Exploring multiplication and division
🎯 In this topic you will
- Recognise multiples of 2, 5, and 10.
- Make and use multiplication and division fact families.
- Multiply single-digit and two-digit numbers by 10.
🧠 Key Words
- array
- commutative
- multiple
- pattern
- sequence
- term
- term-to-term rule
Show Definitions
- array: A way of showing numbers using rows and columns to represent multiplication.
- commutative: A property of multiplication where changing the order of the numbers does not change the answer.
- multiple: A number obtained by multiplying another number by a whole number.
- pattern: A repeated or predictable arrangement of numbers, shapes, or objects.
- sequence: An ordered list of numbers that follows a specific rule.
- term: Each individual number in a sequence.
- term-to-term rule: The rule that describes how to get from one term in a sequence to the next.
Understanding Multiples and Fact Families
K Knowing the pattern of the multiples of a number helps you to remember them or work them out. You can connect multiplication and division facts in a fact family in the same way that you connect addition and subtraction facts.
❓ EXERCISES
1. Draw a ring around all the multiples of $2$.
👀 Show answer
2. Why are multiples of $2$ also even numbers?
👀 Show answer
3. Sort these numbers into the correct place on the Venn diagram.
$45$, $120$, $132$, $401$, $740$, $215$, $805$, $490$, $96$, $387$, $350$, $675$.
👀 Show answer
Multiples of $5$ only: $45$, $215$, $805$, $675$.
Multiples of $10$ (also multiples of $5$): $120$, $740$, $490$, $350$.
Not multiples of $5$ or $10$: $132$, $401$, $96$, $387$.
🧠 Think like a Mathematician
Sofia says that if you are sorting multiples of $2$, $5$ or $10$ in a Venn diagram, the multiples of $10$ always belong in the overlap and it does not matter which two sets of multiples you sort.
Question: Do you agree with Sofia? How do you know?
Show Answer
Yes, Sofia is correct. Every multiple of $10$ is also a multiple of $2$ and a multiple of $5$, because $10 = 2 \times 5$. This means any number that is a multiple of $10$ will always fit into the overlap of the two sets, no matter which two of the sets ($2$, $5$, or $10$) are being compared.
❓ EXERCISES
4. Write the fact family for this array.
👀 Show answer
The array shows $6$ rows and $5$ columns, making $30$ stars.
$6 \times 5 = 30$
$5 \times 6 = 30$
$30 \div 5 = 6$
$30 \div 6 = 5$
5. Monifa wrote the fact family for $3 \times 10$:
$3 \times 10 = 30$, $10 \times 3 = 30$, $30 = 3 \times 10$, $30 = 10 \times 3$,
$30 \div 10 = 3$, $30 = 10 \div 3$, $30 \div 3 = 10$, $30 = 3 \div 10$.
Is Monifa correct?
👀 Show answer
No, Monifa is not completely correct.
The correct fact family is:
$3 \times 10 = 30$
$10 \times 3 = 30$
$30 \div 10 = 3$
$30 \div 3 = 10$
Statements such as $30 = 10 \div 3$ and $30 = 3 \div 10$ are incorrect.
🧠 Think like a Mathematician
How is finding a fact family for a multiplication fact the same as finding a fact family for an addition fact?
How is it different?
Show Answer
Finding a fact family is the same in both cases because you use the same three numbers and show how they are related using different operations.
It is different because addition fact families use addition and subtraction, while multiplication fact families use multiplication and division. In multiplication fact families, the numbers grow more quickly, and division is needed to work backwards instead of subtraction.
❓ EXERCISES
6. Choose three single-digit numbers and three 2-digit numbers to multiply by $10$. Record your multiplications.
👀 Show answer
$3 \times 10 = 30$
$7 \times 10 = 70$
$9 \times 10 = 90$
$12 \times 10 = 120$
$25 \times 10 = 250$
$48 \times 10 = 480$
7. Explain what happens to a single-digit number and a 2-digit number when it is multiplied by $10$.
👀 Show answer
When a number is multiplied by $10$, its digits move one place to the left, making the number ten times larger.
8. A school has $23$ boxes of $10$ pencils. How many pencils does the school have? Show your method.
👀 Show answer
$23 \times 10 = 230$
The school has $230$ pencils.
9. The term-to-term rule is: the next term is $5$ more than the previous term. What are the next four numbers in the sequence?
$6$, $11$, ___, ___, ___, ___
👀 Show answer
The next four numbers are $16$, $21$, $26$, and $31$.