ACARA v9 CONTENT DESCRIPTION “recognise and explain the connection between addition and subtraction as inverse operations, apply to partition numbers and find unknown values in number sentences”
Addition and subtraction are not two separate skills to learn side by side; they are one relationship read in two directions. Put 8 and 5 together and you get 13; take 5 away from 13 and you are back to 8. Each operation undoes the other, which is what mathematicians mean by inverse. Seeing this turns a wall of number facts into a small set of families: once a child knows 8 + 5 = 13, they already know 5 + 8, 13 - 8 and 13 - 5 without learning them separately. This unit makes the relationship visible and then puts it to work finding unknown numbers.
The fact family
Three numbers make a family of four facts. The bar shows why.
One trio, one bar. Reveal the next fact — the same three numbers, rearranged.
One trio makes a whole family
A part-part-whole bar shows why the four facts belong together: the whole sits on top, the two parts beneath it. Read the bar as parts joining and it is addition; read it as the whole splitting and it is subtraction. The three numbers never change — only the direction you read them. This is the single most efficient idea in early number, because it roughly halves how many facts must be memorised and it gives a child a way to recover a fact they have forgotten: if 13 - 5 has slipped their mind, the partner 5 + 8 = 13 brings it straight back.
Build and undo
Every addition has an opposite that puts the counter back. Try it.
Add a number, then press its opposite. The counter returns: that is the inverse.
Every step has an undo
The counter makes inverse physical: add five and the counter climbs; subtract five and it returns to exactly where it began. The opposite operation with the same number always lands you back at the start, which is the property that lets us check work and solve for unknowns. It is the same idea as a door that opens and closes, or steps walked forward and then back. Children who feel this reversibility stop treating subtraction as a brand new and harder thing; it is simply addition walked backwards.
Find the unknown
A box stands for a missing number. Use the inverse to set it free.
What number belongs in the box to make 7 + _ = 16 true?
A box for the missing number
A number sentence with a box, like 7 + _ = 16, is the first algebra a child meets: a symbol standing for a number we do not yet know. The inverse is the key that opens the box. Because something was added to 7 to reach 16, undoing it means subtracting: 16 - 7 = 9. The habit to build is reading the sentence to see what was done to the unknown, then doing the opposite. This is exactly the reasoning that later solves equations, introduced here with friendly numbers and a box instead of a letter.
Check by inverse
To check a subtraction, add the answer back. It should rebuild the start.
Is 16 − 9 = 7 correct? Add the answer back to find out.
Use the inverse to check
Inverse operations give every child a way to mark their own work. After a subtraction, add the answer back: if 16 - 9 = 7 is right, then 7 + 9 must return 16, and it does. If the add-back does not rebuild the start, the answer was wrong and the inverse has caught it. This self-checking is a genuinely powerful habit, far better than hoping an answer is correct, and it works both ways — a doubtful addition can be checked by subtracting one part back out. Good mathematicians check; the inverse is how.
Partition partner
Split a whole into two parts. Fix one part and its partner is decided.
Move the split: the moment you fix one part as 6, the other must be 7, because together they make 13. Knowing the whole and one part finds the other by subtraction.
One part decides the other
When a whole is split into two parts, the parts are not independent: fix one and the other is forced. If 13 is split and one part is 5, the partner can only be 8, because the two must rebuild 13. This is partitioning seen through the inverse — knowing the whole and one part finds the missing part by subtraction. It is the structure behind word problems where a total and one group are known, and it is why a child who understands part-part-whole can set up the calculation themselves rather than guessing whether to add or subtract.
Which operation?
Pick the inverse that frees the box, then read off the answer.
To find the box in 7 + _ = 16, do you add or subtract?
Choose the inverse, find the answer
The final skill ties it together: look at a number sentence, decide which operation acts on the box, and apply its inverse. If the sentence adds to the unknown, subtract to find it; if it subtracts from the unknown, add. Choosing the right inverse is the whole game, and it rests on understanding the relationship rather than memorising a procedure. With fact families, reversibility, boxes, checking and partitioning all in hand, addition and subtraction become a single flexible tool — and the multiplication and division facts of the next Algebra unit follow the very same inverse idea.
Quick self-check
1. Which fact belongs to the same family as 8 + 5 = 13?