B. Number · Operations: Numbers to 10 000, Decimal Tenths, Multiplying and DividingMath Rules: Properties of Operations
B2.1 A1About 35 minutes
Learning goals- I can add or multiply numbers in any order and in any grouping.
- I can use the zero and one rules for adding and multiplying.
- I can split a number to make multiplying easier.
- I can use the opposite operation to check my answer.
Words to know
property
A rule that is always true for an operation, like "you can add in any order".
operation
Something you do to numbers: add (+), subtract (−), multiply (×) or divide (÷).
commutative property
The any order rule: 4 + 9 = 9 + 4 and 4 × 9 = 9 × 4.
associative property
The grouping rule: when you add three numbers, or multiply three numbers, you can choose which two to do first.
distributive property
The splitting rule: 6 × 14 = 6 × 10 + 6 × 4.
inverse operations
Operations that undo each other. Adding and subtracting are inverse operations. So are multiplying and dividing.
Remember from Grade 3
In Grade 3 you learned some multiplication rules: you can multiply in any order, any number times 1 stays the same, any number times 0 is 0, and you can split a hard fact into easier ones. In Grade 4 these rules get real math names, and you will use them with bigger numbers.
Let's start with a puzzle
Can you solve 25 + 87 + 75 in your head in 5 seconds?
Here's a trick: add 25 + 75 = 100 first. Then 100 + 87 = 187. Done!
You just changed the order of the numbers to make it easier. Math has rules, called properties, that tell us when tricks like this are allowed.
Rule 1: Any order (the commutative property)
When you add or multiply two numbers, the order does not change the answer.
- 38 + 15 = 15 + 38 = 53
- 7 × 4 = 4 × 7 = 28
Turn an array on its side and you still have the same number of dots.
Careful: this rule does not work for subtraction or division. 10 − 4 = 6, but 4 − 10 is not 6. 12 ÷ 3 = 4, but 3 ÷ 12 is not 4.
Rule 2: Any grouping (the associative property)
When you add three numbers, or multiply three numbers, you can choose which two to do first. Use brackets ( ) to show what you do first.
- (25 + 75) + 87 = 100 + 87 = 187
- 2 × 9 × 5 → do (2 × 5) first: 10 × 9 = 90
Look for pairs that make 10 or 100. They make the rest easy.
Worked exampleExample 1: Choose a smart grouping
- Look for a friendly pair. 4 × 25 = 100.
- Now multiply what is left: 100 × 13 = 1300.
Rule 3: Zero and one (identity and zero properties)
- Adding 0 does not change a number: 468 + 0 = 468.
- Multiplying by 1 does not change a number: 1 × 357 = 357.
- Multiplying by 0 always gives 0: 999 × 0 = 0. (Zero groups of anything is nothing!)
Rule 4: Splitting (the distributive property)
A hard multiplication can be split into two easier ones. Split one factor into parts, multiply each part, then add.
6 × 14: split 14 into 10 + 4.
6 × 10 = 60 and 6 × 4 = 24. Then 60 + 24 = 84.
Worked exampleExample 2: Split to multiply
- Split 13 into 10 + 3.
- 7 × 10 = 70.
- 7 × 3 = 21.
- Add the parts: 70 + 21 = 91.
Rule 5: Operations are partners (inverse operations)
- Adding and subtracting undo each other. If 156 − 89 = 67, then 67 + 89 = 156.
- Multiplying and dividing undo each other. If 8 × 7 = 56, then 56 ÷ 8 = 7.
- Multiplying is repeated adding: 4 × 6 = 6 + 6 + 6 + 6.
- Dividing is repeated subtracting: 20 ÷ 5 = 4, because you can take 5 away from 20 four times.
Because they are partners, you can check a subtraction with addition, and a division with multiplication.
Worked exampleExample 3: Check with the partner operation
Leo says 523 − 287 = 246. Is he right?
- Check with addition: 246 + 287 = 533.
- 533 is not 523, so Leo made a mistake.
- Try again: 523 − 287 = 236. Check: 236 + 287 = 523. ✓
Leo was not right. 523 − 287 = 236.
Think like a mathematician
When a question looks hard, stop and ask yourself: Can I change the order? Can I group numbers that make 10 or 100? Can I split a number? Trying a new idea, even if it does not work the first time, is how strong math thinkers grow. Explain your choice out loud to your grown-up.
Just for funQ: Why do 3 × 8 and 8 × 3 never argue?
A: Because no matter who goes first, they always end up at the same answer: 24!
Hands-on activityProperty detectives with an array
You need: about 40 small things to arrange, like dry pasta, buttons or cereal (keep small objects away from little brothers and sisters), and a sheet of paper.
- Make an array with 4 rows of 6. Count: 24.
- Turn the paper a quarter turn. Now it is 6 rows of 4. Still 24? That is the any order rule.
- Make 3 rows of 12. Lay a pencil between the 10th and 11th column to split it into 3 × 10 and 3 × 2. Find 30 + 6 = 36. That is the splitting rule.
- Your grown-up says three numbers, like 30 + 46 + 70. Find a friendly pair, then say your answer and which rule you used.
Math talk: "Why can't we turn 10 − 4 around?"
Your turn
37. Adding 0 does not change a number.
58. Multiplying by 1 does not change a number.
0. Zero groups of anything is 0.
4You know 9 × 6 = 54. What is 6 × 9?
54. You can multiply in any order.
516 + 47 + 84 = ? Find a friendly pair.
147. 16 + 84 = 100. Then 100 + 47 = 147.
130. 5 × 2 = 10. Then 10 × 13 = 130.
78 × 12 = ? Split 12 into 10 + 2.
96. 8 × 10 = 80 and 8 × 2 = 16. 80 + 16 = 96.
More practice: warm-up
33 + 8 = 11. What is 8 + 3?
11. Same numbers, different order.
563 − 20 = 43. Write an addition sentence that checks it.
More practice: on your own
1700. Group 4 × 25 = 100 first. Then 100 × 17 = 1700.
26 × 15 = ? Use splitting.
90. 6 × 10 = 60 and 6 × 5 = 30. 60 + 30 = 90.
3Is 10 − 4 the same as 4 − 10? Explain.
No. 10 − 4 = 6, but 4 − 10 does not give 6 (you can't take 10 away from 4 and stay at or above zero). The any order rule only works for adding and multiplying.
398. 245 + 55 = 300. Then 300 + 98 = 398.
5Use a multiplication fact to find 56 ÷ 7.
8. 7 × 8 = 56, so 56 ÷ 7 = 8.
Challenge
1Find 9 × 23 in two different ways.
The answer is 207.
- Split 23: 9 × 20 = 180 and 9 × 3 = 27. 180 + 27 = 207.
- Use 10 groups and take one away: 10 × 23 = 230. Then 230 − 23 = 207.
2Find the missing number: 7 × 14 = 7 × 10 + 7 × □. What is 7 × 14?
□ = 4, because 14 = 10 + 4. 7 × 10 = 70 and 7 × 4 = 28, so 7 × 14 = 98.
3Mia says 468 − 179 = 289. Check her answer with addition.
289 + 179 = 468. ✓ The sum matches the starting number, so Mia is right.
Quiz
1. Which is the same as 6 × 8?
4. What is 4 × 23? (Split 23 into 20 + 3.)
5. Which sentence checks that 37 + 45 = 82?
6. Why is 2 × 9 × 5 easy to solve?
Big idea: The Ontario curriculum (B2.1) asks Grade 4 students to use the properties of operations and the relationships between adding, subtracting, multiplying and dividing to solve problems and check calculations. Your child does not need to memorize the formal names (commutative, associative, distributive), but should know what each rule lets them do.
Common difficulties:
- Using "any order" with subtraction or division. Show that 10 − 4 and 4 − 10 are not the same with real objects.
- Mixing up × 0 and × 1. Say it as groups: "5 groups of 0 cookies" is 0 cookies; "5 groups of 1 cookie" is 5.
- Splitting and forgetting to add the parts. Have your child draw the split array and write both parts before adding.
How to help: When your child finishes a subtraction or division, ask "How could you check it?" Encourage the habit of using the partner operation. Social-emotional learning (A1) is woven in: praise trying a new strategy, even when it doesn't work the first time.