Grade by Grade
C. Algebra · Patterns, Equations, Coding and Modelling

Same Value, Different Ways: Equivalent Relationships to 1000

C2.3 A1About 40 minutes
Learning goals
  • I can show the same number up to 1000 in many equivalent ways, like 346 = 300 + 40 + 6 = 34 tens and 6 ones.
  • I can find a missing number that makes two sides equal.
  • I can use equivalent relationships to make adding easier, like 199 + 45 = 200 + 44.
  • I can find equivalent amounts with money (cents and dollars) and length (centimetres and metres).

Words to know

equivalent
Having the same value, even if it looks different.
regroup
Trade 10 ones for 1 ten, or 10 tens for 1 hundred, or the other way around.
expanded form
A number written as the sum of its place values, like 300 + 40 + 6.
cent (¢)
A small unit of money. 100 cents make 1 dollar.
centimetre (cm) and metre (m)
Units of length. 100 centimetres make 1 metre.

Remember from Grade 2

In Grade 2 you showed numbers to 100 in many ways, like 46 = 40 + 6 = 30 + 16. You also made 100¢ with different coins. Now we do the same thing with numbers up to 1000.

One number, many names

The number 346 can be shown with base-ten blocks: 3 hundreds, 4 tens and 6 ones.

3 hundreds4 tens6 ones346300 + 40 + 6 = 346
3 hundred flats, 4 ten rods and 6 one cubes show 346.

All of these are equivalent to 346:

  • 300 + 40 + 6 (expanded form)
  • 3 hundreds, 4 tens, 6 ones
  • 34 tens, 6 ones (each hundred is 10 tens, so 3 hundreds = 30 tens, plus 4 tens = 34 tens)
  • 2 hundreds, 14 tens, 6 ones (trade 1 hundred for 10 tens)
  • 346 ones
  • 300 + 46, or 200 + 146, or 340 + 6

The number did not change. We just regrouped it.

Keeping both sides balanced

The equal sign means both sides have the same value. If we know one side, we can find a missing number on the other side.

400 + 250 = 500 + □

The left side is 650. So 500 + □ must be 650 too. □ = 150.

Another way to think: 500 is 100 more than 400. So the other number must be 100 less than 250. 250 − 100 = 150.

400250500150= 650= 650100400 + 250 = 500 + 150
Both bars are 650 long. The first number grew by 100, so the second number must shrink by 100: 250 − 100 = 150.

Money and measurement

Equivalent relationships are everywhere in real life.

Money in Canada:

  • 1 dollar ($1, a loonie) = 100¢
  • 1 toonie ($2) = 200¢
  • 4 quarters = 10 dimes = 20 nickels = $1
  • A $5 bill = 500¢

Length:

  • 1 metre (m) = 100 centimetres (cm)
  • 2 m = 200 cm, 5 m = 500 cm
  • 3 m and 40 cm = 300 cm + 40 cm = 340 cm
AmountEquivalent ways
$101 ten-dollar bill = 2 five-dollar bills = 5 toonies = 10 loonies
$11 loonie = 4 quarters = 10 dimes = 20 nickels = 100¢
$31 toonie + 1 loonie = 3 loonies = 12 quarters = 300¢
4 m400 cm
250 cm2 m and 50 cm
Equivalent amounts of money and length.
1 loonie$14 quarters25¢25¢25¢25¢10 dimes10¢10¢10¢10¢10¢10¢10¢10¢10¢10¢20 nickels5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢5¢each row = 100¢
Four equivalent ways to make $1. Every row adds up to 100¢. (Coins are drawn as labelled circles, not to scale.)
Worked example

Example 1: Many ways to show 528

Show 528 in three equivalent ways.

  1. Expanded form: 500 + 20 + 8.
  2. Regroup a hundred into tens: 4 hundreds, 12 tens, 8 ones → 400 + 120 + 8.
  3. All in tens and ones: 5 hundreds = 50 tens, so 52 tens and 8 ones.
  4. Check one: 400 + 120 = 520, and 520 + 8 = 528. ✔

500 + 20 + 8 = 400 + 120 + 8 = 52 tens and 8 ones. All equal 528.

Worked example

Example 2: Find the missing number

Find □: 300 + 200 = 450 + □

  1. Left side: 300 + 200 = 500.
  2. 450 + □ must also be 500.
  3. 450 + 50 = 500, so □ = 50.

□ = 50. Both sides are 500.

Worked example

Example 3: Make it friendly

Add 199 + 345 in your head.

  1. 199 is very close to 200. Move 1 from 345 to 199.
  2. 199 + 345 = 200 + 344. Both sides are equivalent because we only moved 1 from one number to the other.
  3. 200 + 344 = 544.

199 + 345 = 544

Worked example

Example 4: Money

How many quarters are equivalent to $3?

  1. 4 quarters = $1.
  2. $3 is 3 groups of $1, so 4 + 4 + 4 = 12 quarters.
  3. Check by counting by 25s: 25, 50, 75, 100, …, 300. That is 12 counts. (3 × 4 = 12)

12 quarters = $3 = 300¢.

Worked example

Example 5: Length

A snowman is 1 m and 65 cm tall. How many centimetres is that?

  1. 1 m = 100 cm.
  2. 100 cm + 65 cm = 165 cm.

The snowman is 165 cm tall.

Just for fun

Funny but true: Canada's $1 coin is called the loonie because it shows a loon. When the $2 coin came along, people gave it a rhyming nickname: the toonie!

Moving amounts keeps it balanced

When you add, you can take some from one number and give it to the other. The total stays the same: 199 + 345 = 200 + 344. This trick makes mental math much easier.

Your turn

1

Write 472 in expanded form.

2

How many tens are in 350?

3

Find □: 600 + □ = 1000

4

Find □: 250 + 250 = 300 + □

5

How many cents are in $4?

6

How many centimetres are in 3 m?

7

Use a friendly number to add 298 + 125.

Hands-on activity

Make a dollar, measure a metre

You need: some coins (or paper coins you draw), a measuring tape or metre stick, and paper.

Part 1: Money

  1. Make $1 (100¢) using coins in three different ways. For example: 4 quarters; 10 dimes; 2 quarters + 5 dimes.
  2. Write each way as a number sentence, like 25 + 25 + 25 + 25 = 100.

Part 2: Length

  1. With a grown-up, measure a couch, a table or a bed.
  2. Write the length in centimetres (like 185 cm) and in metres and centimetres (1 m 85 cm).

Math talk: "These are equivalent because ___."

Grown-up note: keep small items away from younger children who might put them in their mouths.

More practice: warm-up

1

What is 200 + 30 + 5?

2

How many cents are in $1?

3

How many centimetres are in 1 m?

4

Find □: 500 + □ = 900

5

How many quarters make $1?

More practice: on your own

1

Write 605 in two equivalent ways.

2

Find □: 350 + 150 = 400 + □. Explain.

3

Is 5 hundreds, 3 tens, 9 ones equivalent to 4 hundreds, 13 tens, 9 ones? Explain.

4

Mila has 3 toonies and 2 loonies. Ben has 8 loonies. Do they have the same amount? Explain.

5

A ribbon is 240 cm long. Write the length in metres and centimetres.

Challenge

1

Find □: 499 + 276 = 500 + □. Then find the total.

2

How many dimes are equivalent to $5? Explain.

3

Show 1000 in three different equivalent ways.

Check your understanding

Quiz

1.

Which is equivalent to 463?

2.

Find □: 700 + 200 = 800 + □

3.

How many tens are in 270?

4.

Which is equivalent to $2?

5.

How long is 4 m and 25 cm in centimetres?

6.

Which is the same as 198 + 54?

For parents and teachers

Big idea: The same number can be named many ways (C2.3): 346 is 300 + 40 + 6, 34 tens and 6 ones, or 346 ones. Seeing these equivalent forms helps with regrouping, mental math and measurement and money conversions.

Common difficulties:

  • Counting tens in a 3-digit number. Many children say 350 has 5 tens. It has 5 tens in the tens place, but 35 tens in all.
  • Missing numbers on the left side of =. Encourage finding the value of the complete side first.
  • Mixing up cm and m. Remind them that 100 cm = 1 m, and show both on a measuring tape.

How to help: Count real coins. Measure things with a measuring tape and say the length both ways. Ask, "Can you name this number another way?"