Subtraction as taking away
Some of a group goes away. How many are left, and how do we write that down?
Where you are 12 in Addition and subtraction
Why this exists
A child who can count and who has met putting groups together already knows what happens when a group gets bigger. Things also go the other way: apples get eaten, birds fly off, children get off the bus. Taking away is the name for that, and subtraction is the way to write it down.
It is usually the first meaning of subtraction a child meets, and the easiest to act out, because the child can do it with their own hands: make a pile, move some off, count what stays.
Definition
Taking away means starting with a group, removing some of it, and finding how many are left.
We write it with the minus sign, -, read “minus” or “take away”. The number sentence
is read “seven take away three is four” or “seven minus three equals four”. Each number has a job, and they come in the order the story happens:
- 7 is how many there were at the start,
- 3 is how many were taken away,
- 4, after the equals sign, is how many are left.
With counters, as in Figure 1, the child makes the starting group, crosses out or pushes away the ones that leave, and counts the rest. Crossing out is worth teaching early: the taken ones stay visible, so the child can check both numbers afterwards.
Intuition
Taking away is a before-and-after story. Before, there is one group. Something happens. After, there is a smaller group, and the ones that left are somewhere else.
A tower of cubes shows this well (Figure 2). The child builds a tower of 6, lifts 2 off the top and sets them aside, then counts the tower again: 4. Nothing vanished. The 6 cubes are still on the table, split into the 4 that stayed and the 2 that left. That split is the idea behind parts and the whole, and it is why subtraction and addition are so closely tied: put the 2 back and the tower is 6 again.
Results
Taking away leaves fewer. If you take at least one away, fewer are left than you started with. If you take none away, the group stays the same.
Taking away nothing, and taking away everything. For any number of things,
and the same is true with any other number in place of 5.
Putting back undoes taking away. Take 3 away from 7 and 4 are left; put the 3 back and there are 7 again:
Why it is true
Every result above comes from counting the same objects before and after.
The counters that are left were all in the starting group, so there cannot be more of them than there were at the start. If even one was moved away, the ones left are fewer.
If nothing is moved, every counter is still there, so the count has not changed: 5 - 0 = 5. If every counter is moved, none is left, and “none” is the number zero: 5 - 5 = 0.
Taking away does not destroy anything; it only splits the group into the ones that stayed and the ones that left. Join those two groups again and every counter is back, so the count is the starting number. That is why adding checks subtracting.
Worked examples
Ducks on a pond
Read aloud: “There are 5 ducks on the pond. 2 ducks fly away. How many ducks are still on the pond?”
SolutionThe child draws 5 circles for the ducks, because the story starts with 5. Then they cross out 2 of them, because 2 flew away. The circles not crossed out are the ducks still on the pond: 1, 2, 3.
3 ducks are still on the pond. It is worth asking which number in the sentence is the ducks at the start, which is the ducks that flew, and which is the ducks left, so the sentence stays tied to the story.
Ten birds on a wire
Read aloud: “10 birds sit on a wire. 4 fly away. How many are still on the wire?”
Ten is a full ten frame, so fill all ten boxes. Take away 4 by crossing out the last four counters in the bottom row, as in Figure 4.
The child does not need to count the leftovers one by one. The whole top row is still there, and a full row is 5; one more in the bottom row makes 6.
6 birds are still on the wire. The ten frame turns counting into seeing, which is what makes facts like this one easy later on.
A story for a number sentence
Show the child 9 - 6. Ask: “Can you tell me a story that goes with this? Then find the answer.”
SolutionAny story works if it starts with 9 of something and then 6 of them go away. For example: “There were 9 grapes in my bowl. I ate 6. How many are left?”
Now act it out: 9 counters, push 6 to the side, count the rest: 3.
Going from the sentence to a story is the reverse of the first two examples, and it is how you know the child reads - as something that happens, not just a mark between two numbers.
How many popped?
Read aloud: “There were 8 balloons. Some popped. Now there are 5. How many popped?”
This time we know the start and the end, but not how many went away. Draw 8 balloons. Cross them out one at a time, counting the crosses, and stop as soon as 5 are left (Figure 5).
Three crosses were needed, so 3 balloons popped:
Check. Start with 8, take away 3, and 5 are left, just as the story says. A missing number like this can also be found by thinking “5 and how many more make 8?”, which is the idea of subtraction as a missing part.
Checking by putting back
Read aloud: “Leo had 9 stickers. He gave 3 to his sister. He says he has 5 left. Is he right?”
SolutionCheck by putting back. If Leo really has 5 left, then giving back the 3 should make 9 again. But
and 8 is not 9, so Leo is not right. To find the real answer, take 3 away from 9:
Leo has 6 stickers left. Check. 6 + 3 = 9, the number he started with.
Birds fly off twice
Read aloud: “7 birds are on a fence. 2 fly away. Then 3 more fly away. How many birds are left on the fence?”
SolutionTake the story one step at a time. First 2 fly away:
Then 3 more fly away from those 5:
2 birds are left. Now look at it another way: altogether 2 + 3 = 5 birds flew away, and taking all 5 from 7 at once gives the same answer:
Taking away in two steps leaves the same number as taking the whole lot away at once, and later that becomes a strategy: 13 - 5 can be done as 13 - 3, then take away 2 more (bridging back through ten).
Boundaries
You cannot take away more than you have. With 4 counters on the table there is no way to take away 6. When we count whole things, the number taken away must be no bigger than the number at the start. Later, with negative numbers, 4 - 6 does get an answer, but not by taking things away.
The order matters. 7 - 3 means “start with 7, take away 3”. Swapping the numbers gives 3 - 7, a different question, which with whole things has no answer at all. Addition can be swapped; subtraction cannot. That is its own idea, met a little later.
Taking away is one meaning of minus, not the only one. “Tom has 7 cars and Ava has 3. How many more does Tom have?” is also 7 - 3, but nothing leaves: two groups are compared. The sign and the answer are the same, but the picture is different.
Where this goes next
Once taking away makes sense with counters, the child can learn the facts themselves, subtracting within ten, from pictures and ten frames rather than from a list. On a number line, taking away becomes stepping backwards, which is counting back.
The before-and-after split in the tower picture grows into parts and the whole and number bonds, and from there into fact families, where 4 + 3 = 7 and 7 - 3 = 4 are seen as one set of numbers told four ways.
Exercises
Cross out and count
- Draw 6 circles. Cross out 2. How many are not crossed out? Write the number sentence.
- Look at picture A in the figure below. Write the number sentence it shows.
- Look at picture B. Write the number sentence it shows.
Use counters or a drawing to find each answer.
- 5 - 1
- 8 - 4
- 7 - 0
- 6 - 6
Stories to act out
- “4 frogs sit on a log. 1 frog hops into the water. How many frogs are on the log now?”
- “7 children are on the bus. At the first stop, 4 get off. How many are still on the bus?”
- “You have 8 grapes. You eat 6. How many are left?”
- “There are 3 strawberries on a plate. Dad eats all 3. How many are left?”
- Tell a story that goes with 6 - 2. Then find the answer.
How many went away?
- “There were 8 balloons. Some popped. Now there are 6. How many popped?”
- “Ana’s tower had 7 cubes. She took some off the top. Now it has 2 cubes. How many did she take off?”
Explain
- Jo says 5 - 0 = 0. Is Jo right? How do you know?
- You have 4 counters. Can you take away 6 of them? Show it with counters and say what happens.
- Ben had 7 cars. He gave 2 away. He says he has 4 left. Put the cars back to check. Is he right?
Answers
- 4, and the sentence is 6 - 2 = 4. Six circles, two crossed out, four left.
- 9 - 5 = 4. Nine counters at the start, five crossed out, four left.
- 10 - 3 = 7. A full ten frame is 10; three are crossed out, leaving the top row of 5 and 2 more.
- 4. Five counters, take one away: 5 - 1 = 4.
- 4. Eight counters, push four aside, four stay: 8 - 4 = 4.
- 7. Taking away nothing leaves every counter: 7 - 0 = 7.
- 0. Taking away all six leaves none: 6 - 6 = 0.
- 3 frogs. 4 - 1 = 3.
- 3 children. 7 - 4 = 3.
- 2 grapes. Eight grapes, cross out six, and two are left: 8 - 6 = 2.
- 0 strawberries. All of them were eaten: 3 - 3 = 0.
- Any story that starts with 6 things and then 2 go away, for example “6 ducks on a pond, 2 swim off”. The answer is 6 - 2 = 4.
- 2 balloons. Cross out from 8 one at a time until 6 are left; that takes 2 crosses, so 8 - 2 = 6.
- 5 cubes. From 7, take cubes off one at a time until 2 are left; that is 5 cubes, so 7 - 5 = 2.
- No. Taking away 0 means taking nothing, so all 5 are still there: 5 - 0 = 5. It is 5 - 5 that equals 0.
- No. After taking away all 4 there are none left to take, and 2 more are still needed. With whole things you cannot take away more than you have, so 4 - 6 has no answer in whole numbers. Much later, negative numbers give 4 - 6 an answer, but not by taking things away.
- No. Put the 2 cars back with his 4: 4 + 2 = 6, not 7. He really has 7 - 2 = 5 cars, and putting the 2 back checks it: 5 + 2 = 7.