Pre-Algebra & Math Foundations

The big picture

If math has ever made your stomach drop, you are in the right place, and you are going to be okay. Here is a secret almost no one tells you: you already read big numbers every single day. When you see a price like $4,275 on a car, or 1,200 followers, or 350 miles on a map, your brain is already doing the thing we are about to name. You are not starting from nothing.

This week we look closely at how a number is built, and how to add, subtract, multiply, and divide the everyday numbers we call whole numbers (0, 1, 2, 3, and up). We will go slowly, one small step at a time. Nothing skipped, nothing rushed, and no question is too small.

Key idea: A number is not one big lump. It is made of digits, and where each digit sits tells you what it is worth.

Digits and place value, told with a picture

There are only ten symbols in all of arithmetic: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We call each one a digit. Read that once more if you like: every number you have ever seen, no matter how huge, is spelled with just those ten symbols.

What makes 4 in one spot different from 4 in another spot is its place. Picture a row of labeled buckets, and we drop one digit into each bucket. Reading from the right, the buckets are: ones, tens, hundreds, thousands.

thousands	hundreds	tens	ones
4	2	7	5

So the number 4,275 is really 4 thousands, plus 2 hundreds, plus 7 tens, plus 5 ones. Let us read that out in plain money: four thousand, two hundred seventy-five. The 4 is not just a 4 here. Because it sits in the thousands bucket, it is worth 4,000.

Why does each bucket to the left count for more? Because each step left is ten times bigger than the bucket on its right: ones, then tens (ten ones), then hundreds (ten tens), then thousands (ten hundreds). This ten-times-bigger pattern is why our system is called base ten.

Try it. In the number 6,300, what is the 3 worth? (Answer below in a moment.)

Reading from the right: the last 0 is ones, the next 0 is tens, the 3 is hundreds, the 6 is thousands. So the 3 sits in the hundreds bucket and is worth 300. Nice, that is exactly the skill.

Key idea: To find what a digit is worth, find its bucket (ones, tens, hundreds, thousands) by counting from the right.

Adding, with regrouping shown one tiny step at a time

Adding just means putting amounts together. The one new word is regroup, sometimes called "carry." Here is the whole idea in plain words: a single bucket can only hold the digits 0 through 9. The moment a bucket would need to hold 10 or more, we bundle up ten of them and pass one bundle to the next bucket on the left. That is all carrying is.

Let us add 4,275 + 1,849 together, slowly. We line the numbers up so ones sit under ones, tens under tens, and so on.

	4	2	7	5
+	1	8	4	9

Step 1. Start at the ones bucket (the far right). 5 + 9 = 14. That is more than 9, so it will not fit. We write the 4 and carry the 1 ten over to the tens bucket.

Step 2. Now the tens bucket. 7 + 4 = 11, and do not forget the 1 we just carried, so 11 + 1 = 12. Again too big for one bucket. Write the 2 and carry the 1 to the hundreds.

Step 3. Now the hundreds bucket. 2 + 8 = 10, plus the carried 1 makes 11. Write the 1 and carry the 1 to the thousands.

Step 4. Now the thousands bucket. 4 + 1 = 5, plus the carried 1 makes 6. Nothing to carry this time. Write the 6.

Reading the answer off the buckets: 6,124. Take a second to notice what just happened. You added two four-digit numbers by only ever adding small pairs, one bucket at a time. That is the whole method.

A gentle habit: estimate first. Round each number to something easy: 4,275 is about 4,000 and 1,849 is about 2,000. So the answer should be near 6,000. Our 6,124 is close to 6,000, which tells us we probably did not make a big mistake. Estimating is a kind, quick safety check, not extra work.

Key idea: Add one bucket at a time from the right, and whenever a bucket reaches 10 or more, carry one to the next bucket on the left.

Subtracting, and the idea of borrowing

Subtracting means taking away. The new word here is borrow, and it is just carrying run in reverse. Sometimes the top digit in a bucket is too small to take away from. When that happens, we borrow one from the bucket to its left, which arrives as ten in the current bucket.

Let us do 6,004 - 1,268, slowly.

Step 1. Ones bucket: we want 4 - 8. But 4 is smaller than 8, so we need to borrow. The tens and hundreds here are 0, so we borrow across: think of 6,004 as 5 thousands, 9 hundreds, 9 tens, and 14 ones (we walked one thousand down through the empty buckets). Now the ones bucket holds 14.

Step 2. Ones bucket: 14 - 8 = 6. Write 6.

Step 3. Tens bucket: 9 - 6 = 3. Write 3.

Step 4. Hundreds bucket: 9 - 2 = 7. Write 7.

Step 5. Thousands bucket: 5 - 1 = 4. Write 4.

Reading it off: 4,736. Quick check the friendly way: 6,004 is about 6,000 and 1,268 is about 1,000, so we expect around 5,000. We got 4,736, which is close. Good sign.

If borrowing across zeros felt strange, that is normal. It trips up almost everyone at first. The safe move is to slow down and rewrite the top number bucket by bucket, exactly as we did.

Key idea: If the top digit is too small to subtract, borrow one from the bucket to its left, which becomes ten where you need it.

Multiplying and dividing, in plain words

Multiplication is just fast repeated addition. Saying "6 times 4" is a shortcut for adding 6 four times: 6 + 6 + 6 + 6 = 24. So 6 x 4 = 24. The result of multiplying is called the product.

Division is splitting a total into equal groups. Asking "72 divided by 6" is asking: if I share 72 things fairly into 6 groups, how many land in each group?

Let us reason through 72 divided by 6, slowly.

Step 1. Ask the friendly question: 6 times what gives 72?

Step 2. Try a number. 6 x 10 = 60, which is not quite enough.

Step 3. Try a bit more. 6 x 12 = 72. That is it.

Step 4. So 72 divided by 6 = 12.

Here is a lovely fact: multiplication and division are opposites, so they check each other. To make sure 72 divided by 6 is really 12, multiply back: 6 x 12 = 72. It matches, so we are right. You can always check a division by multiplying, and check a subtraction by adding.

Key idea: Multiplication is repeated adding; division is fair sharing. They undo each other, so each one checks the other.

Bigger multiplication: break it into buckets

How do you multiply 34 x 26 without a calculator? With the buckets again. The number 26 is really 20 + 6, so 34 x 26 means "34 times 20, plus 34 times 6." Multiplying by each part, then adding the pieces, is called the distributive property.

Estimate first: both are about 30, and 30 x 30 = 900, so expect an answer near 900.

Step 1. Multiply by the ones bucket: 34 x 6 = 204.

Step 2. Multiply by the tens bucket: 34 x 20 = 680. The 2 in 26 is worth 2 tens, so we multiply by 20, not by 2.

Step 3. Add the two partial answers: 204 + 680 = 884.

Step 4. Compare: we predicted about 900 and got 884. Close. Good sign.

Why you can never divide by zero

Here is a question that stumps plenty of adults: what is 12 / 0? It has no answer, and the reason is friendly. Every division is secretly a multiplication question. "12 / 0 = ?" asks "0 times what gives 12?" But 0 times anything is 0, never 12. No number works, so we say dividing by zero is undefined.

Watch the pattern too: 12 / 6 = 2, 12 / 4 = 3, 12 / 2 = 6, 12 / 1 = 12. Smaller divisors give ever bigger answers, with no stopping point. And note that 0 / 12 is fine, and equals 0. Only dividing BY zero breaks.

Common misconceptions

• "The bigger-looking number is always worth more." Not by itself. A digit is worth its face value times its bucket. The 3 in 3,000 is worth far more than the 9 in 90, because place decides value.
• "When I carry, I add the carried 1 at the end." Add the carried 1 into the very next bucket you work on, before you move past it, not at the end.
• "You cannot subtract a bigger digit from a smaller one." You can, once you borrow. Borrowing brings ten more into the bucket so the subtraction works.
• "Lining up does not matter." It matters a lot. Always line up ones under ones and tens under tens, or the buckets get mixed and the answer goes wrong.

Recap

• Every number is spelled with ten digits, and each digit's value depends on its place (ones, tens, hundreds, thousands), because we work in base ten.
• Add and subtract one bucket at a time from the right. Carry when a bucket reaches 10 or more; borrow when the top digit is too small.
• Multiplication is repeated addition; division is fair sharing; they undo and check each other.
• Estimate first as a gentle safety check. You did real arithmetic this week, one small step at a time.

Sources

• OpenStax. (2020). 1.1 Introduction to whole numbers. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 1.2 Add whole numbers. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 1.5 Divide whole numbers. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Add and subtract through 1,000,000 [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Multiply and divide multi-digit numbers [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Place value. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Long division. Math is Fun. mathsisfun.com

The big picture

This week has a nickname, PEMDAS, but the idea behind it is simple: it is just an agreement about the order to do things so that everyone in the whole world gets the same answer. That is all it is. An agreement, like "we drive on the right side of the road" so nobody crashes.

Here is why we need it. Look at 2 + 3 x 4. If you add first you get 5 x 4 = 20. If you multiply first you get 2 + 12 = 14. Two different answers from the same expression. That cannot stand, so long ago mathematicians picked one rule and we all follow it. We will learn that rule together, slowly, with the answer to every step shown.

Key idea: The order of operations is an agreed order so a single expression always gives a single answer.

The rule, in plain English first

Read this list slowly. It is the whole rule.

1. Do anything inside Parentheses ( ) first.
2. Then do Exponents (the little raised numbers, like the small 2 in 3²).
3. Then do Multiplication and Division, working left to right.
4. Then do Addition and Subtraction, working left to right.

People remember the first letters with the word PEMDAS, or the sentence "Please Excuse My Dear Aunt Sally." Whatever helps you recall the order is fine.

Two gentle warnings that save people a lot of grief. First, multiply and divide are equal partners, so you do not always multiply before you divide. You do whichever one shows up first as you read left to right, like reading a sentence. Second, add and subtract are also equal partners, same left-to-right rule. Read that once more if you like: M and D are a team, A and S are a team, and inside each team you just go left to right.

Key idea: Parentheses, then exponents, then multiply/divide left to right, then add/subtract left to right.

Why multiply before add? The reason behind the rule

The order is not random, and it is not teachers being fussy. Most of it protects the meaning of our shorthand. Remember from Week 1 that multiplication is repeated addition: 3 x 4 is shorthand for 4 + 4 + 4. So when we write 2 + 3 x 4, we really mean 2 + 4 + 4 + 4, which is 2 + 12 = 14. Doing the multiplication first simply unpacks the shorthand before combining. Adding 2 + 3 first would tear the 3 away from the 4s it was counting.

Exponents outrank multiplication for the same reason, one level up: an exponent is repeated multiplication, an even tighter bundle. The pattern is: the tighter the bundle, the earlier it gets unpacked. Parentheses beat everything because they are our way of saying "treat this part as one package, no matter what."

Could the world have picked different rules? For the left-to-right part, yes. That part is a convention, like driving on the right side of the road. The multiply-before-add part is the piece with a real reason behind it. Either way, everyone agreeing is what lets your calculator, your teacher, and a student on the other side of the world all read 2 + 3 x 4 and land on 14.

What an exponent means

Before we mix everything, let us be sure about exponents, since they trip people up. An exponent is a shorthand for multiplying a number by itself. The little raised number tells you how many copies to multiply.

For example, 3² means "3 times itself, two of them," which is 3 x 3 = 9. It does not mean 3 x 2. That is the single most common exponent slip, so let us say it plainly: 3² is 9, not 6.

One more: 2³ means 2 x 2 x 2. Do it in steps: 2 x 2 = 4, then 4 x 2 = 8. So 2³ = 8.

Key idea: An exponent counts how many copies of the base to multiply together, not the base times the exponent.

Let us work one together, one tiny step per line

Simplify 4 + 3 x (8 - 6)². We will not skip anything.

Step 1. Look for parentheses first. Inside we have 8 - 6.

Step 2. Do it: 8 - 6 = 2. Now the expression reads 4 + 3 x 2².

Step 3. Look for exponents next. We have 2².

Step 4. Do it: 2² = 2 x 2 = 4. Now the expression reads 4 + 3 x 4.

Step 5. Look for multiply or divide next. We have 3 x 4.

Step 6. Do it: 3 x 4 = 12. Now the expression reads 4 + 12.

Step 7. Last, do add or subtract. 4 + 12 = 16.

The answer is 16. Notice the rhythm: at each line we asked "what is the highest-priority thing left?" and did only that one thing. If you had added 4 + 3 first, you would have gotten a different, wrong answer. Following the order keeps everyone honest.

One more, to show the left-to-right rule

Simplify 20 - 12 / 4 + 1. There are no parentheses and no exponents, so we go to multiply/divide.

Step 1. Scan left to right for x or /. The first one is 12 / 4.

Step 2. Do it: 12 / 4 = 3. Now the expression reads 20 - 3 + 1.

Step 3. Now add and subtract, left to right. The leftmost is 20 - 3.

Step 4. Do it: 20 - 3 = 17. Now the expression reads 17 + 1.

Step 5. Do the last one: 17 + 1 = 18.

The answer is 18. See how we did the subtraction before the addition here, only because the subtraction came first reading left to right? That is the left-to-right rule doing its job.

Try it. Simplify (6 + 2) x 5. Work it out, then read on.

Parentheses first: 6 + 2 = 8. Then multiply: 8 x 5 = 40. The answer is 40. Nice work, that is exactly right.

Two more chains, with every move named

Let us simplify 36 / (4 + 2) x 3, saying out loud why each move is legal.

Step 1. Parentheses first: 4 + 2 = 6. The expression becomes 36 / 6 x 3.

Step 2. No exponents, so move to multiply/divide, left to right. The leftmost of those is the division: 36 / 6 = 6. Now the expression reads 6 x 3.

Step 3. Multiply: 6 x 3 = 18. The answer is 18.

Here is why left-to-right matters so much in this one. If you had multiplied first, 6 x 3 = 18, you would then face 36 / 18 = 2. Wrong answer, 2 instead of 18, from the exact same digits. Reading order is not decoration; it decides the result.

Now one with an exponent: 5 + 18 / 3². Predict the size first: 3 squared is 9, and 18 / 9 is small, so expect an answer only a little above 5.

Step 1. No parentheses, so exponents come first: 3² = 3 x 3 = 9. The expression becomes 5 + 18 / 9.

Step 2. Multiply/divide next: 18 / 9 = 2. Now the expression reads 5 + 2.

Step 3. Add last: 5 + 2 = 7.

Compare with the prediction: we expected a little above 5 and got 7. It fits, so we can trust it. That predict-then-check habit catches wrong turns before they cost you.

Parentheses inside parentheses

Sometimes one set of parentheses sits inside another, like 2 x (3 + (8 - 5)). The rule is gentle: work from the innermost pair outward, like peeling an onion from the center.

Step 1. Innermost parentheses first: 8 - 5 = 3. The expression becomes 2 x (3 + 3).

Step 2. The remaining parentheses: 3 + 3 = 6. Now we have 2 x 6.

Step 3. Multiply: 2 x 6 = 12.

Check the size: we doubled a number that had to be bigger than 3, so the answer had to be bigger than 6. We got 12, which fits. Nothing about nesting is new. It is the same rule, applied from the inside out.

Common misconceptions

• "Always multiply before dividing." No. Multiply and divide are equal. Do whichever appears first from the left. Same for add and subtract.
• "3² means 3 times 2." No. It means 3 times itself, 3 x 3 = 9.
• "Parentheses are optional decoration." Parentheses change the answer. (6 + 2) x 5 = 40, but 6 + 2 x 5 = 16. The little curved marks matter.
• "I can do the operations in any order I like as long as I am careful." Being careful is good, but the order is fixed by agreement. Careful in the wrong order still gives a wrong answer.

Recap

• Order of operations is a shared agreement so every expression has exactly one answer.
• The order: Parentheses, Exponents, Multiply/Divide (left to right), Add/Subtract (left to right).
• An exponent means repeated multiplication of the base, not base times exponent.
• At each step, do only the single highest-priority operation left, and write the new, shorter expression.

Sources

• OpenStax. (2020). 2.1 Use the language of algebra. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 2.2 Evaluate, simplify, and translate expressions. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Exponents intro and order of operations [Unit]. In Pre-algebra. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Exponents and order of operations [Unit]. In 6th grade math. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Order of operations - PEMDAS. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Exponents. Math is Fun. mathsisfun.com
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM. nctm.org

The big picture

This week has some big words, factor, multiple, prime, GCF, LCM, but every one of them is a plain, simple idea in disguise. We will meet them one at a time, with real numbers first and the fancy word second. You do not need to memorize definitions to understand these. You just need to see them in action.

Why does this matter in real life? These ideas are the quiet helpers behind adding fractions, splitting things into equal groups, and figuring out when two repeating events line up (like two blinking lights). We are building tools you will lean on in the fraction weeks coming up.

Key idea: Factors and multiples are just two directions from a number: what divides into it, and what it builds up to.

Factors: the numbers that fit evenly

A factor of a number is a number that divides into it evenly, meaning no remainder is left over. Picture 12 cookies. If you can share them into equal rows with none left over, the number of rows is a factor.

Let us list the factors of 12, slowly, by testing each number.

Step 1. Does 1 divide 12 evenly? Yes, 12 / 1 = 12. So 1 is a factor (and so is 12).

Step 2. Does 2 divide 12 evenly? Yes, 12 / 2 = 6. So 2 is a factor (and so is 6).

Step 3. Does 3 divide 12 evenly? Yes, 12 / 3 = 4. So 3 is a factor (and so is 4).

Step 4. Does 5 divide 12 evenly? No, 12 / 5 leaves a remainder. So 5 is not a factor.

Collecting them, the factors of 12 are: 1, 2, 3, 4, 6, 12. Notice they came in pairs (1 and 12, 2 and 6, 3 and 4). That pairing is a nice check that you found them all.

Key idea: A factor divides the number with nothing left over, and factors usually come in pairs.

Multiples: counting by a number

A multiple of a number is what you get by multiplying it by 1, 2, 3, 4, and so on. It is just skip-counting. The multiples of 12 are 12 (that is 12 x 1), 24 (12 x 2), 36 (12 x 3), 48 (12 x 4), and it never stops.

Here is the easy way to keep factors and multiples straight. Factors are smaller-or-equal and hide inside the number. Multiples are larger-or-equal and stretch out from the number. Read that once more if you like: factors go inward, multiples go outward.

Key idea: Multiples are the times-table list of a number, going on forever.

Prime and composite: how many factors?

A prime number is a number greater than 1 that has exactly two factors: just 1 and itself. Nothing else divides it evenly. The first few primes are 2, 3, 5, 7, 11, 13. (Notice 2 is the only even prime, because every other even number can also be divided by 2.)

A composite number has more than two factors. For example, 12 is composite because it has six factors. The number 1 is a special case that is neither prime nor composite, since it has only one factor.

Let us check whether 17 is prime. Try dividing by 2, 3, 4, and so on. None of them go in evenly (17 is odd, not divisible by 3, and so on). The only things that divide 17 are 1 and 17. So 17 is prime.

Key idea: Prime means exactly two factors (1 and itself); composite means more than two.

Divisibility shortcuts: spotting factors fast

Testing every divisor gets slow, so here are the shortcuts everyone actually uses. A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. By 5 if it ends in 0 or 5. By 10 if it ends in 0. And the surprising one: by 3 if its DIGITS add up to a multiple of 3 (by 9 if they add to a multiple of 9).

Try 51. It looks prime to many people. Add the digits: 5 + 1 = 6, a multiple of 3. So 3 divides 51 evenly: 51 = 3 x 17, and 51 is composite after all.

Why does the digit trick work? Because 10 is 9 + 1, and 100 is 99 + 1. The 9 and 99 parts are always divisible by 3, so the only part that can leave a remainder is the digits themselves. That is why their sum is all you need to check.

Key idea: Endings test 2, 5, and 10; digit sums test 3 and 9.

Prime factorization: breaking a number into prime bricks

Every composite number can be written as a product of primes. Think of primes as the building bricks and prime factorization as taking the number apart into its bricks. This is the secret that makes GCF and LCM easy.

Let us break down 12 and 18.

Step 1. Break 12. 12 = 2 x 6, and 6 = 2 x 3. So 12 = 2 x 2 x 3.

Step 2. Break 18. 18 = 2 x 9, and 9 = 3 x 3. So 18 = 2 x 3 x 3.

Now we can see each number as its bricks: 12 is (2, 2, 3) and 18 is (2, 3, 3).

Key idea: Prime factorization rewrites a number as a multiplication of primes, its basic bricks.

A bigger factor tree

Let us break 60 into bricks, naming the reason for each split.

Step 1. 60 is even (it ends in 0), so split off a 2: 60 = 2 x 30.

Step 2. 30 is even too: 30 = 2 x 15. So far, 60 = 2 x 2 x 15.

Step 3. 15 ends in 5, so 5 divides it: 15 = 3 x 5. Both 3 and 5 are prime, so the splitting stops.

Step 4. Collect the bricks: 60 = 2 x 2 x 3 x 5. Check by rebuilding: 2 x 2 = 4, then 4 x 3 = 12, then 12 x 5 = 60. It rebuilds, so the bricks are right.

A comforting fact: no matter which factor you split off first, you end with the same bricks. Every whole number has exactly one prime factorization; that is why primes are called the building blocks of numbers.

GCF and LCM, using the bricks

The greatest common factor (GCF) is the largest number that divides evenly into both numbers. The least common multiple (LCM) is the smallest number that both numbers divide into. With the bricks, both are gentle to find.

We have 12 = 2 x 2 x 3 and 18 = 2 x 3 x 3.

For the GCF, take only the bricks they share.

Step 1. Both have a 2. Take one 2.

Step 2. Both have a 3. Take one 3.

Step 3. Multiply the shared bricks: 2 x 3 = 6. So GCF = 6.

Check: does 6 divide both 12 and 18? 12 / 6 = 2 and 18 / 6 = 3. Yes. Good.

For the LCM, take the most of each brick that appears in either number.

Step 1. The most 2s in either is two of them (12 has two 2s). Take 2 x 2.

Step 2. The most 3s in either is two of them (18 has two 3s). Take 3 x 3.

Step 3. Multiply: 2 x 2 x 3 x 3 = 36. So LCM = 36.

Check: 36 / 12 = 3 and 36 / 18 = 2, and no smaller number works for both. Good. Nicely done, that is a real skill you will use when adding fractions.

Try it. Find the GCF of 16 and 24. Break them into bricks first, then read on.

16 = 2 x 2 x 2 x 2 and 24 = 2 x 2 x 2 x 3. The shared bricks are three 2s: 2 x 2 x 2 = 8. So the GCF is 8. Well done.

One more round, with a beautiful check

Let us do 24 and 60. Bricks first: 24 = 2 x 2 x 2 x 3, and 60 = 2 x 2 x 3 x 5.

Step 1. GCF takes the shared bricks. Both numbers hold two 2s and one 3, so GCF = 2 x 2 x 3 = 12.

Step 2. Check: 24 / 12 = 2 and 60 / 12 = 5, both even splits. Good.

Step 3. LCM takes the most of each brick: three 2s (from 24), one 3, and one 5 (from 60). LCM = 2 x 2 x 2 x 3 x 5 = 120.

Step 4. Check: 120 / 24 = 5 and 120 / 60 = 2, so both numbers fit into 120 evenly. Good.

Now the beautiful check: GCF times LCM is 12 x 120 = 1,440, and 24 x 60 = 1,440 as well. That always happens, because between them the GCF and LCM use exactly the bricks the two numbers hold. A lovely way to catch mistakes.

Why does the LCM rule say "the most of each brick"? Because the LCM must be a multiple of BOTH numbers, so it has to contain each number's full set of bricks inside it. Taking the most of each brick is the smallest way to fit both sets at once.

Common misconceptions

• "Factors and multiples are the same thing." They are opposites. Factors divide into the number (they are inside it); multiples build out from it.
• "1 is prime." No. A prime needs exactly two different factors. The number 1 has only one factor, so it is neither prime nor composite.
• "The GCF is the bigger number and the LCM is the smaller." It is the reverse feel: the GCF is at most the smaller number, and the LCM is at least the larger number.
• "For LCM, use only the shared bricks." That is the GCF. For the LCM you take the most of each brick from either number.

Recap

• A factor divides a number evenly; a multiple is the number times 1, 2, 3, and on.
• A prime has exactly two factors; a composite has more; 1 is neither.
• Prime factorization breaks a number into its prime bricks.
• GCF = multiply the shared bricks. LCM = multiply the most of each brick from either number.

Sources

• OpenStax. (2020). 2.4 Find multiples and factors. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 2.5 Prime factorization and the least common multiple. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Factors and multiples [Unit]. In Pre-algebra. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Prime numbers and composite numbers. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Prime factorization. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Greatest common factor. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Least common multiple. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Divisibility rules. Math is Fun. mathsisfun.com

The big picture

Fractions have made a lot of people feel lost, so if that is you, you are in good company and you are going to be fine. Here is the friendly truth: you already understand fractions. Every time you eat half a sandwich, share a pizza, or say "a quarter past three," you are using fractions. This week we just learn to write down what you already picture in your head.

A fraction is a way to talk about part of a whole thing. We will keep a pizza in mind the entire time, because a pizza cut into equal slices is exactly what a fraction is.

Key idea: A fraction shows part of a whole, like some slices out of a whole pizza.

The two numbers, and what each one does

A fraction is written as one number over another, like ³⁄₄. Each number has a job, and a plain name.

• The bottom number is the denominator. It tells you how many equal slices the whole pizza was cut into. Think "d for down, d for divided into this many pieces."
• The top number is the numerator. It tells you how many of those slices you actually have.

So ³⁄₄ means: the pizza was cut into 4 equal slices, and you have 3 of them. We read it aloud as "three fourths" or "three quarters." Read that once more if you like: bottom = how many pieces in all, top = how many you have.

Key idea: Denominator (bottom) = total equal pieces; numerator (top) = how many you have.

Equivalent fractions: same amount, different slicing

Here is something that surprises people in a good way. The exact same amount of pizza can be written as different fractions, depending on how finely you slice it. Half a pizza is ¹⁄₂. But if you cut each of those two halves in two, you now have 4 small slices and you are holding 2 of them, which is ²⁄₄. Same amount of pizza, different numbers. We call these equivalent fractions.

The rule for making them is gentle: multiply (or divide) the top and the bottom by the same number. Whatever you do to one, do to the other.

Step 1. Start with ¹⁄₂.

Step 2. Multiply top and bottom by 2: top 1 x 2 = 2, bottom 2 x 2 = 4.

Step 3. You get ²⁄₄, which is the same amount.

Key idea: Multiplying or dividing top and bottom by the same number gives an equal fraction.

Why the same-number trick is legal

Why are we allowed to multiply the top and bottom by the same number? Because doing so is secretly multiplying by 1. Watch: multiplying by ²⁄₂ means multiplying by "2 divided by 2," and 2 divided by 2 is just 1. Multiplying anything by 1 leaves its value unchanged. So when ¹⁄₂ becomes ²⁄₄, the amount never moved; only the name did.

Simplifying is the same move run backwards: dividing top and bottom by 6 is dividing the fraction by ⁶⁄₆, which is dividing by 1. That is why the value cannot change in either direction. Renaming a fraction is like exchanging a dollar bill for four quarters: new form, same money.

Simplifying: writing a fraction in its easiest form

Most of the time we like a fraction written as simply as possible, using the fewest, biggest pieces. We call this simplifying, or reducing to lowest terms. To do it, divide the top and bottom by their greatest common factor (the GCF we met last week).

Let us simplify ¹²⁄₁₈.

Step 1. Find the GCF of 12 and 18. From last week, it is 6.

Step 2. Divide the top by 6: 12 / 6 = 2.

Step 3. Divide the bottom by 6: 18 / 6 = 3.

Step 4. The simplified fraction is ²⁄₃.

Check: is ²⁄₃ the same amount as ¹²⁄₁₈? Yes, we only rewrote it with bigger, fewer pieces. If you are not sure what the GCF is, you can also divide by any common factor you spot and repeat until nothing divides both. Both roads reach the same simple fraction.

Key idea: Simplify by dividing top and bottom by their greatest common factor.

One more, two ways, so you can pick your favorite. Simplify ²⁴⁄₃₆. Road 1: use the GCF. Since 24 = 2 x 2 x 2 x 3 and 36 = 2 x 2 x 3 x 3, the shared bricks give a GCF of 2 x 2 x 3 = 12. Divide both by 12: 24 / 12 = 2 and 36 / 12 = 3, so the answer is ²⁄₃. Road 2: chip away with easy factors. Divide both by 2 to get ¹²⁄₁₈, by 2 again to get ⁶⁄₉, then by 3 to get ²⁄₃. Same destination, smaller steps. You are done when nothing but 1 divides both the top and the bottom.

Improper fractions and mixed numbers

Sometimes the top is bigger than the bottom, like ¹¹⁄₄. That is called an improper fraction, and there is nothing wrong with it. It just means you have more than one whole pizza's worth of slices. We can rewrite it as a whole number plus a leftover fraction, called a mixed number.

Let us turn ¹¹⁄₄ into a mixed number.

Step 1. Ask how many whole pizzas 11 quarter-slices make. Since 4 slices make one whole, divide: 11 / 4.

Step 2. 4 goes into 11 two times (that uses 8 slices), with 3 slices left over.

Step 3. So we have 2 whole pizzas and 3 leftover quarter-slices: 2 and ³⁄₄, written 2³⁄₄.

Quick size check: 3 whole pizzas would be 12 quarter-slices, and we only had 11, so the answer had to land just under 3. Our 2³⁄₄ sits exactly there. It fits.

To go the other way, from 2³⁄₄ back to an improper fraction: multiply the whole number by the denominator and add the numerator. 2 x 4 = 8, then 8 + 3 = 11, over the same bottom 4, giving ¹¹⁄₄. It matches, so we did it right.

Try it. Write ⁷⁄₂ as a mixed number. Work it out, then read on.

Divide: 7 / 2 is 3 with a remainder of 1 (3 wholes use 6 halves, 1 half left over). So ⁷⁄₂ = 3¹⁄₂. Nicely done.

Which fraction is bigger? Rename, then compare

Which is more pizza, ³⁄₅ or ⁵⁄₈? Estimate first: both are a little more than one half, so this one is close. When a size estimate cannot decide, rename both fractions with the same denominator and compare the tops. Fifths and eighths both fit into fortieths, because 5 x 8 = 40.

Step 1. Rename ³⁄₅: multiply top and bottom by 8 to get ²⁴⁄₄₀.

Step 2. Rename ⁵⁄₈: multiply top and bottom by 5 to get ²⁵⁄₄₀.

Step 3. Now the pieces are the same size, so just compare the tops: 25 beats 24. So ⁵⁄₈ is bigger, by exactly one fortieth.

See what the renaming did? It turned "which looks bigger" into plain counting of same-size pieces. That is the whole trick.

Fractions live on the number line too

A fraction is not just pizza. It is also an address on the number line. To place ³⁄₄, cut the space between 0 and 1 into 4 equal steps and walk 3 of them. You land past ¹⁄₂ but short of 1, which matches the feel of three quarters.

Improper fractions get addresses beyond 1. Where does ¹¹⁄₄ live? Since 11 / 4 is 2 remainder 3, it sits 3 quarter-steps past the number 2, so it lands between 2 and 3, closer to 3. Estimating a fraction's rough position before computing with it is a quick sanity check that catches many mistakes later.

One careful habit: always check what each tick mark is worth before you trust it. If the space from 0 to 1 is cut into 8 steps, each tick is an eighth, not a tenth.

Common misconceptions

• "A bigger denominator means a bigger fraction." Often the opposite. More slices means smaller slices. ¹⁄₈ is smaller than ¹⁄₂, because eighths are tiny compared to halves.
• "To simplify, subtract from the top and bottom." No. Simplifying uses division by a common factor, never subtraction.
• "An improper fraction is a mistake." It is perfectly valid. It just represents an amount of one whole or more.
• "The slices do not have to be equal." They do. A fraction only makes sense when the whole is cut into equal pieces.
• "You can compare fractions by comparing tops." Only when the bottoms already match. To compare 3/5 and 5/8, rename them as 24/40 and 25/40 first; then the tops tell the truth.

Recap

• The denominator (bottom) is how many equal pieces make the whole; the numerator (top) is how many you have.
• Multiply or divide top and bottom by the same number to get an equivalent fraction.
• Simplify by dividing top and bottom by their GCF.
• An improper fraction (top bigger than bottom) can be rewritten as a mixed number, and back again.

Sources

• OpenStax. (2020). 4.1 Visualize fractions. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Understand fractions [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Fractions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Equivalent fractions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Simplifying fractions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Improper fractions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Mixed fractions. Math is Fun. mathsisfun.com

The big picture

If fractions have ever made you freeze up, you are in good company, and you are going to be okay. Here is the friendly truth: you already add and subtract fractions in real life. When you eat one slice of pizza and then another, you just added fractions. When half a tank of gas drops to a quarter tank, you just subtracted them. We are only going to learn to write down something you already understand.

This week is about combining fractions: putting them together (adding) and taking one away from another (subtracting). We will go slowly, one tiny step at a time. Nothing skipped, and no question is too small.

Key idea: You can only add or subtract fractions when the pieces are the same size. Same-size pieces means the same bottom number.

Why the bottoms must match, told with pizza

A fraction is pizza slices. The bottom number (the denominator) tells you how many equal slices the whole pizza was cut into. The top number (the numerator) tells you how many of those slices you have.

Imagine one pizza cut into 8 equal slices. If you have 3 of those slices, that is ³⁄₈, read out loud as "three eighths." If a friend hands you 2 more slices from the same pizza, you now have 5 slices out of 8, which is ⁵⁄₈.

Notice what did and did not change. The slices are still eighth-size slices, so the bottom stayed 8. You just have more of them, so only the top changed. That is the whole rule for fractions that already match.

Key idea: When the bottoms already match, add (or subtract) the tops and keep the bottom exactly as it is.

Same bottom: let us try one together

Let us work out ²⁄₉ + ⁴⁄₉, slowly.

Step 1. Look at the bottoms. Both are 9. They already match, so we are allowed to add right away. Nice, that is the easy case.

Step 2. Add only the tops. 2 + 4 = 6.

Step 3. Keep the bottom the same. It stays 9.

Step 4. Put it together: ⁶⁄₉.

Step 5. Check if it can be made simpler. Both 6 and 9 can be divided by 3. 6 ÷ 3 = 2, and 9 ÷ 3 = 3. So ⁶⁄₉ is the same amount as ²⁄₃.

What we just did: we added two ninths-size amounts, got six ninths, then wrote that more simply as two thirds. See? That was not so bad.

Different bottoms: the one extra step

Sometimes the bottoms do not match, like ¹⁄₄ + ¹⁄₆. Read that as "one fourth plus one sixth." Fourth-size pieces and sixth-size pieces are different sizes, so we cannot just smash the tops together yet. First we have to rename both fractions so they use the same size piece. That shared bottom is called a common denominator.

Here is a reliable way to find one. We look for a number that both bottoms divide into. The smallest such number is the least common denominator, and it is just the LCM of the two bottoms (that idea from Week 3 comes back to help us here).

Let us do ¹⁄₄ + ¹⁄₆, one small step at a time.

Step 1. Find a shared bottom. Count by 4s: 4, 8, 12. Count by 6s: 6, 12. The first number in both lists is 12. So 12 is our common denominator.

Step 2. Rename the first fraction to have a bottom of 12. To turn 4 into 12 we multiply by 3. Whatever we do to the bottom we must do to the top, so multiply the top by 3 too: ¹⁄₄ = ³⁄₁₂.

Step 3. Rename the second fraction the same way. To turn 6 into 12 we multiply by 2, so the top gets multiplied by 2 as well: ¹⁄₆ = ²⁄₁₂.

Step 4. Now the bottoms match. Add the tops: 3 + 2 = 5. Keep the bottom 12.

Step 5. The answer is ⁵⁄₁₂. Can it be simplified? 5 and 12 share no common factor except 1, so we are done.

What we just did: we renamed both fractions into twelfth-size pieces so they were the same size, and only then added. Renaming a fraction does not change how much it is worth, it just changes how it is written. Read that once more if you like: ¹⁄₄ and ³⁄₁₂ are the exact same amount of pizza.

And why is the renaming legal? Same reason as last week: to turn ¹⁄₄ into ³⁄₁₂ we multiplied by ³⁄₃, and ³⁄₃ is just 1. Multiplying by 1 changes the name, never the amount. Every common-denominator step you will ever take rests on that one quiet fact.

Key idea: To add or subtract unlike fractions, first rename both to a common bottom, then add or subtract the tops.

Subtracting works the very same way

Subtracting is the same recipe, we just take away instead of combine. Let us do ⁵⁄₆ − ¹⁄₃, read as "five sixths minus one third."

Step 1. Find a shared bottom. Count by 6s: 6. Count by 3s: 3, 6. Both reach 6, so 6 is our common denominator. (Sometimes one bottom is already the shared one, which saves work.)

Step 2. The first fraction, ⁵⁄₆, already has a bottom of 6, so it stays as it is.

Step 3. Rename the second. To turn 3 into 6 we multiply by 2, so top and bottom both times 2: ¹⁄₃ = ²⁄₆.

Step 4. Bottoms match now. Subtract the tops: 5 − 2 = 3. Keep the bottom 6.

Step 5. The answer is ³⁄₆. Simplify: both divide by 3, giving ¹⁄₂.

What we just did: we made the pieces the same size, took away, then simplified. That last simplify step is a kindness to your reader (and your future self), so the answer is as tidy as possible.

Mixed numbers: add the same gentle way

Let us add 2¹⁄₂ + 1³⁄₄. Estimate first: two and a half plus almost two should land a little over 4. Now the exact work.

Step 1. Rewrite each mixed number as an improper fraction. For 2¹⁄₂: 2 x 2 + 1 = 5, so it is ⁵⁄₂. For 1³⁄₄: 1 x 4 + 3 = 7, so it is ⁷⁄₄.

Step 2. Find a common denominator. Halves and fourths: 4 works, since 2 fits into 4. Rename ⁵⁄₂ by multiplying top and bottom by 2: ¹⁰⁄₄.

Step 3. The bottoms match now, so add the tops and keep the bottom: 10 + 7 = 17, giving ¹⁷⁄₄.

Step 4. Convert back to a mixed number: 17 / 4 = 4 remainder 1, so 4¹⁄₄.

Step 5. Compare with the estimate. We predicted a little over 4, and 4¹⁄₄ is exactly that. The answer checks out.

One tougher subtraction, start to finish

Try ⁷⁄₈ − ¹⁄₃. Predict first with benchmarks: ⁷⁄₈ is nearly 1, and ¹⁄₃ is a smallish chunk, so expect an answer somewhere around one half.

Step 1. Find the least common denominator. Count by 8s: 8, 16, 24. Count by 3s: 3, 6, 9, 12, 15, 18, 21, 24. The first number both lists share is 24.

Step 2. Rename both fractions: ⁷⁄₈ = ²¹⁄₂₄ (times 3 on top and bottom), and ¹⁄₃ = ⁸⁄₂₄ (times 8 on top and bottom).

Step 3. Subtract the tops, keep the bottom: 21 − 8 = 13, giving ¹³⁄₂₄.

Step 4. Simplify? 13 and 24 share no factor except 1, so it is already in lowest terms.

Step 5. Check against the prediction: ¹³⁄₂₄ is a shade over ¹²⁄₂₄, which is exactly one half. Right where we expected it. Lovely.

Try it yourself

Have a go at ³⁄₈ + ¹⁄₈. Take your time.

The bottoms already match at 8, so add the tops: 3 + 1 = 4, giving ⁴⁄₈. Both 4 and 8 divide by 4, so it simplifies to ¹⁄₂. If you got one half, wonderful, you just did the whole process.

Common misconceptions

• Adding the bottoms too. The most common slip is writing ¹⁄₄ + ¹⁄₄ = ²⁄₈. It is not. The pieces are already quarter-size, so only the count of them changes: ¹⁄₄ + ¹⁄₄ = ²⁄₄ = ¹⁄₂. The bottom names the piece size and stays put when it already matches.
• Changing only the bottom when renaming. If you multiply the bottom by 3, you must multiply the top by 3 too. Otherwise you have changed the amount, not just its clothing.
• Forgetting to simplify. An answer like ⁶⁄₉ is not wrong, but ²⁄₃ is the tidy form. Always glance at the end to see if top and bottom share a factor.

Recap

Fractions add and subtract only when the pieces are the same size, meaning the bottoms match. If they already match, add or subtract the tops and keep the bottom. If they do not, rename both fractions to a common bottom first, then combine, then simplify. You do not need to be a math person to do this. You just did it.

Sources

• OpenStax. (2020). 4.4 Add and subtract fractions with common denominators. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 4.5 Add and subtract fractions with different denominators. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Add and subtract fractions (like denominators) [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Add and subtract fractions (different denominators) [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Adding fractions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Subtracting fractions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Least common multiple. Math is Fun. mathsisfun.com

The big picture

Here is a piece of good news that surprises almost everyone: multiplying fractions is actually easier than adding them. No common denominator needed. If adding fractions felt like a lot of steps, you are about to get a break.

This week we multiply fractions (which really means taking a part of a part) and divide fractions (splitting a fraction into equal shares). We will go slowly and gently, and every step will be shown.

Key idea: To multiply fractions, multiply straight across: tops together, bottoms together. That is the whole rule.

What multiplying a fraction really means

When you see × between fractions, a friendly way to read it is the word "of." So ¹⁄₂ × ¹⁄₄ means "one half OF one fourth." Picture a quarter of a pizza sitting on the table. Now take half of just that piece. It is a smaller sliver, and it turns out to be one eighth of the whole pizza. That is why multiplying two fractions usually gives a smaller number: you are taking a part of a part.

Read that once more if you like: with fractions, "times" often means "of," and taking a part of a part makes things smaller, not bigger. That feels backwards at first, and that is completely normal.

Key idea: "Times" with fractions means "of." A part of a part is smaller than either piece.

Multiplying: let us try one together

Let us work out ²⁄₃ × ⁴⁄₅, slowly.

Step 1. Multiply the two tops. 2 × 4 = 8. That 8 is our new top.

Step 2. Multiply the two bottoms. 3 × 5 = 15. That 15 is our new bottom.

Step 3. Put them together: ⁸⁄₁₅.

Step 4. Check if it simplifies. 8 and 15 share no common factor except 1, so we are done.

What we just did: tops times tops, bottoms times bottoms. No common denominator, no fuss. Nice work.

Why multiply-across works: the grid picture

Here is the why behind the rule. Picture a square pan of brownies. To take ²⁄₃ of ⁴⁄₅, cut the pan into 3 rows and 5 columns. That makes 3 x 5 = 15 equal little pieces, which is exactly why the bottoms multiply: the cuts cross each other.

Now shade 4 of the 5 columns (that is the ⁴⁄₅), and then keep only 2 of the 3 rows of the shaded part (that is taking ²⁄₃ of it). The overlap is a little rectangle of 2 rows by 4 columns, which holds 2 x 4 = 8 pieces. That is why the tops multiply: the overlap is a rows-times-columns count.

So ²⁄₃ × ⁴⁄₅ = ⁸⁄₁₅, and now you know it is not a magic spell. It is a grid.

A fraction of a whole number

What about ²⁄₃ × 12? Estimate first: half of 12 is 6, and two thirds is a bit more than half, so expect a bit more than 6.

Step 1. Write the whole number as a fraction: 12 = ¹²⁄₁.

Step 2. Multiply across. Tops: 2 × 12 = 24. Bottoms: 3 × 1 = 3. That gives ²⁴⁄₃.

Step 3. Simplify: 24 ÷ 3 = 8. So ²⁄₃ of 12 is 8.

Check it the sharing way: a third of 12 is 4, so two thirds is 2 x 4 = 8. Same answer by a different road, and it beats our estimate of "a bit more than 6" by just the right amount.

Simplifying before you multiply (a time saver)

Sometimes the numbers get big. There is a kind shortcut: if a top and a bottom share a factor, you can cancel it before multiplying. Let us do ³⁄₄ × ²⁄₉.

Step 1. Look for a top and a bottom that share a factor. The 3 on top and the 9 on the bottom both divide by 3. And the 2 on top and the 4 on the bottom both divide by 2.

Step 2. Cancel the first pair: 3 becomes 1, and 9 becomes 3.

Step 3. Cancel the second pair: 2 becomes 1, and 4 becomes 2.

Step 4. Now multiply the smaller numbers. Tops: 1 × 1 = 1. Bottoms: 2 × 3 = 6.

Step 5. The answer is ¹⁄₆.

What we just did: we shrank the numbers before multiplying, so the answer came out already simplified. If canceling early feels confusing, you can always skip it, multiply straight across, and simplify at the end. Both roads reach the same place.

Dividing: the flip-and-multiply idea

Dividing by a fraction has a reputation for being hard, but there is one trick that makes it straightforward. To divide by a fraction, you flip the second fraction upside down and then multiply. The flipped fraction has a name, the reciprocal, but do not let the word worry you. The reciprocal of ²⁄₃ is just ³⁄₂. You swap the top and the bottom.

Why does flipping work? Here is a gentle example. "How many halves fit into 3?" You can picture six half-slices fitting into 3 whole pizzas, so the answer is 6. And notice: 3 ÷ ¹⁄₂ is the same as 3 × 2, which is 6. Dividing by a half is the same as multiplying by 2. Flipping turns a division into a multiplication we already know how to do.

Let us do ³⁄₄ ÷ ²⁄₅, one small step at a time.

Step 1. Keep the first fraction exactly as it is: ³⁄₄.

Step 2. Flip the second fraction. ²⁄₅ becomes ⁵⁄₂.

Step 3. Change the ÷ into a ×. Now we have ³⁄₄ × ⁵⁄₂.

Step 4. Multiply straight across. Tops: 3 × 5 = 15. Bottoms: 4 × 2 = 8.

Step 5. The answer is ¹⁵⁄₈. Since the top is bigger than the bottom, we can also write it as a mixed number: 15 ÷ 8 = 1 with 7 left over, so 1⁷⁄₈.

Does the size make sense? We asked how many ²⁄₅-size pieces fit into ³⁄₄. Since ²⁄₅ is a bit less than one half, close to two of them should fit. We got 1⁷⁄₈, just under 2. It fits.

What we just did: keep, flip, multiply. Say it to yourself like a little rhyme: "keep, change, flip." Read that once more if you like.

Key idea: To divide by a fraction, keep the first, flip the second, and multiply.

Why the flip is legal, not just lucky

The flip works because a fraction times its reciprocal is always 1. Check one: ²⁄₅ × ⁵⁄₂ = ¹⁰⁄₁₀ = 1. The tops and bottoms trade places, so everything cancels.

Now remember what a division answer must do. Asking ³⁄₄ ÷ ²⁄₅ means asking: what number times ²⁄₅ gives ³⁄₄? Try the flip answer, ³⁄₄ × ⁵⁄₂ = ¹⁵⁄₈, and test it: ¹⁵⁄₈ × ²⁄₅ = ³⁰⁄₄₀ = ³⁄₄. It lands exactly back on ³⁄₄, so the flipped multiplication really is the division answer. The reciprocal quietly undoes the ²⁄₅, because together they multiply to 1.

That multiply-back test is also your everyday checking habit: after any fraction division, multiply your answer by the divisor and make sure you land on the number you started with.

Mixed numbers multiply too

Try 1¹⁄₂ × ²⁄₃. Predict first: two thirds of one and a half feels like about 1.

Step 1. Convert the mixed number: 1¹⁄₂ = ³⁄₂ (1 x 2 + 1 = 3 over 2).

Step 2. Multiply across: ³⁄₂ × ²⁄₃ gives top 3 × 2 = 6 and bottom 2 × 3 = 6, so ⁶⁄₆.

Step 3. Simplify: ⁶⁄₆ = 1. Exactly the 1 we predicted. (No surprise: ³⁄₂ and ²⁄₃ are reciprocals, so their product had to be 1.)

Try it yourself

Have a go at ¹⁄₂ ÷ ¹⁄₄. Take your time.

Keep ¹⁄₂, flip ¹⁄₄ to ⁴⁄₁, and multiply: ¹⁄₂ × ⁴⁄₁ = ⁴⁄₂ = 2. So one half divided by one fourth is 2, which makes sense: two quarter-pieces fit into one half. Nice, that is exactly right.

Common misconceptions

• Looking for a common denominator when multiplying. You do not need one to multiply or divide. Common denominators are only for adding and subtracting. Multiplying goes straight across.
• Flipping the wrong fraction. When dividing, you flip only the second fraction (the one you are dividing by), never the first. Keep the first one still.
• Expecting multiplication to make things bigger. With whole numbers, times makes things bigger. With fractions less than 1, times makes things smaller, because you are taking a part of a part. That is expected, not a mistake.
• Mixing up dividing BY 2 with dividing by 1/2. Dividing by 2 cuts a number in half. Dividing by 1/2 asks how many halves fit, which doubles it: 3 ÷ ¹⁄₂ = 6.

Recap

To multiply fractions, multiply the tops and multiply the bottoms, then simplify. To divide, keep the first fraction, flip the second, and multiply. No common denominators needed for either one. You just handled one of the trickier parts of pre-algebra.

Sources

• OpenStax. (2020). 4.2 Multiply and divide fractions. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Multiply fractions [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Divide fractions [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Multiplying fractions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Dividing fractions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Reciprocal. Math is Fun. mathsisfun.com
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM. nctm.org

The big picture

You already read decimals every single day, probably more than any other kind of number. Every price tag you have ever seen, like $3.99 or $12.50, is a decimal. And you have been handling decimals at the store your whole life. We are just going to slow down and look at how they work.

This week is about decimals, which are simply another way to write fractions whose bottoms are 10, 100, 1000, and so on. We will read them out loud, line them up, and add, subtract, multiply, and divide them, one gentle step at a time.

Key idea: A decimal is just a fraction in disguise. The digits after the dot are pieces of a whole, in tenths, hundredths, and thousandths.

Reading a decimal, told with money

The little dot is called the decimal point. Everything to the left of it is whole amounts; everything to the right is a part of one whole. Think of a dollar. To the left of the dot are your whole dollars. To the right are cents, which are parts of a dollar.

Just like place value for whole numbers, each spot after the dot has a name. Reading from the dot to the right: the first spot is tenths, the next is hundredths, the next is thousandths.

ones	.	tenths	hundredths
3	.	2	5

So 3.25 is 3 whole ones, plus 2 tenths, plus 5 hundredths. Read out loud, it is "three and twenty-five hundredths," which is exactly three dollars and twenty-five cents. The word "and" is where the dot goes.

Key idea: The spots after the decimal point are tenths, then hundredths, then thousandths. Read the dot as the word "and."

Adding and subtracting: line up the dots

Here is the one habit that makes decimal adding and subtracting safe: line up the decimal points, one directly under the other. When the dots line up, the tenths sit under the tenths and the hundredths under the hundredths, and everything behaves just like whole-number adding.

Let us add 3.4 + 12.75, slowly. A helpful trick: you can add a zero at the end of 3.4 to make it 3.40, which does not change its value but makes the columns even.

	3	.	4	0
+	1 2	.	7	5

Step 1. Line up the dots (done above), and place the dot in the answer directly below the others.

Step 2. Add the hundredths column: 0 + 5 = 5.

Step 3. Add the tenths column: 4 + 7 = 11. Write the 1, carry the 1 to the ones, exactly like whole-number carrying.

Step 4. Add the ones column: 3 + 2 = 5, plus the carried 1 makes 6.

Step 5. Bring down the 1 in the tens place. Reading it off: 16.15.

What we just did: we lined up the dots, filled a gap with a zero, then added column by column like normal. The dot in the answer just drops straight down.

Subtracting decimals: same lineup, plus borrowing

Let us subtract 6.4 − 2.75. Estimate first: 6.4 is about 6.5 and 2.75 is about 3, so expect an answer near 3.5.

Step 1. Line up the dots and even out the columns by writing 6.4 as 6.40.

Step 2. Hundredths column: 0 − 5 will not go, so borrow 1 tenth from the tenths column. The 4 tenths become 3, and the hundredths become 10. Now 10 − 5 = 5.

Step 3. Tenths column: 3 − 7 will not go either, so borrow 1 one. The 6 becomes 5, and the tenths become 13. Now 13 − 7 = 6.

Step 4. Ones column: 5 − 2 = 3. Bring the dot straight down.

Step 5. Read it off: 3.65. Compare with the estimate of about 3.5. Close. Good sign.

Borrowing across the dot feels new, but it is the same Week 1 borrowing: every column is still ten times its neighbor, dot or no dot.

Why sliding the dot works: the times-10 secret

Here is the why behind every decimal shortcut this week. In our number system, each place is worth ten times the place to its right. So multiplying a number by 10 promotes every digit one bucket to the left, which LOOKS like the decimal point sliding one place to the right. Watch: 2.5 × 10 = 25 (the 2 moves from ones to tens, the 5 from tenths to ones). And 3.7 × 100 = 370, a two-bucket promotion.

Dividing by 10 does the reverse: every digit steps one bucket right, so 45.6 ÷ 10 = 4.56. The dot itself never actually moves; the digits change buckets. But the sliding-dot picture is a perfectly good way to remember it.

Multiplying decimals: multiply, then count the dots

Multiplying decimals has a surprising, friendly shortcut. You ignore the dots at first, multiply like whole numbers, and only at the very end do you place the decimal point by counting.

Let us do 0.3 × 0.6, one small step at a time.

Step 1. Ignore the dots for now. Multiply 3 × 6 = 18.

Step 2. Count how many digits are after the dot in BOTH original numbers. 0.3 has one digit after the dot. 0.6 has one digit after the dot. That is 1 + 1 = 2 digits total.

Step 3. In the answer, place the dot so there are 2 digits after it. Starting from 18, we count two places from the right and write 0.18.

Step 4. So 0.3 × 0.6 = 0.18.

Does that feel too small? It should not. Remember 0.3 is a bit less than a half and 0.6 is a bit more than a half, so a part of a part lands around 0.18. That is the same "part of a part is smaller" idea from fractions week.

Key idea: To multiply decimals, multiply as if the dots were not there, then count the total decimal digits in both numbers and put that many after the dot in your answer.

And why does counting decimal places work? Because it is fraction multiplication in disguise. 0.3 is ³⁄₁₀ and 0.6 is ⁶⁄₁₀. Multiply across: tops 3 × 6 = 18, bottoms 10 × 10 = 100, giving ¹⁸⁄₁₀₀ = 0.18. Tenths times tenths make hundredths, one decimal place plus one decimal place makes two. The counting rule is just the bottoms multiplying where you cannot see them.

Dividing decimals: slide the dot to make a whole number

Dividing is easiest when the number you are dividing BY (the one outside the box) is a whole number. If it has a decimal, we slide its dot to the right until it is whole, and we slide the dot in the other number the same number of places. Sliding both the same amount keeps the answer correct.

Let us do 4.8 ÷ 0.6, slowly.

Step 1. The divisor 0.6 has one digit after the dot. Slide its dot one place right to make it a whole number: 0.6 becomes 6.

Step 2. Slide the dot in 4.8 the same one place right: 4.8 becomes 48.

Step 3. Now the problem is 48 ÷ 6.

Step 4. 6 times what gives 48? 6 × 8 = 48, so the answer is 8.

Predict the size, too: 0.6 is a little more than a half, and about eight pieces that size fit into 4.8, so an answer of 8 is sensible, not 0.8 and not 80.

What we just did: we turned a decimal division into a plain whole-number one by sliding both dots the same amount. Read that once more if you like: slide both dots equally, and the answer does not change.

Why is that fair? Write the division as a fraction: 4.8 ÷ 0.6 is ^4.8⁄_0.6. Sliding both dots one place is multiplying top and bottom by 10, giving ⁴⁸⁄₆. And multiplying top and bottom by the same number is the equivalent-fraction move from Week 4: it multiplies the whole thing by ¹⁰⁄₁₀, which is 1. Same value, friendlier numbers. Check the answer too: 8 × 0.6 = 4.8, so the division is right.

Try it yourself

Have a go at 5.2 + 0.75. Take your time and line up the dots.

Write 5.20 under 0.75 with the dots aligned. Add: hundredths 0 + 5 = 5, tenths 2 + 7 = 9, ones 5 + 0 = 5. The dot drops down, giving 5.95. If you got 5.95, wonderful.

Common misconceptions

• Not lining up the dots when adding. If you right-align the digits like whole numbers instead of aligning the decimal points, tenths can land under hundredths and the answer goes wrong. Always stack the dots.
• Forgetting to count decimal places when multiplying. 0.3 times 0.6 is 0.18, not 1.8 and not 18. Count the decimal digits in both factors and place the dot to match.
• Sliding only one dot when dividing. When you slide the divisor's dot to make it whole, you must slide the other number's dot the same number of places, or the answer changes.

Recap

Decimals are fractions with bottoms of 10, 100, and 1000, written after a dot. Read the dot as "and." To add or subtract, line up the dots and go column by column. To multiply, ignore the dots, multiply, then count decimal places. To divide, slide both dots until the divisor is a whole number. You handle these every time you shop, and now you know why they work.

Sources

• OpenStax. (2020). 5.1 Decimals. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 5.2 Decimal operations. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Add and subtract decimals [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Multiply and divide decimals [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Decimals. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Adding decimals. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Multiplying decimals. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Dividing decimals. Math is Fun. mathsisfun.com

The big picture

This week ties together three things you already met, and it is going to make them feel like old friends. Fractions, decimals, and percents are not three different monsters. They are three different outfits for the exact same amount. One half, 0.5, and 50 percent are all the same thing wearing different clothes.

Once you can change one form into another, real life gets easier: a "50 percent off" sign, a batting average of 0.250, and half a pizza all suddenly speak the same language. We will go slowly, and every conversion will be shown step by step.

Key idea: Fractions, decimals, and percents are three ways to write the same value. Learning to switch between them is like learning to translate between three languages that all mean the same thing.

What percent really means

The word percent literally means "per hundred." The little % sign is a shorthand for "out of 100." So 50% simply means 50 out of 100, which is the fraction ⁵⁰⁄₁₀₀, which is one half. Read that once more if you like: percent just means "how many out of a hundred."

Money helps again. 100 cents make a dollar. So 25% of a dollar is 25 cents, which is a quarter, which is ¹⁄₄. You have known these conversions since you first counted coins.

Key idea: Percent means per hundred. 37% is the same as the fraction 37 over 100.

Fraction to decimal: top divided by bottom

A fraction bar secretly means "divide." So ³⁄₄ is really the division 3 ÷ 4. To turn any fraction into a decimal, you divide the top by the bottom.

Let us change ³⁄₄ into a decimal, slowly.

Step 1. Read the fraction bar as divide: 3 ÷ 4.

Step 2. Since 3 is smaller than 4, the answer starts with 0 and a dot: 0.something.

Step 3. Think of 3 as 3.00. Now 30 divided by 4 is 7 (because 4 × 7 = 28) with 2 left over. Write 0.7 so far.

Step 4. Bring down the next 0 to make 20. 20 divided by 4 is exactly 5. Write the 5.

Step 5. So ³⁄₄ = 0.75.

What we just did: we treated the fraction as a division and worked it out to get a decimal. It is comforting to memorize a few common ones: ¹⁄₂ = 0.5, ¹⁄₄ = 0.25, ³⁄₄ = 0.75, ¹⁄₁₀ = 0.1.

Let us do a longer one so the method is solid: ⁵⁄₈. Estimate first: 5 is more than half of 8, so expect a decimal a bit above 0.5.

Step 1. Read the bar as divide: 5 ÷ 8. Since 5 is smaller than 8, think of 5 as 5.000.

Step 2. 50 tenths divided by 8: 8 × 6 = 48, so 6 with 2 left over. Write 0.6.

Step 3. Bring down a zero to make 20. 8 × 2 = 16, so 2 with 4 left over. Write 0.62.

Step 4. Bring down a zero to make 40. 8 × 5 = 40 exactly, nothing left. Write 0.625.

Step 5. So ⁵⁄₈ = 0.625. Check by multiplying back: 0.625 × 8 = 5. It rebuilds the 5, so the division is right, and 0.625 is indeed a bit above 0.5, matching the estimate.

Key idea: To turn a fraction into a decimal, divide the top number by the bottom number.

Some fractions repeat forever, and that is okay

Try turning ¹⁄₃ into a decimal: 1 ÷ 3. Ten tenths divided by 3 is 3 with 1 left over. Bring down a zero: 10 again. Divide by 3: 3 with 1 left over. Again. The remainder keeps coming back, so the digit 3 repeats without end: ¹⁄₃ = 0.333..., where the dots mean the 3s never stop.

This is not a mistake, and your division is not broken. Whenever a remainder repeats, the digits must repeat too, because the same division keeps happening. Decimals that stop, like 0.75, are called terminating; ones that cycle, like 0.333..., are called repeating. Both are perfectly good numbers. (For the record, ²⁄₃ = 0.666..., which people often round to 0.67 in real life.)

Decimal to percent: multiply by 100 (slide the dot two right)

Since percent means per hundred, going from a decimal to a percent means asking "how many hundredths is this?" The shortcut is to multiply by 100, which just slides the decimal point two places to the right.

Let us change 0.75 into a percent, one small step at a time.

Step 1. Take 0.75.

Step 2. Slide the decimal point two places to the right: 0.75 becomes 75.

Step 3. Add the percent sign: 75%.

So 0.75 = 75%. And going the other way, percent to decimal, you slide the dot two places to the LEFT: 75% becomes 0.75. Left and right are opposites, just like the two directions of the conversion.

Why two places? Because percent means per hundred, and multiplying or dividing by 100 shifts every digit two buckets, exactly the times-10 secret from last week done twice.

Watch out for single-digit percents. 7% means 7 out of 100, which is ⁷⁄₁₀₀ = 0.07, not 0.7. Sliding two places past a single digit means a zero must fill the gap. And percents above 100 are legal: 150% = 1.5, meaning one and a half times the whole thing.

Try it. Write 4% and 250% as decimals. For 4%: it is 4 out of 100, so 0.04. For 250%: slide two places left to get 2.5, two and a half times the whole. And a sports one: a batting average of 0.250 is 25%, a hit in one quarter of at-bats.

Key idea: Decimal to percent, slide the dot two places right. Percent to decimal, slide it two places left.

Percent to fraction: put it over 100 and simplify

Because percent means "out of 100," turning a percent into a fraction is as simple as writing it over 100, then simplifying.

Let us change 40% into a fraction, slowly.

Step 1. Write it over 100: ⁴⁰⁄₁₀₀.

Step 2. Look for a common factor. Both 40 and 100 divide by 20.

Step 3. Divide top and bottom by 20: 40 ÷ 20 = 2, and 100 ÷ 20 = 5.

Step 4. So 40% = ²⁄₅.

What we just did: we wrote the percent as a fraction over 100, then simplified it down to its tidy form. Every percent is really just a fraction with a hidden bottom of 100.

One number, every direction

Let us push a single value, ³⁄₅, all the way around the loop, naming each move.

Step 1. Fraction to decimal: divide top by bottom. 3 ÷ 5 = 0.6.

Step 2. Decimal to percent: slide the dot two places right. 0.6 becomes 60%.

Step 3. Check by a second road. Rename ³⁄₅ with a bottom of 100: multiply top and bottom by 20 to get ⁶⁰⁄₁₀₀, which is 60% on sight. Two different roads, same 60%, so we can trust it.

Step 4. Now travel backwards. 60% over 100 is ⁶⁰⁄₁₀₀; divide top and bottom by their GCF of 20 to land back on ³⁄₅. The loop closes.

And decimal to fraction? Read the place value out loud. 0.35 says "thirty-five hundredths," so write ³⁵⁄₁₀₀ and simplify: divide top and bottom by 5 to get ⁷⁄₂₀. Check: 7 ÷ 20 = 0.35. It rebuilds, so the conversion is right.

Key idea: Every conversion has a reverse road, and traveling both directions is the fastest way to check yourself.

The whole loop in one small table

Here are the same amounts written all three ways, so you can see they truly match:

Fraction	Decimal	Percent
1⁄2	0.5	50%
1⁄4	0.25	25%
3⁄4	0.75	75%
1⁄5	0.2	20%

Try it yourself

Change ¹⁄₅ into a decimal and then into a percent. Take your time.

Divide top by bottom: 1 ÷ 5 = 0.2. Then slide the dot two places right to get a percent: 0.2 becomes 20%. So ¹⁄₅ = 0.2 = 20%. If you got 20 percent, wonderful, you traveled the whole loop. For a reverse pass, try 45%: written over 100 it is ⁴⁵⁄₁₀₀, and dividing top and bottom by their GCF of 5 gives ⁹⁄₂₀.

Common misconceptions

• Sliding the dot only one place. Percent means per hundred, so the jump between a decimal and a percent is always two places, not one. 0.5 is 50%, not 5%.
• Sliding the dot the wrong direction. Going to a percent, the number gets bigger, so slide right. Going to a decimal, it gets smaller, so slide left. Picture which answer should be larger to catch yourself.
• Dividing bottom by top for a decimal. A fraction is top divided by bottom. For 3/4 it is 3 divided by 4, giving 0.75, not 4 divided by 3.
• Turning 7% into 0.7. Percent means per hundred, so 7% is 7/100 = 0.07. When the percent has one digit, a zero must fill the second place.

Recap

Fractions, decimals, and percents are one value in three outfits. Fraction to decimal, divide top by bottom. Decimal to percent, slide the dot two places right. Percent to fraction, write it over 100 and simplify. Memorizing a few common ones makes the rest quick. You just learned to translate fluently between all three.

Sources

• OpenStax. (2020). 5.3 Decimals and fractions. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 6.1 Understand percent. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Percentages [Unit]. In Pre-algebra. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Convert fractions to decimals. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Convert decimals to percents. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Convert percents to fractions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Convert fractions to percents. Math is Fun. mathsisfun.com

The big picture

Ratios and proportions sound formal, but you use them constantly without naming them. When you double a recipe, mix juice with the right amount of water, or figure out gas mileage, you are using ratios. This week just gives those everyday habits a name and a tidy way to write them down.

A ratio compares two amounts. A proportion says two ratios are equal, which lets you solve for a missing amount. We will go slowly, one small step at a time, and I will show every step.

Key idea: A ratio compares two things. A proportion sets two ratios equal so you can find a number you do not know yet.

What a ratio is, told with a recipe

Suppose a lemonade recipe uses 2 cups of sugar for every 6 cups of water. The ratio of sugar to water is 2 to 6. You can write that three ways, and they all mean the same thing: "2 to 6," or "2 : 6," or as a fraction ²⁄₆. Read the colon as the word "to."

Just like a fraction, a ratio can be simplified. Both 2 and 6 divide by 2, so 2 : 6 is the same as 1 : 3. That tells us the pattern in the simplest terms: 1 cup of sugar for every 3 cups of water. Read that once more if you like: simplifying a ratio does not change the recipe, it just states it in the smallest whole numbers.

Key idea: A ratio compares two amounts and can be written with the word to, a colon, or a fraction bar. Simplify it like a fraction.

A rate is a ratio with different units

When the two things being compared have different units, we call the ratio a rate. "120 miles in 2 hours" is a rate comparing miles to hours. A unit rate is a rate with a 1 on the bottom, like "miles per 1 hour," which is what a car speedometer shows.

Let us find the unit rate for 120 miles in 2 hours, slowly.

Step 1. Write it as a fraction: ^{120 miles}⁄_{2 hours}.

Step 2. To get a 1 on the bottom, divide both top and bottom by 2. Top: 120 ÷ 2 = 60. Bottom: 2 ÷ 2 = 1.

Step 3. So the unit rate is 60 miles per 1 hour, which we say as 60 miles per hour.

What we just did: we boiled the rate down to "per one hour," which is the number your speedometer would read. Unit rates make prices and speeds easy to compare.

Here is that comparing power at the store. Which is the better buy: 3 pens for $2.40, or 5 pens for $3.75? Estimate first: both work out to somewhere under a dollar a pen, so it is close and worth computing exactly.

Step 1. Find the price per pen for the first pack: 2.40 ÷ 3 = 0.80, so 80 cents per pen.

Step 2. Find it for the second pack: 3.75 ÷ 5 = 0.75, so 75 cents per pen.

Step 3. Compare the unit prices: 75 cents beats 80 cents, so the 5-pack is the better buy, a nickel cheaper per pen.

Notice what the unit rate did: two prices that could not be compared directly (different pack sizes) became two per-one prices that compare at a glance. That is the entire reason unit rates exist.

What a proportion is

A proportion is just a statement that two ratios are equal, like ¹⁄₃ = ²⁄₆. This is powerful because it lets us scale a recipe or a map up or down. If one of the four numbers is missing, we can find it.

The tool for this is cross multiplication. In a true proportion, if you multiply diagonally (top of one times bottom of the other), the two results are equal. Picture drawing an X across the equal sign: the two arms of the X give equal products.

Use it to TEST a proportion, too. Is ²⁄₅ = ⁶⁄₁₅ true? One arm: 2 × 15 = 30. Other arm: 5 × 6 = 30. Equal, so yes, it is a true proportion. You can see the same thing by scaling: 6 is 3 times 2, and 15 is 3 times 5, so the second ratio is the first one multiplied straight through by 3. Two views, one answer.

Key idea: In any true proportion, the cross products are equal. That fact lets us solve for a missing number.

Solving a proportion: let us try one together

Suppose 3 apples cost $2, and we want to know the cost of 12 apples. We set up a proportion with the unknown cost as a box, and we keep the same things on top in both ratios (apples over dollars).

We write ^{3 apples}⁄_{2 dollars} = ^{12 apples}⁄_{box dollars}. Let us solve it slowly, calling the missing number the box.

Step 1. Cross multiply. One arm of the X: 3 × box. The other arm: 2 × 12.

Step 2. Work out the arm we can: 2 × 12 = 24.

Step 3. Now we have 3 × box = 24. Read that as "3 times the mystery number is 24."

Step 4. To undo the "times 3," divide both sides by 3. box = 24 ÷ 3.

Step 5. 24 ÷ 3 = 8. So the box is 8, meaning 12 apples cost $8.

What we just did: we set two ratios equal, drew the X, and solved for the missing price. Does $8 make sense? 12 apples is 4 times as many as 3 apples, and 4 times $2 is $8. It checks. Nice work.

Key idea: To solve a proportion, cross multiply to get a simple equation, then undo the multiplication by dividing.

Why cross multiplication is legal, not magic

Where does the X really come from? It is a shortcut for a completely honest move: multiplying both sides of the equation by both bottoms. Take our proportion ³⁄₂ = ¹²⁄_box and multiply each side by 2 and by box (whatever you do to one side, you do to the other, so the equation stays true).

Step 1. Left side: ³⁄₂ × 2 × box. The 2 on the bottom cancels the 2 we multiplied by, leaving 3 × box.

Step 2. Right side: ¹²⁄_box × 2 × box. The box on the bottom cancels the box we multiplied by, leaving 12 × 2 = 24.

Step 3. So the equation is now 3 × box = 24, with no fractions in sight. That is exactly what the X shortcut wrote down in one stroke.

See why each side ends up as "top times the OTHER side's bottom"? Its own bottom canceled away. Cross multiplication is just this both-bottoms move with the boring cancel steps skipped. Now you know the reason, you can trust the shortcut.

Map scales: proportions you can travel by

A map says 1 inch stands for 20 miles, and two towns sit 3.5 inches apart. How far apart are they really? Keep inches on top in both ratios: ¹⁄₂₀ = ^3.5⁄_box.

Step 1. Cross multiply: 1 × box = 20 × 3.5.

Step 2. Work it out: 20 × 3.5 = 70. So box = 70 miles.

Step 3. Check the size: 3 inches would be 60 miles and 4 inches would be 80, so 70 miles for 3.5 inches sits right where it should, between the two.

The same setup runs backwards. Planning a 90-mile trip? Then 90 ÷ 20 = 4.5, so look for towns about 4.5 inches apart on the map. Keeping the same unit on top in both ratios (inches over miles here) is what makes both directions safe.

Try it yourself

If 2 pizzas feed 5 people, how many pizzas feed 20 people? Set up ²⁄₅ = ^box⁄₂₀ and take your time.

Cross multiply: 5 × box = 2 × 20, so 5 × box = 40. Divide both sides by 5: box = 40 ÷ 5 = 8. So 8 pizzas feed 20 people. If you got 8, wonderful. (Check: 20 people is 4 times 5 people, and 4 times 2 pizzas is 8.)

Common misconceptions

• Mixing up the order. A ratio of sugar to water, 2 : 6, is not the same as water to sugar, 6 : 2. Keep the same thing on top in both ratios of a proportion.
• Adding instead of scaling. If 3 apples cost $2, then 6 apples cost $4 (double), not $5 ($2 plus $3). Ratios scale by multiplying, not by adding a fixed amount.
• Cross multiplying the same side. Cross multiply means diagonally across the equal sign (top of one times bottom of the other), never top times bottom of the same fraction.
• Dividing the wrong way at the end. From 3 × box = 24, the box is found by 24 ÷ 3, not 3 ÷ 24. Undo "times 3" by dividing BY 3, and label your units so apples stay on top in both ratios.

Recap

A ratio compares two amounts and can be simplified like a fraction. A rate compares different units, and a unit rate has a 1 on the bottom. A proportion sets two ratios equal, and cross multiplication turns it into a simple equation you can solve by dividing. You use these every time you cook or shop, and now you can write them down and solve them.

Sources

• OpenStax. (2020). 6.5 Solve proportions and their applications. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Ratios and rates [Unit]. In Pre-algebra. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Proportional relationships [Unit]. In Pre-algebra. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Ratios. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Proportions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Unit price. Math is Fun. mathsisfun.com
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM. nctm.org

The big picture

This is one of the most useful weeks in the whole course, because percentages are everywhere in real life: sales tags, tips at restaurants, sales tax, quiz scores, interest on money. Percentages show up constantly, so they are well worth getting comfortable with.

Last week we learned percent means "per hundred." This week we put it to work: finding a percent OF a number, figuring out discounts and tips, and working out a percent increase or decrease. We will go slowly, one small step at a time.

Key idea: To use a percent in a calculation, first turn it into a decimal, then multiply. "Percent of a number" means multiply.

Finding a percent of a number

The word "of" in math almost always means multiply. So "20% of 50" means 0.20 × 50. The one habit to build is: change the percent to a decimal FIRST (slide the dot two places left), then multiply.

Let us find 20% of 50, slowly.

Step 1. Change the percent to a decimal. Slide the dot in 20% two places left: 20% becomes 0.20.

Step 2. Replace the word "of" with a times sign: 0.20 × 50.

Step 3. Multiply. 0.20 × 50 = 10.

Step 4. So 20% of 50 is 10.

Does that feel right? 10% of 50 would be 5, and 20% is double that, so 10. It checks. What we just did: turned the percent into a decimal, then multiplied. That is the engine behind every percentage problem this week.

Key idea: Percent of a number equals (percent as a decimal) times the number.

A friendly shortcut for 10% and 1%

Here is a trick that makes mental math feel like a superpower. To find 10% of any number, just move its decimal point one place to the left. 10% of 80 is 8. 10% of 45 is 4.5. To find 1%, move the dot two places left. 1% of 80 is 0.8.

Why is this handy? You can build other percents from these. Want 20%? That is two lots of 10%. Want 15%, a common tip? Take 10% and then add half of it. Want 5%? Take half of 10%: on $80, half of 8 is $4. Read that once more if you like: 10% is just the number with its dot slid one place left, and you can build everyday percents from there.

Percent problems in reverse

Sometimes the percent is the mystery, or the whole is. Both reverse problems come from the same sentence: part = percent × whole. Cover the piece you do not know, and the other two tell you what to do.

Mystery percent. What percent of 40 is 10?

Step 1. The part is 10 and the whole is 40. Divide part by whole: 10 ÷ 40 = 0.25.

Step 2. Turn the decimal into a percent: 0.25 becomes 25%. So 10 is 25% of 40.

Step 3. Check: 0.25 × 40 = 10. It rebuilds the part, so the answer is right.

You can also solve it with last week's tool. Write the percent proportion ¹⁰⁄₄₀ = ^box⁄₁₀₀, cross multiply to get 40 × box = 1,000, and divide both sides by 40: box = 25. Same 25%, by the proportion road.

Mystery whole. 15 is 30% of what number?

Step 1. Write the sentence with a box for the whole: 0.30 × box = 15.

Step 2. Undo the "times 0.30" by dividing both sides by 0.30: box = 15 ÷ 0.30 = 50.

Step 3. Check: 30% of 50 is 0.30 × 50 = 15. It matches, so the whole is 50.

The proportion road agrees here too: ¹⁵⁄_box = ³⁰⁄₁₀₀. Cross multiply: 15 × 100 = 1,500, so 30 × box = 1,500, and box = 1,500 ÷ 30 = 50. Two roads, one answer, extra confidence.

Discounts: putting percent to work at the store

A discount is a percent taken OFF a price. Let us find the sale price of a $40 shirt that is 25% off, one small step at a time.

Step 1. First find the discount amount, which is 25% of $40. Change 25% to 0.25, then multiply: 0.25 × 40 = 10. So you save $10.

Step 2. Subtract the savings from the original price: 40 − 10 = 30.

Step 3. The sale price is $30.

What we just did: found the part being taken off, then took it off. There is also a one-step way: if 25% comes off, then 75% stays, so 0.75 × 40 = 30 gives the same answer. Both roads reach $30, so use whichever you prefer.

Tips and tax: putting percent ON

A tip or a sales tax is a percent ADDED to a price. The method mirrors a discount, but we add instead of subtract. Let us find the total on a $30 meal with a 15% tip.

Step 1. Find the tip: 15% of $30. Change 15% to 0.15, then multiply: 0.15 × 30 = 4.5. So the tip is $4.50.

Step 2. Add the tip to the meal: 30 + 4.50 = 34.50.

Step 3. The total is $34.50.

Here is the same tip by mental math, using the 10% trick. Ten percent of $30 is $3.00 (slide the dot). Fifteen percent is 10% plus half of it: 3.00 + 1.50 = 4.50. Same $4.50 as the careful method, computed in your head before the waiter returns. When two roads agree, you can be confident in the answer.

What we just did: found the added amount, then added it on. Tax works exactly the same way, just with the tax percent.

Real receipts often chain the two. Say that $40 shirt is 25% off, and then 8% sales tax lands on the sale price. Estimate first: the sale price will be $30, and 8% of 30 is a bit under a tenth of it, so the total should sit near $33.

Step 1. Discount: 0.25 × 40 = 10, so the sale price is 40 − 10 = 30.

Step 2. Tax on the SALE price: 0.08 × 30 = 2.40.

Step 3. Add the tax: 30 + 2.40 = 32.40. Total: $32.40, close to our $33 guess.

Notice the order mattered: the tax applied to the discounted $30, not the original $40. Reading which number a percent acts on is half the skill.

Key idea: A discount subtracts a percent of the price. A tip or tax adds a percent of the price. Find the part first, then add or subtract.

Percent increase and decrease

Sometimes we want to describe how much something grew or shrank as a percent. The recipe is: find the amount of change, then compare it to the ORIGINAL amount.

Let us find the percent increase if a plant grew from 20 cm to 25 cm, slowly.

Step 1. Find the change: 25 − 20 = 5 cm taller.

Step 2. Compare the change to the original (not the new value): ⁵⁄₂₀.

Step 3. Turn that fraction into a decimal: 5 ÷ 20 = 0.25.

Step 4. Turn the decimal into a percent by sliding two places right: 0.25 becomes 25%.

Step 5. So the plant grew by 25%.

What we just did: measured the change, then asked what fraction of the starting size that change was. The starting amount is always the number you compare against. Read that once more: change over ORIGINAL, then make it a percent.

Decreases work the same way. If a plant shrank from 25 cm back to 20 cm, the change is 5, and 5 ÷ 25 = 0.20, a 20% decrease. Wait, the same 5 cm was a 25% increase going up but only a 20% decrease coming down? Yes, and that is not a mistake. The measuring stick changed: going up we compared 5 to a start of 20, coming down we compared 5 to a start of 25. Percent change always speaks relative to where you started, which is why the direction matters.

Try it yourself

A $60 jacket is 20% off. What is the sale price? Take your time.

Find the discount: 0.20 × 60 = 12, so you save $12. Subtract: 60 − 12 = 48. The sale price is $48. If you got $48, wonderful. (Quick check the other way: 80% stays, and 0.80 × 60 = 48. Same answer.)

Common misconceptions

• Forgetting to convert the percent first. "20% of 50" is 0.20 times 50, not 20 times 50. Slide the dot two places left before multiplying.
• Stopping at the discount amount. The discount ($10 off) is not the sale price. You still have to subtract it from the original to get what you pay.
• Comparing change to the new value. For percent increase or decrease, always divide the change by the ORIGINAL amount, not the final one.
• Treating a percent like a fixed amount. "20% off" is not "$20 off." On a $50 shirt, 20% off saves 0.20 × 50 = $10. A percent is a share of something, so its dollar value depends on the price it acts on.

Recap

Percent of a number means change the percent to a decimal and multiply. A discount subtracts a percent of the price; a tip or tax adds one. For percent increase or decrease, divide the amount of change by the original and turn it into a percent. These are the exact skills you use shopping, tipping, and budgeting, and you just practiced all of them.

Sources

• OpenStax. (2020). 6.2 Solve general applications of percent. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 6.3 Solve sales tax, commission, and discount applications. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Percentages [Unit]. In Pre-algebra. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Rates and percentages [Unit]. In 6th grade math. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Introduction to percents. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Percentage change. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Estimation. Math is Fun. mathsisfun.com

The big picture

Negative numbers can feel strange at first, because you cannot hold negative three apples in your hand. But you already understand negatives from everyday life. A temperature of 5 below zero, owing a friend $10, a basement two floors below the ground: those are all negative numbers. This week we just give them a home and learn to picture them.

The integers are the whole numbers together with their negatives: ...−3, −2, −1, 0, 1, 2, 3... We will meet the number line, learn what a negative really means, and get comfortable comparing them. We will go slowly, and no question is too small.

Key idea: Integers are the counting numbers, their negatives, and zero. A negative number is just an amount on the opposite side of zero.

The number line, your new best friend

Picture a straight road with zero in the middle. To the right of zero are the positive numbers, growing bigger: 1, 2, 3, and on. To the left of zero are the negative numbers, marked with a minus sign: −1, −2, −3, and on. The number line is the single most helpful picture in all of this week, so we will lean on it a lot.

A handy real-life version is a thermometer standing up, or an elevator panel. Zero is ground level. Positive floors go up, negative floors (the basement) go down. When you picture a negative number, imagine walking to the LEFT of zero, or going DOWN below ground.

Key idea: On the number line, positives sit to the right of zero and negatives sit to the left. Zero is the dividing point, belonging to neither side.

What the negative sign means

The minus sign in front of a number, like −7, is read "negative seven." It tells you the number is 7 steps to the LEFT of zero, or 7 below. A useful money picture: positive money is what you HAVE, and negative money is what you OWE. So −7 dollars means you are $7 in debt. Read that once more if you like: a negative number is an amount pointed the opposite way from a positive one.

Every positive number has an opposite, the same distance from zero but on the other side. The opposite of 7 is −7. The opposite of −4 is 4. And the opposite of 0 is just 0, since it sits right in the middle.

Depth works the same way as debt. A snorkeler at 10 feet below the surface is at −10 ft, and a submarine at 50 feet below is at −50 ft. Which number is smaller? The submarine's: −50 < −10, because it sits farther down (farther left on a sideways number line). But which is the bigger DISTANCE from the surface? The submarine's again: |−50| = 50 beats |−10| = 10. Smaller number, bigger depth. Keeping those two questions separate is most of this week.

Key idea: Negative seven means seven to the left of zero. The opposite of a number is its mirror image across zero.

Comparing integers: left is always less

Here is the one rule that keeps comparing integers simple: on the number line, the number farther to the LEFT is always the smaller one, and farther to the RIGHT is always the larger one. This is true even when it feels backwards.

Let us compare −5 and −2, slowly.

Step 1. Find both on the number line. −5 is five steps left of zero. −2 is only two steps left of zero.

Step 2. Which is farther left? −5 is farther left.

Step 3. Farther left means smaller, so −5 < −2. Read that as "negative five is less than negative two."

This surprises almost everyone, so do not feel bad if it feels odd. With debts it makes sense: owing $5 (−5) is a worse position than owing $2 (−2), so −5 is the smaller number. The bigger the debt, the smaller (more negative) the number.

Key idea: The number farther left on the line is smaller. So −5 is less than −2, even though 5 is bigger than 2.

Ordering a whole handful

Let us order −3, 5, 0, −7, and 2 from least to greatest, using the line.

Step 1. Picture (or sketch) each one: −7 sits far left, then −3, then 0 in the middle, then 2, then 5 out right.

Step 2. Read them off left to right: −7, −3, 0, 2, 5. That is the order, because left means less.

Step 3. Check the pattern: all the negatives come first (the most negative leading), zero sits between the signs, and the positives finish. If your ordered list ever mixes a positive in among negatives, that is the signal to re-check.

One trap to dodge while you sketch: check what each tick mark on a number line is worth before you trust it. If the ticks count by 2s, a point two ticks left of zero is −4, not −2, and three ticks left is −6. Just like reading a graph's scale, read the line's scale first.

Absolute value: distance from zero

The absolute value of a number is simply how far it is from zero, ignoring direction. We write it with two straight bars, like |−3|, read as "the absolute value of negative three." Since distance is never negative, the absolute value is always zero or positive.

Let us find |−3|, one small step at a time.

Step 1. Locate −3 on the number line. It is three steps to the left of zero.

Step 2. Count the distance to zero, ignoring the direction: 3 steps.

Step 3. So |−3| = 3.

Likewise |5| = 5, because 5 is five steps from zero. Absolute value strips off the sign and just reports the distance. Think of it as asking "how far?" not "which way?"

Key idea: Absolute value is distance from zero, so it is never negative. |−3| and |3| are both 3.

Distance and position are different questions

Here is a pair that untangles the whole week. Compare −7 and 5 two ways.

Position: which is greater? On the line, 5 sits to the right of −7, so 5 is greater. Write −7 < 5.

Distance: which is farther from zero? |−7| = 7 and |5| = 5, and 7 beats 5, so −7 is farther out.

Both answers are true at once, because they answer different questions. A temperature of 7 below zero is a LOWER temperature than 5 above, but a BIGGER swing away from freezing at zero. When a problem mentions "how far," "how much of a change," or "how big a swing," it wants absolute value. When it asks "which is more or less," it wants position. Naming which question you are answering is the skill.

Crossing zero: counting the gap

The morning temperature is −3 degrees, and by afternoon it reads 4 degrees. How many degrees did it climb? Count the walk on the number line.

Step 1. From −3 up to 0 is 3 steps (that is |−3|).

Step 2. From 0 up to 4 is 4 more steps (that is |4|).

Step 3. Add the two legs: 3 + 4 = 7. The temperature climbed 7 degrees.

When a trip crosses zero, the total distance is the two absolute values added together. Sanity check: the answer, 7, is bigger than either number alone, which is exactly what crossing zero should feel like. (Next week this becomes the arithmetic 4 − (−3) = 7, so you have just previewed it with plain counting.)

Try it. The temperature climbs from −5 degrees to 2 degrees. How big was the climb? From −5 to 0 is 5 steps, and from 0 to 2 is 2 more, so 5 + 2 = 7 degrees. If you got 7, the crossing-zero idea is yours.

Try it yourself

Which is larger, −8 or −3? And what is |−8|? Take your time and picture the line.

On the line, −8 is far to the left and −3 is closer to zero. Farther right is larger, so −3 is the larger number: −3 > −8. And |−8| = 8, since −8 is eight steps from zero. If you got those, wonderful.

Common misconceptions

• Thinking a bigger digit means a bigger number. With negatives it flips: −5 is smaller than −2, because it is farther left. The bigger the negative digit, the smaller the value.
• Making absolute value negative. Absolute value is a distance, so it is never negative. |−3| is 3, not −3.
• Reading the minus sign as subtraction here. In −7 the sign labels the number as negative (its position), it is not asking you to subtract anything.
• Assuming every tick mark is worth 1. Number lines often count by 2s, 5s, or 10s. Check the scale first, or a point two ticks left of zero (worth −4 on a by-2s line) gets misread as −2.

Recap

Integers are the whole numbers, their negatives, and zero. On the number line, positives go right and negatives go left, and the number farther left is always smaller. The opposite of a number is its mirror across zero, and absolute value is its distance from zero, always zero or positive. You already knew negatives from temperatures and debts. Now you can place them and compare them.

Sources

• OpenStax. (2020). 3.1 Introduction to integers. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 7.4 Properties of identity, inverses, and zero. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Negative numbers [Unit]. In 6th grade math. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Number line. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Absolute value. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Adding and subtracting positive and negative numbers. Math is Fun. mathsisfun.com
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM. nctm.org

The big picture

This week has a reputation for being tricky, all those plus and minus signs bumping into each other. But we are going to lean on two friendly pictures, money and the number line, and take it one tiny step at a time. By the end, the rules will feel like common sense rather than magic spells to memorize.

We are adding, subtracting, multiplying, and dividing integers (positive and negative whole numbers). Every rule below will come with a real-life reason, not just a "because I said so." We will go slowly, and no question is too small.

Key idea: Think of positive numbers as money you have and negative numbers as money you owe. Most integer rules make sense the moment you picture money.

Adding integers, told with money

Let us use the money picture. Positive is cash in your pocket; negative is debt.

Same signs (both positive or both negative): add and keep the sign. If you have $3 and gain $5 more, you have $8. So 3 + 5 = 8. If you owe $3 and then owe $5 more, you owe $8, which we write as −3 + (−5) = −8. In both cases you add the amounts and keep the shared sign.

Different signs (one of each): subtract, and take the sign of the bigger amount. Suppose you owe $8 but have $3 cash. Pay what you can: you are still $5 in debt, so −8 + 3 = −5. The debt was bigger, so the answer is negative.

Let us do −8 + 3, one small step at a time.

Step 1. The signs are different (one negative, one positive), so we subtract the smaller amount from the bigger: 8 − 3 = 5.

Step 2. Decide the sign. The 8 (the debt) is bigger, and it is negative, so the answer is negative.

Step 3. So −8 + 3 = −5.

What we just did: matched the cash against the debt, kept the sign of whichever was larger. Read that once more if you like: same signs add, different signs subtract and follow the bigger one.

Key idea: Same signs, add and keep the sign. Different signs, subtract and take the sign of the larger amount.

The number line tells the same story as the money. To add a positive, walk right; to add a negative, walk left. For −8 + 3, start at −8 and take 3 steps right: −7, −6, −5. You land on −5, matching the money answer. For −6 + (−4), start at −6 and walk 4 steps left to −10. Whenever a sign rule feels slippery, walk it out; the line never lies.

Subtracting integers: add the opposite

Subtraction is where most people slip, so here is the single trick that fixes everything: subtracting a number is the same as adding its opposite. We change the subtraction into an addition, flip the sign of the number being subtracted, and then use the adding rules we just learned. Some people say "keep, change, change": keep the first number, change the minus to a plus, change the sign of the second number.

Let us do 5 − (−3), slowly. Read it as "five minus negative three."

Step 1. Keep the first number: 5.

Step 2. Change the subtraction to addition: 5 + ...

Step 3. Change the sign of what follows: −3 becomes +3. Now we have 5 + 3.

Step 4. Add using our rule (same signs): 5 + 3 = 8.

So 5 − (−3) = 8. Why should subtracting a negative make things bigger? Picture debt again: if someone removes a $3 debt from you (takes away a negative), you are $3 better off. Taking away a debt is a gift. Read that once more if you like: minus a negative becomes a plus.

Key idea: To subtract, add the opposite. Two minus signs in a row (minus a negative) turn into a plus.

A longer chain, one operation at a time

Let us simplify −4 + 7 − (−2), working left to right and naming each rule.

Step 1. First, −4 + 7. Different signs, so subtract the amounts: 7 − 4 = 3. The larger amount, 7, is positive, so the result is +3. The expression is now 3 − (−2).

Step 2. Subtracting a negative means add the opposite: 3 − (−2) becomes 3 + 2.

Step 3. Same signs, so add: 3 + 2 = 5. The answer is 5.

Check it with the money story: you owed $4, earned $7 (now $3 ahead), then someone erased a $2 debt you had forgotten (a $2 gift). Ending $5 ahead feels right, so the arithmetic and the story agree.

Multiplying and dividing: just watch the signs

Good news: for multiplying and dividing, you multiply or divide the numbers exactly as usual, and then you only have to figure out the sign. The sign follows one short pattern.

• positive × positive = positive (2 × 3 = 6)
• negative × negative = positive (−2 × −3 = 6)
• positive × negative = negative (2 × −3 = −6)
• negative × positive = negative (−2 × 3 = −6)

The tidy way to remember it: same signs give a positive, different signs give a negative. Division follows the exact same sign pattern.

Let us do −12 ÷ 4, one small step at a time.

Step 1. Divide the numbers as usual, ignoring signs for a moment: 12 ÷ 4 = 3.

Step 2. Check the signs. One is negative (−12) and one is positive (4). Different signs.

Step 3. Different signs give a negative, so the answer is −3.

So −12 ÷ 4 = −3. What we just did: did the plain division, then applied the sign rule. That is the whole method for both multiply and divide.

Always check a division by multiplying back: 4 × (−3) = −12, which rebuilds the number we divided. And a both-negative one: −18 ÷ (−3) = 6, positive because the signs match. Check: (−3) × 6 = −18. It rebuilds again. Decide the sign first, then the size, then multiply back. Three small habits, almost no wrong answers.

Key idea: For multiplying and dividing, same signs make a positive, different signs make a negative.

Why negative times negative is positive

This is the rule everyone is told to memorize, so let us actually earn it. Watch what happens as we multiply −2 by a countdown of numbers, and keep the pattern honest.

3 × (−2) = −6. Then 2 × (−2) = −4. Then 1 × (−2) = −2. Then 0 × (−2) = 0.

Look at the answers: −6, −4, −2, 0. Every time the first number drops by 1, the answer RISES by 2. Now keep the countdown going past zero. The next line must rise by 2 again: (−1) × (−2) = 2. And the next: (−2) × (−2) = 4. Positive answers, forced on us by the pattern. If negative times negative gave a negative, the steady rise would suddenly break for no reason.

The money story agrees. Multiplying by a positive count means gaining copies; by a negative count, removing copies. Removing 3 debts of $2 (that is (−3) × (−2)) leaves you $6 better off: +6. Taking away what drags you down lifts you up.

And division has no choice but to follow along, because division must undo multiplication. Since (−3) × 6 = −18, the division −18 ÷ (−3) has to give back 6, a positive. If same-sign division ever gave a negative, the multiply-back check would fail, and the two operations would stop being partners. One sign rule, shared by both.

Key idea: Negative times negative is positive because the times tables keep a steady pattern across zero, and because removing debts makes you richer.

Try it yourself

Work out −6 + (−4), then −5 × 2. Take your time.

For −6 + (−4): same signs, so add and keep the sign, giving −10 (owe $6, owe $4 more, owe $10). For −5 × 2: multiply to get 10, and different signs make it negative, so −10. If you got −10 for both, wonderful.

One more everyday one: 2 − 7. Rewrite as adding the opposite: 2 + (−7). Different signs, so subtract the amounts (7 − 2 = 5) and follow the larger amount's sign, negative: the answer is −5. Have $2, owe $7, and you are $5 short. The story and the rule agree.

Common misconceptions

• Mixing up the add rules. Same signs add; different signs subtract. A quick money check catches errors: does the answer match having cash versus owing?
• Forgetting to flip when subtracting. Subtracting a negative, like 5 minus negative 3, becomes 5 plus 3. Two minuses in a row make a plus.
• Blending the sign rules. The multiply and divide rule (same signs positive) is not the same as the add rule. Keep them separate: this sign trick is only for times and divide.

Recap

To add integers, same signs add and keep the sign, different signs subtract and follow the larger amount. To subtract, add the opposite (minus a negative becomes plus). To multiply or divide, do the arithmetic and then apply the sign rule: same signs give positive, different signs give negative. Picture money and the number line whenever a rule feels slippery. You just handled the part of pre-algebra that trips up the most people.

Sources

• OpenStax. (2020). 3.2 Add integers. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 3.3 Subtract integers. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 3.4 Multiply and divide integers. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Add and subtract negative numbers [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Multiply and divide negative numbers [Unit]. In Arithmetic. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Adding and subtracting positive and negative numbers. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Multiplying negatives makes a positive. Math is Fun. mathsisfun.com

The big picture

If the word algebra has ever made your stomach drop, you are in the right place, and you are going to be okay. Here is a secret almost no one tells you: you already do algebra in your head all the time. Every time you work out the change from a $20 bill, double a recipe, or split a bill three ways, that is algebra. You are not starting from nothing. You are just going to learn to write down something you already know how to do.

This week we meet the variable (a letter that stands for a number) and learn to build and evaluate expressions. We will go slowly, one small step at a time. Nothing skipped, nothing rushed.

Key idea: A variable is just a letter standing in for a number we do not know yet. Algebra is arithmetic with a placeholder.

A variable is a mystery box

Picture a small box with a closed lid. Inside is a number, but we cannot see it yet. To talk about that hidden number, we give the box a name, usually a letter like x. That letter has a fancy name, a variable, but do not let the word worry you. A variable is just the name of a box with a number hiding inside. Read that once more if you like.

You already use this idea. When a recipe says "use as many eggs as people," the phrase "as many as people" is a stand-in for a number you will fill in later. That is a variable in plain English.

Key idea: A variable, like x, is a placeholder for a number. Different letters can name different mystery boxes.

Reading the shorthand of algebra

Algebra uses a few quiet shorthands that trip people up only because no one explains them out loud. Let us name them plainly.

• 2x means "2 times x." When a number sits right next to a letter, it means multiply. There is an invisible times sign between them.
• The 2 in 2x is called the coefficient. It is just the number multiplying the variable.
• A term is a single piece like 2x, or 5, or y. Terms are separated by plus and minus signs.
• An expression is a bunch of terms combined, like 2x + 1. An expression has no equals sign; it is just a recipe waiting for a number.

Read that once more if you like: a number stuck to a letter means multiply, and an expression is just a little recipe.

Key idea: 2x is 2 times x. A coefficient is the number in front; a term is one piece; an expression combines terms with plus and minus.

Evaluating an expression: let us try one together

To evaluate an expression means to put a number into the box and see what comes out. It is exactly like a vending machine: you put a value in for x, and the expression gives you a result.

Let us evaluate 2x + 1 when x = 5, slowly.

Step 1. Remember that 2x means "2 times x." So we really have (2 times x) + 1.

Step 2. We were told x is 5. So everywhere we see x, we gently put a 5 in its place: (2 times 5) + 1.

Step 3. Do the multiply first (order of operations from Week 2). 2 times 5 is 10. Now we have 10 + 1.

Step 4. Do the add. 10 + 1 is 11.

That is the answer: 11. Take a second to notice what just happened. You took 2x + 1, put in a number, and got a real answer. That is the whole skill. You just did algebra.

Key idea: To evaluate, replace the variable with its number, then follow the order of operations.

Two quick variations to make the habit sturdy. First, order of operations still rules: evaluate 10 − 2x when x = 3. Substitute: 10 − (2 times 3). Multiply first: 2 times 3 is 6. Then subtract: 10 − 6 = 4. (Subtracting first, 10 − 2 = 8, then multiplying would give the wrong 24.)

Second, expressions can hold two different mystery boxes. Evaluate 2a + 3b when a = 4 and b = 2. Substitute each letter with its own number: (2 times 4) + (3 times 2). Multiply both: 8 + 6. Add: 14. Each variable keeps its own value, like two labeled boxes holding different numbers.

Writing expressions from words

A big use of algebra is turning a sentence into an expression. The trick is to translate a few key words. Let x be the mystery number.

• "5 more than a number" becomes x + 5 (more than means add).
• "3 less than a number" becomes x − 3 (less than means subtract, and the order flips: the 3 comes off x).
• "double a number" becomes 2x (double means times 2).
• "a number split into 4 equal parts" becomes ^x⁄₄ (split means divide).

Let us try one together. "Four more than twice a number." Take it in pieces: "twice a number" is 2x, and "four more than" that adds 4, giving 2x + 4. What we just did: translated an everyday phrase, chunk by chunk, into symbols.

Combining like terms (tidying an expression)

Sometimes an expression has repeats we can gather up. Like terms are terms with the exact same variable part, such as 3x and 2x (both have an x). You can combine them by adding their coefficients, just like counting objects: 3 apples plus 2 apples is 5 apples, so 3x + 2x = 5x.

But you cannot combine 3x and 4, because one is a number of x-boxes and the other is a plain number, like apples and oranges. So 3x + 4 stays as it is; it is already as tidy as it gets.

Let us simplify 3x + 2x + 4, one small step at a time.

Step 1. Spot the like terms: 3x and 2x both have x.

Step 2. Add their coefficients: 3 + 2 = 5, giving 5x.

Step 3. The plain number 4 has nothing to combine with, so it stays.

Step 4. The tidy result is 5x + 4.

Key idea: Combine like terms by adding their coefficients. Terms with different variable parts (or a plain number) cannot be combined.

The distributive property: multiplying a package

What does 3(x + 2) mean? A number touching parentheses means multiply, so it is 3 copies of the whole package (x + 2). Write the copies out: (x + 2) + (x + 2) + (x + 2). Gather the like terms: three x's make 3x, and three 2s make 6. So 3(x + 2) = 3x + 6. Handing the 3 to EACH thing inside the parentheses is called distributing, the same distributive property from Week 1's multiplication, now wearing algebra clothes.

The classic slip is to multiply only the first thing: 3(x + 2) is NOT 3x + 2. Catch it with the test-a-number habit: try x = 4. The original: 3(4 + 2) = 3 × 6 = 18. Our answer: 3(4) + 6 = 12 + 6 = 18. They match. The slip: 3(4) + 2 = 14. No match, so the slip is exposed.

Key idea: Distribute means multiply every term inside the parentheses. Test any simplification by plugging in an easy number.

Test with a number: the algebra safety net

That checking move deserves its own name, because it works on almost everything this week. Claimed that 3x + 2x = 5x? Test x = 2: the left side gives 6 + 4 = 10, and the right gives 5 × 2 = 10. Match. Wondering whether 3x + 4 can become 7x? Test x = 2: the left is 6 + 4 = 10, but 7x is 14. No match, so no, they are different expressions.

One friendly caution: avoid testing with 0 or 1, since too many different expressions happen to agree there. A plain number like 2 or 3 does the honest work. Estimate, substitute, compare: the same check-yourself rhythm as every other week, now for algebra.

Try it yourself

Evaluate 3x − 2 when x = 4, and simplify 5y + 3y. Take your time.

For 3x − 2 with x = 4: replace x with 4 to get (3 times 4) − 2, then multiply first, 12 − 2 = 10. For 5y + 3y: like terms, add coefficients, 5 + 3 = 8, giving 8y. If you got 10 and 8y, wonderful, you just did algebra.

Common misconceptions

• Reading 2x as "twenty-something." 2x is not a two-digit number; it means 2 times x. When x is 5, 2x is 10.
• Combining unlike terms. 3x + 4 is not 7x. You can only combine terms that share the same variable, like 3x and 2x.
• Flipping less than the wrong way. "3 less than a number" is x minus 3, not 3 minus x. The number you start with comes first.
• Distributing to only the first term. 3(x + 2) means 3 times EVERYTHING inside: 3x + 6, never 3x + 2. Test x = 4 and the wrong version gives 14 instead of 18.

Recap

A variable is a letter standing for a hidden number, like a labeled box. An expression is a recipe of terms with no equals sign. To evaluate, replace the variable with its number and follow order of operations. Translate word phrases piece by piece, and tidy expressions by combining like terms. You already did this kind of thinking with recipes and change; now it has symbols. You just did algebra.

Sources

• OpenStax. (2020). 2.2 Evaluate, simplify, and translate expressions. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 10.1 Add and subtract polynomials. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Variables & expressions [Unit]. In Pre-algebra. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Variables & expressions [Unit]. In 6th grade math. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Introduction to algebra. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Algebra - basic definitions. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Like terms. Math is Fun. mathsisfun.com

The big picture

This is the week where all the pieces come together, and it is genuinely exciting: you are going to SOLVE equations, which means finding the hidden number. If that sounds hard, remember you have solved for unknowns your whole life. "I have $12 and the movie costs $15, how much more do I need?" That is solving an equation. We are just going to write it down neatly.

An equation is a math sentence with an equals sign, like x + 3 = 8. Solving it means finding the value of the variable that makes the sentence true. We will go slowly, one small step at a time, and I will show every move.

Key idea: An equation is a balance scale. Whatever you do to one side, you must do to the other to keep it balanced.

The balance scale, the heart of everything

Picture an old-fashioned balance scale, the kind with two trays. The equals sign is the middle post, and the two sides must always weigh the same. If you add weight to one tray, you must add the same weight to the other, or the scale tips. If you remove weight from one, you remove it from the other.

That is the one rule behind all equation solving: whatever you do to one side, do the exact same thing to the other side. Read that once more if you like. Everything else this week is just that idea applied carefully.

Our goal each time is to get the variable all alone on one side, which we call isolating the variable. Once x sits by itself, the number on the other side is our answer.

Key idea: Keep the scale balanced by doing the same operation to both sides, working to get the variable alone.

Undoing with opposite operations

To get the variable alone, we peel away the numbers attached to it by using the OPPOSITE operation. Addition and subtraction are opposites; multiplication and division are opposites. To undo a plus, we subtract. To undo a times, we divide. Think of it like untying knots in reverse order.

Key idea: Undo addition with subtraction, and undo multiplication with division. Opposite operations cancel each other.

One-step equations: let us try one together

Let us solve x + 3 = 8, slowly. Read it as "some number plus 3 equals 8."

Step 1. Look at what is happening to x. It has a +3 stuck to it. To free x, we undo the +3 by doing the opposite: subtract 3.

Step 2. To keep the scale balanced, subtract 3 from BOTH sides: x + 3 − 3 = 8 − 3.

Step 3. Simplify the left side. The +3 and −3 cancel, leaving just x.

Step 4. Simplify the right side. 8 − 3 = 5.

Step 5. So x = 5.

Let us check it, because checking feels great. Put 5 back into the original: 5 + 3 = 8. True. So x = 5 is correct. What we just did: undid the +3 on both sides to get x alone. You can always check an answer by putting it back in.

Now a times example. Solve 4x = 20, read as "4 times some number equals 20."

Step 1. The x is being multiplied by 4. To undo a times, we divide by 4.

Step 2. Divide BOTH sides by 4: ^4x⁄₄ = ²⁰⁄₄.

Step 3. On the left, the 4 on top and the 4 on the bottom cancel, leaving x.

Step 4. On the right, 20 ÷ 4 = 5.

Step 5. So x = 5. Check: 4 times 5 is 20. True. Nice work.

Negatives and division join the party

The balance idea does not care whether the numbers are friendly. Solve x + 7 = 3.

Step 0. Predict first: we are taking away more than we have, so x should land below zero.

Step 1. Undo the +7 by subtracting 7 from both sides: x = 3 − 7.

Step 2. Use last week's rule for 3 − 7: different signs, subtract the amounts (7 − 3 = 4) and keep the sign of the larger, which is negative. So x = −4.

Step 3. Check by substituting back: −4 + 7 = 3. True, so x = −4 is right. A negative answer is perfectly fine; the scale stayed balanced the whole time.

Now one where x is being divided: solve ^x⁄₃ = 6, read "a number split into 3 equal parts is 6 per part."

Step 1. The x is divided by 3. Undo that with the opposite: multiply both sides by 3.

Step 2. Left side: the divide-by-3 and times-3 cancel, leaving x. Right side: 6 × 3 = 18.

Step 3. So x = 18. Check: 18 ÷ 3 = 6. True. All four operations now have an undo you can use.

Two-step equations: undo in reverse order

A two-step equation has two things happening to x, like 2x + 3 = 11. Read it as "2 times a number, plus 3, equals 11." The gentle strategy: undo the addition or subtraction FIRST, then undo the multiplication or division. It is like taking off your shoes before your socks, the reverse of how they went on.

Let us solve 2x + 3 = 11, one small step at a time.

Step 1. First undo the +3. Subtract 3 from both sides: 2x + 3 − 3 = 11 − 3.

Step 2. Simplify. Left: the +3 and −3 cancel, leaving 2x. Right: 11 − 3 = 8. Now we have 2x = 8.

Step 3. Now undo the times 2. Divide both sides by 2: ^2x⁄₂ = ⁸⁄₂.

Step 4. Simplify. Left: the 2s cancel, leaving x. Right: 8 ÷ 2 = 4.

Step 5. So x = 4.

Check it: put 4 back into the original. 2 times 4 is 8, plus 3 is 11. True. So x = 4 is correct. What we just did: peeled off the +3 first, then the times 2, one layer at a time. Read that once more if you like: subtract or add first, then multiply or divide.

Why that order? Think about how the expression was built. Following order of operations, x was FIRST multiplied by 2, and THEN 3 was added on top. Unwrapping runs the story backwards: the last thing added is the first thing removed. Same as getting dressed and undressed: shoes went on after socks, so shoes come off first.

Key idea: For two-step equations, undo addition or subtraction first, then undo multiplication or division. Always do each step to both sides.

One more, with a guess before the algebra

Solve 3x − 5 = 16. Before touching the tools, guess the size: 3x must be around 21 (since taking 5 away left 16), so x should be near 7. Now the careful version.

Step 1. Undo the −5 by adding 5 to both sides: 3x − 5 + 5 = 16 + 5.

Step 2. Simplify both sides. Left: the −5 and +5 cancel, leaving 3x. Right: 16 + 5 = 21. So 3x = 21.

Step 3. Undo the times 3 by dividing both sides by 3: x = 21 ÷ 3 = 7.

Step 4. Check in the original: 3 × 7 = 21, minus 5 is 16. True, and it matches the guess we made before starting. Guess, solve, check: the full safety routine.

Equations also come from everyday sentences. "I have $12, and the movie costs $15. How much more do I need?" becomes 12 + x = 15. Subtract 12 from both sides: x = 3. You need $3 more, which you likely knew in your head. Now the head-math and the paper-math are the same machine.

Try it yourself

Solve x − 4 = 6, then solve 3x + 1 = 10. Take your time and keep both sides balanced.

For x − 4 = 6: undo the −4 by adding 4 to both sides, giving x = 10. Check: 10 − 4 = 6, true. For 3x + 1 = 10: subtract 1 from both sides to get 3x = 9, then divide both sides by 3 to get x = 3. Check: 3 times 3 is 9, plus 1 is 10, true. If you got x = 10 and x = 3, wonderful. See? You just solved algebra equations.

Common misconceptions

• Changing only one side. The scale must stay balanced. If you subtract 3, subtract it from BOTH sides, not just one.
• Using the same operation instead of the opposite. To undo a +3 you subtract 3, not add 3. Opposites are what cancel.
• Doing the two steps in the wrong order. In 2x + 3 = 11, undo the +3 first, then the times 2. Handling the multiplication too early makes the numbers messy.

Recap

An equation is a balance scale, and solving means getting the variable alone by doing the same thing to both sides. Undo operations with their opposites: subtract to undo adding, divide to undo multiplying. For two-step equations, undo the plus or minus first, then the times or divide. Always check by putting your answer back in. You have been solving for unknowns your whole life; now you can do it on paper, neatly and with confidence.

Sources

• OpenStax. (2020). 2.3 Solving equations using the subtraction and addition properties of equality. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 8.1 Solve equations using the subtraction and addition properties of equality. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 8.2 Solve equations using the division and multiplication properties of equality. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). One-step and two-step equations & inequalities [Unit]. In Pre-algebra. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Equations & inequalities introduction [Unit]. In Pre-algebra. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Solving equations. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Introduction to algebra. Math is Fun. mathsisfun.com

The big picture

Geometry is one of the friendliest topics in math because you can SEE it. Every measurement this week answers a real question you have probably asked: How much fencing do I need? How much carpet covers this room? How much water fills this box? If those feel practical, that is exactly the point.

This week we measure shapes three ways: perimeter (the distance around), area (the flat space inside), and volume (the space a solid holds). We will go slowly, with a picture and a real-life reason for each. No question is too small.

Key idea: Perimeter is the fence around a shape, area is the carpet covering its floor, and volume is the water that fills a solid.

Perimeter: the distance all the way around

Perimeter is the total distance around the outside of a flat shape. Picture walking along every edge of a garden and adding up how far you walked. To find any perimeter, you simply add up the lengths of all the sides. That is it.

Let us find the perimeter of a rectangle that is 5 m long and 3 m wide, slowly.

Step 1. A rectangle has four sides: two long ones and two short ones. Here the long sides are 5 m each and the short sides are 3 m each.

Step 2. Add all four sides: 5 + 3 + 5 + 3.

Step 3. Work left to right: 5 + 3 = 8, then 8 + 5 = 13, then 13 + 3 = 16.

Step 4. So the perimeter is 16 m.

What we just did: walked around the rectangle adding each side. There is a shortcut for rectangles, add the length and width then double it: (5 + 3) × 2 = 8 × 2 = 16 m, the same answer. Perimeter is measured in plain length units like meters or feet.

Key idea: Perimeter is the sum of all the side lengths, measured in length units like m or ft.

Area: how much flat space fits inside

Area is the amount of flat surface inside a shape, like how much carpet you would need to cover a floor. We measure it in square units (square meters, square feet), because we are counting little 1-by-1 squares that tile the inside.

For a rectangle, the area is length times width, written A = length × width. Picture the rectangle divided into a grid of little squares: the length tells you how many squares across, and the width tells you how many rows, so multiplying gives the total count.

Let us find the area of that same 5 m by 3 m rectangle, one small step at a time.

Step 1. Write the rule: area = length × width.

Step 2. Put in the numbers: area = 5 × 3.

Step 3. Multiply: 5 × 3 = 15.

Step 4. Attach the right unit. Because it is area, the unit is square meters: 15 square meters.

What we just did: counted how many 1-meter squares fit inside, which is 15. The little word "square" in the unit is your reminder that area covers a surface. Read that once more if you like: perimeter uses plain units, area uses square units.

A triangle is half of a rectangle, so its area is A = ¹⁄₂ × base × height. For a base of 6 and height of 4: ¹⁄₂ × 6 × 4 = ¹⁄₂ × 24 = 12 square units.

Where does that one half really come from? Draw a rectangle 6 wide and 4 tall, then draw one diagonal, corner to corner. The diagonal cuts the rectangle into two triangles that match exactly, so each triangle owns exactly half of the rectangle's 24 squares: 12 each. The formula is just that picture written in symbols. No halving, and you would be counting the whole rectangle by mistake.

Key idea: Area of a rectangle is length times width, measured in square units. A triangle is half of that: one half times base times height.

An L-shaped room: two roads to one area

Real floors are not always neat rectangles. Say a room is an 8 m by 5 m rectangle with a 3 m by 2 m corner bite taken out of it. What is the floor area?

Road 1: subtract the bite. The full rectangle would be 8 × 5 = 40 square meters. The bite removes 3 × 2 = 6 square meters. So the floor is 40 − 6 = 34 square meters.

Road 2: split the L into two rectangles and add. One slab is 8 m by 3 m: 8 × 3 = 24. The remaining piece is 5 m by 2 m: 5 × 2 = 10. Together: 24 + 10 = 34 square meters.

Both roads land on 34, and that agreement is the check. Area behaves like carpet: it adds when you join pieces and subtracts when you cut pieces away, so any honest split of the shape must total the same.

Same carpet, different fence

Can two shapes hold the same area but need different amounts of fencing? Yes, and it shows why area and perimeter are truly different measurements. A 1 by 12 rectangle has area 1 × 12 = 12, and perimeter 1 + 12 + 1 + 12 = 26. A 3 by 4 rectangle also has area 3 × 4 = 12, but its perimeter is 3 + 4 + 3 + 4 = 14. A 2 by 6 splits the difference: area 2 × 6 = 12 again, perimeter 2 + 6 + 2 + 6 = 16. Same carpet, very different fence. Skinny shapes spend a lot of edge on a little inside. So never assume one measurement tells you the other; compute the one the question actually asks for.

Volume: how much a solid holds

Volume is the amount of space inside a three-dimensional solid, like how much water fills a fish tank. We measure it in cubic units (cubic meters, cubic feet), because now we are stacking little 1-by-1-by-1 cubes to fill the space.

For a box shape (a rectangular prism), the volume is length times width times height: V = length × width × height. It is the area of the bottom (length times width) multiplied by how tall the stack goes (height).

Let us find the volume of a box that is 5 m long, 3 m wide, and 2 m tall, slowly.

Step 1. Write the rule: volume = length × width × height.

Step 2. Put in the numbers: volume = 5 × 3 × 2.

Step 3. Multiply two at a time. First 5 × 3 = 15.

Step 4. Then 15 × 2 = 30.

Step 5. Attach the unit. Because it is volume, the unit is cubic meters: 30 cubic meters.

What we just did: found how many 1-meter cubes fill the box, which is 30. Notice the pattern in the units: length is plain, area is square, volume is cubic. That matches one direction, then two, then three.

The layer picture makes the formula feel inevitable. A fish tank is 6 ft long, 2 ft wide, and 3 ft tall. The bottom layer holds 6 × 2 = 12 one-foot cubes. The tank stacks 3 such layers, so it holds 12 × 3 = 36 cubic feet. And the grouping does not matter: 2 × 3 = 6 and 6 × 6 = 36 gives the same count, a handy way to double-check by multiplying in a different order.

Key idea: Volume of a box is length times width times height, measured in cubic units.

Try it yourself

A rectangle is 8 cm long and 2 cm wide. Find its perimeter and its area. Take your time and watch the units.

Perimeter: add all sides, 8 + 2 + 8 + 2 = 20 cm. Area: length times width, 8 × 2 = 16 square cm. If you got 20 cm and 16 square cm, wonderful. Notice perimeter came out in plain cm and area in square cm. And if that same 8 by 2 rectangle were the floor of a box 3 cm tall, the volume would stack three 16-square-cm layers: 16 × 3 = 48 cubic cm.

Common misconceptions

• Mixing up perimeter and area. Perimeter is the distance around (add the sides); area is the space inside (multiply length by width). If you need fencing, that is perimeter; if you need carpet, that is area.
• Forgetting the square or cubic in the units. Area is in square units and volume is in cubic units. Writing 15 meters for an area instead of 15 square meters loses important meaning.
• Forgetting the one half for a triangle. A triangle is half a rectangle, so its area is one half times base times height, not just base times height.
• Mixing units in one formula. A shelf 2 m long and 30 cm deep is not "2 times 30." Convert first: 200 cm × 30 cm = 6,000 square cm (which is 0.6 square meters, since 2 × 0.3 = 0.6). Same units in, sensible units out.

Recap

Perimeter is the distance around a shape, found by adding all the sides, in plain length units. Area is the flat space inside, found by length times width for a rectangle, in square units. Volume is the space a solid holds, found by length times width times height for a box, in cubic units. Picture fencing, carpet, and water, and the three ideas stay separate and clear. You just measured the world three different ways.

Sources

• OpenStax. (2020). 9.4 Use properties of rectangles, triangles, and trapezoids. In Prealgebra 2e. OpenStax. openstax.org
• OpenStax. (2020). 9.6 Solve geometry applications: Volume and surface area. In Prealgebra 2e. OpenStax. openstax.org
• Khan Academy. (n.d.). Intro to area and perimeter [Unit]. In Basic geometry and measurement. Khan Academy. khanacademy.org
• Khan Academy. (n.d.). Volume [Unit]. In Basic geometry and measurement. Khan Academy. khanacademy.org
• Math is Fun. (n.d.). Perimeter. Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). What is area? Math is Fun. mathsisfun.com
• Math is Fun. (n.d.). Volume. Math is Fun. mathsisfun.com

The big picture

This is our very last week together, and it is a gentle one. This week is about data, the math word for numbers collected from the real world. You meet data every day: your phone's screen-time report, the five-day weather forecast, the points your team scored this season. All of that is data, and by the end of this lesson you will be able to read it with real confidence.

Here is the plan. First we meet four numbers that describe any list of data: the mean, the median, the mode, and the range. Then we practice reading bar graphs and line graphs. At the very end, we look back at how far you have come, because it is farther than you might think. One small step at a time, nothing rushed, and no question is too small.

Key idea: Data is a collection of numbers. This week you learn four numbers that describe a data set and two kinds of graphs that picture it.

Why we summarize data

Imagine a friend asks, "How did your bowling season go?" You would not recite all twenty scores one by one. You would say something like "I usually scored around 90." That one sentence is a summary, and that is exactly the job of the mean, median, mode, and range. Each one squeezes a long list of numbers into one small, honest number. A summary number is like a one-sentence book report: it does not tell you everything, but it tells you a lot, fast.

Key idea: Mean, median, mode, and range each turn a whole list of numbers into one number that describes it.

The mean: share everything out equally

The mean is what most people call the average. Here is the friendliest way to picture it: pooling and sharing. Suppose three friends have 2, 5, and 5 candies. If they pour all their candy into one pile and then deal it back out equally, how many does each person get? That equal share is the mean.

Let us find it together, slowly.

Step 1. Add the first two numbers: 2 + 5 = 7. (We are building the big shared pile.)

Step 2. Add the next number: 7 + 5 = 12. The whole pile holds 12 candies.

Step 3. Count how many numbers there are. Here there are 3.

Step 4. Divide the pile by the count: 12 ÷ 3 = 4. (Read that aloud as "twelve divided by three is four.")

The mean is 4. If the friends shared equally, each would get 4 candies. What we just did: added everything up, then divided by how many numbers there were. That is the whole recipe. Nice, that was the trickiest of the four measures, and you have already done it.

Two gentle warnings that surprise almost everyone at first. First, the mean does not have to be one of the original numbers. Second, the mean is allowed to be a decimal, like 6.33. Nothing has gone wrong when that happens.

Try it: Find the mean of 4, 6, and 8. Answer: 4 + 6 = 10, then 10 + 8 = 18. There are 3 numbers, so 18 ÷ 3 = 6. The mean is 6.

Key idea: Mean = add up every value, then divide by how many values there are.

The median: the value in the middle

Picture a group of kids lining up by height for a photo. The kid standing exactly in the middle of the line is the median. The median of a data set is the middle value, but there is one rule people forget: the numbers must be in order first. A crowd is not a line until it lines up.

Let us find the median of 9, 2, 7, one tiny step at a time.

Step 1. Put the numbers in order from least to greatest: 2, 7, 9.

Step 2. Point to the middle value. With three numbers, the middle one is the second: 7.

The median is 7. Notice that the 9 sitting first in the original list did not matter at all. Order first, then look in the middle.

What if there are two middle numbers? That happens whenever the list has an even count, and the median is then the mean of those two middle values. Let us do 3, 1, 7, 5 together.

Step 1. Order the numbers: 1, 3, 5, 7.

Step 2. Find the two middle values: 3 and 5.

Step 3. Add them: 3 + 5 = 8.

Step 4. Divide by 2: 8 ÷ 2 = 4.

The median is 4. And yes, 4 is not even in the list. That is allowed, and you handled it.

Where people get stuck: grabbing the middle of the unsorted list. For 9, 2, 7 it is tempting to pick 2 because it sits in the middle position. Sort first, every single time: 2, 7, 9, so the median is 7, not 2.

Try it: Find the median of 8, 3, 5. Answer: in order the list is 3, 5, 8, and the middle value is 5. The median is 5.

Key idea: Median = sort the list, then take the middle value (or the mean of the two middle values if the count is even).

The mode: the most popular value

The mode is the value that shows up most often. Think of the most common shoe size in your class, or the flavor ordered most at an ice cream shop. There is no adding and no dividing here, only counting, so if the mean felt heavy, this one is a rest stop.

Example with numbers first: for the shoe sizes 6, 7, 7, 8, the size 7 appears twice and every other size appears once. The mode is 7.

Two comfort notes. If every value appears exactly once, the set has no mode, and that is a perfectly fine answer. If two values tie for most appearances, the set has two modes. You are describing the data, not forcing it to behave.

Key idea: Mode = the value that appears most often. MOde sounds like MOst.

The range: how spread out the data is

The range answers one question: how far is it from the smallest value to the largest one? Picture the tallest kid and the shortest kid on a playground. The range is the gap between them.

Let us find the range of 4, 8, 6, 4, 8, 8, slowly.

Step 1. Find the largest value: 8.

Step 2. Find the smallest value: 4.

Step 3. Subtract: 8 − 4 = 4. (Read that aloud as "eight minus four is four.")

The range is 4. A small range means the numbers huddle close together. A big range means they are spread far apart.

Try it: Find the range of 2, 9, 5. Answer: the largest is 9 and the smallest is 2, so the range is 9 − 2 = 7.

Key idea: Range = largest value − smallest value.

One data set, all four measures

Let us describe the data set 4, 8, 6, 4, 8, 8 every way we know, one measure at a time. This is the same kind of question you will see on the quiz, so we will go slowly.

Mean. Add one number at a time: 4 + 8 = 12, then 12 + 6 = 18, then 18 + 4 = 22, then 22 + 8 = 30, then 30 + 8 = 38. There are 6 values, so the mean is 38 ÷ 6, which is about 6.33. A decimal, and that is okay.

Median. Order the list: 4, 4, 6, 8, 8, 8. Six values means two middles: 6 and 8. Add them: 6 + 8 = 14. Divide by 2: 14 ÷ 2 = 7. The median is 7.

Mode. The value 8 appears three times, more than any other value. The mode is 8.

Range. Largest minus smallest: 8 − 4 = 4. The range is 4.

What we just did: we summarized one messy list four different ways, and each answer tells us something slightly different about it. That is real statistics, and you did it yourself.

A friendly way to keep the four names straight:

• MOde sounds like MOst: the value that appears most often.
• MEDian is the MIDdle, like the median strip in the middle of a road.
• The mean is the "meanest" one because it makes you work the hardest: add everything, then divide.
• The range tells you how far the data ranges, from smallest to largest.

Reading bar graphs and line graphs

A bar graph compares categories using bars, and a taller bar means a bigger number, exactly like taller stacks of blocks. Suppose a class counts its pets:

Pet	Count
Cats	6
Dogs	8
Fish	3

On the bar graph, the dog bar stands tallest because 8 is the biggest count, and the fish bar is shortest because 3 is the smallest. That is all a bar graph is: counts you can see at a glance.

A line graph shows how something changes over time. It is made of dots connected by a line. Suppose the temperature is 50 degrees on Monday, 55 on Tuesday, and 60 on Wednesday. Each day gets a dot, and the line climbs uphill from left to right, which tells you at a glance that it is getting warmer. Line going up means growing. Line going down means shrinking. A flat line means staying the same.

Here is a gentle four-step routine for reading any graph.

Step 1. Read the title, so you know what the graph is about.

Step 2. Read the labels along the bottom and the side, so you know what each direction measures.

Step 3. Check the scale, meaning how much each gridline is worth. Does it count by 1s? By 5s? By 100s?

Step 4. Only now answer the question.

Where people get stuck: skipping Step 3 and assuming every gridline is worth 1. If the gridlines count by 5s (0, 5, 10, 15) and a bar reaches the third line above 0, the bar shows 15, not 3. The scale is the single most common trap in graph questions, and now you know to check it first.

Try it: The gridlines count by 5s, and a bar reaches the fourth line above 0. What value does the bar show? Answer: 4 × 5 = 20. (Read that aloud as "four times five is twenty.")

Key idea: Bars compare categories, lines show change over time, and the scale tells you what each gridline is worth.

Look how far you have come

Sixteen weeks ago, some of these topics may have made your stomach drop. Look at what you can do now.

• Weeks 1 to 3: read big numbers by place value, follow the order of operations, and find factors, multiples, GCF, and LCM.
• Weeks 4 to 6: understand fractions and add, subtract, multiply, and divide them.
• Weeks 7 to 8: work with decimals and translate between fractions, decimals, and percents.
• Weeks 9 to 10: use ratios, solve proportions, and handle real percent problems like tips and discounts.
• Weeks 11 to 12: make friends with negative numbers and compute with them.
• Weeks 13 to 14: read variables and expressions, and solve one-step and two-step equations.
• Week 15: find perimeter, area, and volume.
• Week 16, today: describe data and read graphs.

That list is the entire foundation that Algebra 1 expects, and you built it one small step at a time. If any topic still feels wobbly, that is completely normal. Wobbly does not mean broken. Revisit that week, work the examples slowly, and be kind to yourself while you do. You do not need to be a "math person" to keep going, because there is no such thing. There are only people who practice patiently, and you have been doing exactly that all term.

Common misconceptions

• "The mean has to be one of the numbers in the list." It does not. The mean of 4, 8, 6, 4, 8, 8 is about 6.33, which appears nowhere in the list, and that is fine.
• "You can find the median without sorting." Sorting is the heart of the median. Put the values in order first, every single time.
• "The mode is the biggest number." The mode is the most frequent number, not the largest. In 2, 3, 3, 9 the mode is 3, not 9.
• "The range is the largest number." The range is a subtraction: largest − smallest. For 2, 9, 5 the range is 9 − 2 = 7, not 9.
• "Every gridline on a graph is worth 1." Check the scale before anything else. Gridlines often count by 5s, 10s, or 100s.

Recap

• Mean: add all the values, then divide by how many there are.
• Median: put the values in order, then take the middle one. If there are two middles, average them.
• Mode: the value that appears most often. MOde sounds like MOst.
• Range: largest value − smallest value.
• To read a graph: title first, then the axis labels, then the scale, and only then the answer.
• You finished the entire course. Take a moment and be proud. You earned this.

Sources

• OpenStax. (2020). Averages and probability. In Prealgebra 2e. openstax.org
• Math is Fun. (n.d.). Finding a central value. mathsisfun.com
• Math is Fun. (n.d.). The range (statistics). mathsisfun.com
• Math is Fun. (n.d.). Bar graphs. mathsisfun.com
• Math is Fun. (n.d.). Line graphs. mathsisfun.com
• Khan Academy. (n.d.). Data and statistics [Grade 6 unit]. khanacademy.org