Subtraction as the difference
Two groups sit side by side and nothing is taken away. How do you say how many more one has than the other?
Where you are 12 in Addition and subtraction
Why this exists
A child’s first subtraction is taking away: seven birds sit on a fence, three fly off, four are left. Something leaves, and we count what stays.
Then a different question turns up. Mia has seven shells and Sam has four. How many more does Mia have? Nobody gives anything away. Both piles stay on the table the whole time. And yet the answer is still 7 - 4 = 3.
This lesson is about that second meaning of the minus sign. It answers “how far apart are these two amounts?”, and the answer has a name, the difference. Children use it every day (who has more crayons, which tower is taller, how many more days until a birthday), and it is the kind of subtraction that later measures lengths, reads bar graphs and finds distances on a number line.
Definition
The difference between two amounts is how many more the bigger one has. It is also how many fewer the smaller one has: that is the same gap, seen from the other side. When the two amounts are the same, the difference is 0.
We find it by subtracting the smaller number from the bigger one:
Read aloud, “seven minus four equals three” now says two things at once: Mia has 3 more shells than Sam, and Sam has 3 fewer shells than Mia.
The words that call for it are “how many more”, “how many fewer”, “how many less”, “how much taller” or “longer”, and “how far apart”.
Intuition
The picture behind the difference is matching. Line the two groups up, one row above the other, and give every counter in the short row a partner in the long row. This is the same pairing a child uses to decide which group is bigger at all (One-to-one correspondence and Comparing quantities); now we also count what is left over.
In Figure 1 each of Sam’s 4 shells has a partner from Mia’s row. Three of Mia’s shells have no partner, and those three are exactly how many more she has. Read from the bottom row instead, Sam is missing a partner for three of Mia’s shells, so he has three fewer. One gap, two ways to say it.
With a child, the language that works is buddies: “Every one of Sam’s shells gets a buddy. Which shells have no buddy?” Lining up matters. Rows that start at the same place, with counters the same distance apart, make the gap easy to see; a ten frame builds that lining up in.
Results
The difference. For two amounts, the bigger minus the smaller counts how many more the bigger one has, and also how many fewer the smaller one has.
For Mia and Sam, 7 - 4 = 3: Mia has 3 more shells and Sam has 3 fewer. Two equal amounts have a difference of 0.
Why it is true
Look only at the long row after matching. It splits into two parts: the counters that have a partner and the counters that do not. There is one partnered counter for each counter in the short row, so that part has exactly as many as the short row, 4.
So the long row is a Parts and the whole picture: the whole is 7, one part is 4, and the other part is the unknown gap. Finding a missing part of a whole is what subtraction does. Put a hand over the 4 partnered counters and what still shows is 7 - 4.
Nothing was really taken away. The hand only hides the counters that already matched, and Sam’s row never moves. That is why a take-away sentence can answer a question where nobody takes anything: covering the matched part of the bigger group works just like removing it. And each unmatched counter is one partner the short row is missing, so the same number also says how many fewer the short row has.
Worked examples
Children and apples
There are 6 children and 4 apples. Does every child get an apple? How many more children are there than apples?
SolutionAct it out, or draw it: put the children in a row and give each child one apple until the apples run out. Four children get an apple. Two children are left with none, so not every child gets one.
The children with no apple are the gap, so there are 2 more children than apples. As a subtraction, the bigger group is the children and the smaller is the apples:
Said the other way round, there are 2 fewer apples than children, which is exactly how many more apples we would need.
Two ten frames
The top ten frame has dark counters. The bottom ten frame has light counters. How many fewer light counters are there than dark counters?
A full ten frame holds ten, so the top frame, with one empty box, has 9. The bottom frame has its top row full, 5.
Because both frames fill boxes in the same order, every light counter already sits right under a dark one. The shaded boxes in the bottom frame are the places where a dark counter has no light partner. There are four of them.
So there are 4 fewer light counters than dark ones. It is equally true that there are 4 more dark counters than light, and that 9 is 4 more than 5.
Two trains of cubes
A child builds two trains of cubes and lines them up at one end. How many more cubes are in the long train than in the short train?
The long train has 12 cubes and the short one has 8. Up to the dashed line the two trains match cube for cube, so the answer is the shaded part that sticks out past the short train.
There are three good ways to count it, and they must agree.
Count the shaded cubes: 4.
Count on from the short train: start at 8 and count the cubes past the line, “nine, ten, eleven, twelve”. That is four counts, the same move as counting on, and it answers 8 + \square = 12 with \square = 4. Thinking of a subtraction as a missing addend is its own lesson, subtraction as a missing part.
Cover the matched part: hide the first 8 cubes of the long train and count what shows:
The long train has 4 more cubes. Counting on is usually the quickest here, because the two numbers are close together.
A frog on a number line
A frog sits on 3 and a bug sits on 8. The frog hops one space at a time. How many hops does it need to reach the bug?
Each hop moves the frog one step to the right. From 3 the hops land on 4, 5, 6, 7 and 8, so there are five hops.
We count the hops, not the marks. The marks from 3 to 8 are six numbers, but the frog starts on the first one without hopping to it; every hop is one space of the gap.
So 8 is 5 more than 3. On a number line, the difference is how far apart two numbers sit, which is how the same idea works long after counters are put away.
When the rows match
Pia has 5 socks and Omar has 5 socks. How many more socks does Pia have than Omar?
SolutionLine them up and every one of Pia’s socks gets a partner from Omar’s. Nothing is left over, so the gap is empty:
Pia has 0 more socks than Omar; they have the same number. A difference of 0 is how subtraction says “the same”, and it is a real answer, not a missing one.
Both groups grow
Ana has 5 stickers and Leo has 8. Then each of them gets 2 new stickers. How many more does Leo have now?
SolutionBefore the new stickers, Leo has 8 - 5 = 3 more.
After, Ana has 5 + 2 = 7 and Leo has 8 + 2 = 10, and
Leo still has 3 more. The matching picture shows why without any counting: the old pairs stay paired, Ana’s two new stickers pair up with Leo’s two new stickers, and the same 3 of Leo’s old stickers are still the only ones with no partner. So when both groups get the same number of new things, the difference stays exactly what it was. The same holds when both groups lose the same number, as long as each group has that many to lose. That fact becomes a handy trick later, when a subtraction like 13 - 9 can be shifted to the easier 14 - 10.
Boundaries
Take the smaller from the bigger. With whole numbers the difference is always bigger minus smaller, whichever amount the question names first. If Sam has 4 and Mia has 7, then “how many more does Sam have?” has the answer “none; he has 3 fewer”. A sentence like 4 - 7 has no answer in counters on a table.
It compares how many, not how big. Three elephants and five mice: there are 2 more mice. A row that is spread out can look longer while holding fewer, so the comparison is made by pairing one to one, never by which row reaches further.
Count the same kind of thing. 2 bags of apples and 5 apples cannot be compared until we know how many apples are in a bag. The two amounts have to count the same unit.
Nothing leaves. In a comparison the minus sign does not mean anything went away. Both amounts are still there after the subtraction; the answer describes the gap between them.
Where this goes next
Here the two amounts are known and the gap is the question. In stories about how many more and how many fewer, the gap is known and one of the amounts is missing: “Leo has 3 more than Ana, and Ana has 5. How many does Leo have?” The same matching picture solves those too.
The difference is also how children read a bar graph or pictograph (“how many more children chose dogs than cats?”), how they compare two lengths in cubes, and how they will subtract bigger numbers by counting up from one to the other. The four facts, two additions and two subtractions, that live in one part-part-whole picture are gathered in Fact families.
Much later, the gap between two numbers on a number line becomes distance, measured with absolute value so that it works even when the smaller number is subtracted first.
Exercises
Line them up
- There are 5 dogs and 3 bones. Give each dog one bone. How many dogs have no bone? How many more dogs are there than bones?
- There are 6 red beads and 2 blue beads. How many more red beads are there than blue beads?
- There are 8 children and 5 chairs. How many fewer chairs are there than children?
- There are 4 cats and 4 hats. How many more cats are there than hats?
Write the subtraction
Say each one as a minus sentence, then answer the question.
- Ella has 10 crayons. Max has 6 crayons. How many more crayons does Ella have?
- One tower has 9 cubes. Another tower has 4 cubes. How many more cubes are in the taller tower?
- Jo read 11 books. Kai read 6 books. How many fewer books did Kai read?
- A red train has 15 cars. A blue train has 9 cars. How many more cars does the red train have?
Count on or hop
- A frog sits on 2. A bug sits on 9. How many hops does the frog need to reach the bug?
- Count on from 6 to 10. How many more is 10 than 6?
- Sam has 14 marbles. Lee has 10 marbles. How many more marbles does Sam have?
- Ana has 7 stickers and Leo has 10. Then each of them gets 3 more stickers. How many more stickers does Leo have now?
How do you know?
- Ben says, “7 is 3 more than 4, so 4 is 3 fewer than 7.” Is he right? Show it with counters.
- Mia has 6 grapes and Sam has 9. Nobody eats any grapes. Why can we still use a minus sign to say how many more grapes Sam has?
- One row has 5 buttons spread far apart. Another row has 7 buttons bunched close together. Which row has more buttons, and how many more? How do you know?
- Think of two groups that are 2 apart. Now find a different pair of groups that are also 2 apart. What do you notice?
Answers
- 2 dogs have no bone, so there are 2 more dogs than bones. Three dogs get a bone and two are left over: 5 - 3 = 2.
- 4 more red beads. Pair each blue bead with a red one; four red beads are left: 6 - 2 = 4.
- 3 fewer chairs. Five children sit down and three are still standing: 8 - 5 = 3.
- 0 more. Every cat gets a hat and nothing is left over: 4 - 4 = 0. They are the same.
- 10 - 6 = 4, so Ella has 4 more crayons. Counting on from 6: seven, eight, nine, ten.
- 9 - 4 = 5, so the taller tower has 5 more cubes. Stand the towers side by side and five cubes stick up above the short one.
- 11 - 6 = 5, so Kai read 5 fewer books. Counting on from 6 to 11 takes five counts.
- 15 - 9 = 6, so the red train has 6 more cars. From 9, one more makes 10 and five more make 15, six in all.
- 7 hops, since 9 - 2 = 7. The frog lands on 3, 4, 5, 6, 7, 8 and 9: seven landings, one for each hop.
- 4 more, since 10 - 6 = 4. Counting on from 6: seven, eight, nine, ten.
- 4 more marbles, since 14 - 10 = 4. Fourteen is ten and four, so past Lee’s ten Sam has four extra.
- 3 more. Now Ana has 7 + 3 = 10 and Leo has 10 + 3 = 13, and 13 - 10 = 3. The new stickers pair up with each other, so the gap from before, 10 - 7 = 3, does not change.
- Yes. Put 7 counters in one row and 4 below, matched in pairs. Three counters in the top row have no partner, so 7 is 3 more. The bottom row is short of partners for those same three, so 4 is 3 fewer. It is one gap, read from each row.
- Line up the grapes and pair each of Mia’s with one of Sam’s. Cover Sam’s 6 matched grapes with a hand and 3 still show: 9 - 6 = 3. The minus sign counts what is left of the bigger group after the matched part is hidden, so nothing has to be eaten.
- The row of 7 has 2 more buttons, even though the row of 5 looks longer. Move the buttons so each of the 5 has one partner from the 7; two buttons from the 7 have no partner. How far a row stretches does not change how many are in it: 7 - 5 = 2.
- Many answers work, for example 5 and 3, or 9 and 7: 5 - 3 = 2 and 9 - 7 = 2. Any two groups where the bigger one has two extra when they are matched are 2 apart, so there are lots of pairs, and you can make a new one by adding the same number of counters to both groups.