Addition as putting together
There are some here and some more over there. How many are there when we put them all together?
Where you are 12 in Addition and subtraction
Why this exists
A child who can count a set of objects can already find out how many there are when two groups are pushed together: push them together and count everything. Addition begins as a name for that act. It turns “three here and four there, how many now?” into something you can say, write down and later work out without touching anything.
Once the act has a name and a sign, it can be talked about. 3 + 4 = 7 is a sentence about any three things joined with any four things: ducks, blocks, children on a bus. That is the first time a child writes a fact that is true about more than the objects in front of them.
Definition
Adding is putting groups together and finding how many there are in all. The two amounts being joined are the parts and the amount we get is the total, sometimes called the sum.
We write it with two signs:
- the plus sign + joins the parts. Read 3 + 4 as “three plus four”, or simply “three and four”.
- the equals sign = says that the two sides are the same amount. Read 3 + 4 = 7 as “three and four is the same amount as seven”.
Joining happens in two kinds of situation, and children meet both. In one, something arrives: there were three ducks and two more swim over. In the other, nothing moves at all: there are four red blocks and three blue blocks, and we ask how many blocks there are altogether. Both are the same addition, because both ask for the size of one group made from two.
Intuition
To a child, adding is first a thing the hands do. Make the first group, make the second, slide them into one pile and count the pile starting from one. That is “counting all”, and it is the right place to start: it is slow, but it is plainly the answer to the question.
Fingers work just as well as counters (three fingers up, then four more), and so does a quick drawing of dots. The words that signal joining are small ones: and, more, in all, altogether, now. It helps to say them while the counters move.
The equals sign deserves care from the very first day. Children who only ever see 3 + 4 = 7 can come to read = as “and the answer is”. It is better read as a balance: whatever is on one side is the same amount as what is on the other. Writing the same fact as 7 = 3 + 4 now and then keeps that meaning alive, and the equal sign as the same amount builds on it.
Results
Joining keeps every object. The total is the count of everything in both parts, each object counted once.
The order of the parts does not change the total.
Joining nothing changes nothing.
The total is never smaller than either part. With whole numbers, putting together makes more, or the same when one part is empty.
Why it is true
Every result above comes from what counting is. When we count the joined pile, each object gets exactly one counting word, so the last word we say is how many objects there are (One-to-one correspondence, How counting tells how many). The objects of the first group are all still in the pile, and so are the objects of the second, so the total counts them all, once each.
The pile does not care which group went in first. Swapping the order of the parts changes the order we might count in, never which objects are there, and a count does not depend on the order. That is why 2 + 6 and 6 + 2 must agree.
An empty group puts no objects into the pile, so the count is what it was. And since joining never removes an object, the pile has at least as many as either group had on its own.
Worked examples
Ducks on a pond
Read aloud: “Three ducks are on the pond. Two more ducks swim over. How many ducks are on the pond now?”
SolutionPut out three counters for the ducks already there. Then put out two more for the ducks that swim over, and push them next to the first three. The question asks how many are on the pond now, after the two arrive, so it asks for the whole pile.
Count the pile from the start: one, two, three, four, five. There are five ducks. As a number sentence:
The 3 is the ducks that were there, the 2 is the ducks that came, and the 5 is all of them.
Blocks of two colors
Read aloud: “Mia has four red blocks and three blue blocks. How many blocks does she have altogether?”
SolutionNothing arrives in this story. The blocks are all there already, and the word altogether asks us to think of them as one group. Snap the four red blocks into a train and add the three blue ones to the end. Count the train: seven blocks.
The colors help the child see the two parts inside the total. The total counts blocks, not colors: Mia has seven blocks, four of them red and three blue.
Five and three on a ten frame
Read aloud: “Fill the top row of the ten frame. Now put three counters in the bottom row. How many counters are there?”
SolutionThe top row of a ten frame holds five, so a child who knows that a full row is five does not need to count those counters one by one. They can start at five and count only the bottom row: six, seven, eight. That is a first taste of counting on.
The frame shows more than the total. There are two empty boxes, so eight is two away from ten. Seeing a number as “five and some more” or “ten take away some” is what later makes adding quick without counting.
The same train from both ends
Read aloud: “Sam has two apples and gets six more. Lena has six apples and gets two more. Who has more apples, or do they have the same?”
SolutionBuild Sam’s apples as a train: two dark cubes, then six light ones. Count it: eight. Now turn the same train around. It reads six light, then two dark, which is Lena’s story, and it is still the very same eight cubes. They have the same: eight apples each.
This is worth pausing on with a child, because it is the first shortcut in arithmetic. To find 2 + 6 you can start at the bigger part, six, and count on two: seven, eight. That move, counting on, is only allowed because the order of the parts never changes the total.
Adding nothing
Read aloud: “There are four fish in the bowl. We do not put in any more fish. How many fish are in the bowl?”
SolutionPut four counters on one plate for the fish in the bowl. The other plate is for the fish we put in, and it stays empty. Tip the empty plate onto the full one and nothing changes. There are still four.
A group with nothing in it is still a group, and zero is the number that says how many are in it. Adding it is allowed, and it leaves the amount alone.
From a number sentence to a story
Read aloud: “Make up a story for 1 + 4 = 5, and show it with counters.”
SolutionThe sentence says: one thing, joined with four things, makes five things. Any story with one group of one and one group of four works. For example: “One cat is on the rug. Four more cats come to sit with her. Now there are five cats on the rug.”
To show it, put out one counter, then four more, push them together and count: five. The story, the counters and the sentence all say the same thing, and a child who can move freely between the three understands what + and = mean. The same fact can also be written the other way round,
because the equals sign only says the two sides are the same amount.
Boundaries
We count the total as one kind of thing. Three dogs and two cats make five animals, not five dogs. When the parts are different things, the total has to be named by what they share.
The two groups must not overlap. If three children want apples and four want pears, there are seven children only if nobody wants both. A child who is in both groups would be counted twice.
Joining never makes less. With whole numbers the total is at least as big as each part, so finding a smaller amount needs a different action and a different sign: taking away.
Starting from the bigger part is a choice, not a rule. Counting on from six to find 2 + 6 works only because 2 + 6 and 6 + 2 are the same amount. The story itself may start with the smaller number.
Where this goes next
Joining has a partner that runs it backwards: taking away starts from a total and removes a part. Looking at the joined group and asking which two parts made it is the idea of parts and the whole, and listing every way to split one number leads to Number bonds.
The adding itself becomes quicker in stages. Counting all gives way to counting on, small totals become known facts through adding within five and adding within ten, and ten becomes a landmark in making ten. Stories grow into addition and subtraction word problems, and the fact that order and grouping never matter is named, much later, in the properties of addition.
Exercises
Look at the pictures
Use the pictures for Exercises 1 to 3. For each one, ask: “How many dark? How many light? How many in all?” Then write the number sentence.
- Picture A.
- Picture B.
- Picture C.
Stories to act out
Read each story aloud and show it with counters or cubes.
- Two birds sit on a fence. Three more birds land on the fence. How many birds are on the fence now?
- Ana has four crayons. Ben gives her one more crayon. How many crayons does Ana have now?
- There are three red balloons and three yellow balloons. How many balloons are there in all?
- Five children are on the bus. Four more children get on. How many children are on the bus now?
- Jo has six shells. She looks for more but does not find any. How many shells does Jo have?
- Seven frogs sit on a log and three frogs swim in the water. How many frogs are there altogether?
Number sentences
Find the number that goes in the box. Show it with counters if you like.
- 1 + 3 = \square
- 5 + 2 = \square
- 0 + 6 = \square
- 4 + 5 = \square
- \square = 2 + 7
Explain
- Show 3 + 4 with counters. How do you know there are seven in all?
- Tom says 3 + 6 and 6 + 3 are different because the numbers are in a different order. Is he right? Show it with cubes.
- Make up a story for 4 + 2 = 6.
- You put three counters together with some more counters. Can you end up with fewer than three? How do you know?
Answers
- 2 + 3 = 5. Two dark and three light; count them all to get five.
- 4 + 4 = 8. Four dark and four light make eight.
- 1 + 5 = 6. One dark and five light make six.
- Five birds. 2 + 3 = 5: two on the fence, three land, count all five.
- Five crayons. 4 + 1 = 5: one more than four is five.
- Six balloons. 3 + 3 = 6: the colors are the two parts, and the total counts balloons.
- Nine children. 5 + 4 = 9: start from five and count on four: six, seven, eight, nine.
- Six shells. 6 + 0 = 6: she joins nothing to her six shells, so the amount does not change.
- Ten frogs. 7 + 3 = 10: frogs on the log and frogs in the water are one group of frogs.
- 4. 1 + 3 = 4.
- 7. 5 + 2 = 7: five, then six, seven.
- 6. 0 + 6 = 6: an empty group joined with six leaves six.
- 9. 4 + 5 = 9: four counters and five counters, pushed together and counted, make nine.
- 9. 9 = 2 + 7: the box is on the left, but the equals sign still says both sides are the same amount, and two and seven make nine.
- Seven. Put out three counters and four counters, push them together and count every counter once: one to seven. The last number said is how many there are.
- No. Build three dark cubes and six light cubes as one train and count nine. Turn the train around and it shows 6 + 3, still the same nine cubes.
- Any story joining a group of four with a group of two, for example: “Four children are on the swings. Two more children come to swing. Now six children are on the swings.”
- No. Joining keeps all three counters and adds the others, so there are at least three. There are exactly three only if you add none.