Subtraction as an unknown addend

Why this exists

A child who has learned taking away can find 13 - 9 by making 13, removing 9 one at a time and counting what is left. It works, but it is slow, and with that many to move or count backwards it is easy to lose track.

There is a second way to see the same question. Instead of taking 9 away, ask: “9 and how many more make 13?” Now the child starts at 9 and counts up: 10, then 11, 12, 13. That is four steps, and four is the answer. A subtraction has turned into an addition with a missing part, and children usually know their addition facts before their subtraction.

Some stories ask for a missing part from the start. “I have 8 stickers and I want 11. How many more do I need?” Nothing is being taken away, yet the answer is 11 - 8. Seeing subtraction as a missing part is what lets a child recognize that this, too, is a subtraction.

Definition

Subtraction finds a missing part. When we know a whole and one of its two parts, subtracting the known part from the whole gives the other part. So

asks the same thing as

Read the first as “nine take away four” and the second as “four and what make nine”. Both have the same answer, 5. The box stands for a number we have not found yet, and it can sit anywhere in the sentence.

A number bond shows why. The whole, 9, sits in the top circle. One part, 4, sits below it, and the other part is empty. Taking 4 away from 9 leaves exactly that other part. Joining 4 with that other part makes 9. Either way, the empty circle is the number we want.

Intuition

Children who already know parts and the whole have most of this idea. A whole is made of two parts, and if you know the whole and one part, the other part is fixed. Taking away and “what goes with” are two ways of looking at the same picture.

To a child the missing part usually shows up as a hiding game. Show 8 counters, cover some with a cup, and let the child see the 3 that are still out. “How many are under the cup?” The child is finding 8 - 3, but the question in their head is “3 and what make 8?”

Two ways of finding the missing part come naturally. With counters, the child starts from the part they can see and adds counters one at a time until there are 8, counting the ones they add. On a number line, they start at the part and hop forward to the whole, counting the hops. Both are counting on, used to find a gap rather than a total. The strategy has its own lesson later, counting up to find a difference, for bigger numbers.

Results

Worked examples

Boundaries

The known part cannot be bigger than the whole. 6 + \square = 4 has no whole-number answer, because 6 is already more than 4 and joining makes more, never less. It is the same question as 4 - 6, which cannot be done by taking away either. Negative numbers give it an answer much later.

The whole is the number we start from. 9 - 4 asks “4 and what make 9?”, not “9 and what make 4?” Swapping the numbers in a subtraction changes the question, which is why subtraction has an order.

Counting up is always correct, but not always quick. It is short when the two numbers are close, as in 13 - 9. When the number taken away is small, as in 13 - 2, counting back is shorter. Both find the same missing part, because there is only one.

Where this goes next

With this idea a child can use every addition fact they know to subtract. The facts themselves grow from subtracting within ten to the facts within twenty, and stopping at ten works in both directions: counting up to 10 first, as in the fourth example, or taking away down to 10 first, as in bridging back through ten.

The four facts that one whole and its two parts give, two additions and two subtractions, are gathered into fact families. Stories where the missing number is the change or the start are problems with the change missing and problems with the starting amount missing. For bigger numbers, counting up becomes a written method, counting up to find a difference, and in shops it is how change is counted out. Much later, the same thinking solves equations with a letter where the box is, in one-step equations.

Exercises

Find the missing part

Use the bonds for Exercises 1 to 3. What number goes in the box? Then say the take-away sentence.

1. Bond A.
2. Bond B.
3. Bond C.

Think addition to subtract

For each one, ask “what goes with the smaller number to make the bigger one?”

4. 8 - 5
5. 10 - 6
6. 9 - 7
7. 3 + \square = 10
8. 14 - 9
9. 12 - 8

Stories and puzzles

Read each one aloud and show it with counters, fingers or a number line.

10. “Ella has 5 blocks. She needs 9 blocks to build her tower. How many more blocks does she need?”
11. “There are 8 pets in the shop. 3 are cats. The rest are dogs. How many dogs are there?”
12. “There are 11 counters in all. You can see 6. The rest are under a cup. How many are under the cup?”
13. “I am thinking of a number. I take away 4 and I have 6 left. What is my number?”

Explain

14. To find 10 - 8, Lily counts up from 8: “9, 10.” She says the answer is 2. How does she know?
15. Max says 9 - 4 = 5 because 4 + 5 = 9. Show with counters why that works.
16. To find 15 - 13, would you take away 13 one at a time, or count up from 13? Which is quicker, and why? What is the answer?
17. Can any number of counters go in the box in 6 + \square = 4? How do you know?