Elementary Mathematics: Arithmetic & Fractions

The big picture

Today we learn to count and to tell which number is bigger. This is fun because you use it every day. You count your toys, your snacks, and your friends!

What counting means

To count means to say numbers in order, one at a time. You say 1, 2, 3, 4, 5. Each number is one more than the number before it.

Picture 3 apples on a table. You add 1 more apple. Now you say 4. You counted up by one.

Key idea: Each number is just one more than the one before it.

Counting up and counting back

You can count two ways. Counting up makes numbers bigger: 6, 7, 8, 9.

Counting back makes numbers smaller: 9, 8, 7, 6. A rocket counts back: 3, 2, 1, blast off!

Try it. Count back from 5. You say 5, 4, 3, 2, 1, 0. Good job!

Key idea: Up means bigger. Back means smaller.

Skip counting is faster

Skip counting means counting in jumps. It is faster than counting by ones.

• Count by 2s: 2, 4, 6, 8, 10.
• Count by 5s: 5, 10, 15, 20, 25.
• Count by 10s: 10, 20, 30, 40, 50.

Say you have coins worth 5 cents each. Skip count by 5s to add them fast!

Key idea: Skip counting jumps by the same number each time.

Which is bigger? Comparing numbers

To compare means to tell which number is bigger, or if they are the same. We use three signs.

• The greater than sign is >. It means the first number is bigger. Example: 8 > 3.
• The less than sign is <. It means the first number is smaller. Example: 4 < 9.
• The equal sign is =. It means both numbers are the same. Example: 6 = 6.

Here is a fun trick. The open, wide side is a hungry mouth. It wants to eat the bigger number! In 8 > 3, the open side faces the 8.

Key idea: The open mouth of the sign always faces the bigger number.

Using a number line

A number line is a line with numbers in order. On a number line, bigger numbers are to the right. Smaller numbers are to the left.

Which is bigger, 7 or 5? Start at 5 and count up: 5, 6, 7. You had to go up to reach 7. So 7 is bigger. We write 7 > 5.

See how 7 sits to the right of 5? That is another way to know 7 is bigger.

Key idea: On a number line, farther right means bigger.

Putting numbers in order

To order numbers means to line them up from small to big, or big to small. From least to greatest means smallest first.

Let us order 8, 3, and 6. Find the smallest first. That is 3. Next is 6. Biggest is 8. In order: 3, 6, 8.

Here is one more. Order 10, 2, 7, 5 from least to greatest. Smallest to biggest: 2, 5, 7, 10.

Key idea: Least to greatest means smallest number first.

Count real things, one at a time

Counting works best when you touch each thing as you say its number. Try it with real objects right now.

Put 6 crayons in a row. Touch the first one and say one. Touch the next and say two. Keep going until you touch the last one and say six.

Touching as you count stops the two most common mistakes: counting one thing twice, or skipping one.

Here is one more helpful rule. The last number you say is the answer. If you touch the last crayon and say six, then there are 6 crayons. You do not have to count again.

Key idea: Touch each thing once, and the last number you say is the total.

Skip counting practice

Fill in the missing numbers. Say the whole list out loud, then check.

• By 2s: 2, 4, 6, ___, 10, ___, 14. The answers are 8 and 12.
• By 5s: 5, 10, ___, 20, ___, 30. The answers are 15 and 25.
• By 10s: 10, ___, 30, ___, 50, 60. The answers are 20 and 40.

Why does skip counting matter? It saves time. Counting 6 nickels one cent at a time takes 30 counts. Skip counting by 5s takes only 6: 5, 10, 15, 20, 25, 30.

You already skip count without noticing. Counting pairs of shoes goes by 2s. Counting fingers on hands goes by 5s.

Key idea: Skip counting saves time when things come in equal groups.

Comparing bigger numbers

What if the numbers are too big to draw on a number line? Then compare one place at a time, starting from the left.

Compare 47 and 52.

1. Look at the tens first. 47 has 4 tens. 52 has 5 tens.
2. 5 tens is more than 4 tens, so 52 is bigger.
3. Write 52 > 47. You could also write 47 < 52. Both say the same thing.

Now compare 63 and 68. The tens are the same, because both have 6 tens. So move right to the ones. 3 is less than 8, so 63 is smaller. Write 63 < 68.

Here is a tricky one. Compare 9 and 21. Do not let the big-looking 9 fool you. 21 has 2 tens, and 9 has no tens at all, so 21 is bigger. Write 21 > 9.

Key idea: Compare the biggest place first. Only look at the next place if that one ties.

Try ordering these

Put each set in order from least to greatest. Work it out first, then check.

• 14, 9, 21. Find the smallest: 9. Then 14. Then 21. Answer: 9, 14, 21.
• 30, 13, 3, 31. Answer: 3, 13, 30, 31.
• 55, 45, 54, 44. Answer: 44, 45, 54, 55.

That last set is tricky on purpose. Check the tens first. Both 44 and 45 have 4 tens, so compare the ones: 4 is less than 5. Both 54 and 55 have 5 tens, so 54 comes first.

If you got that one wrong, do not worry. Mixed-up digits trip up almost everybody.

Key idea: When numbers look alike, slow down and compare one place at a time.

Watch out for

• Do not let the sharp point trick you. The point touches the smaller number. The open mouth faces the bigger number.
• For whole numbers, more digits means bigger. 100 is bigger than 99.
• When you count back, remember the numbers get smaller, not bigger.
• Counting the same object twice. Touch each one as you go.
• Comparing the ones place first. Always start with the biggest place.

Recap

• Counting means saying numbers in order. Each one is one more.
• Touch each object once, and the last number you say is the total.
• Skip counting jumps by 2s, 5s, or 10s to count fast.
• The signs > and < and = tell us which number is bigger, smaller, or the same.
• On a number line, bigger numbers are to the right.
• To compare bigger numbers, check the biggest place first.
• Least to greatest puts the smallest number first.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Kindergarten: Counting and cardinality. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 1: Number and operations in base ten. Common Core State Standards for Mathematics. thecorestandards.org
3. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 2: Number and operations in base ten. Common Core State Standards for Mathematics. thecorestandards.org
4. Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children (NCEE 2014-4005). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
5. The Math Learning Center. (n.d.). Number line. Math Apps. mathlearningcenter.org
6. National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. National Academies Press. nap.edu
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we learn about place value. Place value is a clever trick that lets us write any number using just ten little symbols. It is fun because it makes big numbers easy!

Digits are our building blocks

A digit is one of the ten symbols we use to write numbers. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Just like a few letters can make many words, these ten digits can make every number in the world.

Key idea: There are only ten digits, from 0 to 9.

Where a digit sits matters

Place value means how much a digit is worth because of where it sits. The same digit can mean different amounts.

Look at the digit 5. In 5, it means five. In 50, it means fifty. In 500, it means five hundred. Same digit, but its place changed its worth!

Key idea: A digit is worth more when it sits farther to the left.

The places

Starting from the right, the places are ones, then tens, then hundreds, then thousands.

• The ones place is on the far right. It counts single things.
• The tens place is next. Each one here is worth 10.
• The hundreds place comes next. Each one here is worth 100.

Each place is 10 times bigger than the place to its right. This is called our base-ten system. It is called base ten because we have ten fingers to count on!

Key idea: Each place to the left is 10 times bigger.

Reading a number by its places

Let us look at 256. We can put each digit in its place.

Hundreds	Tens	Ones
2	5	6

1. The 2 is in the hundreds place. It is worth 2 hundreds, which is 200.
2. The 5 is in the tens place. It is worth 5 tens, which is 50.
3. The 6 is in the ones place. It is worth 6 ones, which is 6.

Key idea: Read each digit by the place it sits in.

The mighty zero

Zero is a special digit. It is a placeholder. It holds a place open so the other digits stay where they belong.

Look at 408. The 0 in the tens place tells us there are no tens. But it keeps the 4 in the hundreds place and the 8 in the ones place. Without the zero, 408 would turn into 48. That is a very different number!

Key idea: Zero holds a place open so the other digits stay put.

Expanded form

Expanded form means writing a number as the sum of its place values. It shows the number stretched out.

For 256, the expanded form is 200 + 50 + 6.

Let us check. 200 + 50 is 250. Then 250 + 6 is 256. It matches! Checking your work is a smart habit.

Key idea: Expanded form breaks a number into its place value parts.

Bigger numbers and the comma

Here is a bigger number: 1,437.

• The 1 is in the thousands place. It is worth 1,000.
• The 4 is in the hundreds place. It is worth 400.
• The 3 is in the tens place. It is worth 30.
• The 7 is in the ones place. It is worth 7.

So 1,437 = 1,000 + 400 + 30 + 7. We use a small comma to help us read big numbers. We put it three digits from the right. It groups the thousands so the number is easier to read.

Key idea: A comma every three digits makes big numbers easy to read.

Build numbers with bundles

Place value is easiest to see when you can touch it. Grab a big handful of straws, toothpicks, or craft sticks.

Count out 10 sticks and wrap a rubber band around them. That bundle is now one ten. Make three bundles. That is 3 tens, or 30.

Now put 4 loose sticks beside your 3 bundles. You have 3 tens and 4 ones, which is 34.

Try a few more.

• 5 bundles and 2 loose sticks is 52.
• 1 bundle and 8 loose sticks is 18.
• 6 bundles and 0 loose sticks is 60.

Notice the last one. Zero loose sticks does not mean you skip the ones place. It means you write a 0 there.

Key idea: A ten is just ten ones bundled together.

Ten of these makes one of those

Here is the rule that makes the whole system work. Whenever you collect 10 of something, you trade it in for 1 of the next bigger thing.

• 10 ones trade for 1 ten.
• 10 tens trade for 1 hundred.
• 10 hundreds trade for 1 thousand.

Try it with your sticks. Make 10 bundles of ten, then put a big band around all 10. That giant bundle is 1 hundred, and it holds 100 sticks.

This trading rule is exactly what you use later when you carry in addition and borrow in subtraction. Understand the trade, and those steps stop feeling like magic.

Key idea: Every 10 of one place trades for 1 of the next place left.

Name the place

For each number, say what the named digit is worth. Work it out, then check.

• In 372, what is the 7 worth? It sits in the tens place, so it is worth 70.
• In 372, what is the 3 worth? It sits in the hundreds place, so it is worth 300.
• In 806, what is the 8 worth? Hundreds place, so 800.
• In 806, what is the 0 worth? It is worth 0 tens, but it holds the tens place open.
• In 5,140, what is the 5 worth? Thousands place, so 5,000.

A helpful habit is to point at the digit and count places from the right: ones, tens, hundreds.

Key idea: Find the place first, then say what the digit is worth.

Expanded form practice

Write each number in expanded form. Then add your parts back up to check.

• 84 becomes 80 + 4. Check: 80 + 4 = 84.
• 619 becomes 600 + 10 + 9. Check: 600 + 10 = 610, and 610 + 9 = 619.
• 307 becomes 300 + 7. There are no tens, so we leave that part out.
• 2,450 becomes 2,000 + 400 + 50. There are no ones this time.

Look at 307 again. In expanded form we do not write 0, because adding 0 changes nothing. But in the number itself we must write the 0, or 307 would collapse into 37.

Key idea: Skip the zero parts in expanded form, but never skip the 0 in the number.

Watch out for

• Do not forget the zero. It keeps the other digits in the right place.
• The far right place is ones, not tens. Always start counting places from the right.
• In expanded form, use the place value amount, like 50, not just the digit 5.
• Reading 1,437 as one four three seven. Read it as one thousand, four hundred thirty-seven.

Recap

• The ten digits are 0 through 9.
• Place value tells how much a digit is worth by where it sits.
• The places from the right are ones, tens, hundreds, thousands.
• Ten of one place always trades for one of the next place left.
• Zero is a placeholder that holds a spot open.
• Expanded form shows a number as the sum of its parts, like 200 + 50 + 6.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 2: Number and operations in base ten. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 4: Number and operations in base ten. Common Core State Standards for Mathematics. thecorestandards.org
3. The Math Learning Center. (n.d.). Number pieces. Math Apps. mathlearningcenter.org
4. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
5. Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children (NCEE 2014-4005). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
6. The Math Learning Center. (n.d.). Number frames. Math Apps. mathlearningcenter.org
7. National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. National Academies Press. nap.edu

The big picture

Today we learn to add bigger numbers. Adding puts groups together to find how many in all. It is useful when you count all your toys or all your treats!

What adding means

To add means to put groups together. If you have 2 apples and get 3 more, you now have 5 apples.

The answer to an adding problem is called the sum. The numbers we add are called addends. In 2 + 3 = 5, the addends are 2 and 3, and the sum is 5.

Key idea: Adding puts groups together to find the total.

Line up the places

To add big numbers, we stack them so the ones are under the ones and the tens are under the tens. This lineup is called a column.

Let us add 23 + 14. We line them up like this:

  2 3
+ 1 4
-----

1. Add the ones first: 3 + 4 = 7. Write 7 in the ones place.
2. Add the tens next: 2 + 1 = 3. Write 3 in the tens place.
3. The sum is 37.

Key idea: Always start adding from the ones place on the right.

When a column is 10 or more, we carry

Sometimes a column adds up to 10 or more. Then we carry. To carry means to move a group of ten to the next place on the left. This is also called to regroup.

Let us add 27 + 15.

  2 7
+ 1 5
-----

1. Add the ones: 7 + 5 = 12. That is more than 9! 12 is 1 ten and 2 ones.
2. Write the 2 in the ones place. Carry the 1 ten to the top of the tens place.
3. Add the tens: 1 (carried) + 2 + 1 = 4. Write 4 in the tens place.
4. The sum is 42.

Key idea: When a column reaches 10, carry the ten to the next place.

Adding three-digit numbers

The same steps work for bigger numbers. Let us add 148 + 126.

  1 4 8
+ 1 2 6
-------

1. Ones: 8 + 6 = 14. Write 4, carry 1 ten.
2. Tens: 1 (carried) + 4 + 2 = 7. Write 7.
3. Hundreds: 1 + 1 = 2. Write 2.
4. The sum is 274.

Key idea: Work one column at a time, right to left, and carry when needed.

Estimating to check

To estimate means to make a smart guess that is close to the real answer. It helps you check if your sum makes sense.

For 27 + 15, think 27 is close to 30 and 15 is close to 15. 30 + 15 is about 45. Our real answer 42 is close to 45, so it looks right!

Key idea: Estimate first to check that your answer is reasonable.

Two carries in a row

Sometimes you carry more than once in the same problem. Let us work through 356 + 287 together, slowly.

  3 5 6
+ 2 8 7
-------

1. Ones: 6 + 7 = 13. That is 1 ten and 3 ones. Write 3 in the ones place. Carry the 1 above the tens.
2. Tens: 1 (carried) + 5 + 8 = 14. That is 14 tens, which is 1 hundred and 4 tens. Write 4 in the tens place. Carry the 1 above the hundreds.
3. Hundreds: 1 (carried) + 3 + 2 = 6. Write 6.
4. The sum is 643.

Let us check with estimating. 356 is close to 350 and 287 is close to 300. 350 + 300 is 650. Our answer 643 is close to 650, so it looks right.

Key idea: You can carry in more than one column in the same problem.

Two mistakes, and how to fix them

Here are the two mistakes that show up most often. Seeing them written out helps you avoid them.

Mistake 1: forgetting the carried ten.

  2 7        2 7
+ 1 5      + 1 5
-----      -----
  3 2        4 2
(wrong)    (right)

In the wrong one, the writer added 2 + 1 = 3 in the tens and forgot the carried ten. The right answer needs 1 + 2 + 1 = 4.

Mistake 2: columns not lined up.

  1 2 4      1 2 4
+   3 6    +   3 6
(sloppy)   (lined up)

If the 6 sits under the 2 instead of under the 4, every column is off, and the answer will be far too big. Write on lined paper turned sideways, so the lines make neat columns for you.

Key idea: Most addition mistakes are a missed carry or crooked columns.

Adding by jumping on a number line

Column adding is not the only way. You can also jump along a number line, and this way is great for mental math.

To find 47 + 25, start at 47.

1. Jump 20 first. 47 becomes 67.
2. Now jump 5 more. 67 becomes 72.
3. So 47 + 25 = 72.

You can split the jumps any way you like. Some people jump to a friendly ten first: 47 plus 3 lands on 50, then 22 more makes 72. Same answer.

Key idea: Jumping tens first and then ones is a fast way to add in your head.

Practice set

Work these out on paper, then check.

• 36 + 45 = 81. Ones: 6 + 5 = 11, write 1 and carry 1. Tens: 1 + 3 + 4 = 8.
• 58 + 27 = 85. Ones: 8 + 7 = 15, write 5 and carry 1. Tens: 1 + 5 + 2 = 8.
• 209 + 96 = 305. Ones: 9 + 6 = 15, write 5 and carry 1. Tens: 1 + 0 + 9 = 10, write 0 and carry 1. Hundreds: 1 + 2 = 3.
• 174 + 118 = 292. Ones: 4 + 8 = 12, write 2 and carry 1. Tens: 1 + 7 + 1 = 9. Hundreds: 1 + 1 = 2.

Got them all? Wonderful. Missed one? Find the column where things went wrong. Almost always it is a carry.

Key idea: When an answer is off, check each column one at a time to find the slip.

Watch out for

• Do not forget to add the carried ten. It is easy to leave it out.
• Keep your columns lined up neat, ones under ones.
• Always start on the right, in the ones place.
• Writing the whole two-digit total in one column, like putting 12 in the ones place.

Recap

• Adding puts groups together. The answer is the sum.
• Line up ones under ones and tens under tens.
• Start adding from the ones place on the right.
• When a column reaches 10, carry the ten to the next place.
• You can carry in more than one column.
• Estimate to check your answer.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 2: Number and operations in base ten. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Number and operations in base ten. Common Core State Standards for Mathematics. thecorestandards.org
3. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
4. Fuchs, L. S., Newman-Gonchar, R., Schumacher, R., Dougherty, B., Bucka, N., Karp, K. S., Woodward, J., Clarke, B., Jordan, N. C., Gersten, R., Jayanthi, M., Keating, B., & Morgan, S. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades (WWC 2021006). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
5. The Math Learning Center. (n.d.). Number line. Math Apps. mathlearningcenter.org
6. The Math Learning Center. (n.d.). Number pieces. Math Apps. mathlearningcenter.org
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we learn to subtract bigger numbers. Subtracting takes some away to find how many are left. It helps when you share treats or count what is gone!

What subtracting means

To subtract means to take some away. If you have 5 apples and eat 2, you have 3 apples left.

The answer to a subtraction is called the difference. In 5 - 2 = 3, the difference is 3.

Key idea: Subtracting takes away to find what is left.

Line up the places

Just like adding, we stack the numbers so ones are under ones and tens are under tens. The bigger number goes on top.

Let us subtract 48 - 23.

  4 8
- 2 3
-----

1. Subtract the ones first: 8 - 3 = 5. Write 5.
2. Subtract the tens: 4 - 2 = 2. Write 2.
3. The difference is 25.

Key idea: Start subtracting from the ones place on the right.

When the top digit is too small, we borrow

Sometimes the top digit is too small to take away from. Then we borrow. To borrow means to take one ten from the next place and give it to the ones. This is also called to regroup.

Let us subtract 42 - 15.

  4 2
- 1 5
-----

1. Look at the ones: 2 - 5. We cannot take 5 from 2. So we borrow.
2. Take 1 ten from the 4 tens. Now the tens place has 3. Give that ten to the ones, so 2 becomes 12.
3. Now subtract the ones: 12 - 5 = 7. Write 7.
4. Subtract the tens: 3 - 1 = 2. Write 2.
5. The difference is 27.

Key idea: If the top digit is too small, borrow a ten from the next place.

Borrowing with three-digit numbers

The same steps work for bigger numbers. Let us subtract 253 - 128.

  2 5 3
- 1 2 8
-------

1. Ones: 3 - 8. Too small, so borrow. Take 1 ten from the 5 tens (now 4 tens). The 3 becomes 13.
2. Now 13 - 8 = 5. Write 5.
3. Tens: 4 - 2 = 2. Write 2.
4. Hundreds: 2 - 1 = 1. Write 1.
5. The difference is 125.

Key idea: Borrow only from the place right next door, on the left.

Check by adding back

Adding and subtracting are opposites. We call them inverse operations. That means you can check by adding the answer back.

We found 42 - 15 = 27. To check, add 27 + 15. That is 42, our starting number. It matches, so we are right!

Key idea: To check subtraction, add the answer to the number you took away.

The mistake almost everyone makes

Here is the single most common subtraction mistake. When the top digit is smaller, some people just flip the digits around and subtract the other way.

  4 2        4 2
- 1 5      - 1 5
-----      -----
  3 3        2 7
(wrong)    (right)

In the wrong one, the writer looked at 2 - 5, decided it was impossible, and did 5 - 2 = 3 instead. That changes the problem completely.

You cannot flip. Instead you borrow. Take 1 ten from the 4 tens, so 2 becomes 12, and then 12 - 5 = 7. The tens are now 3 - 1 = 2. The answer is 27.

A quick check catches the mistake. Does 33 + 15 equal 42? No, it equals 48. Does 27 + 15 equal 42? Yes. So 27 is right.

Key idea: Never flip the digits. If the top is too small, borrow.

Borrowing across a zero

What happens when the place next door is a 0? You cannot borrow from nothing, so you go one more place over. Let us do 302 - 148.

  3 0 2
- 1 4 8
-------

1. Ones: 2 - 8. Too small, so we need to borrow. But the tens place is 0, so there is nothing there to take.
2. Go to the hundreds. Take 1 hundred from the 3, leaving 2 hundreds. Give that hundred to the tens, so the tens become 10.
3. Now borrow 1 ten from those 10 tens, leaving 9 tens. Give it to the ones, so the 2 becomes 12.
4. Ones: 12 - 8 = 4. Write 4.
5. Tens: 9 - 4 = 5. Write 5.
6. Hundreds: 2 - 1 = 1. Write 1.
7. The difference is 154.

Check it: 154 + 148 = 302. It matches.

This one takes practice. Go slowly and write down the new digits as you change them, so you do not lose track.

Key idea: If the next place is 0, borrow from the place beyond it first.

Subtract by counting up

Here is a different way that many people find easier in their head. Instead of taking away, count up from the smaller number to the bigger one.

To find 72 - 68, start at 68 and count up to 72: 69, 70, 71, 72. That is 4 jumps, so 72 - 68 = 4.

This works well when the two numbers are close together. Try 100 - 97. Counting up from 97 gives 98, 99, 100. That is 3 jumps, so the answer is 3. Much faster than borrowing twice!

Cashiers use this trick. If something costs 7 dollars and you pay 10, they count up: 8, 9, 10. Your change is 3 dollars.

Key idea: When numbers are close, count up instead of taking away.

Practice set

Work these on paper, then check both the answer and the reason.

• 63 - 27 = 36. Borrow: 3 becomes 13, and 13 - 7 = 6. Tens: 5 - 2 = 3.
• 85 - 41 = 44. No borrowing needed here.
• 200 - 76 = 124. Borrow across the zero, just like the worked example.
• 431 - 158 = 273. Ones: 11 - 8 = 3. Tens: 12 - 5 = 7. Hundreds: 3 - 1 = 2.

For each one, add your answer back to check. If it does not match the starting number, look for the column where the borrow went wrong.

Key idea: Adding your answer back catches almost every subtraction slip.

Watch out for

• Do not just take the small digit from the big one. If the top is smaller, you must borrow.
• When you borrow, remember the next place goes down by one.
• Keep your columns lined up neat.
• Trying to borrow from a 0. Go one more place over first.

Recap

• Subtracting takes away. The answer is the difference.
• Line up ones under ones and start on the right.
• If the top digit is too small, borrow a ten from the next place.
• Never flip the digits around instead of borrowing.
• If the next place is 0, borrow from the place beyond it.
• Check your answer by adding it back.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 2: Number and operations in base ten. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Number and operations in base ten. Common Core State Standards for Mathematics. thecorestandards.org
3. Fuchs, L. S., Newman-Gonchar, R., Schumacher, R., Dougherty, B., Bucka, N., Karp, K. S., Woodward, J., Clarke, B., Jordan, N. C., Gersten, R., Jayanthi, M., Keating, B., & Morgan, S. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades (WWC 2021006). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
4. Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to intervention for elementary and middle schools (NCEE 2009-4060). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
5. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
6. The Math Learning Center. (n.d.). Number line. Math Apps. mathlearningcenter.org
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we learn about multiplication. Multiplication is a fast way to add equal groups. It is fun because it saves you lots of counting!

What multiplication means

To multiply means to add equal groups quickly. We use the sign x, which means times.

Say you have 3 bags. Each bag has 2 apples. That is 3 groups of 2. You can add: 2 + 2 + 2 = 6. Or you can multiply: 3 x 2 = 6. Both give 6 apples!

Key idea: Multiplication is adding the same number over and over.

The parts of a multiplication

The numbers we multiply are called factors. The answer is called the product.

In 3 x 2 = 6, the factors are 3 and 2, and the product is 6.

Key idea: Factors are the numbers you multiply. The product is the answer.

Multiplication is repeated addition

Repeated addition means adding the same number again and again. This is exactly what multiplication does.

4 x 5 means 5 added 4 times: 5 + 5 + 5 + 5 = 20. So 4 x 5 = 20.

You can also skip count to find it. Count by fives four times: 5, 10, 15, 20. Same answer!

Key idea: You can find a product by skip counting the groups.

Equal groups

Equal groups means every group has the same amount. Multiplication only works with equal groups.

3 groups of 4 crayons is 3 x 4 = 12 crayons. Each group must have 4. If the groups are different sizes, we cannot multiply them together like this.

Key idea: Multiplication needs groups that are all the same size.

Arrays help us see it

An array is a picture of rows and columns, like dots lined up in a neat box. Arrays make multiplication easy to see.

Picture 2 rows of dots with 4 dots in each row:

* * * *
* * * *

Count them: there are 8 dots. That shows 2 x 4 = 8. The rows are one factor and the dots in a row are the other factor.

Key idea: An array shows rows times columns.

The order does not matter

Multiplication has a special rule called commutative. It means you can switch the order and get the same product.

3 x 4 = 12 and 4 x 3 = 12. Both are 12! So if you know one, you know the other.

Key idea: You can flip the factors and the product stays the same.

The rules for 0 and 1

• Any number times 0 is 0. 5 x 0 = 0. Zero groups means nothing at all.
• Any number times 1 is itself. 5 x 1 = 5. One group of five is just five.

Key idea: Times 0 is always 0, and times 1 keeps the number the same.

Build arrays with real objects

Get 12 small things: buttons, cereal pieces, coins, or blocks. Now try arranging all 12 into neat rows.

• 2 rows of 6. That shows 2 x 6 = 12.
• 3 rows of 4. That shows 3 x 4 = 12.
• 4 rows of 3. That shows 4 x 3 = 12.
• 6 rows of 2. That shows 6 x 2 = 12.
• 1 row of 12. That shows 1 x 12 = 12.

Five different arrays, and every one has 12 objects. Look at the 3 rows of 4 and the 4 rows of 3. They are the same array, just turned sideways. That is the commutative rule you can hold in your hands.

Now try the same thing with 7 objects. You can only make 1 row of 7 or 7 rows of 1. Some numbers just do not spread out into neat boxes.

Key idea: Turning an array sideways shows why 3 x 4 and 4 x 3 match.

Equal groups are all around you

Once you start looking, equal groups are everywhere.

• An egg carton has 2 rows of 6. That is 2 x 6 = 12 eggs.
• A car has 4 wheels. Three cars have 3 x 4 = 12 wheels.
• One hand has 5 fingers. Four hands have 4 x 5 = 20 fingers.
• A week has 7 days. Three weeks have 3 x 7 = 21 days.

Try making up one of your own. Look around, find something that comes in equal groups, and write the multiplication for it.

Key idea: Anything that comes in equal groups is a multiplication waiting to happen.

Work these out together

For each one, draw or picture the groups first. Then find the product.

• 5 plates with 3 grapes on each. Draw 5 circles with 3 dots in each. Skip count: 3, 6, 9, 12, 15. So 5 x 3 = 15.
• 4 rows of 6 chairs. Skip count by 6: 6, 12, 18, 24. So 4 x 6 = 24.
• 2 boxes with 8 crayons each. Double 8. So 2 x 8 = 16.
• 6 spiders with 8 legs each. Skip count by 8: 8, 16, 24, 32, 40, 48. So 6 x 8 = 48.

Notice that skip counting always works, even when you do not remember the fact. It is slower, but it never lets you down.

Key idea: If you forget a product, skip count the groups and you will find it.

A mistake worth naming

A very common slip is adding when you meant to multiply.

Wrong: 3 x 4 = 7. Right: 3 x 4 = 12.

The wrong answer came from adding 3 + 4. Multiplying means 3 groups of 4, so you add 4 three times: 4 + 4 + 4 = 12.

Here is a quick test. Except when 0 or 1 is involved, a product should be bigger than either factor. If your answer is small, check whether you added by accident.

Key idea: 3 x 4 means three groups of four, not three plus four.

Watch out for

• Groups must be equal to multiply. Do not multiply groups of different sizes.
• Times 0 is 0, not the number itself. 7 x 0 = 0.
• Remember 3 x 4 and 4 x 3 give the same answer.
• Adding the factors instead of multiplying them.

Recap

• Multiplication is a fast way to add equal groups.
• The factors are what you multiply. The product is the answer.
• An array shows multiplication as rows and columns.
• You can flip the factors and get the same product.
• If you forget a product, skip count the groups.
• Times 0 is 0. Times 1 keeps the number the same.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Operations and algebraic thinking. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 2: Operations and algebraic thinking. Common Core State Standards for Mathematics. thecorestandards.org
3. The Math Learning Center. (n.d.). Number frames. Math Apps. mathlearningcenter.org
4. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
5. Fuchs, L. S., Newman-Gonchar, R., Schumacher, R., Dougherty, B., Bucka, N., Karp, K. S., Woodward, J., Clarke, B., Jordan, N. C., Gersten, R., Jayanthi, M., Keating, B., & Morgan, S. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades (WWC 2021006). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
6. National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. National Academies Press. nap.edu
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we practice multiplication facts for 2s, 5s, and 10s. Knowing these by heart makes math quick and easy. It feels great to just know the answer!

What a multiplication fact is

A multiplication fact is a small times problem you can remember, like 2 x 3 = 6. When you know a fact by heart, you do not have to count it out each time.

You already know some! 2 x 1 = 2 and 5 x 1 = 5 are facts. We will learn many more.

Key idea: A fact is a times problem you know right away.

The 2s: just double it

To multiply by 2 means to double a number. To double means to make two of the same and add them.

• 2 x 1 = 2
• 2 x 2 = 4
• 2 x 3 = 6
• 2 x 4 = 8
• 2 x 5 = 10

See the pattern? The answers are 2, 4, 6, 8, 10. Those are the counting-by-twos numbers!

Key idea: Times 2 just means double the number.

Even numbers

An even number is a number you can split into two equal groups with none left over. 2, 4, 6, 8, and 10 are even.

All the answers in the 2s are even. That is a handy clue!

Key idea: Every answer in the 2s times table is an even number.

The 5s: skip count by fives

To multiply by 5, just skip count by fives. Skip count means to jump by the same number each time.

• 5 x 1 = 5
• 5 x 2 = 10
• 5 x 3 = 15
• 5 x 4 = 20
• 5 x 5 = 25

Here is a fun pattern. The answers always end in 5 or 0: 5, 10, 15, 20, 25. Just like counting nickels!

Key idea: The 5s always end in a 5 or a 0.

The 10s: the easiest of all

To multiply by 10, just put a 0 at the end of the other number. That is it!

• 10 x 1 = 10
• 10 x 2 = 20
• 10 x 3 = 30
• 10 x 4 = 40
• 10 x 5 = 50

Key idea: Times 10 puts a zero at the end of the number.

Forgot a fact? Skip count to find it

If you forget a fact, you can always find it. Say you forget 5 x 4. Skip count by fives four times: 5, 10, 15, 20. The answer is 20!

Or use a pattern you know. A pattern is a rule that repeats. Since the 5s end in 5 or 0, you know 5 x 4 will end in 0.

Key idea: You can always skip count to figure out a fact you forget.

Build the 4s from the 2s

Here is a shortcut that saves real work. To multiply by 4, double the number and then double it again.

• 4 x 3: double 3 to get 6, then double 6 to get 12. So 4 x 3 = 12.
• 4 x 5: double 5 to get 10, then double 10 to get 20. So 4 x 5 = 20.
• 4 x 7: double 7 to get 14, then double 14 to get 28. So 4 x 7 = 28.

This works because 4 is just 2 twos. It means every 4s fact is really two easy doubling steps.

The 3s have a helper too. To multiply by 3, double the number and add one more group. For 3 x 6, double 6 to get 12, then add 6 to get 18.

Key idea: Hard facts get easy when you build them from facts you already know.

The 9s finger trick

Hold both hands up in front of you with all 10 fingers spread out. Number them 1 to 10 from left to right.

To find 9 x 4, bend down your fourth finger. Now count the fingers still standing: 3 to the left of the bent one, and 6 to the right. That gives 36. And 9 x 4 = 36!

Try 9 x 7. Bend the seventh finger. There are 6 fingers to the left and 3 to the right. That is 63, and 9 x 7 = 63.

There is a pattern hiding here too. In every 9s answer, the two digits add up to 9. Look: 3 + 6 = 9, and 6 + 3 = 9, and 1 + 8 = 9.

Key idea: For the 9s, bend one finger and read the fingers on each side.

Practice set

Say each answer out loud, then check. If one is slow, that is a fact worth practicing tomorrow.

• 2 x 7 = 14. Double 7.
• 5 x 6 = 30. Skip count: 5, 10, 15, 20, 25, 30.
• 10 x 8 = 80. Add a zero to 8.
• 2 x 9 = 18. Double 9.
• 5 x 9 = 45. It ends in 5, which fits the rule.
• 4 x 8 = 32. Double 8 to get 16, then double again.

Key idea: Check your answer against the pattern, not just your memory.

How to practice facts

Facts stick best in short, cheerful bursts. Five minutes a day beats an hour once a week.

• Pick just three facts you find hard. Not twenty. Three.
• Practice those three for a few minutes.
• Tomorrow, practice the same three plus one new one.
• Play them as a game: say them while bouncing a ball or climbing stairs.

Getting a fact wrong is not a problem. It just tells you which one to practice next. Every mathematician you have ever heard of had to learn these too.

Key idea: A few minutes a day on a few facts works better than long, tiring sessions.

Watch out for

• Times 2 is double, not plus 2. 2 x 5 = 10, not 7.
• The 5s end in 5 or 0. If you get 5 x 3 = 16, that is wrong, because it must end in 5 or 0.
• Times 10 adds one zero, not two. 10 x 4 = 40, not 400.
• Trying to learn every fact at once. Pick a few at a time.

Recap

• A fact is a times problem you know by heart.
• Times 2 means double the number, and the answers are even.
• The 5s end in 5 or 0, like counting nickels.
• Times 10 puts a zero at the end.
• Build the 4s by doubling twice, and use the finger trick for the 9s.
• If you forget, skip count to find the answer.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Operations and algebraic thinking. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 4: Operations and algebraic thinking. Common Core State Standards for Mathematics. thecorestandards.org
3. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
4. Institute of Education Sciences. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades. What Works Clearinghouse. ies.ed.gov
5. The Math Learning Center. (n.d.). Number frames. Math Apps. mathlearningcenter.org
6. NRICH. (n.d.). Primary students. NRICH Mathematics. University of Cambridge. nrich.maths.org
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we learn about division. Division shares things into equal groups. It is useful when you split treats fairly with friends!

What division means

To divide means to share into equal groups. We use the sign for division to write it.

Say you have 6 cookies and 3 friends. You share them fairly. Each friend gets 2 cookies. We write 6 divided by 3 = 2.

Key idea: Division shares a group into equal parts.

The parts of a division

The answer to a division is called the quotient. It tells how many are in each equal share.

In 6 divided by 3 = 2, the quotient is 2. Each friend gets 2 cookies.

Key idea: The quotient is the answer to a division problem.

Making equal shares

Equal shares means every group gets the same amount, with none left over. That is what makes sharing fair.

Let us share 8 blocks between 2 kids. Give one to each, then one to each, until they are gone. Each kid ends up with 4. So 8 divided by 2 = 4.

Key idea: Fair sharing means each group gets an equal amount.

Division and multiplication are opposites

Division and multiplication are inverse operations. That means they undo each other, like opposites.

You know 3 x 2 = 6. So 6 divided by 3 = 2, and 6 divided by 2 = 3. They all use the same three numbers: 2, 3, and 6.

Key idea: Multiplying and dividing undo each other.

Fact families

A fact family is a group of facts that use the same numbers. The numbers 2, 3, and 6 make one fact family.

• 2 x 3 = 6
• 3 x 2 = 6
• 6 divided by 3 = 2
• 6 divided by 2 = 3

If you know one fact in the family, you can find the others!

Key idea: A fact family uses the same three numbers in four facts.

Turn division into a missing-factor question

Here is a great trick. To solve a division, ask a multiplication question with a missing factor. A missing factor is the number you are trying to find.

To find 12 divided by 4, ask: 4 times what equals 12? Count: 4, 8, 12. That is three fours. So the missing factor is 3, and 12 divided by 4 = 3.

Key idea: To divide, ask what number times the divisor makes the total.

Two ways to share

Division answers two different questions, and both use the same math.

Question 1: how many in each group? You have 12 stickers and 3 friends. How many does each friend get? Deal them out one at a time, like cards. Each friend gets 4. So 12 divided by 3 = 4.

Question 2: how many groups? You have 12 stickers and want to give each friend 3. How many friends can you give to? Make piles of 3 until the stickers run out. You get 4 piles. So 12 divided by 3 = 4.

Notice the answer is the same both times, but the story is different. In one, 4 is the size of each share. In the other, 4 is the number of shares.

Key idea: Division can mean how many in each group, or how many groups.

What if it does not share evenly?

Real life is not always tidy. Suppose you have 13 cookies and 4 friends.

Deal them out. Each friend gets 3 cookies, and that uses 12 cookies. There is 1 cookie left over.

That leftover has a name. It is called the remainder. We say 13 divided by 4 = 3 with a remainder of 1.

Try one more. Share 17 marbles among 5 kids. Each gets 3, using 15. The remainder is 2.

How do you check? Multiply the answer by the number of groups, then add the remainder. Here that is 3 x 5 = 15, plus 2 more, which gives 17. It matches.

Key idea: A remainder is what is left over after the equal shares are made.

Work these out

Use the missing-factor question for each one. Say what times the divisor makes the total.

• 15 divided by 5. Ask: 5 times what is 15? Count 5, 10, 15. That is 3 fives, so the answer is 3.
• 24 divided by 4. Ask: 4 times what is 24? Count 4, 8, 12, 16, 20, 24. That is 6 fours, so the answer is 6.
• 18 divided by 2. Ask: 2 times what is 18? Half of 18 is 9, so the answer is 9.
• 20 divided by 3. Ask: 3 times what is close to 20? 3 x 6 = 18, and 2 are left. So 6 with a remainder of 2.

Key idea: Turning a division into a times question makes it much easier.

Dividing by 1 and by itself

Two division rules are worth knowing by heart.

• Any number divided by 1 is itself. 7 divided by 1 = 7. If there is only one group, that group gets everything.
• Any number divided by itself is 1. 7 divided by 7 = 1. If there are 7 groups and 7 things, each group gets exactly one.

There is one more rule that surprises people. You cannot divide by 0. Sharing 6 cookies among 0 friends does not make sense, because there is nobody to share with.

Key idea: Dividing by 1 keeps the number, and dividing by itself gives 1.

Watch out for

• The shares must be equal. If groups have different amounts, the sharing is not fair.
• Order matters in division. 6 divided by 2 is not the same as 2 divided by 6.
• Use the fact family. If you know the times fact, the division is easy.
• Forgetting the remainder when things do not share evenly.

Recap

• Division shares a group into equal parts.
• The quotient is the answer.
• Division and multiplication are opposites.
• A fact family uses the same three numbers.
• A remainder is what is left over when the sharing is not even.
• To divide, ask a missing-factor multiplication question.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Operations and algebraic thinking. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 4: Operations and algebraic thinking. Common Core State Standards for Mathematics. thecorestandards.org
3. Institute of Education Sciences. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades. What Works Clearinghouse. ies.ed.gov
4. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
5. The Math Learning Center. (n.d.). Number frames. Math Apps. mathlearningcenter.org
6. NRICH. (n.d.). Primary students. NRICH Mathematics. University of Cambridge. nrich.maths.org
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we learn what a fraction is. A fraction is a part of a whole thing. It is fun because it helps us share pizza, pie, and candy fairly!

What a fraction means

A fraction is a way to name part of a whole. Think of one pizza cut into equal slices. One slice is a fraction of the whole pizza.

We write a fraction with two numbers, one on top and one on the bottom, like 1 over 2.

Key idea: A fraction names part of a whole.

The top and bottom numbers

The bottom number is the denominator. It tells how many equal parts the whole is cut into.

The top number is the numerator. It tells how many of those parts we have.

In the fraction 1 over 2, the denominator is 2 (the whole has 2 parts) and the numerator is 1 (we have 1 part).

Key idea: The bottom shows the total parts. The top shows how many we have.

One half

One half, written 1 over 2, means one of two equal parts. If you cut a sandwich into 2 equal pieces and take one, you have one half.

Picture a circle split down the middle into 2 equal parts. Color one part. That colored part is 1 over 2.

Key idea: One half is one of two equal parts.

The parts must be equal

Equal parts means all the pieces are the same size. Fractions only work when the parts are equal.

If you cut a cookie into one big piece and one tiny piece, those are not halves. Halves must be the same size.

Key idea: Fraction parts must all be the same size.

Reading some fractions

Let us read a few together.

• 1 over 3 means 1 part out of 3 equal parts. We say one third.
• 1 over 4 means 1 part out of 4 equal parts. We say one fourth, or one quarter.
• 3 over 4 means 3 parts out of 4 equal parts. We say three fourths.

The bigger the bottom number, the more pieces the whole is cut into, so each piece is smaller. One fourth is smaller than one half!

Key idea: A bigger bottom number means smaller pieces.

When a fraction makes one whole

When the top and bottom numbers are the same, the fraction equals one whole. A whole means the complete thing, all the parts together.

If a pie has 4 parts and you have all 4, that is 4 over 4, which is 1 whole pie. Yum!

Key idea: When the top equals the bottom, you have one whole.

Fold paper to make fractions

Get a plain sheet of paper. This is the best way to feel what fractions really are.

1. Fold the paper in half and open it back up. You made 2 equal parts. Each one is 1 over 2.
2. Fold it in half again, the same way. Open it. Now there are 4 equal parts. Each one is 1 over 4.
3. Fold once more. Open it. Now there are 8 equal parts, and each is 1 over 8.

Look at your paper. Every fold doubled the number of parts, and every fold made each part smaller.

Now color 3 of the 8 parts. You have colored 3 over 8 of the paper. Color 3 more, and you have 6 over 8 colored.

Key idea: More folds means more pieces, and each piece gets smaller.

Which piece is bigger?

Look back at your folded paper and compare the sizes.

• 1 over 2 is the biggest of the three. Half a page is a lot.
• 1 over 4 is smaller than 1 over 2.
• 1 over 8 is the smallest of all.

So 1 over 2 is bigger than 1 over 4, which is bigger than 1 over 8. The bottom numbers got bigger, but the pieces got smaller.

This surprises a lot of people, because with whole numbers 8 is bigger than 2. Here is the way to remember it. A bigger bottom number means you are sharing the same thing with more people, so everyone gets less.

Key idea: Sharing with more people means smaller pieces for everyone.

Fractions of a group

Fractions do not only work on pizzas and paper. They also work on groups of things.

Put out 6 buttons. Split them into 2 equal piles of 3. One pile is 1 over 2 of the buttons, which is 3 buttons.

Now split the 6 buttons into 3 equal piles of 2. One pile is 1 over 3 of the buttons, which is 2 buttons.

Try one on your own. If you have 8 crayons and take 1 over 4 of them, how many is that? Split 8 into 4 equal piles. Each pile has 2, so 1 over 4 of 8 is 2 crayons.

Key idea: A fraction of a group means splitting the group into equal piles.

Name each one

Picture each shape and name the fraction. Then check.

• A circle cut into 4 equal parts, with 3 colored. That is 3 over 4.
• A rectangle cut into 6 equal parts, with 1 colored. That is 1 over 6.
• A square cut into 2 equal parts, with 2 colored. That is 2 over 2, which is 1 whole.
• A circle cut into 8 equal parts, with 5 colored. That is 5 over 8.

The steps are always the same. Count all the equal parts to get the bottom. Count the colored parts to get the top.

Key idea: Count all the parts for the bottom, then count the shaded ones for the top.

Watch out for

• The parts must be equal. Different-sized pieces are not fractions.
• A bigger bottom number does not mean a bigger piece. It means smaller pieces.
• The top counts the parts you have. The bottom counts all the parts.
• Counting only the uncolored parts for the bottom. The bottom counts every part.

Recap

• A fraction names part of a whole.
• The denominator (bottom) shows the total equal parts.
• The numerator (top) shows how many parts you have.
• Parts must be equal in size.
• A fraction can also name part of a group of objects.
• When the top equals the bottom, the fraction is one whole.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Number and operations, fractions. Common Core State Standards for Mathematics. thecorestandards.org
2. Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade (NCEE 2010-4039). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
3. The Math Learning Center. (n.d.). Fractions. Math Apps. mathlearningcenter.org
4. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 2: Geometry. Common Core State Standards for Mathematics. thecorestandards.org
5. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
6. NRICH. (n.d.). Primary students. NRICH Mathematics. University of Cambridge. nrich.maths.org
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we add and subtract fractions that have the same bottom number. It is easy and fun. You just work with the top numbers!

Fractions with the same bottom

Like fractions are fractions that have the same denominator. That means the bottom numbers match.

1 over 4 and 2 over 4 are like fractions, because both have a 4 on the bottom. They both come from a whole cut into 4 equal parts.

Key idea: Like fractions have the same bottom number.

Adding: add the tops, keep the bottom

To add like fractions, we add the tops and keep the bottom the same. The bottom does not change because the pieces are still the same size.

Let us add 1 over 4 plus 2 over 4.

1. Add the top numbers: 1 + 2 = 3.
2. Keep the bottom the same: 4.
3. The answer is 3 over 4.

Picture a pizza cut into 4 slices. You have 1 slice, then get 2 more. Now you have 3 slices out of 4, which is 3 over 4.

Key idea: To add, add the tops and keep the bottom the same.

Why the bottom stays the same

The bottom tells the size of the pieces. When you add fourths to fourths, the pieces are still fourths. So the bottom stays 4.

Think of it like adding apples. 1 apple plus 2 apples is 3 apples. The word apple does not change. Here, fourths do not change either.

Key idea: The bottom stays the same because the pieces stay the same size.

Subtracting: take away the tops

To subtract like fractions, we take away the top numbers and keep the bottom the same.

Let us do 3 over 5 minus 1 over 5.

1. Subtract the tops: 3 - 1 = 2.
2. Keep the bottom the same: 5.
3. The answer is 2 over 5.

You had 3 pieces out of 5 and ate 1. Now you have 2 out of 5, which is 2 over 5.

Key idea: To subtract, take away the tops and keep the bottom.

When the answer is one whole

Remember, when the top and bottom match, you have one whole. So sometimes adding fractions makes a whole.

Let us add 1 over 3 plus 2 over 3. Add the tops: 1 + 2 = 3. Keep the bottom: 3. The answer is 3 over 3, which is 1 whole!

That makes sense. If a pie has 3 parts and you gather all 3, you have the whole pie.

Key idea: If the tops add up to the bottom, the answer is one whole.

One more example

Let us try 2 over 6 plus 3 over 6.

1. Add the tops: 2 + 3 = 5.
2. Keep the bottom: 6.
3. The answer is 5 over 6.

Key idea: Same steps work for any like fractions.

The mistake to avoid

Here is the mistake that catches nearly everyone at first. When adding fractions, some people add the bottoms too.

Wrong: 1 over 4 plus 2 over 4 = 3 over 8.

Right: 1 over 4 plus 2 over 4 = 3 over 4.

Why is the first one wrong? Think about the pizza. You had 1 slice out of 4, then got 2 more slices. Nobody cut the pizza again, so it still has 4 slices in it. The bottom cannot change.

Here is a way to catch this yourself. The wrong answer, 3 over 8, is smaller than what you started with. Adding should never make things smaller. If your answer shrinks, check the bottom.

Key idea: Adding fractions should make the amount bigger, never smaller.

Draw the bars

Drawing helps a lot. Draw a long rectangle and cut it into 6 equal parts with lines.

Now shade 2 parts. That is 2 over 6. Shade 3 more with a different color. That is 3 over 6.

Count all your shaded parts: 5. So 2 over 6 plus 3 over 6 = 5 over 6. The picture shows exactly why the bottom stayed at 6. You never changed how many parts the bar was cut into.

Try subtracting on a drawing too. Shade 5 of the 6 parts, then erase 2. You have 3 shaded left, so 5 over 6 minus 2 over 6 = 3 over 6.

Key idea: Drawing a bar shows why only the top number changes.

Taking a fraction away from a whole

What if you start with a whole thing and take part away? The trick is to rewrite the whole as a fraction first.

Say you have a whole pie and eat 2 over 5 of it. How much is left?

1. Rewrite 1 whole as 5 over 5, because the pie is cut into fifths.
2. Now subtract: 5 over 5 minus 2 over 5.
3. Take away the tops: 5 - 2 = 3. Keep the bottom: 5.
4. The answer is 3 over 5 of the pie is left.

The rule is to write the whole with the same bottom number as the fraction you are taking away. If the fraction is in eighths, write the whole as 8 over 8.

Key idea: Rewrite 1 whole using the same bottom number, then subtract as usual.

Practice set

Work each one out, then check.

• 2 over 7 plus 4 over 7 = 6 over 7. Tops: 2 + 4 = 6.
• 5 over 8 minus 3 over 8 = 2 over 8. Tops: 5 - 3 = 2.
• 3 over 10 plus 7 over 10 = 10 over 10, which is 1 whole.
• 1 whole minus 1 over 4 = 3 over 4. Rewrite 1 as 4 over 4 first.
• 6 over 9 minus 6 over 9 = 0 over 9, which is just 0. You took away everything you had.

If you got the last one, well done. Zero is a perfectly good fraction answer.

Key idea: Every one of these problems uses the same two steps: change the top, keep the bottom.

Watch out for

• Do not add the bottom numbers. Only add the tops. 1 over 4 plus 2 over 4 is 3 over 4, not 3 over 8.
• These easy steps only work when the bottoms are the same.
• When the top equals the bottom, remember that is one whole.
• Forgetting to rewrite 1 whole as a fraction before subtracting from it.

Recap

• Like fractions have the same bottom number.
• To add, add the tops and keep the bottom the same.
• To subtract, take away the tops and keep the bottom the same.
• The bottom stays the same because the pieces stay the same size.
• To subtract from 1 whole, rewrite the whole with that same bottom number.
• When the top equals the bottom, the answer is one whole.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 4: Number and operations, fractions. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Number and operations, fractions. Common Core State Standards for Mathematics. thecorestandards.org
3. Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade (NCEE 2010-4039). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
4. The Math Learning Center. (n.d.). Fractions. Math Apps. mathlearningcenter.org
5. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 5: Number and operations, fractions. Common Core State Standards for Mathematics. thecorestandards.org
6. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
7. Institute of Education Sciences. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades. What Works Clearinghouse. ies.ed.gov

The big picture

Today we learn about decimals. A decimal is another way to show a part of a whole. It is handy for money and for measuring!

What a decimal is

A decimal is a number that shows parts smaller than one, using a little dot. That dot is called the decimal point.

The number 0.5 is a decimal. The dot separates the whole numbers on the left from the parts on the right.

Key idea: A decimal uses a dot to show parts of a whole.

The tenths place

The first place right after the dot is the tenths place. It means the whole is cut into 10 equal parts.

So 0.1 is one tenth, the same as the fraction 1 over 10. And 0.3 means 3 tenths, or 3 over 10.

Picture a chocolate bar with 10 equal squares. One square is 0.1 of the bar.

Key idea: The first place after the dot counts tenths.

Connecting decimals and fractions

Decimals and fractions are two ways to name the same part. Look how they match.

• 0.1 is the same as 1 over 10.
• 0.5 is the same as 5 over 10, which is one half.
• 0.7 is the same as 7 over 10.

Key idea: A decimal in tenths matches a fraction over 10.

The hundredths place

The second place after the dot is the hundredths place. It means the whole is cut into 100 equal parts.

So 0.01 is one hundredth, the same as 1 over 100. And 0.25 means 25 hundredths, or 25 over 100. That is like 25 cents out of a whole dollar!

Key idea: The second place after the dot counts hundredths.

Comparing decimals

To compare decimals, line up the decimal points and look at the tenths place first.

Which is bigger, 0.7 or 0.3? Both have a 0 for the whole. Look at the tenths: 7 is bigger than 3. So 0.7 is bigger.

Here is another. Compare 0.6 and 0.6. The tenths are the same, so they are equal.

Key idea: To compare, line up the dots and check the tenths first.

A worked example with money

Money uses decimals every day. Half of a dollar is 0.50 dollars, or 50 cents. A quarter is 0.25 dollars, or 25 cents.

Which is more, 0.50 or 0.25? Look at the tenths: 5 is bigger than 2. So 0.50 is more. Fifty cents beats twenty-five cents!

Key idea: Decimals help us read and compare money.

Ten dimes make a dollar

Money is the easiest way to feel what tenths and hundredths mean.

A dollar is the whole. A dime is one tenth of it, because 10 dimes make a dollar. So one dime is 0.10 dollars.

A penny is one hundredth of a dollar, because 100 pennies make a dollar. So one penny is 0.01 dollars.

Try counting some out.

• 3 dimes is 0.30 dollars, or 30 cents.
• 7 dimes is 0.70 dollars, or 70 cents.
• 4 pennies is 0.04 dollars, or 4 cents.
• 2 dimes and 5 pennies is 0.25 dollars, or 25 cents.

Key idea: A dime is a tenth of a dollar, and a penny is a hundredth.

Decimals on a number line

Picture a number line from 0 to 1. Now cut the space between them into 10 equal steps.

Each step is one tenth. The marks are 0.1, 0.2, 0.3, and so on, all the way up to 1.0.

Where does 0.5 sit? Exactly halfway, because 5 tenths is half of 10 tenths. Where does 0.9 sit? Almost all the way to 1, with one step to go.

Between which two marks does 0.35 sit? It is bigger than 0.3 and smaller than 0.4, so it sits between those two, closer to the middle.

Key idea: Decimals fit between the whole numbers on a number line.

A tricky comparison

Here is a comparison that fools a lot of people. Which is bigger, 0.5 or 0.35?

Many people say 0.35, because 35 looks bigger than 5. But that is wrong.

Line up the decimal points and check the tenths place first. In 0.5 there are 5 tenths. In 0.35 there are only 3 tenths. Since 5 tenths beats 3 tenths, 0.5 is bigger.

Money makes this obvious. 0.5 dollars is 50 cents. 0.35 dollars is 35 cents. Fifty cents is more.

A helpful move is to write 0.5 as 0.50 so both have the same number of places. Then compare 50 to 35, and it is clear.

Key idea: Compare tenths first, and add a zero so both numbers have the same number of places.

Practice set

Answer each one, then check.

• Write 4 over 10 as a decimal. Answer: 0.4.
• Write 0.8 as a fraction. Answer: 8 over 10.
• Which is bigger, 0.6 or 0.45? Answer: 0.6, because 6 tenths beats 4 tenths.
• Write 63 cents as a decimal part of a dollar. Answer: 0.63.
• Which is bigger, 0.20 or 0.2? Answer: they are equal. The extra zero adds nothing.

Key idea: Decimals, fractions, and money are three ways of saying the same thing.

Watch out for

• Line up the decimal points before you compare. Do not just count digits.
• The first place after the dot is tenths, not hundredths.
• 0.5 and 0.50 are the same amount. Extra zeros at the end do not change the value.
• Thinking 0.35 is bigger than 0.5 because it has more digits.

Recap

• A decimal uses a dot to show parts of a whole.
• The first place after the dot is tenths.
• The second place after the dot is hundredths.
• Decimals in tenths match fractions over 10.
• A dime is a tenth of a dollar, and a penny is a hundredth.
• To compare, line up the dots and check the tenths first.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 4: Number and operations, fractions. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 5: Number and operations in base ten. Common Core State Standards for Mathematics. thecorestandards.org
3. Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade (NCEE 2010-4039). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
4. The Math Learning Center. (n.d.). Number line. Math Apps. mathlearningcenter.org
5. The Math Learning Center. (n.d.). Money pieces. Math Apps. mathlearningcenter.org
6. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we learn to count money. We will use pennies, nickels, dimes, and quarters. It is fun because you can buy your favorite treats!

Cents and dollars

A cent is the smallest unit of money we count. A dollar is worth 100 cents. So it takes 100 cents to make one dollar.

We use a cent sign or a dollar sign to write money. Ten cents can be written as 10 cents or as 0.10 dollars.

Key idea: One dollar is the same as 100 cents.

Meet the coins

Here are the four coins and how much each is worth.

• A penny is worth 1 cent.
• A nickel is worth 5 cents.
• A dime is worth 10 cents.
• A quarter is worth 25 cents.

A dime is small but it is worth more than a big nickel. Size does not tell you the value!

Key idea: Penny is 1, nickel is 5, dime is 10, quarter is 25.

Counting a group of coins

To count coins, start with the biggest ones and add as you go. A smart trick is to skip count.

Say you have 3 dimes. Skip count by tens: 10, 20, 30. So 3 dimes are 30 cents.

Say you have 4 nickels. Skip count by fives: 5, 10, 15, 20. So 4 nickels are 20 cents.

Key idea: Skip count coins that are the same to add them fast.

Counting mixed coins

What if you have different coins? Add them one group at a time. Let us count 1 quarter, 2 dimes, and 1 penny.

1. Start with the quarter: 25 cents.
2. Add 2 dimes. That is 20 more. 25 + 20 = 45 cents.
3. Add 1 penny. That is 1 more. 45 + 1 = 46 cents.
4. The total is 46 cents.

Key idea: Add the biggest coins first, then the smaller ones.

Making the same amount different ways

You can make the same amount with different coins. Here are three ways to make 25 cents.

• 1 quarter is 25 cents.
• 2 dimes and 1 nickel is 10 + 10 + 5 = 25 cents.
• 5 nickels is 5 + 5 + 5 + 5 + 5 = 25 cents.

Key idea: Different coins can add up to the same amount.

Writing money with a dollar sign

When we have a dollar or more, we use a dollar sign and a decimal point. The numbers after the dot are the cents.

One dollar and fifty cents is written as 1.50 dollars. The 1 is the dollar and the 50 is the cents. Two quarters make that 50 cents!

Key idea: After the dot in a money amount comes the cents.

Count a real handful

If you can find some real coins, spread them out on a table. If not, picture this handful: 2 quarters, 3 dimes, 1 nickel, and 4 pennies.

The trick is to sort first. Put all the same coins in their own little pile. Then count pile by pile, biggest to smallest.

1. Quarters: 25, 50. That is 50 cents so far.
2. Dimes: 60, 70, 80. Now we are at 80 cents.
3. Nickel: 85. Now 85 cents.
4. Pennies: 86, 87, 88, 89. The total is 89 cents.

Notice how we never started over. We kept one running total and just kept counting up. Sorting first is what makes that possible.

Key idea: Sort the coins into piles, then count from biggest to smallest without restarting.

Making change by counting up

Change is the money you get back when you pay more than something costs.

Say a pencil costs 65 cents and you pay with 1 dollar. How much change do you get?

Count up from 65 to 100.

1. From 65, add a nickel to reach 70.
2. From 70, add 3 dimes to reach 100.
3. That is 5 + 30 = 35 cents of change.

Counting up is easier than subtracting in your head, and it is exactly what a shopkeeper does when handing you coins.

Key idea: To find change, count up from the price to the amount you paid.

Which pile is worth more?

Do not judge by how many coins there are. Judge by the total value.

• 6 pennies, or 1 dime? The dime wins. 6 cents versus 10 cents.
• 3 nickels, or 1 quarter? The quarter wins. 15 cents versus 25 cents.
• 2 dimes and 1 nickel, or 1 quarter? They tie. Both are 25 cents.
• 4 dimes, or 1 quarter and 2 nickels? The dimes win. 40 cents versus 35 cents.

A big pile of pennies can be worth less than one small dime. That is a surprise the first time you notice it.

Key idea: More coins does not mean more money. Add the values.

Practice set

Find each total, then check.

• 3 quarters and 2 pennies. Answer: 75 + 2 = 77 cents.
• 1 dollar, 1 dime, and 3 nickels. Answer: 100 + 10 + 15 = 125 cents, written 1.25 dollars.
• 5 dimes and 5 nickels. Answer: 50 + 25 = 75 cents.
• You buy a sticker for 40 cents and pay with 2 quarters. Answer: count up from 40 to 50, so your change is 10 cents.

Key idea: Money problems are just adding and counting up, with names on the numbers.

Watch out for

• A dime is worth more than a nickel, even though it is smaller.
• Skip count by the right number: fives for nickels, tens for dimes.
• Remember 100 cents make a dollar, not 10 cents.
• Counting how many coins instead of how much they are worth.

Recap

• One dollar is 100 cents.
• Penny is 1, nickel is 5, dime is 10, quarter is 25 cents.
• Sort coins into piles, then count from biggest to smallest.
• To find change, count up from the price to what you paid.
• Write money with a dollar sign and a dot before the cents.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 2: Measurement and data. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 4: Number and operations, fractions. Common Core State Standards for Mathematics. thecorestandards.org
3. The Math Learning Center. (n.d.). Money pieces. Math Apps. mathlearningcenter.org
4. Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children (NCEE 2014-4005). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
5. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
6. NRICH. (n.d.). Primary students. NRICH Mathematics. University of Cambridge. nrich.maths.org
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we learn to measure length and to tell time. Measuring tells us how long or how far. Telling time tells us when. Both help us every day!

What measuring means

To measure means to find out how big something is using a tool. We measure length, which is how long something is from one end to the other.

We use a ruler or a tape to measure length. Think of measuring how long your pencil is.

Key idea: Measuring tells us how long or how far something is.

Inches and centimeters

We measure length with units. An inch is one unit. A centimeter is another, smaller unit. Both are marked on a ruler.

A centimeter is smaller than an inch. So a crayon might be about 3 inches long, or about 8 centimeters long. Same crayon, different units!

Key idea: Inches and centimeters are units we use to measure length.

Start at zero

Here is a super important rule. Always line up the end of the object with the 0 on the ruler, not the 1.

If your pencil starts at 0 and ends at 5, it is 5 units long. If you start at 1 by mistake, your answer will be wrong.

Key idea: Always start measuring from the 0 mark.

Feet are made of inches

When something is long, we can use a foot. One foot is made of 12 inches.

So if a ribbon is 12 inches long, that is the same as 1 foot. If it is 24 inches, that is 2 feet, because 12 + 12 = 24.

Key idea: There are 12 inches in one foot.

Telling time on a clock

A clock with hands has two pointers. The short hand points to the hour. The long hand points to the minutes.

When the long hand points straight up to the 12, it is an even hour. If the short hand points to 3 and the long hand points to 12, the time is 3 oclock.

Key idea: The short hand shows the hour and the long hand shows the minutes.

Half past the hour

An hour is made of 60 minutes. Half of 60 is 30. So halfway through an hour is 30 minutes.

When the long hand points straight down to the 6, it is half past the hour. That means 30 minutes have gone by.

If the short hand is between 4 and 5, and the long hand is on the 6, the time is half past 4, which we can say as 4:30.

Key idea: When the long hand points to 6, it is half past the hour.

Guess first, then measure

Here is a game that makes you a better measurer. Before you use the ruler, guess.

Pick up a spoon. Guess how many inches long it is. Write your guess down. Now measure it. How close were you?

Try it with five things around you: a book, your shoe, a fork, a phone, and your hand.

At first your guesses will be way off, and that is completely normal. After a few tries, your guesses get much better, because your brain starts to remember what an inch feels like.

Here are two helpers. The top part of most adult thumbs is about 1 inch. Your pinky fingernail is about 1 centimeter across.

Key idea: Guessing before you measure teaches your brain the size of a unit.

Reading between the marks

What if an object does not end right on a number? Look at the small lines between the numbers.

On an inch ruler, the longest small line between two numbers marks the half inch. If your crayon ends at that line between 3 and 4, it is 3 and a half inches long.

On a centimeter ruler, the small lines are millimeters, and 10 of them make one centimeter.

When something lands between marks, say the closer one. If the pencil ends just past the 6, it is about 6 inches. The word about is honest and useful.

Key idea: Use the small lines between numbers, and say about when it lands in between.

Minutes go around by fives

Look at a clock face. The numbers 1 through 12 are for hours, but they also mark the minutes, five at a time.

• Long hand on the 1 means 5 minutes past.
• Long hand on the 2 means 10 minutes past.
• Long hand on the 3 means 15 minutes past, which we call quarter past.
• Long hand on the 6 means 30 minutes past, or half past.
• Long hand on the 9 means 45 minutes past, or quarter to the next hour.

So to read the minutes, skip count by 5s around the face. If the long hand is on the 4, count 5, 10, 15, 20. That is 20 minutes past.

Key idea: Skip count by 5s around the clock face to read the minutes.

How much time has passed?

Sometimes you need to know how long something took. Count forward from the start time.

Recess starts at 10:15 and ends at 10:45. How long is it? Count up from 15 to 45 minutes, which is 30 minutes.

Here is a harder one. A movie starts at 2:30 and ends at 4:00. Count in chunks. From 2:30 to 3:00 is 30 minutes. From 3:00 to 4:00 is 60 more minutes. Together that is 90 minutes, or 1 hour and 30 minutes.

Counting in chunks up to the next whole hour first makes these much easier.

Key idea: To find how much time passed, count up to the next whole hour first.

Watch out for

• Start your ruler at 0, not at 1.
• The short hand is the hour, the long hand is the minutes. Do not mix them up.
• One foot is 12 inches, and one hour is 60 minutes.
• Reading the long hand on the 4 as 4 minutes. It means 20 minutes.

Recap

• Measuring tells how long something is.
• We measure length with inches and centimeters.
• Always start measuring from the 0 mark.
• There are 12 inches in a foot and 60 minutes in an hour.
• Skip count by 5s around the clock face to read minutes.
• The short hand shows the hour, and half past means the long hand points to 6.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 2: Measurement and data. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Measurement and data. Common Core State Standards for Mathematics. thecorestandards.org
3. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 1: Measurement and data. Common Core State Standards for Mathematics. thecorestandards.org
4. The Math Learning Center. (n.d.). Math clock. Math Apps. mathlearningcenter.org
5. Frye, D., Baroody, A. J., Burchinal, M., Carver, S. M., Jordan, N. C., & McDowell, J. (2013). Teaching math to young children (NCEE 2014-4005). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
6. NRICH. (n.d.). Primary students. NRICH Mathematics. University of Cambridge. nrich.maths.org
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we learn about shapes. We will name them, find the distance around them, and find how much space they cover. Shapes are all around you!

What a flat shape is

A 2D shape is a flat shape you can draw on paper. 2D means it has two directions, across and up and down, but no thickness.

A drawing of a square on paper is a 2D shape. A block you can hold is not flat, so it is different.

Key idea: A 2D shape is flat, like a shape drawn on paper.

Sides and corners

A side is one straight edge of a shape. A corner is where two sides meet. A corner is also called a vertex.

Here are some shapes and their sides and corners.

• A triangle has 3 sides and 3 corners.
• A square has 4 sides that are all the same length and 4 corners.
• A rectangle has 4 sides and 4 corners, but the sides are not all equal.

Key idea: A side is a straight edge, and a corner is where two sides meet.

Perimeter is the distance around

The perimeter is the distance all the way around a shape. To find it, add up the lengths of all the sides.

Picture a square garden where each side is 3 feet long. Add all 4 sides: 3 + 3 + 3 + 3 = 12. The perimeter is 12 feet.

Think of walking all the way around the edge of the garden. The distance you walk is the perimeter.

Key idea: Perimeter is the distance around, found by adding all the sides.

A perimeter example with a rectangle

Let us find the perimeter of a rectangle. It is 5 across the top and bottom, and 2 up each side.

1. Top is 5 and bottom is 5.
2. Left side is 2 and right side is 2.
3. Add them all: 5 + 5 + 2 + 2 = 14.
4. The perimeter is 14 units.

Key idea: Add every side, even the ones that are the same.

Area is the space inside

The area is how much space is inside a shape. We measure area in square units, which are little squares that fill the shape.

For a rectangle, we find area by multiplying the length by the width. That is faster than counting every little square.

Key idea: Area is the space inside a shape.

An area example

Let us find the area of a rectangle that is 4 long and 3 wide.

1. Multiply the length by the width: 4 x 3.
2. 4 x 3 = 12.
3. The area is 12 square units.

You can check by picturing 3 rows of 4 little squares. Count them and you get 12 squares. It matches!

Key idea: Area of a rectangle is length times width.

Perimeter and area are different

Perimeter is the distance around the edge. Area is the space inside. They are not the same!

Perimeter is like the fence around a yard. Area is like the grass that fills the yard.

Key idea: Perimeter goes around the edge, area fills the inside.

More shapes to know

Shapes are named by how many sides they have. Once you know the pattern, new names are easy.

• A pentagon has 5 sides and 5 corners.
• A hexagon has 6 sides and 6 corners. A honeycomb is full of them.
• An octagon has 8 sides. A stop sign is an octagon.
• A circle is special. It has no straight sides and no corners at all.

Go look around the room for these. A window is usually a rectangle. A slice of pizza is close to a triangle. A clock face is a circle.

Key idea: Count the sides, and the name usually follows.

Perimeter of an odd shape

Not every shape is a neat square. When the sides are all different, you still do the same thing: add every side.

Picture a shape with sides of 3, 5, 2, 4, and 6 units, all the way around.

1. Start at one corner and go around in order.
2. 3 + 5 = 8.
3. 8 + 2 = 10.
4. 10 + 4 = 14.
5. 14 + 6 = 20.
6. The perimeter is 20 units.

The trick is to go around in order and mark each side as you use it. Skipping a side or counting one twice is the usual mistake.

Key idea: Walk around the shape in order and mark each side as you add it.

Area by counting squares

You can always find area by counting little squares, and this is a good way to check your multiplying.

Draw a rectangle on graph paper that is 5 squares across and 3 squares down. Now count the squares inside.

Row 1 has 5 squares. Row 2 has 5. Row 3 has 5. Skip count: 5, 10, 15. There are 15 squares.

Now multiply: 5 x 3 = 15. Same answer! Multiplying is just a fast way of counting the rows of squares.

Key idea: Area by multiplying and area by counting squares always agree.

Same perimeter, different area

Here is something surprising. Two shapes can have the same perimeter but hold very different amounts of space.

• A rectangle 1 wide and 5 long: perimeter is 1 + 5 + 1 + 5 = 12. Area is 1 x 5 = 5 square units.
• A rectangle 2 wide and 4 long: perimeter is 2 + 4 + 2 + 4 = 12. Area is 2 x 4 = 8 square units.
• A rectangle 3 wide and 3 long: perimeter is 3 + 3 + 3 + 3 = 12. Area is 3 x 3 = 9 square units.

All three need the same length of fence, but the square one holds the most grass. Farmers and builders really do think about this.

Key idea: The same perimeter can hold different amounts of area.

Practice set

Work each one out, then check.

• A square with sides of 6. Perimeter: 6 + 6 + 6 + 6 = 24. Area: 6 x 6 = 36 square units.
• A rectangle 7 long and 2 wide. Perimeter: 7 + 7 + 2 + 2 = 18. Area: 7 x 2 = 14 square units.
• A triangle with sides of 4, 5, and 6. Perimeter: 4 + 5 + 6 = 15.

Key idea: For perimeter you add, and for the area of a rectangle you multiply.

Watch out for

• For perimeter, add every single side. Do not skip any.
• For area of a rectangle, multiply. Do not add.
• Do not mix up perimeter (around) and area (inside).
• Forgetting that a rectangle has two long sides and two short ones.

Recap

• A 2D shape is flat, with sides and corners.
• Shapes are named by how many sides they have.
• Perimeter is the distance around, found by adding all sides.
• Area is the space inside, measured in square units.
• Area of a rectangle is length times width, which matches counting squares.
• Two shapes can share a perimeter but hold different areas.

Sources

1. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Measurement and data. Common Core State Standards for Mathematics. thecorestandards.org
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Geometry. Common Core State Standards for Mathematics. thecorestandards.org
3. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 2: Geometry. Common Core State Standards for Mathematics. thecorestandards.org
4. The Math Learning Center. (n.d.). Geoboard. Math Apps. mathlearningcenter.org
5. The Math Learning Center. (n.d.). Pattern shapes. Math Apps. mathlearningcenter.org
6. NRICH. (n.d.). Primary students. NRICH Mathematics. University of Cambridge. nrich.maths.org
7. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org

The big picture

Today we solve word problems and read graphs. A word problem is a math story. Graphs are pictures of numbers. Both help us use math in real life!

What a word problem is

A word problem is a math question told as a little story. Your job is to find the numbers and figure out what to do with them.

Here is one. Sam has 3 apples. Mom gives him 2 more. How many apples does Sam have now?

Key idea: A word problem is a math story you solve step by step.

Choosing the right operation

An operation is a math action like adding, subtracting, multiplying, or dividing. To solve a word problem, you pick the right one.

In the apple story, Sam gets more apples, so we add. 3 + 2 = 5. Sam has 5 apples.

Key idea: First decide which operation the story needs.

Clue words help you choose

Clue words are words in the story that hint at which operation to use. They are like little helpers.

• Words like in all, more, and together often mean add.
• Words like left, fewer, and take away often mean subtract.
• Words like each and groups of often mean multiply.
• Words like share and split evenly often mean divide.

Key idea: Look for clue words to pick the operation.

A subtraction word problem

Let us try one. Mia has 8 stickers. She gives 3 to her friend. How many are left?

1. Find the numbers: 8 and 3.
2. See the clue word: left means subtract.
3. Subtract: 8 - 3 = 5.
4. Mia has 5 stickers left.

Key idea: Read carefully, find the numbers, then do the math.

A multiplication word problem

Here is one more. There are 3 baskets. Each basket has 4 eggs. How many eggs in all?

1. Find the numbers: 3 baskets and 4 eggs each.
2. The clue word each with equal groups means multiply.
3. Multiply: 3 x 4 = 12.
4. There are 12 eggs in all.

Key idea: Equal groups in a story often means multiply.

Reading a bar graph

A bar graph uses bars to show how many. A taller bar means more. A shorter bar means less.

Say a bar graph shows fruit picked. The apple bar reaches 5 and the pear bar reaches 3. That means 5 apples and 3 pears were picked. To compare, we see 5 is more than 3, so more apples were picked.

Key idea: In a bar graph, a taller bar means more.

Reading a picture graph

A picture graph uses little pictures to show how many. Each picture stands for one thing. You count the pictures to find the number.

If a picture graph shows 4 star pictures for Monday and 2 star pictures for Tuesday, then 4 stars were earned Monday and 2 on Tuesday. Monday had more, because 4 is more than 2.

Key idea: In a picture graph, count the pictures to find the number.

A four-step plan that always works

When a word problem looks confusing, slow down and use these four steps every time.

1. Read it twice. The first read is for the story. The second read is for the numbers.
2. Circle the numbers and underline the question at the end.
3. Draw a quick picture or say the story in your own words.
4. Solve, then check whether your answer makes sense.

That third step matters more than people expect. If you can draw it, you almost always know what to do with the numbers.

Key idea: Read twice, circle the numbers, draw it, then solve.

Clue words can trick you

Clue words are helpful, but they are not a rule. Look at this problem.

Jo has 12 marbles. That is 5 more than Ben has. How many does Ben have?

The words say more, so it is tempting to add and answer 17. But read it again. Jo has more, so Ben must have fewer. The right move is 12 - 5 = 7. Ben has 7 marbles.

Check it: does Jo having 12 mean she has 5 more than 7? Yes, because 7 + 5 = 12. It works.

So use clue words as a hint, and then ask yourself whether the answer should be bigger or smaller than what you started with.

Key idea: Ask whether your answer should be bigger or smaller before you trust a clue word.

A two-step problem

Some problems need two steps. Do not panic. Just take one step at a time.

Ana bakes 4 trays of muffins with 6 muffins on each tray. She gives away 9 muffins. How many are left?

1. Step 1. How many muffins in all? Equal groups means multiply: 4 x 6 = 24.
2. Step 2. How many after giving some away? Take away means subtract: 24 - 9 = 15.
3. Ana has 15 muffins left.

The clue that there are two steps is that the question at the end asks about something you cannot get from a single calculation. Write down the answer to step 1 before you start step 2, so you do not lose it.

Key idea: Write down the answer to step one before you move to step two.

Reading the numbers on a graph

Bar graphs do not always count by ones. Look at the numbers along the side before you read any bar.

Say the side of a graph is marked 0, 2, 4, 6, 8, 10. Each step is 2, not 1.

If a bar reaches the third mark up, that is not 3. Skip count: 2, 4, 6. That bar means 6.

Picture graphs can do the same thing. Sometimes a key at the bottom says that each picture stands for 5 things. Then 4 pictures means 4 x 5 = 20, not 4.

Always find the scale or the key first. It takes three seconds and saves the whole answer.

Key idea: Check the scale or key before reading any bar or picture.

Watch out for

• Read the whole story before you choose an operation.
• Clue words are helpers, but always check that your answer makes sense.
• On a graph, look at the numbers on the side to read each bar correctly.
• Stopping after step one in a two-step problem.

Recap

• A word problem is a math story you solve step by step.
• Read twice, circle the numbers, draw it, then solve and check.
• Clue words hint at the operation, but check whether the answer should grow or shrink.
• Some problems need two steps, so write down the first answer.
• A bar graph shows how many with bars, and you must check the scale.
• A picture graph shows how many with pictures, and each one may stand for more than one.

Sources

1. Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., Koedinger, K. R., & Ogbuehi, P. (2012). Improving mathematical problem solving in grades 4 through 8 (NCEE 2012-4055). What Works Clearinghouse Practice Guide. Institute of Education Sciences. ies.ed.gov
2. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Standards for mathematical practice. Common Core State Standards for Mathematics. thecorestandards.org
3. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 3: Measurement and data. Common Core State Standards for Mathematics. thecorestandards.org
4. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Grade 4: Operations and algebraic thinking. Common Core State Standards for Mathematics. thecorestandards.org
5. Institute of Education Sciences. (2021). Assisting students struggling with mathematics: Intervention in the elementary grades. What Works Clearinghouse. ies.ed.gov
6. National Council of Teachers of Mathematics. (2023). Procedural fluency in mathematics. NCTM Position Statements. nctm.org
7. NRICH. (n.d.). Primary students. NRICH Mathematics. University of Cambridge. nrich.maths.org