Multiplying and dividing integers
🎯 In this topic you will
- Multiply and divide with positive and negative integers
- Multiply and divide integers, especially when both are negative
- Understand that brackets, indices, and operations follow a particular order
🧠 Key Words
- brackets
- conjecture
- inverse
- investigate
- product
Show Definitions
- brackets: Symbols used in mathematics to group parts of an expression, usually to show which operations to do first.
- conjecture: A statement believed to be true based on patterns or reasoning, but not yet proven.
- inverse: An operation that reverses the effect of another, such as subtraction undoing addition.
- investigate: To explore or examine a mathematical idea or problem in detail to understand it better.
- product: The result you get when two or more numbers are multiplied together.
✖️ Multiplying positive and negative integers
3 × 4 = 3 + 3 + 3 + 3 = 12
In a similar way, –3 × 4 = –3 + –3 + –3 + –3 = –12.
5 × 2 = 2 + 2 + 2 + 2 + 2 = 10
In a similar way, 5 × –2 = –2 + –2 + –2 + –2 + –2 = –10.
➗ Division as the inverse of multiplication
Division is the inverse operation of multiplication.
3 × 4 = 12 So 12 ÷ 4 = 3.
This is also true when you divide a negative integer by a positive integer.
–3 × 4 = –12 So –12 ÷ 4 = –3.
The product of a positive and a negative integer is always negative.
For example, 6 × –9 = –54 and –4 × 12 = –48.
The same is true for division. When one integer is positive and the other is negative, the answer is negative. For example, –48 ÷ 6 = –8 and 63 ÷ –7 = –9.
❓ EXERCISES
1. Work out:
- a) $3 \times -2$
- b) $5 \times -7$
- c) $10 \times -4$
- d) $6 \times -6$
👀 Show answer
b) $-35$
c) $-40$
d) $-36$
2. Work out:
- a) $-15 \div 3$
- b) $-30 \div 6$
- c) $-24 \div 4$
- d) $27 \div -9$
👀 Show answer
b) $-5$
c) $-6$
d) $-3$
3. Work out the missing numbers.
- a) $9 \times \square = -18$
- b) $5 \times \square = -30$
- c) $-2 \times \square = -14$
- d) $-8 \times \square = -40$
👀 Show answer
b) $-6$
c) $7$
d) $5$
4. Work out the missing numbers.
- a) $-12 \div \square = -3$
- b) $18 \div \square = -9$
- c) $\square \div 4 = -4$
- d) $\square \div 10 = -2$
👀 Show answer
b) $-2$
c) $-16$
d) $-20$
5. The product of two integers is $-10$.
Find the possible values of the two integers.
👀 Show answer
6. Copy and complete this multiplication table.
| × | $-3$ | $-5$ |
|---|---|---|
| $5$ | ||
| $7$ |
👀 Show answer
| × | $-3$ | $-5$ |
|---|---|---|
| $5$ | $-15$ | $-25$ |
| $7$ | $-21$ | $-35$ |
7. Estimate the answers to these calculations by rounding to the nearest whole number.
- a) $-3.2 \times 6.8$
- b) $9.8 \times -5.35$
- c) $-16.1 \div 1.93$
- d) $7.38 \div -1.86$
👀 Show answer
b) $10 \times -5 = -50$
c) $-16 \div 2 = -8$
d) $7 \div -2 = -3.5 \approx -4$
8. Estimate the answers to these calculations by rounding the numbers.
- a) $-53 \times 39$
- b) $32 \times -61$
- c) $-38 \times 9.3$
- d) $493 \div -5.1$
👀 Show answer
b) $30 \times -60 = -1800$
c) $-40 \times 9 = -360$
d) $500 \div -5 = -100$
9. Work out these calculations. Do the calculation in the brackets first.
- a) $3 \times (-6 + 2)$
- b) $-4 \times (-1 + 7)$
- c) $5 \times (-2 - 4)$
- d) $-2 \times (3 - (-7))$
👀 Show answer
b) $-4 \times 6 = -24$
c) $5 \times (-6) = -30$
d) $-2 \times 10 = -20$
10.
- a) Copy and complete these divisions. For example, $-20 \div 2 = -10$.
- b) Can you add any more lines to the diagram? You must divide by a positive integer. The answer must be an integer.
- c) Draw a similar diagram with $-28$ in the centre.
- d) Compare your answer to part c with a partner’s. Do you agree?
👀 Show answer
b) No other positive integers divide $-20$ to give whole number answers.
c) Possible answers for $-28$: divide by $2$, $4$, $7$, $14$ → gives $-14$, $-7$, $-4$, $-2$
d) Discuss with your partner if they used the same divisors or different ones.
11. In these diagrams, the integer in a square is the product of the integers in the circles next to it. For example, $-3 \times 4 = -12$.
Copy and complete the diagrams.
👀 Show answer
Left square: $-3 \times 2 = -6$.
Right square: $4 \times -5 = -20$.
b) Top right circle: $-18 \div 6 = -3$.
Bottom right circle: $-12 \div -3 = 4$.
Bottom square: $-5 \times 4 = -20$.
Left square: $6 \times -5 = -30$.
🧠 Think like a Mathematician
Question: Can you complete a diagram using integers that satisfy all given operations? How many solutions are possible?
Equipment: Pencil, paper
- Copy the diagram below, placing circles in the corners and squares on the sides.
- The numbers inside the circles must be integers.
- The number in each square is the result of subtracting the left circle from the right circle.
- Use the known values in the squares to find numbers that satisfy the relationships.
- Explore whether more than one valid solution exists.
Follow-up Questions:
Show Answers
- 1: One possible solution is top-left = $2$, top-right = $-5$, bottom-right = $6$, bottom-left = $−4$.
- 2: Yes, multiple solutions are possible. Adding a constant to all four circle values keeps the differences the same.
- 3: Choose any starting value for one circle (e.g., top-left), then use the subtraction relationships to find the others. The system is underdetermined, so you can generate a family of solutions using the same relative differences.
🍬 Learning Bridge
Now that you've practised multiplying and dividing with both positive and negative numbers, it's time to take things a step further. You'll explore what happens when both numbers are negative — and learn how to solve problems that follow a specific order, using brackets, indices, and operation rules to guide you.
➕➖ Multiplying and dividing integers
You can add and subtract any two integers.
For example:
2 + –4 = –2 –2 + –4 = –6 –2 – 4 = –6 –2 – –4 = 2
You can also multiply and divide a negative integer by a positive one.
For example:
2 × –9 = –18 –6 × 3 = –18 –18 ÷ 3 = –6 20 ÷ –5 = –4
In this section you will investigate how to multiply or divide any two integers. You will use number patterns to do this.
🧠 Think like a Mathematician
Question: What patterns can you find in sequences of multiplications involving negative integers?
Equipment: Pencil, paper, calculator
- Study this multiplication sequence:
- $-3 \times 4 = -12$
- $-3 \times 3 = -9$
- $-3 \times 2 = -6$
- a) Copy the sequence and write six more terms. Use a pattern to fill in the answers.
- b) Describe the patterns in the sequence.
- Now try this second multiplication sequence:
- $-5 \times 4 = -20$
- $-5 \times 3 = -15$
- $-5 \times 2 = -10$
- c) Copy the sequence and write six more terms. Describe any patterns in the sequence.
- d) In the sequences in parts a and c, you have some products of two negative integers. What can you say about the product of two negative integers?
- e) Make up a sequence of your own like the ones in a and c.
- f) Share your answers to parts d and e with a partner. Are your partner’s sequences correct?
Follow-up Questions:
Show Answers
- 1: Each time the second factor decreases by 1, the product increases by 3 (for −3 × ...), or by 5 (for −5 × ...). The results form arithmetic sequences.
- 2: The product of two negative numbers is a positive number. For example, $-3 \times -2 = 6$.
- 3: Example sequence: $-4 \times 5 = -20$, $-4 \times 4 = -16$, ..., leading to $-4 \times -1 = 4$, $-4 \times -2 = 8$, etc.
❓ EXERCISES
12. Work out these multiplications.
- a) $5 \times -2$
- b) $-5 \times 2$
- c) $-5 \times -2$
- d) $-2 \times -5$
👀 Show answer
b) $-10$
c) $10$
d) $10$
13. Work out these multiplications.
- a) $-6 \times -4$
- b) $-7 \times -7$
- c) $-10 \times -6$
- d) $-8 \times -11$
👀 Show answer
b) $49$
c) $60$
d) $88$
14. Copy and complete this multiplication table.
👀 Show answer
Middle row: $-3 \times -5 = 15$, already given: $-3 \times 3 = -9$, $-3 \times -8 = 24$
Bottom row: already given: $-6 \times -5 = 30$, $-6 \times 3 = -18$, $-6 \times -8 = 48$
15. Work out:
- a) $(3 + 5) \times -4$
- b) $(-3 + -5) \times -6$
- c) $-4 \times (5 - 8)$
- d) $-6 \times (-2 - -7)$
💡 Tip
Do the calculation in brackets first.
👀 Show answer
b) $-8 \times -6 = 48$
c) $-4 \times -3 = 12$
d) $-6 \times 5 = -30$
16. Round these numbers to the nearest whole number to estimate the answer.
- a) $3.9 \times -6.8$
- b) $-11.2 \times 2.95$
- c) $(-6.1)^2$
- d) $(-4.88)^2$
👀 Show answer
b) $-11 \times 3 = -33$
c) $(-6)^2 = 36$
d) $(-5)^2 = 25$
17.a) Put these multiplications into groups based on the answers:
$3 \times -4$, $-6 \times -2$, $12 \times 1$, $-4 \times -3$, $2 \times -6$, $-12 \times -1$
b) Find one more product to put in each group.
👀 Show answer
$-6 \times -2 = 12$, $12 \times 1 = 12$, $-4 \times -3 = 12$, $-12 \times -1 = 12$
Extra: $-3 \times -4 = 12$
Group 2 (negative results):
$3 \times -4 = -12$, $2 \times -6 = -12$
Extra: $-1 \times 12 = -12$
18. These are multiplication pyramids.
Each number is the product of the two numbers below it. For example, in a, $2 \times -4 = -8$.
Copy and complete the multiplication pyramids.
👀 Show answer
b) Middle: $-3 \times 5 = -15$, $5 \times -1 = -5$ → Top: $-15 \times -5 = 75$
c) Middle: $-4 \times -5 = 20$, $-5 \times -2 = 10$ → Top: $20 \times 10 = 200$
19.a) Draw a multiplication pyramid like those in Question 8, with the integers $-2$, $3$ and $-5$ in the bottom row, in that order. Complete your pyramid.
b) Is this idea correct?
Idea: If you change the order of the bottom numbers, the number at the top of the pyramid is the same.
Test this idea by changing the order of the numbers in the bottom row of your pyramid.
👀 Show answer
b) Try reordering (e.g. $3, -5, -2$ → $3 \times -5 = -15$, $-5 \times -2 = 10$, then $-15 \times 10 = -150$). Result is different, so this idea is not correct.
20. Find the missing numbers in these multiplications.
- a) $-3 \times \square = -12$
- b) $-5 \times \square = 45$
- c) $\square \times -6 = 24$
- d) $\square \times -10 = 80$
👀 Show answer
b) $-9$
c) $-4$
d) $-8$
🧠 Think like a Mathematician
Question: How can multiplication statements be written as divisions? Can you spot consistent rules when dividing by negative integers?
Equipment: Pencil, paper
🔎 Reasoning Tip
Conjecture: A conjecture is a possible value based on what you already know.
- A multiplication can be written as a division. For example:
$5 \times 8 = 40$ can be written as:
$40 \div 8 = 5$ or $40 \div 5 = 8$ - a) Here is a multiplication: $-4 \times 6 = -24$
Write it as a division in two different ways. - b) Write a multiplication of a positive integer and a negative integer. Then write it as a division in two different ways.
- c) Here is a multiplication: $-7 \times -2 = 14$
Write it as a division in two different ways. - d) Write a multiplication of two negative integers. Then write it as a division in two different ways.
- e) Can you make a conjecture about the answer when you divide an integer by a negative integer? Test your conjecture.
- f) Compare your answer with a partner’s. Have you made the same conjectures?
Follow-up Questions:
Show Answers
- 1:$-24 \div 6 = -4$ and $-24 \div (-4) = 6$
- 2: When dividing by a negative number, the result changes sign. For example, $12 \div (-3) = -4$
- 3:Conjecture: Dividing a positive number by a negative number gives a negative result; dividing two negative numbers gives a positive result.
❓ EXERCISES
21. Work out these divisions.
- a) $18 \div -6$
- b) $-28 \div -4$
- c) $30 \div -6$
- d) $-30 \div -10$
- e) $42 \div -6$
- f) $-24 \div -4$
- g) $60 \div -5$
- h) $-25 \div -5$
👀 Show answer
b) $7$
c) $-5$
d) $3$
e) $-7$
f) $6$
g) $-12$
h) $5$
22. Here are three multiplication pyramids. Copy and complete each pyramid.
💡 Tip
Remember, division is the inverse of multiplication so you will divide as you work down the pyramid.
👀 Show answer
b) Bottom left: $12 \div -2 = -6$ → Bottom right: $-8 \div -2 = 4$ → Top: $12 \times -8 = -96$
c) Middle: $-200 \div -20 = 10$ → Bottom middle: $-20 \div -4 = 5$ → Bottom left: $10 \div 5 = 2$
23. Work out:
- a) $(3 \times -4) \div -2$
- b) $(2 - 20) \div -3$
- c) $(-3 + 15) \div -4$
- d) $24 \div (2 \times -4)$
👀 Show answer
b) $-18 \div -3 = 6$
c) $12 \div -4 = -3$
d) $24 \div -8 = -3$
24. Find the value of $x$.
- a) $x \div -4 = 8$
- b) $x \div -3 = -15$
- c) $16 \div x = -2$
- d) $-15 \div x = 3$
👀 Show answer
b) $45$
c) $-8$
d) $-5$
25. Round these numbers to the nearest whole number to estimate the answer.
- a) $-8.75 \div 2.8$
- b) $18.1 \div -5.9$
- c) $-28.2 \div -3.8$
- d) $-35.2 \div -6.9$
👀 Show answer
b) $18 \div -6 = -3$
c) $-28 \div -4 = 7$
d) $-35 \div -7 = 5$
26. Round these numbers to the nearest 10 to estimate the answer.
- a) $-48 \times -29$
- b) $-18.1 \times 61.5$
- c) $-71.4 \div -11.8$
- d) $-99.4 \div 19$
👀 Show answer
b) $-20 \times 60 = -1200$
c) $-70 \div -10 = 7$
d) $-100 \div 20 = -5$