Revisiting multiplication and division
🎯 In this topic you will
- Create a multiplication square.
- Read multiplication and division facts from a multiplication square.
- Use a multiplication square to generate number sequences.
🧠 Key Words
- commutative
- diagonal
- extend
- product
Show Definitions
- commutative: A property of multiplication where changing the order of numbers does not change the answer (for example, $3 \times 4 = 4 \times 3$).
- diagonal: A line or pattern that runs from one corner to the opposite corner of a square or grid, often showing repeated values in a multiplication square.
- extend: To make something longer or larger, such as continuing a number pattern or expanding a multiplication square.
- product: The result you get when two or more numbers are multiplied together.
🔢 Exploring the Multiplication Square
A multiplication square helps you see all of the products in one place. You will use a multiplication square to help you find multiplication and division facts, fact families, and sequences. There are many patterns in the multiplication tables.
❓ EXERCISES
$1.$ Write the missing products in the highlighted squares.
👀 Show answer
$4 \times 2 = 8$
$3 \times 8 = 24$
$6 \times 3 = 18$
$7 \times 9 = 63$
$8 \times 6 = 48$
$9 \times 4 = 36$
$2.$ When you draw a diagonal line on the multiplication square, from the multiplication symbol in the top left-hand corner to the bottom right-hand corner, what can you say about the products that the line passes through?
👀 Show answer
🧠 Think like a Mathematician
Question: Which numbers are shown on the multiplication square only once? Explain why.
👀 Show answer
- The numbers that appear only once are the square numbers, such as $1, 4, 9, 16, 25, 36, 49, 64, 81,$ and $100$.
- This happens because these numbers come from multiplying a number by itself (for example $3 \times 3 = 9$).
- All other products appear twice in the table because multiplication is commutative: $a \times b = b \times a$. These matching pairs sit symmetrically on opposite sides of the diagonal.
- Square numbers lie exactly on the diagonal, so they have no matching partner — which is why each one appears only once.
❓ EXERCISES
$3$. Use a multiplication grid to help you find the numbers in the fact families for $28$ and $72$. Write each fact family.
👀 Show answer
$28$:
$4\times7=28$
$7\times4=28$
$28\div7=4$
$28\div4=7$
$72$:
$8\times9=72$
$9\times8=72$
$72\div9=8$
$72\div8=9$
$4$. Extend each sequence. What is the term-to-term rule for each sequence?
a. $18,27,36,45,54,63,72$
b. $18,23,28,33,38,43,48$
c. $83,91,99,107,115,123,131$
d. $135,126,117,108,99,90,81$
👀 Show answer
a) add $9$
b) add $5$
c) add $8$
d) subtract $9$
$5$. The term-to-term rule is subtract $8$. Start at $108$. What are the next five numbers?
👀 Show answer
$100,\ 92,\ 84,\ 76,\ 68$