C. Algebra · Patterns, Variables, Coding and ModellingSame Amount, Different Ways: Equivalent Relationships to 100
C2.3About 40 minutes
Learning goals- I can show the same number in many equivalent ways, like 46 = 40 + 6 = 30 + 16.
- I can use the turn-around idea: 30 + 25 = 25 + 30.
- I can use equivalent relationships to make adding easier, like 9 + 6 = 10 + 5.
- I can find equivalent amounts with money and measurements.
Words to know
equivalent
Having the same value.
decompose
To break a number into parts, like 46 = 40 + 6.
compose
To put parts together to make a number.
relationship
How numbers or amounts are connected.
One number, many names
People might call you by your name, by a nickname, or the one who loves hockey. All of those mean you!
Numbers are the same. 46 can be written as:
- 40 + 6
- 30 + 16
- 50 − 4
- 23 + 23
They all have the same value. They are equivalent.
Breaking numbers into tens and ones
When you decompose a number, you break it into parts. Here are some equivalent ways to show 46 with tens and ones:
| Tens | Ones | Number sentence |
|---|
| 4 | 6 | 46 = 40 + 6 |
| 3 | 16 | 46 = 30 + 16 |
| 2 | 26 | 46 = 20 + 26 |
| 1 | 36 | 46 = 10 + 36 |
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The turn-around idea
You can add numbers in any order and get the same total.
30 + 25 = 25 + 30
Both are 55. This means you can start with whichever number is easier. For 3 + 48, start with 48 and count on 3: 49, 50, 51.
Worked exampleMake a ten to add
Use an equivalent number sentence to add 9 + 6.
- 9 is very close to 10. It needs just 1 more.
- Take 1 from the 6 and give it to the 9.
- 9 + 6 becomes 10 + 5. Nothing was lost. We only moved 1.
- 10 + 5 = 15.
9 + 6 = 10 + 5 = 15. These are equivalent, and 10 + 5 is easier!
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9 red and 6 blue counters. Move 1 blue counter up to the red row and you have 10 and 5. The total is still 15.Worked exampleBigger numbers
Use the same idea to add 38 + 25.
- 38 needs 2 more to make 40.
- Take 2 from the 25. That leaves 23.
- 38 + 25 becomes 40 + 23.
- 40 + 23 = 63.
Equivalent amounts of money
Money gives lots of chances to find equivalent amounts. All of these are worth 50¢:
- 2 quarters (25¢ + 25¢)
- 5 dimes (10¢ + 10¢ + 10¢ + 10¢ + 10¢)
- 1 quarter, 2 dimes and 1 nickel (25¢ + 10¢ + 10¢ + 5¢)
- 10 nickels
And 100¢ is worth the same as one loonie ($1), which is 4 quarters or 10 dimes.
Worked exampleEqual money
Ava has 3 dimes and 1 nickel. Ben has 1 quarter and 2 nickels. Do they have the same amount?
- Ava: 10 + 10 + 10 = 30, then 30 + 5 = 35¢.
- Ben: 25, then 25 + 5 + 5 = 35¢.
- 35¢ = 35¢.
Yes. Both have 35¢. 10 + 10 + 10 + 5 = 25 + 5 + 5.
Just for funFunny but true: Canada's $1 coin is called the loonie because it has a loon on it. So when the $2 coin came along, people called it the toonie: two + loonie! 🦆
Equivalent measurements
Equivalent relationships help with measuring too. A strip of paper 40 cm long could be cut into:
- 20 cm + 20 cm
- 10 cm + 30 cm
- 10 cm + 10 cm + 10 cm + 10 cm
The pieces are different, but together they still measure 40 cm.
Your turn
18. 4 tens and 18 ones is 58.
3True or false? 20 + 45 = 45 + 20
True. You can add in any order. Both are 65.
4Make an easier sentence: 9 + 7 = 10 + ___
6. Move 1 from the 7 to the 9.
5Name two ways to make 25¢ with coins.
Many answers, for example: 1 quarter; 2 dimes and 1 nickel; 5 nickels; 1 dime and 3 nickels.
6True or false? 70 − 10 = 30 + 30
Hands-on activityCoin match-up
You need: real or paper coins (nickels, dimes, quarters) and a loonie if you have one.
- A grown-up makes an amount with coins, like 1 quarter and 1 dime (35¢).
- You make the same amount using different coins. (Example: 3 dimes and 1 nickel.)
- Write it as a number sentence: 25 + 10 = 10 + 10 + 10 + 5.
- Try amounts up to 100¢. How many ways can you make 100¢?
Bonus: use Lego or blocks. Build 46 with 4 tens and 6 ones. Then break a ten apart. What do you have now?
Math talk
Use the word equivalent: 25¢ + 25¢ is equivalent to 5 dimes, because both are 50¢.
More practice: warm-up
3True or false? 5 + 8 = 8 + 5
5 dimes. 10, 20, 30, 40, 50.
5How many quarters make 100¢?
4 quarters. 25, 50, 75, 100.
More practice: on your own
1Write 72 in three equivalent ways using tens and ones.
Examples: 70 + 2, 60 + 12, 50 + 22. Each time a ten is broken into 10 ones.
2Make it easier, then add: 19 + 8
27. Move 1 from the 8 to the 19: 19 + 8 = 20 + 7 = 27.
3Is 2 quarters + 1 dime equivalent to 6 dimes?
Yes. 25 + 25 + 10 = 60¢ and 6 dimes = 60¢.
4Fill in: 45 + 30 = 30 + ___
45. Turning the numbers around does not change the total.
5A 60 cm ribbon is cut into 2 pieces. One piece is 25 cm. How long is the other piece?
35 cm. 25 + 35 = 60. (25 + 5 = 30, then 30 + 30 = 60, so 5 + 30 = 35.)
Challenge
1Fill in to make it true: 48 + 27 = 50 + ___
25. Move 2 from 27 to 48: 48 + 2 = 50 and 27 − 2 = 25. Check: both sides are 75.
2Mia says 63 = 50 + 3. Is she right? If not, fix it.
No. 50 + 3 = 53. She lost a ten. It should be 63 = 50 + 13 (or 60 + 3).
3Find 3 different ways to make 100¢ using only quarters and dimes.
Examples: 4 quarters; 10 dimes; 2 quarters and 5 dimes (50 + 50). Check each adds to 100.
Quiz
1. Which is equivalent to 54?
3. 9 + 5 is equivalent to...
4. Which coins are worth the same as 2 quarters?
6. A 50 cm stick is broken into 2 pieces. One piece is 30 cm. The other is...
Children explore equivalent ways to show numbers up to 100: decomposing into tens and ones, adding in any order, making a ten to add more easily, and equal money and measurement amounts.
Common mistakes
- Losing a ten when regrouping (writing 63 = 50 + 3). Use tens and ones blocks, Lego or bundles of 10 straws.
- Moving an amount in compensation but forgetting to take it from the other number (9 + 6 = 10 + 6). Ask: Where did the extra 1 come from?
- Counting coins as 1 each. Sort coins first and count the biggest values first.
How to help
- Play shop with real coins.
- Bundle 10 straws or popsicle sticks with elastics for tens.
- Ask How else could we make that number?