The laws of arithmetic
🎯 In this topic you will
- Use the laws of arithmetic to simplify calculations.
- Understand the order of operations and use brackets to change the order of operations.
🧠 Key Words
- associative rule
- commutative rule
- order of operations
- brackets
- distributive rule
Show Definitions
- associative rule: A property of addition and multiplication stating that the grouping of numbers does not change the result, such as $(a+b)+c=a+(b+c)$.
- commutative rule: A property of addition and multiplication stating that changing the order of numbers does not change the result, such as $a+b=b+a$.
- order of operations: A set of rules that determines the correct sequence for performing calculations in a mathematical expression.
- brackets: Symbols such as $( )$ used in expressions to indicate that the enclosed calculations must be performed first.
- distributive rule: A property that allows multiplication to be applied to each term inside brackets, such as $a(b+c)=ab+ac$.
📏 Following Rules in Mathematics
A rule tells you what you can and cannot do. In mathematics, rules help us decide the correct way to carry out calculations so that everyone gets the same answer.
➗ Why Order Matters in Calculations
I n mathematics, there is a rule about the order of operations. This rule tells us to perform multiplication and division before addition and subtraction when solving a calculation.
🧠 Different Ways to Calculate
A run and Marcus are both using the numbers 5, 6 and 7 in their calculations. Arun wants to multiply 6 and 7 first and then add the result to 5, which gives the calculation $5 + 6 \times 7 = 47$. Marcus wants to add 5 and 6 first and then multiply the result by 7. In this section, you will learn how to write Marcus’s calculation correctly.
💡 Quick Math Tip
Brackets Change the Order: In a calculation, brackets tell you which part must be solved first. Without brackets, multiplication and division are done before addition and subtraction, but brackets can change this order and produce a different result.
❓ EXERCISES
1. Each learner is thinking of a number. Draw a diagram and write a calculation to show how to work out their numbers.
a. Tariq is thinking of a number. He adds $7$ to his number, then divides by $10$. His answer is $1$. What number is Tariq thinking of?
b. Sonja is thinking of a number. She adds $5$ to her number, then divides by $2$. Her answer is $6$. What number is Sonja thinking of?
c. Pierre is thinking of a number. He multiplies his number by $3$, then subtracts $2$. His answer is $4$. What number is Pierre thinking of?
d. Lan is thinking of a number. She divides her number by $3$, then adds $11$. Her answer is $14$. What number is Lan thinking of?
👀 Show answer
b. $(x + 5) \div 2 = 6$. So $x + 5 = 12$ and $x = 7$.
c. $3x - 2 = 4$. So $3x = 6$ and $x = 2$.
d. $(x \div 3) + 11 = 14$. So $x \div 3 = 3$ and $x = 9$.
2. Calculate.
a. $(5 + 2) \times 3$
b. $(3 \times 6) + 4$
c. $3 \times (8 - 5)$
d. $(8 - 6) \times 4$
e. $(3 + 7) \div 10$
f. $(12 + 6) \div 3$
One of the calculations gives the same answer even if the brackets are removed. Which calculation is it? Check your answers with your partner.
👀 Show answer
b. $22$
c. $9$
d. $8$
e. $1$
f. $6$
The calculation that gives the same answer without brackets is b because multiplication is done before addition anyway.
3. Are the following statements true or false? If a statement is false, write it out correctly.
a. $6 + 3 \times 4 = 18$
b. $(6 + 3) \times 4 = 36$
c. $(6 + 3) \times 4 = 18$
👀 Show answer
b. True because $6 + 3 = 9$ and $9 \times 4 = 36$.
c. False. Correct statement: $(6 + 3) \times 4 = 36$.
4. Put brackets in these calculations to make them correct.
a. $6 + 2 \times 5 = 40$
b. $3 + 4 \times 2 + 4 = 42$
c. $3 \times 4 + 2 = 18$
d. $4 + 3 + 2 \times 2 = 18$
👀 Show answer
b. $(3 + 4) \times (2 + 4) = 42$
c. $3 \times (4 + 2) = 18$
d. $(4 + 3 + 2) \times 2 = 18$
5. Use these numbers together with brackets and operation signs to make the target number.
Example: $3, 4, 6$ Target $42$ Answer $(3 + 4) \times 6$
a. $2, 5, 5$ Target $35$
b. $5, 7, 10$ Target $20$
c. $2, 5, 14$ Target $18$
👀 Show answer
b. $(10 - 5) \times 4 = 20$ (one possible answer using the numbers creatively)
c. $(14 + 2) + 2 = 18$ (one possible valid expression)
6. $42 \times 24$ is equivalent to $42 \times 2 \times 12$. Find three more ways to multiply $42 \times 24$. Write your answers in the form $42 \times \square \times \square$. Choose one way to do the calculation.
👀 Show answer
7. Use the distributive rule to calculate the following showing all the stages of your working.
a. $5 \times (70 + 1)$
b. $6 \times (60 - 3)$
c. $7 \times (90 + 2)$
d. $8 \times (40 - 3)$
👀 Show answer
b. $6 \times (60 - 3) = 6 \times 60 - 6 \times 3 = 360 - 18 = 342$
c. $7 \times (90 + 2) = 7 \times 90 + 7 \times 2 = 630 + 14 = 644$
d. $8 \times (40 - 3) = 8 \times 40 - 8 \times 3 = 320 - 24 = 296$
8. Use the distributive rule to work out these calculations. Show all your working.
a. $3 \times 67$
b. $8 \times 93$
c. $7 \times 48$
d. $9 \times 79$
👀 Show answer
b. $8 \times 93 = 8 \times (90 + 3) = 720 + 24 = 744$
c. $7 \times 48 = 7 \times (40 + 8) = 280 + 56 = 336$
d. $9 \times 79 = 9 \times (70 + 9) = 630 + 81 = 711$
9. Are the following statements true or false? Explain your decisions to your partner.
a. $8 + 5 - 7 = 8 + 7 - 5$
b. $2 \times (3 + 4) = 2 \times 3 + 4$
c. $10 \times 5 \div 2 = 10 \times (5 \div 2)$
👀 Show answer
b. False because $2 \times (3 + 4) = 14$ but $2 \times 3 + 4 = 10$.
c. True because $10 \times 5 \div 2 = 50 \div 2 = 25$ and $10 \times (5 \div 2) = 10 \times 2.5 = 25$.
🧠 Think like a Mathematician
Challenge: You have a set of cards containing the numbers $1, 2, 3, 4$ and the operation symbols $+$, $-$, $\times$, $\div$, as well as brackets $(\ )$.
You may use as many of these cards as you like to try to make the numbers from $11$ to $20$.
You are not allowed to make two-digit numbers. For example, $12 + 3$ is not allowed because $12$ is a two-digit number card.
Your goal is to use all four number cards ($1,2,3,4$) in a calculation.
Question: How many numbers between $11$ and $20$ can you make using all four number cards in the calculation?
Show Answers
- Many numbers from $11$ to $20$ can be created by combining $1,2,3,4$ with operations and brackets.
- Examples include:
- $(4 \times 3) + (2 \times 1) = 14$
- $(4 + 3) \times (2 + 1) = 21$ (too large, but shows how results change with brackets)
- $(4 \times 3) + (2 + 1) = 15$
- $(4 + 3) \times 2 + 1 = 15$
- $(4 \times 3) + 2 + 1 = 15$
- Students should explore systematically to find which targets between $11$ and $20$ are possible.