The laws of arithmetic
🎯 In this topic you will
- Apply the associative and commutative laws to add and multiply numbers in different ways.
- Use the distributive law to rewrite a multiplication fact as the sum of two related multiplication facts.
- Perform calculations using the correct order of operations: multiplication and division before addition and subtraction.
🧠 Key Words
- associative law
- commutative law
- distributive law
- decompose
- regroup
Show Definitions
- associative law: A property stating that when adding or multiplying, the way numbers are grouped does not change the result.
- commutative law: A property stating that the order of numbers in addition or multiplication does not affect the total or product.
- distributive law: A rule showing how multiplication can be applied across addition or subtraction, such as \(a(b + c) = ab + ac\).
- decompose: To break a number into parts to make a calculation easier.
- regroup: To rearrange or combine numbers in a different structure to simplify a calculation.
Working Out Calculations
S ometimes the best way to work out calculations is to use mental methods or pencil and paper jottings.
Helpful Calculation Rules
T here are some rules that can make calculations easier.
A Simple Addition Example
F or example, solve $7 + 38$:
Let's Calculate: It is easier to start with 38 and then add 7. Using the commutative rule, we can write: $7 + 38 = 38 + 7 = 45$
Thinking Further
C an you think of any other rules to help you with calculations?
💡 Quick Math Tip
Use Friendly Numbers: Sometimes it helps to round a number to the nearest ten, do a quick calculation, and then adjust. For example, instead of adding $9 + 38$ directly, think of $9$ as $10 - 1$. So you can do $38 + 10 = 48$, then subtract $1$ to get $47$.
❓ EXERCISES
1. Use the digits $2$, $5$, $6$ and $7$ once each time to complete this calculation in four different ways.
Work out the answer to each calculation.
What do you notice?
Check your results with your partner.
👀 Show answer
$(2 \times 5 \times 6 \times 7)$, $(2 \times 5 \times 7 \times 6)$, $(2 \times 6 \times 5 \times 7)$, etc.
All yield the same product because multiplication is commutative and associative.
2. Sofia wrote these statements.
Write true or false for each one. Explain those that are false.
a. $88 + 16 = 16 + 88$
b. $18 \div 6 = 6 \div 18$
c. $34 \times 16 = 16 \times 34$
d. $56 - 6 = 6 - 56$
👀 Show answer
b. False — $18 \div 6 = 3$ but $6 \div 18 = \frac{1}{3}$.
c. True — multiplication is commutative.
d. False — subtraction is not commutative. $56 - 6 = 50$ but $6 - 56 = -50$.
3. Copy and complete these calculations.
a. $17 \times 2$
b. $25 \times 9 \times 4 = 25 \times \square \times 9$
👀 Show answer
4. Use the distributive law to help you work out these calculations. Show your working.
a. $36 \times 8$
b. $48 \times 7$
c. $19 \times 6$
👀 Show answer
b. $48 \times 7 = (50 - 2)\times 7 = 350 - 14 = 336$
c. $19 \times 6 = (20 - 1)\times 6 = 120 - 6 = 114$
5. Use the associative law to help you work out these calculations. Show your working.
a. $50 \times 16 \times 2$
b. $25 \times 17 \times 4$
c. $15 \times 17 \times 6$
👀 Show answer
b. $25 \times (17 \times 4) = 25 \times 68 = 1700$
c. $15 \times (17 \times 6) = 15 \times 102 = 1530$
6. Calculate.
a. $6 + 7 \times 9$
b. $14 - 2 \times 7$
c. $54 + 9 \div 3$
👀 Show answer
b. $14 - (2 \times 7) = 14 - 14 = 0$
c. $54 + (9 \div 3) = 54 + 3 = 57$
7. Use $+$, $-$, $\times$ and $\div$ to copy and complete these number sentences.
a. $4 \square 6 \square 3 = 6$
b. $5 \square 6 \square 2 = 28$
c. $5 \square 9 \square 3 = 8$
d. $8 \square 2 \square 4 = 0$
👀 Show answer
b. $5 \times 6 - 2 = 28$
c. $5 + 9 - 3 = 11$ (or other valid expressions giving $8$ depending on interpretation)
d. $(8 - 2) \times 4 = 24$ or $8 \div 2 - 4 = 0$ (valid solution: $8 \div 2 - 4 = 0$)
8. Here are five multiplication calculations.
$54 \times 6$ $22 \times 3$ $41 \times 5$ $19 \times 4$ $37 \times 6$
Show how you would do each calculation.
Explain your methods to your partner.
Did you make the same decisions?
👀 Show answer
$54 \times 6 = 324$
$22 \times 3 = 66$
$41 \times 5 = 205$
$19 \times 4 = 76$
$37 \times 6 = 222$
9. Here are three number cards.
Sofia, Arun and Zara each choose a card.
They each multiply the number on their card by $5$ using a different method.
• Sofia says, ‘I multiplied my number by $10$ to give $210$ and then divided by $2$. ’
• Arun says, ‘I halved my number and doubled $5$ to calculate $21 \times 10$. ’
• Zara says, ‘I multiplied $40$ by $5$ and then subtracted $2$ lots of $5$. ’
a. Which number did Sofia, Arun and Zara choose?
b. Which of these methods of multiplying by $5$ would you choose?
Explain your decision.
👀 Show answer
Arun: card $21$ (since halving $21$ gives $10.5$ and doubling $5$ gives $10$, producing the reinterpretation $21 \times 5$).
Zara: card $38$ (since $40 \times 5 = 200$, then subtracting $2 \times 5 = 10$ gives $190$).
Method choice varies — sample explanation: multiplying by $10$ then halving is often the quickest.
🧠 Think like a Mathematician
Numbers and operations
Write down two different 2-digit numbers:
- multiply one of the numbers by $5$
- add $36$
- multiply by $20$
- add the other 2-digit number
- subtract $720$
Choose different starting numbers. What happens? Does it always work for any starting number?
You will show you are generalising and convincing when you notice and explain what happens.
Follow-up Questions:
Show Answers
- 1: No matter which two 2-digit numbers you choose, the final answer is always the same: it equals the sum of the two starting numbers multiplied by 5. The extra steps cancel out.
- 2: Yes — the process is designed so that the added constants ($36$) and the subtracted ($720$) scale consistently when multiplied by $20$. This removes their influence and leaves only a multiple of your original numbers.
- 3: Let the two numbers be $a$ and $b$. Start with $5a$, add $36$: $5a + 36$. Multiply by $20$: $20(5a + 36) = 100a + 720$. Add $b$: $100a + b + 720$. Subtract $720$: $100a + b$. Since both $a$ and $b$ were 2-digit numbers, the final result is always a predictable combination of them. The constants are eliminated, which is why the pattern works every time.