ACARA v9 CONTENT DESCRIPTION “multiply and divide one- and two-digit numbers, representing problems using number sentences, diagrams and arrays, and using a variety of calculation strategies”
Multiplication begins as a shortcut for adding the same number again and again. Four plates with three apples each is 3 and 3 and 3 and 3, and rather than write all that out we say four threes, or 4 times 3. The word that matters is equal: every group holds the same amount, and that is exactly what lets a single multiplication stand in for a whole string of additions. Division is the same picture read backwards, asking how a total breaks into equal parts — so the two operations grow from one idea.
The array
Set the rows and the columns. The dots are the product, and the grid never lies.
3 rows of 4 is 12. Read it down the columns and it is 4 groups of 3 — the same 12.
The array shows it all at once
An array lines those equal groups up into neat rows and columns, and the whole product becomes visible at a glance. Three rows of four dots is twelve dots, and you can read it as three fours across or four threes down. That is why the array makes the order of the factors stop mattering: give it a quarter turn and rows become columns, yet the count never changes. Seeing 3 times 4 and 4 times 3 land on the same twelve is a child’s first proof that multiplication can be done in either order.
Equal groups
Add one equal group at a time and watch the product climb by the same step.
2 = 2, the same as 1 × 2. Each new group adds another 2.
Building a product group by group
Watching a product grow one group at a time keeps the meaning fastened to the number. Start with a single group of five, then two groups make ten, three make fifteen, and each new group lifts the total by the same step of five. The repeated addition underneath and the multiplication on top are one fact in two outfits, and feeling the total jump by equal amounts is precisely what readies a child to skip count. Build the product before naming it, and the name has something solid to hold on to.
Skip counting
Each hop is one more group. The hop you land on is that many times the step.
Counting by 2s, the first jump lands on 2 — that is 1 × 2.
The times table is skip counting
A times table is nothing more exotic than skip counting written down. Counting by threes — three, six, nine, twelve — touches every multiple of three in order, and each landing is one more jump, so the fifth jump sits on five threes. Drawn on a number line the table becomes a row of evenly spaced hops, and the answer to 5 times 3 is simply where the fifth hop finishes. Memorising the tables can come later; meeting them first as equal jumps is what makes the memorising make sense.
The fact family
One array, four sentences. Know any one and the other three come free.
The 3 by 4 array holds all four facts: two multiplications and two divisions built from 3, 4 and 12.
Multiplication and division undo each other
Every array carries four number sentences inside it, and together they make one fact family. The three-by-four array says 3 times 4 is 12 and 4 times 3 is 12, and read the other way it says 12 shared into 3 rows is 4, and 12 split into groups of 4 is 3. Multiplication and division undo each other, so a child who knows one of the four facts is handed the other three for nothing. Owning the family of 3, 4 and 12 means those two divisions never arrive as strangers.
Share or group
Both readings of a division reach the same answer from opposite directions.
Sharing: deal 12 into 3 groups and each group gets 4.
Division shares or groups
Division wears two faces that arrive at the same answer. Sharing deals a total out one at a time into a fixed number of groups and asks how many land in each — twelve counters into three bowls gives four each. Grouping instead asks how many equal groups of a fixed size the total makes — twelve counters taken three at a time makes four groups. The trio 12, 3 and 4 stays together whichever way you read it, which is why both questions are written as 12 divided by 3, and why both are honestly division.
Split to multiply
Break the two-digit number by place value, multiply each part, then recombine.
13 is 10 and 3. Multiply each piece by 4, then add.
Split a big number to multiply it
Once a factor grows past the times tables, place value lends a hand again. To work out 4 times 13, split the 13 into 10 and 3, multiply each part, then add: 4 tens is 40, 4 threes is 12, and together that is 52. The array picture makes this plain, because a four-by- thirteen block can be sliced into a four-by-ten block beside a four-by-three block. Breaking one factor into friendly pieces turns a hard multiplication into two easy ones, and it is the seed of the written methods that Year 4 will grow.
Quick self-check
1. What is 4 × 3?
2. What is 6 × 5?
3. Share 20 counters equally among 4 children. How many does each child get?
4. A fact family is built from 6 × 4 = 24. Which division fact belongs to it?