = 16 gives 8 - 3 = 5, and 5 > 4 is true. x = 14 gives 7 - 3 = 4, and 4 > 4 is false, so 14 itself is not a solution.

Worked example 5: a negative coefficient. Solve -3x + 7 ≥ 22 and graph the solution.

Step 1. Subtract 7: -3x ≥ 15.

Step 2. Divide by -3 and reverse the sign: x ≤ -5.

Step 3. Check: x = -6 gives 18 + 7 = 25, and 25 ≥ 22 is true. x = 0 gives 7, and 7 ≥ 22 is false. x = -5 gives 15 + 7 = 22, and 22 ≥ 22 is true, so -5 is included and the circle is closed.

Worked example 6: warming up. At dawn it is -2 °C, and the temperature rises 1.5 °C an hour. When will it be warmer than 7 °C?

Step 1. -2 + 1.5h > 7.

Step 2. Add 2: 1.5h > 9. Divide by 1.5: h > 6. No flip, because 1.5 is positive, even though the story began below zero.

Step 3. Check: after 7 hours it is -2 + 10.5 = 8.5 °C, warmer than 7. After exactly 6 hours it is 7 °C, not warmer. So it is warmer than 7 °C after more than 6 hours.

A second check: the boundary and a test point

Illustrative Mathematics teaches a method that avoids the flip rule altogether, and it makes an excellent check. Solve the matching equation to find the boundary, then test one number on either side.

Worked example 7: -3x + 7 ≥ 22 by boundary and test point.

Step 1. Solve -3x + 7 = 22: -3x = 15, so x = -5. This is the boundary, and because the sign is ≥, the boundary itself is a solution.

Step 2. Test a number on one side, x = 0: -3 × 0 + 7 = 7, and 7 ≥ 22 is false. So the numbers on that side of -5, the ones greater than -5, are not solutions.

Step 3. The solutions are on the other side: x ≤ -5, the same answer the flip rule gave.

What matters here: An inequality's answer is a whole set of numbers, so check it at more than one place: one value inside the set, one outside, and the boundary itself to decide between an open and a closed circle.

Common misconceptions

Forgetting the flip. From -6h < -60, dividing by -6 gives h > 10, not h < 10. A test value catches it: at h = 5 the battery is 50 percent, nowhere near below 20.

Flipping when you should not. The sign reverses only when you multiply or divide by a negative number. It does not reverse when you subtract, when the answer is negative, or when a negative number appears somewhere else in the problem, as in -2 + 1.5h > 7.

"At least means greater than." At least 17 boxes includes 17, so it is ≥ and the circle is closed. More than 17 would leave 17 out.

Ignoring the story. b ≥ 16 2/3 is the algebra's answer, but the club sells whole boxes. The answer to the question is at least 17 boxes.

Pulling it together

• At least means ≥, at most means ≤, more than means >, and fewer than means <.
• Solve an inequality like an equation: add, subtract, multiply or divide both sides by the same number.
• Multiplying or dividing both sides by a negative number reverses the sign. Nothing else does.
• The solution is a set. Graph it with a closed circle when the boundary is included and an open circle when it is not, and a ray in the direction of the solutions.
• Check with a value inside the set, a value outside, and the boundary, and read the answer back into the situation, where only whole numbers of boxes or games may make sense.

The next module leaves algebra for geometry, starting with scale drawings and maps, where a single proportion turns centimetres on paper into kilometres on the ground.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Expressions and Equations (7.EE.B.4b). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 6, Lesson 15: Efficiently solving inequalities. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 6, Lesson 14: Finding solutions to inequalities in context. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 2.7, Solve linear inequalities. Elementary Algebra 2e. OpenStax. openstax.org

One inch, two thousand feet

Some of the best-known maps made by the U.S. Geological Survey, the 7.5-minute topographic quadrangles, are printed at a scale of 1:24,000. Suppose you spread one out on the kitchen table before a hike and lay a piece of string along the trail you want to walk. The string measures 3 1/4 inches. How long is the hike?

By the end of this lesson you will be able to answer that in feet and in miles, and to explain why the same map multiplies areas by a far larger number than 24,000.

This lesson covers Common Core standard 7.G.A.1: solving problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

What a scale says

A scale drawing is a picture in which every length is the same fixed fraction of the real length. The scale tells you that fraction, and maps and plans write it in three common ways:

A ratio with no units works in any unit, which is why it is useful: 1 inch on the map is 24,000 inches on the ground, and 1 centimetre on the map is 24,000 centimetres on the ground. The relationship between map length and ground length is proportional, and 24,000 is its constant of proportionality.

The trail, end to end

Worked example 1: the hike on the 1:24,000 map.

Step 1. Read the scale in one unit: 1 inch on the map is 24,000 inches on the ground.

Step 2. Change the ground unit to something sensible. There are 12 inches in a foot, so 24,000 inches = 24,000 ÷ 12 = 2,000 feet. The scale says 1 inch to 2,000 feet.

Step 3. Multiply by the map length: 3 1/4 × 2,000 = 3.25 × 2,000 = 6,500 feet.

Step 4. Change to miles, since a mile is 5,280 feet: 6,500 ÷ 5,280 = 1.23 miles, to two decimal places.

Step 5. Check with a benchmark. Three inches would be 6,000 feet, a little over a mile; 3 1/4 inches should be a bit more than that. It is. The hike is about 6,500 feet, or about 1 1/4 miles.

Key idea: A scale is a unit rate. Turn it into "1 unit on paper stands for this many real units", multiply the measured length by it, and convert units last.

A bar scale, and going the other way

Many maps print a bar instead of a ratio, because a bar stays correct when the map is enlarged or shrunk on a photocopier or a screen.

Worked example 2: a road measured against a bar scale. Each section of the bar is 2 cm long and stands for 5 km. The road measures 7.4 cm.

Step 1. Unit rate: 2 cm stands for 5 km, so 1 cm stands for 5 ÷ 2 = 2.5 km.

Step 2. Multiply: 7.4 × 2.5 = 18.5 km.

Step 3. Check against the picture: the road reaches a little short of the 20 km mark, and 18.5 km is a little short of 20.

Worked example 3: from real life onto paper. You want to draw a soccer pitch 100 m long and 64 m wide at a scale of 1 cm to 8 m. How big is the drawing?

Step 1. Going from real to drawing, divide by the unit rate: each 8 m becomes 1 cm.

Step 2. Length: 100 ÷ 8 = 12.5 cm. Width: 64 ÷ 8 = 8 cm.

Step 3. Check by going back: 12.5 × 8 = 100 m and 8 × 8 = 64 m. The drawing is 12.5 cm by 8 cm, which fits on a page.

Change one thing, length to area, and the multiplier changes

Now the second example of the procedure, where one change, asking for an area instead of a length, alters the answer completely.

Worked example 4: the area of a lake. On a map at 1:25,000, a lake is roughly a rectangle 3 cm long and 2 cm wide. What is its real area?

Step 1. Unit rate: 1 cm on the map is 25,000 cm on the ground, and 25,000 cm = 250 m. So 1 cm stands for 250 m.

Step 2. Real length: 3 × 250 = 750 m. Real width: 2 × 250 = 500 m.

Step 3. Real area: 750 × 500 = 375,000 square metres, which is 0.375 square kilometres.

A tempting shortcut goes wrong here. The map area is 3 × 2 = 6 square centimetres, and multiplying that by 250 gives 1,500 square metres, which is 250 times too small. The reason is in the picture: one square centimetre of map is a square 250 m by 250 m on the ground, so it stands for 250 × 250 = 62,500 square metres, not 250. Check: 6 × 62,500 = 375,000, the same answer as Step 3.

The point: If lengths are multiplied by a scale factor k, areas are multiplied by k × k. Safest of all: convert the lengths first, then multiply them to get the area.

Worked example 5: a floor plan. A bedroom plan uses 1 inch to 2 feet. The room is drawn 6 inches by 5 inches.

Step 1. Real length: 6 × 2 = 12 feet. Real width: 5 × 2 = 10 feet.

Step 2. Real area: 12 × 10 = 120 square feet. The drawing's area is 6 × 5 = 30 square inches.

Step 3. So each square inch of plan stands for 120 ÷ 30 = 4 square feet, which is 2 × 2, the length factor squared.

Redrawing at a different scale

Worked example 6: shrink the bedroom plan. The plan must fit a smaller page at 1 inch to 3 feet. What are the new drawing's dimensions?

Step 1. Go through the real room, not straight from drawing to drawing: the room is 12 feet by 10 feet.

Step 2. At 1 inch to 3 feet, divide by 3: 12 ÷ 3 = 4 inches, and 10 ÷ 3 = 3 1/3 inches.

Step 3. Check the effect on area. The new drawing is 4 × 3 1/3 = 13 1/3 square inches, against 30 before. Each length became 2/3 of its old size (2 feet per inch became 3), and 13 1/3 ÷ 30 = 4/9, which is 2/3 × 2/3.

Going through the real measurements is the reliable route, because it uses each scale once, in the direction it was written.

Common misconceptions

"Multiply the area by the scale." Areas scale by the square of the length factor: a map at 1 cm to 250 m makes each square centimetre stand for 62,500 square metres. The lake's area was 375,000 square metres, not 1,500.

"1:24,000 means 1 inch is 24,000 feet." A ratio with no units uses the same unit on both sides: 1 inch is 24,000 inches, which is only 2,000 feet. Reading it as feet makes every distance twelve times too long.

"A bigger second number means a bigger drawing." It is the reverse. At 1 cm to 2 m a 20 m garden is 10 cm long; at 1 cm to 5 m it is only 4 cm. The more real distance each centimetre stands for, the smaller the drawing.

"Scale drawings are always smaller than the real thing." A scale of 20:1 enlarges: a 6 mm ant is drawn 120 mm long. The same proportional reasoning works in both directions.

What you now know

• A scale is a unit rate between drawing length and real length; a ratio such as 1:24,000 uses the same unit on both sides.
• Drawing to real: multiply by the scale; real to drawing: divide. Convert units at the end, and check with a benchmark.
• A bar scale is read by measuring the bar, and stays correct when a map is resized.
• Areas are multiplied by the square of the length scale factor, so find the real lengths first and multiply them.
• To redraw at a new scale, go through the real measurements, then apply the new scale.

The next lesson draws triangles from measurements, and asks a question with a surprising answer: do three measurements always pin down one triangle?

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Geometry (7.G.A.1). thecorestandards.org
2. U.S. Geological Survey. (n.d.). What is a topographic map? USGS Frequently Asked Questions. Retrieved September 24, 2026, for the 1:24,000 quadrangle series. usgs.gov
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 1, Lesson 8: Scale drawings and maps. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Illustrative Mathematics. (n.d.). Grade 7, Unit 1, Lesson 6: Scaling and area. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com

Thirty triangles in a stack

On Monday, a teacher gives each of thirty students an envelope holding the same three measurements, 5 cm, 6 cm and 7 cm, and asks each of them to draw the triangle with those side lengths and cut it out. When the class stacks the thirty triangles, they match exactly, some of them only after being flipped over. Maya says this proves something general: three measurements always fix a triangle.

On Tuesday the envelopes hold three angles, 50°, 60° and 70°. This time the stack is a mess. Every triangle has the same shape, but they come in every size, from a triangle you could cover with a stamp to one the size of a notebook. Sam says Maya was wrong: three measurements sometimes fix a triangle and sometimes do not, and occasionally they give exactly two.

Who is right? This lesson lays out the evidence for each position and then settles the dispute the only way geometry can, by constructing the triangles. It covers Common Core standard 7.G.A.2: drawing geometric shapes with given conditions, with a focus on constructing triangles from three measures of angles or sides, and noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

Maya's evidence: some sets of three always give one triangle

Two triangles count as the same triangle if one can be moved, turned or flipped to sit exactly on the other; the word for that is congruent. A set of conditions determines a unique triangle when every triangle that meets them is congruent to every other.

Worked example 1: three sides, 5 cm, 6 cm and 7 cm, with ruler and compass.

Step 1. Draw the longest side first: a segment AB 7 cm long.

Step 2. Set the compass to 5 cm, put its point on A, and draw an arc. Every point on that arc is 5 cm from A.

Step 3. Set the compass to 6 cm, put its point on B, and draw an arc crossing the first one. Every point on this arc is 6 cm from B.

Step 4. The arcs cross at a point C, which is 5 cm from A and 6 cm from B. Join AC and BC.

The arcs also cross at a second point, below AB, but the triangle there is the same triangle flipped over. So three sides give exactly one triangle, and that is what Monday's stack showed.

Worked example 2: two sides and the angle between them. Sides of 5 cm and 9 cm with a 50° angle between them.

Step 1. Draw a 9 cm segment and, with a protractor, a 50° angle at one end.

Step 2. Mark the point 5 cm along the new ray.

Step 3. There is only one way to close the triangle: join that point to the far end of the 9 cm segment. Anyone following the same steps gets the same triangle, so these conditions determine one triangle.

Worked example 3: two angles and the side between them. Angles of 40° and 75° at the two ends of a 9 cm side.

Step 1. Draw the 9 cm side. Draw a 40° ray at one end and a 75° ray at the other, on the same side of the segment.

Step 2. The two rays can cross at only one point, which is the third corner. One triangle.

Step 3. Its third angle is fixed too, because the angles of a triangle add to 180°: 180 - 40 - 75 = 65°.

So Maya has real evidence: three sides, two sides with the angle between them, and two angles with the side between them each give exactly one triangle, whoever draws it.

Sam's evidence: other sets give many triangles, or none, or two

Worked example 4: three angles. Angles of 50°, 60° and 70° add to 180°, so a triangle exists. But nothing sets its size. Draw a 50° angle and a 60° angle at the ends of a 3 cm segment, and the rays meet to make a small triangle whose third angle is 70°. Do the same on a 12 cm segment and you get a large triangle with the same three angles. There are infinitely many, all the same shape and different sizes. Tuesday's stack was not a mistake.

And if three angles add to anything other than 180°, such as 50°, 60° and 80°, which add to 190°, there is no triangle at all.

Worked example 5: three sides that cannot meet. Try sides of 2 cm, 3 cm and 6 cm. Draw the 6 cm side, then swing a 2 cm arc from one end and a 3 cm arc from the other. The arcs never meet: together the two short sides reach only 2 + 3 = 5 cm, and the gap is 6 cm. No triangle.

This is the triangle inequality: the two shorter sides must add to more than the longest side. Check 3 cm, 4 cm and 7 cm: 3 + 4 = 7, exactly equal, and the two short sides can only lie flat along the long one, with no room to rise into a corner. Equal is not enough; the sum must be more.

Worked example 6: exactly two triangles. Draw a 40° angle at A. Mark B 7 cm along one ray. Now the third condition: side BC must be 5 cm, and C must lie somewhere on the other ray.

Step 1. Every point 5 cm from B lies on a circle around B, so swing a 5 cm arc from B.

Step 2. The arc crosses the lower ray at two points, C1 and C2. Both triangles, ABC1 and ABC2, have a 40° angle at A, a 7 cm side AB and a 5 cm side BC.

Step 3. They are not congruent. Measure them and the third side AC is about 3.2 cm in one and about 7.5 cm in the other, and no flipping makes one sit on the other. These three measurements determine two triangles.

Change the 5 cm to other lengths and the count changes. The shortest distance from B to the lower ray is about 4.5 cm. An arc of 4 cm is too short to reach the ray, so there is no triangle. An arc of 6 cm crosses twice, giving two triangles again. An arc of 8 cm, longer than AB, crosses the ray only once, because its other crossing falls behind A, off the ray, so there is one triangle. The difference from Worked example 2 is where the angle sits: here the 40° angle is not between the two given sides.

Remember: It is not the number of measurements that decides, it is which ones and where they sit. An angle between two given sides fixes the triangle; an angle next to only one of them may not.

What settles it

Put all the evidence in one table. Each row was tested by construction above.

So the dispute ends with both students partly right, in a precise way. Maya is right for three kinds of condition: three sides, two sides with the angle between them, and two angles with the side between them. Sam is right that "three measurements" on its own settles nothing: three angles never fix the size, impossible numbers give no triangle, and two sides with an angle not between them can give two.

Why this matters: Builders, engineers and anyone following a plan need to know whether the measurements on it fix the shape. A triangle made of three rigid bars cannot change shape, which is why triangles appear in bridges, roof trusses and bicycle frames, and why the question "how many triangles fit?" has practical answers.

Quick decisions

Worked example 7: how many triangles? Decide for each set, and give the reason.

• Sides 4 cm, 5 cm, 8 cm: 4 + 5 = 9, which is more than 8. Exactly one.
• Sides 6 cm, 6 cm, 11 cm: 6 + 6 = 12, which is more than 11. Exactly one, a tall thin isosceles triangle.
• Sides 7 cm, 10 cm, 3 cm: 7 + 3 = 10, exactly equal. None; the sides lie flat.
• Angles 90°, 60°, 30°: they add to 180°. Infinitely many, all the same shape.
• A 40° angle, a 7 cm side along one ray, and an 8 cm side opposite the angle: exactly one, as in Worked example 6.

Common misconceptions

"Any three lengths make a triangle." The two shorter sides must add to more than the longest. Sticks of 1 cm, 1 cm and 3 cm cannot close up.

"Three angles fix a triangle." They fix the shape but not the size. A 50°, 60°, 70° triangle can be any size at all.

"A flipped triangle is a different triangle." A mirror image can be turned over to sit exactly on the original, so it is congruent and counts as the same triangle here.

"Two sides and an angle always give one triangle." Only when the angle is between the two sides. With the angle next to only one of them, the arc may cross twice, once, or not at all.

Recap

• Conditions determine a unique triangle when every triangle that meets them is congruent to every other.
• Three sides give exactly one triangle if the two shorter add to more than the longest, and none otherwise.
• Two sides with the angle between them, or two angles with the side between them, give exactly one triangle.
• Three angles adding to 180° give infinitely many triangles of one shape; any other total gives none.
• Two sides and an angle not between them can give none, one or two triangles, and a compass arc shows which.
• When in doubt, construct: the arcs and rays of an honest construction settle the question.

The next lesson moves from flat shapes to solid ones, and asks what shapes appear when you slice a box or a pyramid.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Geometry (7.G.A.2). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 7, Lesson 7: Building polygons (Part 2). IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 7, Lesson 10: Drawing triangles (Part 2). IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 9.3, Use properties of angles, triangles, and the Pythagorean theorem. Prealgebra 2e. OpenStax. openstax.org

A stick of butter and a paperweight

Slice a stick of butter straight across and the cut face is a rectangle. Slice again two centimetres further along and you get exactly the same rectangle, and the same again all the way to the end. Now slice a pyramid-shaped glass paperweight straight across, parallel to its base. The cut face is a rectangle too, but a slice higher up gives a smaller one, and near the top the rectangle shrinks almost to a point.

The flat face that a straight slice exposes is called a cross-section, or plane section. This lesson compares the cross-sections of two solids side by side, and covers Common Core standard 7.G.A.3: describing the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

The two solids

A right rectangular prism is a box: six rectangular faces, with every corner a right angle. A right rectangular pyramid has a rectangular base and four triangular faces that meet at a single top point, the apex, which sits directly above the centre of the base.

Slices of a box

Worked example 1: a cereal box. A cereal box stands 30 cm tall on a base 20 cm long and 8 cm deep. Describe the slices parallel to each pair of faces.

Step 1. Parallel to the base, a level slice at any height is a rectangle exactly like the base: 20 cm by 8 cm, with area 20 × 8 = 160 square cm.

Step 2. Parallel to the front, a vertical slice is a rectangle like the front face: 20 cm by 30 cm, area 600 square cm.

Step 3. Parallel to the side, a vertical slice is a rectangle like the side face: 8 cm by 30 cm, area 240 square cm.

Step 4. Notice that the height of the slice does not matter. A box is the same all the way through in each of these directions, so every slice parallel to a face is a copy of that face.

Worked example 2: slices that are not rectangles.

Step 1. Cut a small corner off the box with one straight slice. The knife crosses three faces, the three that meet at that corner, so the cut face has three sides: a triangle.

Step 2. Tilt the knife and cut right across the box so that the slice crosses all four upright edges. Opposite sides of the cut lie on opposite, parallel faces, so the cut is a parallelogram. If you tilt it straight along one edge direction, it is a rectangle longer than the side face.

Step 3. Tilt it more cleverly and the slice can cross five faces or all six, making a pentagon or a hexagon. A cube sliced through the midpoints of six of its edges gives a hexagon.

In short: Each side of a cross-section lies on a different face of the solid. So a slice has at most as many sides as the solid has faces, and a box, with six faces, can give a triangle, a quadrilateral, a pentagon or a hexagon, but never a heptagon.

Slices of a pyramid

Worked example 3: level slices of a pyramid. A pyramid has a base 8 cm by 6 cm and is 12 cm tall. Describe the slices parallel to the base at heights of 3 cm, 6 cm and 9 cm.

Step 1. Every level slice is a rectangle the same shape as the base, but it shrinks steadily toward the apex, reaching a point at 12 cm.

Step 2. The shrinking is proportional. At height h, the slice is (12 - h)/12 of the base's length and width, because that fraction of the pyramid's height is still above it.

Step 3. Work each height:

Step 4. Read the last column. Halfway up, the slice is half as long and half as wide, so its area is 1/2 × 1/2 = 1/4 of the base's: 12 is a quarter of 48. This is the scale-drawing rule from two lessons ago: each slice is a scale drawing of the base, and area scales by the square of the length factor.

Worked example 4: upright slices of the same pyramid.

Step 1. A vertical slice straight through the apex, parallel to the 8 cm edges, is a triangle. Its base is 8 cm and its height is the pyramid's height, 12 cm, so its area is 1/2 × 8 × 12 = 48 square cm.

Step 2. A vertical slice through the apex parallel to the 6 cm edges is a triangle with base 6 cm and height 12 cm: area 1/2 × 6 × 12 = 36 square cm.

Step 3. Move the vertical slice sideways so that it is still parallel to an edge of the base but misses the apex. It now has a flat top, where it meets the sloping face nearest to it: a trapezoid, wide at the bottom and narrower at the top.

Step 4. Cut a small piece off near the apex at a slant and the slice crosses three faces: a triangle. A steep slant can cross the base and all four triangular faces, giving a pentagon, but a pyramid has only five faces, so no slice of it is a hexagon.

The two side by side

Read down the columns and one difference explains almost everything. A prism is the same all the way through, so its slices keep their size. A pyramid narrows to a point, so its slices shrink in proportion to their distance from the apex, and its upright slices through the apex come to a point too.

The upshot: To predict a cross-section, ask two questions: which faces does the slice cross, and does the solid stay the same size along the slice's direction or narrow toward an apex?

A slice through the Great Pyramid

Worked example 5: halfway up the Great Pyramid of Giza. The pyramid's square base measures about 230.3 metres on each side, and it was originally about 146.6 metres tall. Imagine it sliced level at half that height. How big is the cross-section?

Step 1. Halfway up means half of the height, 146.6 ÷ 2 = 73.3 metres, is still above the slice, so the slice is 1/2 of the base's size.

Step 2. Side of the square slice: 230.3 ÷ 2 = 115.15 metres, about 115 metres.

Step 3. Area: 115.15 × 115.15 = 13,259.5 square metres, against 230.3 × 230.3 = 53,038.1 square metres for the base. The slice halfway up has only a quarter of the base's area.

That is why most of a pyramid's stone is in its lower part: the slices near the bottom are far bigger than the ones near the top.

Common misconceptions

"Every slice of a box is a rectangle." Slices parallel to a face are rectangles, but cutting across a corner gives a triangle, and tilted slices can give parallelograms, pentagons and hexagons.

"A level slice of a pyramid is a triangle." A slice parallel to the base is a rectangle the same shape as the base. The triangles come from upright slices through the apex.

"Halfway up, the slice has half the area." It has half the length and half the width, so a quarter of the area: 12 square cm against the 48 of the base.

"A slice could have any number of sides." Each side lies on a different face, so a box's slices have at most 6 sides and a pyramid's at most 5.

What to remember

• A cross-section is the flat face exposed by a straight slice through a solid.
• A box's slices parallel to a face are copies of that face, at every position.
• A pyramid's slices parallel to the base are rectangles shaped like the base and scaled by the fraction of the height above them; their area scales by the square of that fraction.
• Upright slices of a pyramid are triangles through the apex and trapezoids parallel to a base edge away from it.
• A slice has at most as many sides as the solid has faces: 6 for a box, 5 for a pyramid, and a corner cut gives a triangle in both.

The next module stays with geometry but turns to angles and circles: finding unknown angles with equations, and the number pi.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Geometry (7.G.A.3). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 7, Lesson 11: Slicing solids. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Wikipedia contributors. (n.d.). Cross section (geometry). In Wikipedia. Retrieved September 24, 2026. en.wikipedia.org
4. Wikipedia contributors. (n.d.). Great Pyramid of Giza. In Wikipedia. Retrieved September 24, 2026, for the base of about 230.3 m and original height of about 146.6 m. en.wikipedia.org

Scissors open at 35 degrees

Open a pair of scissors so that the blades make an angle of 35°. What angle do the two handles make with each other? A student works it out like this: "The blades and handles form a straight line, and a straight line is 180°, so the handles make 180 - 35 = 145°."

Open a real pair of scissors a little way and look. The handles spread apart about as far as the blades do, nothing like the 145° the student found. The arithmetic 180 - 35 = 145 is right, so the mistake must be in which angle it measures. Finding it exactly is the start of this lesson.

This lesson covers Common Core standard 7.G.B.5: using facts about supplementary, complementary, vertical and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Tracing the 145

Each blade of a pair of scissors runs, roughly straight, through the pivot into the opposite handle, so you can model the scissors as two straight lines crossing at the pivot. Two crossing lines make four angles.

Call the blade angle a. The student's 145° is angle b, between one blade and the opposite handle: a and b sit side by side on a straight line, so they add to 180°. But the handles are the two lines on the far side of the pivot, and the angle between them is c, directly across from a. That angle is not 145°. It is 35°, the same as the blades. The student answered a correct question, but not the one that was asked.

So what?: Most angle mistakes are not arithmetic mistakes. They come from using the right fact on the wrong pair of angles, so the first step is always to say which angles you are relating and why.

Four relationships

Two more totals are useful. Adjacent angles that fill a straight line add to 180°, and angles that go all the way around a point add to 360°.

Why are vertical angles always equal? In the figure, a and b lie on a straight line, so a + b = 180°. Angles b and c also lie on a straight line, the other one, so c + b = 180°. Both a and c are therefore 180° - b, and so a = c. Nothing about 35° was special: the argument works for any two crossing lines.

Writing and solving angle equations

Worked example 1: vertical angles with expressions. Two lines cross. One angle measures (2x + 10)° and the angle opposite it measures (3x - 20)°. Find x and the angles.

Step 1. Opposite angles at a crossing are vertical, so they are equal: 2x + 10 = 3x - 20.

Step 2. Subtract 2x from both sides: 10 = x - 20. Add 20: x = 30.

Step 3. The angles are 2 × 30 + 10 = 70° and 3 × 30 - 20 = 70°. They are equal, as vertical angles must be.

A common wrong route adds them to 180: (2x + 10) + (3x - 20) = 180, so 5x - 10 = 180 and x = 38. The check exposes it: the angles would be 86° and 94°, which are not equal, so they cannot be vertical angles.

Worked example 2: complementary angles. Two angles fill a right angle together. One measures x° and the other (2x + 6)°.

Step 1. Complementary angles add to 90°: x + 2x + 6 = 90, which is 3x + 6 = 90.

Step 2. Subtract 6: 3x = 84. Divide by 3: x = 28.

Step 3. The angles are 28° and 2 × 28 + 6 = 62°. Check: 28 + 62 = 90.

Worked example 3: three angles on a straight line. A straight line is split into three adjacent angles of 45°, x° and 2x°.

Step 1. Angles on a straight line add to 180°: 45 + x + 2x = 180.

Step 2. Combine like terms: 45 + 3x = 180. Subtract 45: 3x = 135. Divide: x = 45.

Step 3. The angles are 45°, 45° and 90°. Check: 45 + 45 + 90 = 180.

Worked example 4: around a point. Four angles fit round a point with no gaps: 90°, 110°, x° and (x + 20)°.

Step 1. Angles around a point add to 360°: 90 + 110 + x + x + 20 = 360.

Step 2. Combine: 2x + 220 = 360. Subtract 220: 2x = 140. Divide: x = 70.

Step 3. The last two angles are 70° and 90°. Check: 90 + 110 + 70 + 90 = 360.

Worked example 5: two facts in one figure. Two lines cross. One angle is 4x° and the angle next to it, on the same straight line, is (x + 30)°. Find all four angles.

Step 1. Adjacent angles on a straight line are supplementary: 4x + x + 30 = 180, so 5x + 30 = 180.

Step 2. Subtract 30: 5x = 150. Divide: x = 30. The two angles are 4 × 30 = 120° and 30 + 30 = 60°.

Step 3. The other two angles are vertical to these, so they are also 120° and 60°. Check: 120 + 60 + 120 + 60 = 360, the total around the crossing point.

Worth holding on to: Name the relationship first, then write the equation it gives: equal for vertical angles, a sum of 90° for complementary angles, 180° on a straight line, 360° around a point. Check that your angles satisfy the relationship you used.

Two more wrong answers, traced

Wrong answer 2: "The complement of 35° is 145°." That is the supplement. Complementary angles add to 90°, so the complement of 35° is 90 - 35 = 55°. A memory aid many students use: c comes before s in the alphabet, and 90 comes before 180.

Wrong answer 3: "These two angles look like they make a straight line, so they add to 180°." Two angles add to 180° only when their outer sides really do form a straight line. A drawing that is not to scale can make a 170° bend look straight. Use what the figure states, such as a line marked as straight or angles marked as equal, not what it looks like.

Common misconceptions

"Vertical angles add to 180°." Vertical angles are equal. It is adjacent angles on a straight line that add to 180°.

"Vertical means up and down." In angle work, vertical refers to the vertex, the crossing point the two angles share. Vertical angles can open sideways, like the scissors' blade and handle angles.

"Complementary and supplementary mean the same thing." Complementary angles make a right angle, 90°; supplementary angles make a straight angle, 180°.

"x is the answer." In an angle equation, x is often only a step. The question usually asks for the angles, such as 2x + 10 = 70°, so finish by substituting.

Putting it together

• Two crossing lines make two pairs of vertical angles, which are equal because each is supplementary to the same angle.
• Adjacent angles on a straight line add to 180°; angles filling a right angle add to 90°; angles around a point add to 360°.
• To find an unknown angle, name the relationship, write the equation, solve it, substitute to get the angles, and check the relationship holds.
• A correct calculation applied to the wrong pair of angles, like the scissors' 145°, is the commonest error.

The next lesson moves from angles to circles and follows the long story of the number pi.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Geometry (7.G.B.5). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 7, Lesson 3: Nonadjacent angles. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 7, Lesson 5: Using equations to solve for unknown angles. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 9.3, Use properties of angles, triangles, and the Pythagorean theorem. Prealgebra 2e. OpenStax. openstax.org

A short book from Syracuse

Around 250 BC, the Greek mathematician Archimedes, who lived in the city of Syracuse on the island of Sicily, is thought to have written a short work now called Measurement of a Circle. What survives of it is only three propositions. One proves that a circle's area equals the area of a right triangle whose two shorter sides are the circle's radius and its circumference. Another traps the number we now call pi between 3 10/71 and 3 1/7. Those two results are the whole of this lesson, and you will rebuild both of them with nothing more than string, a pizza and a little arithmetic.

This lesson covers Common Core standard 7.G.B.4: knowing the formulas for the area and circumference of a circle and using them to solve problems, and giving an informal derivation of the relationship between the circumference and the area.

Around the outside: circumference

The distance around a circle is its circumference. The distance across it through the centre is its diameter, and half of that, from the centre to the edge, is its radius. Wrap a string round a soup can, then lay the string along the can's diameter, and you will find it goes a little more than three times. Do the same with a bike wheel, a plate or a coin and you get the same answer: every circle's circumference is the same number of diameters long.

That number is pi, written with the Greek letter π. Its decimal begins 3.14159 and never ends or repeats; the Swiss scientist Johann Heinrich Lambert proved in 1768 that pi is not equal to any fraction. Because circumference divided by diameter is always pi,

C = πd, and since d = 2r, also C = 2πr.

This is a proportional relationship, circumference against diameter, with pi as its constant of proportionality.

Worked example 1: how far one turn of a wheel goes. A bike wheel is 26 inches across. How far does the bike travel in one full turn of the wheel, and how many turns does it take to ride a mile, which is 63,360 inches?

Step 1. One turn rolls out one circumference: C = π × 26 = 26π inches, about 81.68 inches.

Step 2. Turns in a mile: 63,360 ÷ 81.68 = 775.7, so about 776 turns.

Step 3. Check with an estimate: pi is a bit more than 3, so one turn is a bit more than 78 inches, and 63,360 ÷ 78 is a bit over 800. Slightly fewer than 800 turns fits.

How Archimedes trapped pi

Archimedes could not measure string precisely enough to pin pi down, so he reasoned with polygons instead. Picture a regular hexagon drawn inside a circle, with its corners on the circle. Its six sides are each exactly one radius long, so its perimeter is 6r, which is 3d. The circle bulges outside the hexagon, so the circumference is more than 3 diameters: pi is more than 3. Now draw a square around the outside of the circle. Its perimeter is 4d, and the circle fits inside it, so pi is less than 4.

Polygons with more sides hug the circle more closely. Archimedes worked all the way up to polygons with 96 sides, inside and outside the circle, and proved that

3 10/71 < π < 3 1/7, that is, 3.1408 < π < 3.1429, to four decimal places.

The upper bound, 3 1/7 = 22/7, is still used today as a handy fraction for pi, and 3.14 is the handy decimal. Both are approximations. When a problem wants an exact answer, leave pi in it: the circumference of the wheel is exactly 26π inches, and 81.68 inches is a rounded value.

The core of it: Pi is the same for every circle because it is a ratio, circumference divided by diameter. Every circle is a scaled copy of every other, so the ratio never changes.

Inside: the area, from slices of pizza

Now the second result. Cut a round pizza of radius r into 8 equal slices, called sectors. Lay the top four slices point down in a row, then slot the bottom four point up into the gaps between them.

The new shape is bumpy, but it is close to a parallelogram. Its height is about one radius, r, because each slice runs from the centre to the crust. Its length along the bottom is made of the crust of four slices, which is half the circumference: half of 2πr, which is πr.

Cut the pizza into 16 slices, then 100, and the bumps flatten out; the shape gets closer and closer to a rectangle of height r and length πr. The area of that rectangle is length times height:

A = πr × r = πr².

This is the same result Archimedes proved. His triangle has shorter sides r and C, so its area is 1/2 × C × r = 1/2 × 2πr × r = πr². The slices and the triangle are two pictures of one fact: the area of a circle is half its circumference times its radius.

Bottom line: Circumference measures around the outside, C = 2πr, in units of length. Area measures the inside, A = πr², in square units. The slices connect them: A = 1/2 × C × r.

Worked examples

Worked example 2: which pizza is the better deal? A 12-inch pizza costs $10 and a 16-inch pizza costs $15. Pizzas are sold by their diameter.

Step 1. Radii: 6 inches and 8 inches.

Step 2. Areas: π × 6² = 36π = 113.1 square inches, and π × 8² = 64π = 201.1 square inches.

Step 3. The big pizza has 64 ÷ 36 = 1.78 times as much pizza, far more than the 16 ÷ 12 = 1.33 you might guess from the diameters.

Step 4. Price per square inch: $10 ÷ 113.1 = 8.8 cents and $15 ÷ 201.1 = 7.5 cents. The 16-inch pizza is the better deal.

Worked example 3: from circumference to area. A tree trunk measures 188.5 cm around. How wide is it, and what is the area of its cross-section?

Step 1. Diameter: d = C ÷ π = 188.5 ÷ π = 60.0 cm, to one decimal place.

Step 2. Radius: 60 ÷ 2 = 30 cm.

Step 3. Area: π × 30² = 900π, about 2,827 square centimetres.

Worked example 4: area straight from circumference. A circle has circumference 31.4 cm. Using 3.14 for pi, find its area.

Step 1. Radius: r = C ÷ (2π) = 31.4 ÷ 6.28 = 5 cm.

Step 2. Area by Archimedes' relationship: A = 1/2 × C × r = 1/2 × 31.4 × 5 = 78.5 square cm.

Step 3. Check with the usual formula: 3.14 × 5² = 3.14 × 25 = 78.5.

Worked example 5: exact or rounded? Find the area of a circular rug of radius 3 feet exactly, then to the nearest tenth, then with 22/7.

Step 1. Exactly: A = π × 3² = 9π square feet.

Step 2. Rounded: 9 × 3.14159 = 28.27, so 28.3 square feet.

Step 3. With 22/7: 9 × 22/7 = 198/7 = 28 2/7, about 28.29. Each approximation lands close to 9π, and 9π is the only one that is exact.

Common misconceptions

Mixing up radius and diameter. C = πd uses the diameter and A = πr² uses the radius. Putting a 12-inch diameter into the area formula gives 144π, four times the real area of 36π.

"r² means 2r." It means r × r. For r = 3, r² is 9, not 6; the area of that circle is 9π, not 6π, which is its circumference.

"Pi is exactly 3.14" or "exactly 22/7". Both are approximations. Archimedes proved pi is a little less than 22/7, and its decimal never ends.

"Doubling the radius doubles the area." Area uses r × r, so doubling r multiplies the area by 2 × 2 = 4. The circumference, which uses r once, only doubles.

The takeaway

• Pi is the ratio of any circle's circumference to its diameter, about 3.14159, the same for every circle and never exactly a fraction.
• Circumference: C = πd = 2πr. It is proportional to the diameter, and one turn of a wheel rolls one circumference.
• Archimedes trapped pi between 3 10/71 and 3 1/7 using 96-sided polygons; 22/7 and 3.14 are useful approximations.
• Area: A = πr². Cutting a circle into thin slices and rearranging them makes a near-rectangle of height r and length πr, which is why A = 1/2 × C × r.
• Give exact answers in terms of pi when asked, and check rounded answers with an estimate using 3.

The next module uses these formulas on shapes made of several pieces, such as a running track or a window with a round top.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Geometry (7.G.B.4). thecorestandards.org
2. Heath, T. L. (Ed. and Trans.). (1897). The works of Archimedes, including Measurement of a Circle, Propositions 1 and 3. Cambridge University Press.
3. Wikipedia contributors. (n.d.). Measurement of a Circle. In Wikipedia. Retrieved September 24, 2026. en.wikipedia.org
4. Illustrative Mathematics. (n.d.). Grade 7, Unit 3, Lesson 8: Relating area to circumference. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
5. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 9.5, Solve geometry applications: Circles and irregular figures. Prealgebra 2e. OpenStax. openstax.org

A 400-metre lap

Lane 1 of a standard running track is exactly 400 metres long, measured along a line close to the inside edge. On that line, each curved end of the track is a half-circle with a radius of 36.80 metres. Two questions, then. How long is each straight part of the track? And how much ground, grass and all, lies inside that 400-metre line?

Neither answer comes from a single formula, because a track is not a rectangle and not a circle. It is both at once. That is what makes it a good problem to start with.

This lesson covers Common Core standard 7.G.B.6 for area: solving real-world and mathematical problems involving the area of two-dimensional objects composed of triangles, quadrilaterals and other polygons, together with the circle formulas of 7.G.B.4.

Solving the track

Worked example 1: the length of each straight.

Step 1. Split the lap into pieces: two straights and two half-circles.

Step 2. The two half-circles together make one whole circle of radius 36.80 m, so together they are C = 2πr = 2 × π × 36.80 = 231.22 m, to two decimal places. Each half-circle is 115.61 m.

Step 3. The straights are what is left of the 400 m: 400 - 231.22 = 168.78 m for the two, so each straight is 168.78 ÷ 2 = 84.39 m.

Step 4. Check: 2 × 84.39 + 2 × 115.61 = 168.78 + 231.22 = 400.00 m.

Worked example 2: the area inside the line.

Step 1. Split the area the same way: a rectangle in the middle and a half-circle at each end.

Step 2. The rectangle is as long as a straight, 84.39 m, and as wide as a circle's diameter, 2 × 36.80 = 73.60 m. Its area is 84.39 × 73.60 = 6,211.1 square metres.

Step 3. The two half-circles make one whole circle: π × 36.80² = 4,254.5 square metres.

Step 4. Add: 6,211.1 + 4,254.5 = 10,465.6 square metres, about one hectare, which is 10,000 square metres.

Step 5. Estimate to check: a rectangle 160 m by 74 m, the track's full length and width, would be about 11,800 square metres, and the rounded ends must take something off that. About 10,500 fits.

What matters here: A composite shape is solved by naming its pieces. Draw the dividing lines, find each piece with a formula you know, and add, keeping track of which lengths each piece shares with its neighbours.

Adding pieces

Worked example 3: a window with a round top. A window is a rectangle 1.2 m wide and 1.5 m tall, with a half-circle on top whose diameter is the 1.2 m width. What is its area?

Step 1. Rectangle: 1.2 × 1.5 = 1.80 square metres.

Step 2. Half-circle: the diameter is 1.2 m, so the radius is 0.6 m. Area = 1/2 × π × 0.6² = 1/2 × π × 0.36 = 0.5655 square metres.

Step 3. Total: 1.80 + 0.5655 = 2.37 square metres, to the nearest hundredth.

Worked example 4: a wall shaped like a house. The end wall of a shed is a rectangle 8 m wide and 3 m tall, topped by a triangle 8 m wide and 2 m tall. It has a door 0.9 m wide and 2 m tall. How much wall needs painting?

Step 1. Rectangle: 8 × 3 = 24 square metres.

Step 2. Triangle: 1/2 × 8 × 2 = 8 square metres.

Step 3. Door, to be left unpainted: 0.9 × 2 = 1.8 square metres.

Step 4. Paint: 24 + 8 - 1.8 = 30.2 square metres.

Worked example 5: a sprinkler in a corner. A sprinkler in the corner of a square lawn sprays a quarter-circle of radius 6 m. How much lawn does it water?

Step 1. A whole circle of radius 6 m has area π × 6² = 36π.

Step 2. A quarter of it: 36π ÷ 4 = 9π, about 28.3 square metres.

Subtracting holes

Worked example 6: a tile with a hole. A square tile 30 cm on each side has a circular hole 20 cm across cut out of it. What area of tile is left?

Step 1. Square: 30 × 30 = 900 square cm.

Step 2. Hole: radius 10 cm, area π × 10² = 100π, about 314.16 square cm.

Step 3. Left: 900 - 314.16 = 585.84 square cm, about 585.8.

Worked example 7: a path round a pond. A circular pond is 8 m across, and a path 1 m wide runs all the way around it. What is the area of the path?

Step 1. The pond has radius 4 m. The pond and path together make a circle of radius 4 + 1 = 5 m.

Step 2. Big circle: π × 5² = 25π. Pond: π × 4² = 16π.

Step 3. Path: 25π - 16π = 9π, about 28.3 square metres. The ring is the big circle with the small one taken away.

Notice that the path has the same area as the quarter-circle lawn in Worked example 5. Two very different shapes can have equal areas, and keeping the answer in terms of pi, 9π in both, makes that easy to see.

Key idea: Add the areas of pieces that sit side by side; subtract the area of anything cut out. A ring is a big circle minus a small one, and a wall with a door is the wall minus the door.

Checking by splitting another way

An L-shaped room can be split into a 6 m by 4 m rectangle and a 3 m by 2 m rectangle, for 24 + 6 = 30 square metres. A classmate who splits it with a different line gets two different rectangles, but the same total, 30 square metres, because the floor has not changed. Splitting a second way is one of the best checks there is for a composite area, and it catches the commonest slip: using one length for two pieces, or leaving a strip out.

Common misconceptions

"Two half-circles means two circles." The track's two ends are halves of one circle. Counting each as a whole circle adds a whole extra circle, giving 14,720 square metres instead of 10,465.6.

Using the diameter as the radius. The window's half-circle has a 1.2 m diameter, so its radius is 0.6 m. Using 1.2 m as the radius makes the half-circle four times too big.

Mixing up area and perimeter of a curved piece. A quarter-circle of radius 6 m has area 9π square metres, but its curved edge is a quarter of the circumference, 3π metres. They are different measurements with different units.

"A ring's area is the width times the circumference." Subtract the circles instead: 25π - 16π = 9π. Multiplying the 1 m width by either circle's circumference, 8π or 10π, gives the wrong answer.

Looking back

• A composite area is found by splitting the shape into pieces with known formulas and adding them: the track is a rectangle plus one whole circle.
• Holes and cut-outs are subtracted: a tile minus its hole, a wall minus its door, a big circle minus a small one for a ring.
• Parts of circles use fractions of the circle formulas: half-circles and quarter-circles take half and a quarter of πr² for area and of 2πr for the curved edge.
• Perimeters of composite shapes add only the outside edges, such as two straights and two half-circles for a track.
• Check by splitting a second way, by estimating, and by keeping answers in terms of pi until the last step.

The next lesson goes from flat areas to solid shapes: the surface area and volume of prisms.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Geometry (7.G.B.4, 7.G.B.6). thecorestandards.org
2. Wikipedia contributors. (n.d.). Running track. In Wikipedia. Retrieved September 24, 2026, for the lane 1 length of 400.000 m and radius of 36.80 m. en.wikipedia.org
3. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 9.5, Solve geometry applications: Circles and irregular figures. Prealgebra 2e. OpenStax. openstax.org
4. Illustrative Mathematics. (n.d.). Grade 7, Unit 3, Lesson 9: Applying area of circles. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com

A tent for two

A two-person tent is shaped like a prism lying on its side. Each end is a triangle 2.4 m across the bottom and 1.6 m tall, each sloping side runs 2 m from the ridge down to the ground, and the tent is 3 m long. How much air does it hold, and how much fabric does it take, floor included?

The first question asks for a volume and the second for a surface area. They sound alike and they are not: one measures the space inside, in cubic metres, and the other the skin around it, in square metres.

This lesson covers Common Core standard 7.G.B.6 for solids: solving real-world and mathematical problems involving the volume and surface area of three-dimensional objects composed of cubes and right prisms.

Volume: base area times length

A right prism has two identical, parallel ends, called its bases, joined by rectangles that meet the bases at right angles. A box is a prism with rectangular bases; the tent is a prism with triangular bases. You saw in the slicing lesson that every slice of a prism parallel to its base is a copy of the base. That is the key to its volume.

Think of the prism as a stack of slices 1 unit thick. Each slice has the same area as the base, B, so each holds B cubic units, and a prism h units long holds h of them:

V = B × h, where B is the area of the base and h is the length of the prism between its bases.

For a box, the base is a rectangle, B = length × width, so V = length × width × height. For any other prism you first find the area of its base, whatever shape it is.

The tent, end to end

Worked example 1: the air inside the tent.

Step 1. Identify the base. The shape that repeats all the way along the tent is the triangle, so the triangle is the base, even though the tent stands on a rectangle.

Step 2. Area of the base: B = 1/2 × 2.4 × 1.6 = 1.92 square metres.

Step 3. Length between the bases: 3 m.

Step 4. Volume: V = 1.92 × 3 = 5.76 cubic metres.

Check with an estimate: a box 2.4 m by 1.6 m by 3 m would hold 11.52 cubic metres, and a triangle is half of the rectangle around it, so half of that, 5.76, is right.

Worked example 2: the fabric for the tent. Surface area is the total area of every face. Unfold the tent in your head into its net, the flat pattern of its faces, and list them.

Step 1. The two triangle ends: 2 × 1.92 = 3.84 square metres.

Step 2. The two sloping sides are rectangles as long as the tent and as wide as the slope: 2 × (2 × 3) = 12 square metres.

Step 3. The floor: 2.4 × 3 = 7.2 square metres.

Step 4. Surface area: 3.84 + 12 + 7.2 = 23.04 square metres. Without the floor, the tent itself takes 15.84 square metres of fabric.

Notice which lengths went where. The height of the triangle, 1.6 m, was used for the triangle's area only. The sloping sides used their own width, 2 m. Using 1.6 m for the sloping sides is a common slip that makes the tent too small to meet at the ridge.

The point: Volume is base area times length, in cubic units. Surface area is the sum of every face of the net, in square units. List the faces in a table so none is missed or counted twice.

The same steps on a box and a trough

Worked example 3: the cereal box again. The cereal box from the slicing lesson is 20 cm long, 8 cm deep and 30 cm tall.

Step 1. Volume: 20 × 8 × 30 = 4,800 cubic centimetres.

Step 2. Surface area: the box has three pairs of equal faces, 20 by 8, 20 by 30 and 8 by 30, so SA = 2 × (160 + 600 + 240) = 2 × 1,000 = 2,000 square centimetres.

Worked example 4: how many litres in a trough? A water trough is a prism whose ends are trapezoids: 40 cm across the top, 20 cm across the bottom and 15 cm deep. The trough is 100 cm long.

Step 1. Base area: the trapezoid's area is the average of its parallel sides times its height, (40 + 20) ÷ 2 × 15 = 30 × 15 = 450 square centimetres.

Step 2. Volume: 450 × 100 = 45,000 cubic centimetres.

Step 3. Litres. The U.S. National Institute of Standards and Technology describes the litre as a special name for the cubic decimetre, a cube 10 cm on each side, which is 10 × 10 × 10 = 1,000 cubic centimetres. So 45,000 ÷ 1,000 = 45 litres.

Solids built from prisms

Worked example 5: a step block. A wooden step for a stage is made of a block 60 cm long, 30 cm deep and 20 cm tall, with a second block 30 cm long, 30 cm deep and 20 cm tall glued on top of one end.

Step 1. Volume of the bottom block: 60 × 30 × 20 = 36,000 cubic centimetres.

Step 2. Volume of the top block: 30 × 30 × 20 = 18,000 cubic centimetres.

Step 3. Total: 36,000 + 18,000 = 54,000 cubic centimetres, which is 54 litres of wood.

Check another way: the step is also a prism, with an L-shaped base made of a 60 by 20 rectangle and a 30 by 20 rectangle, area 1,200 + 600 = 1,800 square centimetres, running 30 cm deep: 1,800 × 30 = 54,000. The same answer.

For surface area, a solid built from pieces needs more care, because where two blocks touch, those faces are hidden and are not painted or covered. Add only the faces on the outside.

Change the shape: more space, less cardboard

Worked example 6: two gift boxes. Box A is a cube 5 cm on each side. Box B is 10 cm by 6 cm by 2 cm. Which holds more, and which takes more cardboard?

Box B looks bigger, since it is twice as long, and it takes more cardboard. But box A holds more. Volume and surface area can move in opposite directions: a flat, spread-out shape has a lot of surface for the space inside, and a chunky, cube-like shape has less. That is why packaging designers, who pay for cardboard and sell space, care about both numbers.

Remember: Knowing a prism's volume does not tell you its surface area, or the other way round. Work each out from the dimensions.

Common misconceptions

"The base is the face it sits on." The base of a prism is the shape that repeats all along it. The tent's base is its triangle, and using the floor rectangle as B, 7.2 × 3, gives a volume more than three times too big.

Mixing up square and cubic units. Surface area covers, so it is in square units; volume fills, so it is in cubic units. The cereal box has 2,000 square centimetres of cardboard and holds 4,800 cubic centimetres.

Using the triangle's height for the sloping faces. The tent's sloping sides are 2 m wide, not 1.6 m. The height of a triangle and the length of its slanted side are different lengths.

"Doubling one dimension doubles the surface area." It doubles the volume, since V = B × h. The surface area grows by less, because the end faces stay the same size.

Where this leaves us

• A right prism has two identical parallel bases; every slice parallel to them is a copy of the base.
• Volume of any right prism: V = B × h, the area of the base times the length between the bases, in cubic units.
• Surface area: the sum of the areas of all the faces, found by listing the net in a table, in square units.
• A litre is a cubic decimetre, 1,000 cubic centimetres.
• For solids built from prisms, add the volumes of the pieces; for surface area, count only the faces on the outside.
• Two prisms with the same or similar volumes can need very different amounts of material, and the more cube-like shape needs less.

The next module leaves geometry for chance: what a probability is, and how to test one with a coin and a die.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Geometry (7.G.B.6). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 7, Lesson 12: Volume of right prisms. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 7, Lesson 14: Surface area of right prisms. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 9.6, Solve geometry applications: Volume and surface area. Prealgebra 2e. OpenStax. openstax.org
5. National Institute of Standards and Technology. (n.d.). SI units: Volume. U.S. Department of Commerce. Retrieved September 24, 2026. nist.gov

Ten thousand tosses in a camp in Denmark

In April 1940 John Kerrich, a mathematics lecturer from South Africa, was visiting his wife's family in Copenhagen when German troops invaded Denmark. He spent the rest of the Second World War interned in a camp at Hald Ege, near Viborg. With a fellow internee, Eric Christensen, he used the time for an experiment that mathematicians still quote: the two men tossed a coin 10,000 times and recorded every result. They got 5,067 heads.

Everyone knows that the chance of heads is one-half. Kerrich wanted to see what that statement means in practice. Does a fair coin give exactly half heads? In 10 tosses? In 10,000? His answer, and the one you will get from your own coin, is the subject of this lesson.

This lesson covers Common Core standards 7.SP.C.5, probability as a number from 0 to 1; 7.SP.C.6, approximating a probability from the long-run relative frequency of data and predicting frequencies from a probability; and 7.SP.C.7, developing probability models, uniform and not, and comparing them with observed frequencies.

A number from 0 to 1

A probability is a number from 0 to 1 that says how likely an event is. An event that cannot happen has probability 0. An event that must happen has probability 1. An event as likely to happen as not has probability 1/2, and the nearer a probability is to 1, the more likely the event.

Probabilities can be written as fractions, decimals or percents: 1/2, 0.5 and 50 percent say the same thing. A probability can never be less than 0 or more than 1, so an answer of 1.3 or -0.2 is always a mistake.

Theoretical probability: counting equally likely outcomes

All the possible results of a chance process make up its sample space. One roll of a die has the sample space 1, 2, 3, 4, 5, 6. When every outcome is equally likely, which is what a fair die or coin means, the probability of an event is

P(event) = (number of outcomes in the event) ÷ (total number of outcomes).

This is a uniform probability model: each outcome gets the same probability.

Worked example 1: a 3 or a 6. What is the probability of rolling a 3 or a 6 with one fair die, and about how many times would that happen in 600 rolls?

Step 1. Outcomes in the event: 3 and 6, which is 2 outcomes out of 6.

Step 2. P(3 or 6) = 2/6 = 1/3.

Step 3. Prediction: 1/3 of 600 is 200. The Common Core standard uses exactly this example, and adds the important words: roughly 200 times, but probably not exactly 200.

Worked example 2: picking a student at random. A class has 28 students, 15 of them girls, and one of them is Jane. The teacher picks one student at random for a job.

Step 1. Each of the 28 students is equally likely, so P(Jane) = 1/28, about 0.036: unlikely.

Step 2. P(a girl) = 15/28, about 0.54: a little better than even.

Worked example 3: the complement. What is the probability of not rolling a 6?

Step 1. Not a 6 means a 1, 2, 3, 4 or 5: 5 outcomes out of 6, so P(not a 6) = 5/6.

Step 2. Notice that P(6) + P(not a 6) = 1/6 + 5/6 = 1. An event and its complement, everything else, always add to 1, so P(not A) = 1 - P(A).

In short: Theoretical probability comes from reasoning about equally likely outcomes, before anything is tossed or rolled. It tells you what to expect in the long run, not what the next toss will be.

Experimental probability: what really happened

The other way to find a probability is to run the process many times and count. The experimental probability, or relative frequency, is

(number of times the event happened) ÷ (number of trials).

Kerrich's 5,067 heads in 10,000 tosses give an experimental probability of 5,067 ÷ 10,000 = 0.5067, very close to 0.5 but not equal to it. Wikipedia's article on Kerrich prints 2,000 of his results as ones and zeros. Counting the heads in the first 10, 20, 100 and more of those printed results gives this table:

Worked example 4: reading the table.

Step 1. In the first 10 printed results the relative frequency was 0.4, and in the first 100 it was 0.44, a long way from 0.5 in both cases.

Step 2. From the 1,000th result to the 2,000th, the running relative frequency stayed between about 0.49 and 0.51.

Step 3. The number of heads never needed to equal half the tosses. Across all 2,000 printed results there were 1,014 heads, 14 more than half, and yet the proportion, 0.507, was closer to 0.5 than it was in the first 100. The proportion settles down even though the count wanders.

This settling down in the long run is called the law of large numbers, and demonstrating it was the point of Kerrich's experiment.

Why this matters: A probability is a prediction about the long run, not a promise about the next few trials. Small experiments wander; large ones settle close to the theoretical probability, when the model is right.

When the outcomes are not equally likely

Kerrich and Christensen also made a lopsided "coin", a wooden disk partly coated in lead, and tossed that. It landed one particular way about 70 percent of the time. No amount of counting sides could have predicted that 70 percent. The only way to find it was by experiment, and the model it gives, about 0.7 for one side and 0.3 for the other, is a probability model that is not uniform.

Worked example 5: a model from data. Suppose the lead disk gave 350 of one side in its first 500 tosses. Find the experimental probability and predict the count in 2,000 tosses.

Step 1. 350 ÷ 500 = 0.7.

Step 2. In 2,000 tosses, about 0.7 × 2,000 = 1,400 of that side, and about 600 of the other, with the usual wandering around those numbers.

When a model and the data disagree badly, something is wrong with the model or with the experiment. If a die came up 6 on 300 of 600 rolls, far above the 100 or so a fair die should give, you would suspect the die, not the arithmetic.

Your turn: a coin and a die

Get a coin and a die and do this now; it takes about ten minutes.

Worked example 6: one student's run, and how to read yours. Toss the coin 20 times and roll the die 30 times, tallying the results. Here is how to read them, using one possible set of results as an example: 12 heads in 20 tosses, and five 6s in 30 rolls.

Step 1. Experimental probability of heads: 12 ÷ 20 = 0.6. The theoretical value is 0.5; a difference of 0.1 in only 20 tosses is ordinary, as the first 10 of Kerrich's printed results showed.

Step 2. Experimental probability of a 6: 5 ÷ 30 = 0.167, which happens to match 1/6 = 0.167 closely. With another 30 rolls it might be 3 or 8 sixes instead.

Step 3. Pool your results with a friend's, or keep going to 100 tosses, and recompute. The relative frequencies usually move closer to 0.5 and 1/6.

Common misconceptions

"After five heads, tails is due." A coin has no memory: the chance of heads on the next toss is still 1/2. Streaks happen more often than people expect. In the 2,000 results of Kerrich's printed on Wikipedia there is a run of 12 heads in a row.

"A fair coin gives exactly half heads." It gives about half, over many tosses. Kerrich's fair coin gave 5,067 heads in 10,000, not 5,000.

"Probability 1/6 means one 6 in every six rolls." It means about one in six over a long run. In six rolls you may get none or three.

"Experimental and theoretical probability must match, or the maths is wrong." In a short run they usually differ. A large, persistent difference, though, is evidence that the model does not fit, as with the lead-coated disk.

What to carry forward

• A probability is a number from 0 (impossible) to 1 (certain); 1/2 means as likely as not.
• With equally likely outcomes, P(event) = outcomes in the event ÷ total outcomes, a uniform model. P(not A) = 1 - P(A).
• Experimental probability is the relative frequency from data: times it happened ÷ trials.
• In short runs the relative frequency wanders; in long runs it settles near the true probability. Kerrich's coin gave 0.44 in the first 100 printed results and 0.5067 over all 10,000 tosses.
• Predict frequencies by multiplying: about 1/3 of 600 rolls, 200, will show a 3 or a 6, though probably not exactly 200.
• When outcomes are not equally likely, like the lead-coated disk, the model has to come from data.

The next lesson combines events: two coins, a coin and a die, or a spinner spun twice, and four ways to count the outcomes.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Statistics and Probability (7.SP.C.5-7). thecorestandards.org
2. Wikipedia contributors. (n.d.). John Edmund Kerrich. In Wikipedia. Retrieved September 24, 2026, for the internment, the 5,067 heads in 10,000 tosses, the lead-coated disk and the 2,000 printed results. en.wikipedia.org
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 8, Lesson 4: Estimating probabilities through repeated experiments. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Marecek, L., Anthony-Smith, M., and Honeycutt Mathis, A. (2020). Section 5.5, Averages and probability. Prealgebra 2e. OpenStax. openstax.org

Four tools for one kind of question

Toss a coin and roll a die at the same time. What is the chance of heads with an even number? Roll two dice: why does 7 come up so much more often than 12? Toss three coins: how likely is exactly two heads? And a question from the Common Core standard itself: if 40 percent of blood donors have type A blood, what is the probability that it takes at least 4 donors to find one with type A?

Each of these is a compound event, an event made of more than one step. This lesson compares four tools for them, an organised list, a table, a tree diagram and a simulation, and shows which tool suits which question. It covers Common Core standard 7.SP.C.8: finding probabilities of compound events using organised lists, tables, tree diagrams and simulation, with the rule that the probability of a compound event is the fraction of outcomes in the sample space for which the event happens.

The four tools side by side

The first three tools count the outcomes of the sample space exactly, and work when the outcomes are equally likely. The fourth replaces counting with experiment. Now the four questions, one tool each.

An organised list: a coin and a die

Worked example 1: heads with an even number.

Step 1. List the outcomes in order, all the heads first, then all the tails: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6. That is 12 equally likely outcomes, which is 2 × 6.

Step 2. The event "heads and an even number" is H2, H4 and H6: 3 outcomes.

Step 3. P(heads and even) = 3/12 = 1/4.

Step 4. From the same list, P(tails and a 6) = 1/12, since only T6 fits.

A table: two dice

Worked example 2: the sums of two dice. Put the first die down the side and the second across the top, and write the sum in each cell.

Step 1. The table has 6 × 6 = 36 cells, and each is an equally likely outcome.

Step 2. A sum of 7 appears on the diagonal from bottom left to top right: 6 cells. So P(sum of 7) = 6/36 = 1/6.

Step 3. A sum of 12 appears once, when both dice show 6. So P(sum of 12) = 1/36. The standard's example "rolling double sixes" is this single cell.

Step 4. The same table answers other questions at a glance: P(a double) = 6/36 = 1/6, and P(sum of 10 or more) = (3 + 2 + 1)/36 = 6/36 = 1/6.

This is why 7 comes up so much more often than 12. The eleven sums from 2 to 12 are not equally likely, because different numbers of cells make them. Counting sums instead of cells, and saying P(7) = 1/11, is a classic mistake.

The upshot: Find the probability of a compound event by counting outcomes of the sample space, not results that merely sound different. A sum of 7 is six outcomes; a sum of 12 is one.

A tree diagram: three coins

Worked example 3: exactly two heads, and at least one. A tree has one level of branches for each coin. The first coin branches to H and T, each of those branches to H and T for the second coin, and each of those again for the third.

Step 1. Following the tree from left to right gives 2 × 2 × 2 = 8 paths, the 8 rows above.

Step 2. Exactly two heads: HHT, HTH and THH, so P(exactly two heads) = 3/8.

Step 3. At least one head: every path except TTT, so P(at least one head) = 7/8. It is quicker to use the complement: 1 - P(no heads) = 1 - 1/8 = 7/8.

Worked example 4: a lunch menu. A cafe picks your lunch at random from 3 sandwiches (turkey, cheese, egg) and 2 drinks (juice, milk). What is the probability of turkey with juice?

Step 1. A tree with 3 branches, then 2 on each, has 3 × 2 = 6 paths: turkey-juice, turkey-milk, cheese-juice, cheese-milk, egg-juice, egg-milk.

Step 2. P(turkey with juice) = 1/6.

A simulation: the blood donors

The donor question is harder to count, because donors keep coming until one has type A, so the number of steps is not fixed. A simulation acts the situation out with a random tool whose probabilities match it.

Worked example 5: at least 4 donors to find type A.

Step 1. Choose the tool. Random digits 0 to 9 are equally likely, so let 0, 1, 2 and 3 stand for a type A donor. Four digits out of ten is 40 percent, matching the question. The digits 4 to 9 stand for a donor who is not type A.

Step 2. One trial: read random digits until you meet a 0, 1, 2 or 3, and count how many digits that took. That is the number of donors needed.

Step 3. Repeat. Here are 20 trials made with a computer's random digits:

Step 4. Count the trials that needed at least 4 donors: 8, 5, 4, 5 and 4, which is 5 trials. The estimate is 5/20 = 0.25.

Step 5. Check against a count. Needing at least 4 donors means the first three donors are all not type A. Think of every possible string of three digits, 000 to 999: there are 1,000 of them, equally likely. The strings with no 0, 1, 2 or 3 in them use only the six digits 4 to 9 in each place, so there are 6 × 6 × 6 = 216 of them. The exact probability is 216/1,000 = 0.216.

The simulation's 0.25 is close to 0.216, and more trials would usually bring it closer. That is the trade: a simulation is easy to run for almost any question, but it gives an estimate whose accuracy depends on how many trials you run.

So what?: When a question is easy to count, count it exactly with a list, a table or a tree. When it is hard to count, simulate it, and run enough trials to trust the estimate.

Common misconceptions

"All the sums of two dice are equally likely." There are 11 sums but 36 equally likely outcomes, and a sum of 7 has six of them. P(7) = 1/6, not 1/11.

"A coin and a die give 2 + 6 = 8 outcomes." Each coin result pairs with each die result, so there are 2 × 6 = 12. Multiply the choices at each step, do not add them.

"(1, 6) and (6, 1) are the same outcome." On two different dice they are different outcomes, two separate cells in the table. Counting them once makes every sum except the doubles look less likely than it is.

"A simulation gives the exact answer." Twenty trials gave 0.25 against the exact 0.216. A simulation estimates, and more trials make a better estimate.

Summing up

• A compound event has more than one step. Its probability is the fraction of the equally likely outcomes in the sample space that make it happen.
• An organised list suits small sample spaces; a table suits two steps; a tree diagram suits three or more steps.
• The number of outcomes is the product of the choices at each step: 2 × 6 = 12, 6 × 6 = 36, 2 × 2 × 2 = 8.
• Translate everyday words into outcomes: double sixes is one cell; at least one head is every path but TTT.
• A simulation uses random digits, dice or coins with matching probabilities, repeated many times, to estimate a probability that is hard to count.

The next module turns from chance to data: what a random sample of a population can tell you, and how far to trust it.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Statistics and Probability (7.SP.C.8). thecorestandards.org
2. Illustrative Mathematics. (n.d.). Grade 7, Unit 8, Lesson 7: Simulating multi-step experiments. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
3. Illustrative Mathematics. (n.d.). Grade 7, Unit 8, Lesson 3: What are probabilities? IM 6-8 Math. Kendall Hunt. im.kendallhunt.com
4. Illowsky, B., and Dean, S. (2023). Section 3.5, Tree and Venn diagrams. Introductory Statistics 2e. OpenStax. openstax.org

2.38 million answers, and the wrong winner

In 1936 the American magazine The Literary Digest mailed ten million questionnaires asking people whom they would vote for as president: Franklin D. Roosevelt, the president, or the Republican Alf Landon. About 2.38 million came back, an enormous total for any poll. From them the magazine predicted that Landon would win, with 57.08 percent of the popular vote. In November, Roosevelt won every state except Maine and Vermont.

That same year George Gallup's American Institute of Public Opinion asked about 50,000 people, fewer than one for every forty the Digest heard from, and predicted a Roosevelt victory. How can a far smaller sample get the winner right when a gigantic one gets it wrong?

This lesson covers Common Core standards 7.SP.A.1 and 7.SP.A.2: using a sample to gain information about a population, understanding that such conclusions are valid only if the sample is representative and that random sampling tends to produce representative samples, and using several samples to gauge how far off an estimate might be.

The question at school

The same argument happens on a smaller scale in any school. Suppose the student council wants to know whether the school's 600 students like a new lunch menu. Two council members disagree about how to find out.

Priya wants as many answers as possible. She plans to stand at the cafeteria door on Monday and ask everyone who comes through, which should give about 300 answers. "More answers means a more accurate result," she says.

Leo wants to put all 600 names on a list, pick 40 of them at random, and ask only those 40. "Forty picked fairly will tell us more than three hundred picked by who happened to walk past," he says.

Both have a real point. Before hearing the evidence, a few words. The population is the whole group the question is about: all 600 students. A sample is the part of it you actually ask. A sample is representative if it is like the population in the ways that matter to the question.

Priya's evidence: bigger samples vary less

There is a population small enough to check completely: the 52 words of the Preamble to the United States Constitution, which begins "We the People of the United States". Counting every letter, the 52 words have 268 letters, so the population's mean word length is 268 ÷ 52 = 5.15 letters, to two decimal places.

Now pretend you do not know that, and estimate it from random samples, picking words with a random number generator so that every word has the same chance.

Worked example 1: samples of 4 words against samples of 16.

Step 1. One random sample of 4 words was: States, Union, ordain, establish. Their lengths are 6, 5, 6 and 9, so the sample mean is 26 ÷ 4 = 6.5 letters.

Step 2. Ten random samples of 4 words gave means of 6.50, 5.25, 4.25, 4.00, 2.75, 4.25, 5.25, 7.25, 6.00 and 3.50 letters. They range from 2.75 to 7.25.

Step 3. Ten random samples of 16 words gave means of 4.38, 4.94, 5.75, 5.44, 5.31, 5.50, 4.38, 5.56, 5.63 and 5.63 letters. They range only from 4.38 to 5.75.

Step 4. Compare with the true 5.15. Both sets of samples scatter around it, but the samples of 16 stay much closer. That is Priya's point, and it is true: larger random samples give estimates that vary less from sample to sample.

Leo's evidence: a biased sample misleads however big it is

Worked example 2: a sample chosen by a rule. Suppose that instead of choosing at random, you picked the words that look important, the 16 capitalised ones: We, People, United, States, Order, Union, Justice, Tranquility, Welfare, Blessings, Liberty, Posterity, Constitution, United, States and America.

Step 1. Their mean length is about 6.94 letters.

Step 2. That is 16 words, the same size as the random samples in Worked example 1, yet it is further from the true 5.15 than any of the ten random samples of 16. Every one of those stayed between 4.38 and 5.75.

Step 3. The reason is the rule. Capitalised words are nouns and names, which tend to be long, and the short words, such as the, of, to and and, never get in. A sample chosen by a rule that favours some members of the population is biased, and making it bigger does not fix it: take more capitalised words from the whole Constitution and you would get a more precise estimate of the wrong number.

That is what happened to the Literary Digest. It drew its names from its own readers and from lists of car owners and telephone users, groups that in the middle of the Great Depression were far wealthier than most Americans and far more likely to vote for Landon. And it counted only the people who chose to send the form back; later research found that this non-response bias, people who strongly disliked Roosevelt being more eager to reply, was the biggest cause of the error. Two biases, and 2.38 million answers could not cancel either of them.

Priya's cafeteria plan has the same weakness on a small scale. Students who bring lunch from home, or who skip lunch, or who are absent on Monday, would never be asked, and their opinions of the new menu may be quite different from those of the students who buy it.

Worth holding on to: Bias comes from how a sample is chosen; variability comes from how big it is. A bigger sample reduces variability but never removes bias.

What settles it

Both students are right about something, and the evidence says which comes first. A sample has to be representative before its size helps, and the reliable way to make it representative is a random sample, in which every member of the population has the same chance of being chosen. Random choice cannot favour any group, because it does not know which group anyone is in. Once the sample is random, bigger is better, exactly as Priya said.

So the verdict is Leo's method with Priya's ambition: pick at random from the whole list, and ask as many as you can manage. A random sample of 40 beats a biased sample of 300; a random sample of 100 beats a random sample of 40.

The core of it: Choose randomly, so the sample is representative; then choose as many as you can, so the estimate varies less.

Using a random sample

Worked example 3: estimating the whole school. The council numbers the 600 students from 1 to 600 and uses a random number generator to pick 40 of them. Of the 40, 26 say they like the new menu.

Step 1. The sample proportion: 26 ÷ 40 = 0.65, or 65 percent.

Step 2. Estimate for the population: 65 percent of 600 is 0.65 × 600 = 390 students.

Worked example 4: how far off might it be? The Common Core standard asks you to gauge the error by taking more samples of the same size. Suppose three more random samples of 40 found 22, 28 and 24 students who liked the menu.

Step 1. As percents: 22 ÷ 40 = 55 percent, 28 ÷ 40 = 70 percent, and 24 ÷ 40 = 60 percent, alongside the first sample's 65 percent.

Step 2. The four samples range from 55 to 70 percent. So the true percent for the school is probably somewhere around there, and a single sample of 40 could easily be 5 or 10 points away from it.

Step 3. Say so in the report: "About 55 to 70 percent of students like the new menu, roughly 330 to 420 of the 600." An estimate with an honest range is more useful than a single number that sounds exact.

Common misconceptions

"A bigger sample is always better." Only among fair samples. The Digest's 2.38 million biased answers lost to Gallup's 50,000.

"A random sample means any sample I did not plan carefully." Random has a precise meaning: every member has the same chance of being chosen, as with names drawn from a hat or a random number generator. Asking whoever walks past is a convenience sample, and a sign-up poll is a volunteer sample; both are usually biased.

"A random sample gives the exact answer." Random samples vary. Ten random samples of 4 words gave ten different means, and even samples of 16 ranged from 4.38 to 5.75.

"If a sample is biased, the whole method of sampling is useless." The fault is in how that sample was chosen, not in sampling. Gallup's much smaller sample picked the right winner.

The short version

• A population is the whole group a question is about; a sample is the part you examine. Conclusions about the population are valid only if the sample is representative.
• Random sampling, where every member has the same chance, tends to give representative samples. Convenience and volunteer samples are usually biased.
• Bias comes from the way a sample is chosen and is not cured by size; the 1936 Literary Digest poll had 2.38 million answers and the wrong winner.
• Larger random samples vary less: random samples of 16 Preamble words stayed much closer to the true mean than samples of 4.
• Estimate a population value by scaling up the sample, and take several samples of the same size to see how far off the estimate might be.

The last lesson uses samples and whole data sets to compare two populations, by their centres and their spreads.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Statistics and Probability (7.SP.A.1, 7.SP.A.2). thecorestandards.org
2. Wikipedia contributors. (n.d.). The Literary Digest. In Wikipedia. Retrieved September 24, 2026, for the 1936 poll: ten million mailed, 2.38 million replies, the 57.08 percent prediction, and Gallup's sample of 50,000. en.wikipedia.org
3. U.S. National Archives and Records Administration. (n.d.). The Constitution of the United States: A transcription. Retrieved September 24, 2026, for the text of the Preamble. archives.gov
4. Illustrative Mathematics. (n.d.). Grade 7, Unit 8, Lesson 14: Sampling in a fair way. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com

Sixty-six degrees and eighty-one

In San Diego, the normal high temperature on a January day is 66.4 °F. In Honolulu it is 80.5 °F. Those numbers come from the U.S. Climate Normals that the National Oceanic and Atmospheric Administration (NOAA) publishes for weather stations across the country: averages over the thirty years from 1991 to 2020, one value for each month. Here are the normal highs for all twelve months at the airport station in each city.

Honolulu is warmer; you can see that at a glance. The question this lesson answers is how to say how much warmer, in a way that also takes account of how much each city's temperatures vary, and how to compare two groups when the difference is not obvious at all.

This lesson covers Common Core standards 7.SP.B.3, assessing the overlap of two distributions with similar variabilities by expressing the difference between their centres as a multiple of a measure of variability, and 7.SP.B.4, using measures of centre and variability from random samples to draw informal comparative inferences about two populations.

Centre and spread for each city

Worked example 1: the means and medians.

Step 1. San Diego's twelve values add to 850.7, so the mean is 850.7 ÷ 12 = 70.9 °F, to one decimal place.

Step 2. Honolulu's add to 1,015.1, so its mean is 1,015.1 ÷ 12 = 84.6 °F.

Step 3. For the medians, put each set in order. With 12 values, the median is halfway between the 6th and 7th. San Diego: 66.0, 66.2, 66.4, 67.0, 68.8, 69.5, 70.7, 71.7, 74.6, 75.3, 77.2, 77.3, so the median is (69.5 + 70.7) ÷ 2 = 70.1 °F. Honolulu: 80.5, 80.5, 81.2, 81.8, 83.1, 84.1, 84.8, 86.9, 86.9, 88.1, 88.4, 88.8, so the median is (84.1 + 84.8) ÷ 2 = 84.45 °F.

The means and medians agree closely in each city, which tells you that neither set is badly lopsided.

Worked example 2: the mean absolute deviation. The mean absolute deviation, or MAD, is the average distance of the values from their mean. It says how far a typical month is from the city's average.

Step 1. For San Diego, find each month's distance from the mean of 70.89, ignoring the sign: 4.49, 4.69, 3.89, 2.09, 1.39, 0.81, 4.41, 6.41, 6.31, 3.71, 0.19 and 4.89.

Step 2. Those distances add to 43.28, and 43.28 ÷ 12 = 3.61. San Diego's MAD is about 3.6 °F.

Step 3. The same steps for Honolulu give distances adding to 32.70, and 32.70 ÷ 12 = 2.73, so its MAD is about 2.7 °F.

Both cities are steady through the year, with similar, small spreads: a typical month is only about 3 degrees from the yearly average.

How far apart, measured in MADs

Here are both sets on one scale. Each dot is one month; dots for months with almost the same value are stacked.

Worked example 3: the gap between the centres.

Step 1. Difference between the means: 84.59 - 70.89 = 13.7 °F.

Step 2. Express it in MADs, the way the Common Core standard does: 13.7 ÷ 3.61 = 3.8 of San Diego's MADs, or 13.7 ÷ 2.73 = 5.0 of Honolulu's. Either way, the centres are about 4 to 5 typical deviations apart.

Step 3. Read the dot plot. The warmest month in San Diego, August at 77.3 °F, is cooler than the coolest months in Honolulu, 80.5 °F. The two distributions do not overlap at all.

That is what a gap of several MADs looks like. When two groups have similar spreads, a difference in centres of about 2 MADs is clearly visible on a plot, and at 4 or 5 MADs the groups barely touch or not at all.

Bottom line: A difference between two means only means something next to the spread. Divide the difference by a typical MAD: a large result says the groups are clearly different; a small one says they overlap a lot.

Three comparisons side by side

Now the same test on two more pairs. The first uses two founding documents from the National Archives: the 52 words of the Preamble to the Constitution, and the 45 words of the First Amendment, which begins "Congress shall make no law". The second puts San Diego beside a city with a very different climate, Minneapolis, whose normal highs run from 23.6 °F in January to 83.4 °F in July.

Worked example 4: two texts that are nearly alike. The Preamble's words have 268 letters, a mean of 5.15 letters per word and a MAD of 2.27. The First Amendment's have 222 letters, a mean of 4.93 and a MAD of 2.73.

Step 1. Difference in means: 5.15 - 4.93 = 0.22 letters.

Step 2. In MADs: 0.22 ÷ 2.27 is about 0.1. The difference is a tenth of a typical deviation.

Step 3. Conclusion: both texts are full of short words such as of, the and to, and long ones such as establishment and Tranquility. Their distributions overlap almost completely, so there is no meaningful sense in which one uses longer words than the other.

Worked example 5: when the spreads are very different. San Diego's mean high is 15.5 °F above Minneapolis's. Divided by San Diego's MAD that is 4.3, which sounds like a clear separation; divided by Minneapolis's MAD, 18.7, it is only 0.8. The method gives two very different answers because it assumes the two groups have similar variabilities, and these do not. Minneapolis's highs spread from 23.6 to 83.4 °F, straddling every one of San Diego's months, so the real story is the spread. Minneapolis has a winter and a summer; San Diego barely has either.

What matters here: The difference in means divided by a MAD is a fair comparison only when the two spreads are similar. When they are not, describe the spreads first; they are often the more important difference.

Comparing populations from random samples

Often you cannot measure whole populations, and have to compare samples instead, as the standard's example does when it asks whether a seventh-grade science book uses longer words than a fourth-grade one.

Worked example 6: samples of 10 words from each text. Pretend you could not count every word, and took random samples of 10 words from each document instead. Three pairs of samples, chosen with a random number generator, gave these means:

Step 1. Check the third pair. The Preamble sample was We, the, a, insure, common, general, and, this, the, America: 2 + 3 + 1 + 6 + 6 + 7 + 3 + 4 + 3 + 7 = 42 letters, a mean of 4.2. The First Amendment sample was respecting, an, establishment, or, abridging, of, assemble, petition, a, of: 57 letters, a mean of 5.7.

Step 2. The pairs disagree about which text uses longer words, and pair 3 points the opposite way from the true means, 5.15 and 4.93.

Step 3. That is exactly what to expect when the true difference is tiny compared with the variability: random samples scatter by more than the gap between the populations. For the two cities, whose difference is several MADs, almost any pair of random samples would put Honolulu ahead.

So an informal inference from samples should always say how big the difference is next to the spread, and whether several samples agree, before claiming that one population differs from another.

Common misconceptions

"Different means prove the groups are different." Means of 5.15 and 4.93 letters differ, but by a tenth of a MAD; the word lengths of the two texts are, for practical purposes, the same.

"The mean tells you what a place is like." San Diego and Minneapolis have mean highs 15.5 degrees apart, but the bigger difference is that Minneapolis's months range over 60 degrees and San Diego's over about 11.

"The MAD is the range." The range is the largest value minus the smallest: 77.3 - 66.0 = 11.3 °F for San Diego. The MAD is the average distance from the mean, 3.6 °F. One extreme value changes the range a lot and the MAD only a little.

"One pair of samples settles the comparison." Pair 3 of the word samples pointed the wrong way. Look at the size of the difference relative to the spread, and at several samples.

Pulling it together

• Describe each group with a centre, the mean or median, and a spread, such as the mean absolute deviation, the average distance of the values from the mean.
• When two groups have similar spreads, divide the difference in their means by a typical MAD. Several MADs, like the 3.8 to 5.0 between San Diego and Honolulu, means little or no overlap; a fraction of a MAD, like the 0.1 between the two texts, means almost complete overlap.
• When the spreads are very different, as with San Diego and Minneapolis, compare the spreads first; one ratio of centre to spread can mislead.
• Random samples from two populations let you compare them informally, but when the true difference is small next to the variability, samples can point either way.
• Real data sets, such as NOAA's climate normals, reward the same questions every time: what is typical, how much does it vary, and how far apart are the groups compared with that variation?

That is the end of the seventh-grade year. The same tools, proportional reasoning, equations, geometry measurement, probability and data, are the ones the eighth-grade year builds on.

Sources

1. National Governors Association Center for Best Practices, and Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics: Grade 7, Statistics and Probability (7.SP.B.3, 7.SP.B.4). thecorestandards.org
2. NOAA National Centers for Environmental Information. (2021). U.S. Climate Normals 1991-2020: Monthly normals [Data set], stations USW00023188, San Diego Lindbergh Field, and USW00022521, Honolulu International Airport, variable MLY-TMAX-NORMAL. San Diego file; Honolulu file
3. NOAA National Centers for Environmental Information. (2021). U.S. Climate Normals 1991-2020: Monthly normals [Data set], station USW00014922, Minneapolis/St Paul Airport. ncei.noaa.gov
4. U.S. National Archives and Records Administration. (n.d.). The Bill of Rights: A transcription. Retrieved September 24, 2026, for the text of the First Amendment. archives.gov
5. Illustrative Mathematics. (n.d.). Grade 7, Unit 8, Lesson 18: Comparing populations using samples. IM 6-8 Math. Kendall Hunt. im.kendallhunt.com