Counting on to add and counting back to subtract
Eight counters are hiding under a cup and three more sit beside it. Do we have to tip them all out and count from one, or can we start at eight? And when two of seven are taken away, can we find what is left just by saying the numbers backwards?
Where you are 7 in Adding and subtracting within 20
When you reach for it
This lesson teaches two moves that mirror each other. Counting on adds by starting at one part and counting forwards through the other. Counting back takes away by starting at the whole and counting backwards, one number for each thing taken away. Both rest on the same fact about counting, and both are quick for the same reason: only a few numbers have to be said.
Counting on to add. A child who adds by joining starts by counting all: make both groups, push them together, and count every object from one. That always works, but most of the counting goes on a group whose size is already known. If the cup holds eight, counting “one, two, three, up to eight” only tells you eight again.
Counting on skips that part. Say the number of the first group, then keep counting through the second group only. For 8 + 3 that is “eight ... nine, ten, eleven”. It grows straight out of counting on from a number, where a child starts a count at five instead of at one, and it is the first way of adding that does not need every object in front of you.
Reach for it when one part is small, one, two or three, sometimes four, and the other part is already known as a number: it is written down, it is hidden under a cup, or it is a group the child sees at a glance, like a full row of a ten frame or a whole hand of five fingers. The small part is the one you count, so it goes second. When the small part comes first, as in 2 + 9, the child turns the sum around and counts on from 9. That turning around is allowed, and the reason is the most important idea on the adding side of this lesson; it is explained under “Starting from the bigger number” in the next section.
Counting back to subtract. A child who understands taking away can find 7 - 2 by laying out seven counters, pushing two aside and counting the ones that stay. That always works, but it needs counters, and it needs the child to count the leftovers again from one.
Counting back does the same job with no objects at all. The child starts at the number there is to begin with, 7, and says the numbers backwards, one for each thing taken away: “6, 5.” The last number said, 5, is how many are left. It rests on saying the numbers backwards from any starting number, which a child practices first as a counting skill on its own.
It is the quick choice when the number taken away is small: 1, 2 or 3, sometimes 4. Then there are only a few numbers to say, and the child can keep track of them on their fingers. Stories where a few things leave a bigger group, such as “12 birds, 3 fly away”, have exactly this shape.
When neither is quick. Both moves cost one counting word for every object counted on or taken away. When both parts of a sum are big, as in 7 + 8, counting on still gives the right total, but even from the bigger number it takes seven counts and the tracking gets shaky. Quicker ways take over there: doubles, near doubles and making ten. When the number taken away is big, counting back gets slow in the same way; the last worked example and the Boundaries section show what to do then.
The procedure
Counting on to add
Take 5 + 2: five dark counters and two light ones, as in Figure 1.
Say the first number. Counting the five dark counters one by one would end on the word “five”, because the last word said in a count is how many there are (How counting tells how many). So saying “five” does the whole of that count in one word. Many children stretch it, “fiiive”, or pat the group as they say it, which is a good sign: they mean all five, not just the last counter.
Carry on counting through the second group. The first light counter is the sixth counter in the pile, so it gets the word “six”, and the next one gets “seven”. Each new counter gets exactly the word it would have got if we had counted all from one. The only thing skipped is saying the words for counters we already knew about. That is why the first finger goes up with “six” and not with “five”: “five” belongs to the dark counters, which are already counted, and the fingers count only the light ones.
Stop when the second group is used up. To know when to stop, the child keeps a second count running: how many words have been said since “five”. With counters in view, the counters keep track. With a written sum like 5 + 2 there is nothing to touch, so children put up one finger for each word: “six” (one finger), “seven” (two fingers). Two fingers are up and the second part is two, so they stop. This double count, one count of words and one count of fingers, is the hard part for a young child, and it is why counting on is kept for small second parts.
The last word is the total. The count ended on “seven”, and a count that gives every counter one word ends on how many there are. So
Starting from the bigger number. For 2 + 9, counting on from 2 means nine counts, “three, four, five, six, seven, eight, nine, ten, eleven”, with nine fingers to keep track of. Counting on from 9 takes two: “ten, eleven”. Both land on eleven, but the second is far quicker and much harder to get wrong. So children learn to start from the bigger number, and it is fair for a child to ask why that is allowed when the sum says 2 first.
The answer goes back to what adding means. 2 + 9 asks how many there are when a group of two and a group of nine are put together into one pile. The pile does not remember which group went in first: the same eleven objects are in it either way. And a count does not depend on the order in which things are touched (Counting in any order), so counting the pile starting with the nine gives the same number as counting it starting with the two. Counting on from 9 is simply counting that same pile, big group first. The same holds in a story: if Leo has 2 stickers and is given 9 more, the story’s order tells us what happened, but it does not change how many stickers he has at the end.
A cube train makes this something a child can hold (Figure 2). Snap two dark cubes and nine light cubes into one train. Read from the left it is 2 + 9. Turn the train around and it reads 9 + 2. Nothing was added and nothing came off, so the totals must agree:
Nothing in that argument used the numbers two and nine, so it holds for any two parts: the order of the parts never changes the total. That fact gets its own lesson later, order does not change a sum. For now it is what gives a child permission to pick the easier starting point.
Counting back to subtract
Start at the whole. The count begins at the number you start with, because that is how many there are before anything goes. In 7 - 2 the start is 7. The number being taken away is never where the count starts; it only says how many numbers to say. This is where counting back differs from counting on: in adding either part may go first, but in taking away the two numbers do different jobs, so there is no turning the question around.
Say one number back for each one taken away. Taking one thing away leaves one fewer, and the number just before 7 is the number that is one fewer than 7, which is 6. So the first word is “6”, not “7”. Take another away and there is one fewer again, 5. Each number said backwards is how many are left after one more has gone.
Figure 3 shows why this is so. Number seven counters 1 to 7 from the left. Take away the last one: counter 7 is gone, and the last counter still there is number 6, so 6 are left. Take away the next: now the last counter is number 5. Counting back is naming the last counter left, one removal at a time. That is why the start is not said as one of the steps: it is the count before anything has gone, just as “five” in counting on is the count already done.
Keep track of how many numbers you have said. The child must stop after saying exactly as many numbers as were taken away. One finger per number said does this. For 7 - 2: “6” (one finger), “5” (two fingers), and two fingers means stop. The fingers count the things taken away, while the words say how many are left. Two counts run side by side, one forwards on the fingers and one backwards in words, and holding both at once is the real work of the method for a young child. That is a good reason to keep the number taken away small at first.
The last number said is the answer.
On a number line, each number said is a hop back. Start at 7 and make one hop to the left for each thing taken away. Where the last hop lands is the answer. The child counts hops, not marks: the mark at 7 is where they stand before anything is taken, and it is not a hop. Figure 4 shows 7 - 2 as two hops, 7 to 6 and 6 to 5.
Two small cases follow at once. Taking away 1 is saying one number back, so the answer is the number just before: 9 - 1 = 8. Taking away 0 means saying no numbers at all, so the answer is where you started: 9 - 0 = 9.
Worked examples
Read each problem aloud. Let the child use counters, cubes or fingers. The first five examples count on; the last five count back.
Two more ducks
Read aloud: “Six ducks are on the pond. Two more ducks swim over. How many ducks are on the pond now?”
SolutionThe six ducks are already known, so there is no need to count them. Say “six” for all of them. Then count the two that swim over, one finger for each: “seven” (one finger), “eight” (two fingers). Two fingers are up, the same as the two new ducks, so stop. The last word was “eight”.
There are eight ducks on the pond. A child who is unsure can check by counting all eight from one, and should get the same answer: counting on is only a shorter way of doing that count.
Hiding under a cup
Read aloud: “Eight counters are hiding under this cup. Here are three more. How many counters are there in all?”
The eight are out of sight, so counting all from one is not possible. This is where counting on earns its place: the child does not need to see the eight, only to know there are eight. Touch the cup and say “eight”. Then touch each counter beside it: “nine, ten, eleven”. Three counters were touched, so the count is over, and the last word was “eleven”.
The count went past ten on the way, and that is no problem: the words simply keep going. Lift the cup and count everything from one to check. There are eleven.
Hops on a number line
Read aloud: “Put your finger on 6. Hop forward three times. Where do you land?”
On a number line each number is a place, and one hop forward moves to the next number. Start with a finger on 6. The start is not a hop: it plays the part of saying “six”. Hop to 7 (one), to 8 (two), to 9 (three). Three hops, and the finger is on 9.
The number line shows the two counts of counting on side by side. The numbers landed on, 7, 8 and 9, are the words said. The hops, one, two, three, are what the fingers kept track of.
Two ways to start
Read aloud: “Work out 4 + 7. Try it starting at 4. Then try it starting at 7.”
Starting at 4, count on seven: “five, six, seven, eight, nine, ten, eleven”. That is seven fingers, more than one hand, and it is easy to lose count. Starting at 7, count on four: “eight, nine, ten, eleven”, four fingers. Both land on eleven (Figure 7).
Doing it both ways once is worth the time: the child sees for themselves that the answer does not change, and feels how much shorter the second way is. Four is about as many as counting on handles comfortably. When the smaller part is bigger than that, a different idea usually helps more, for example making ten: 7 + 4 is 7 + 3, which makes ten, and one more.
How many more to get there?
Read aloud: “Maya has 6 shells. She wants to have 9. How many more shells does she need to find?”
SolutionThis time the total is known and the number to add is not: 6 + \square = 9. Counting on still works, but it stops in a different place. Say “six”, then count on, putting up a finger for each word, until the word is “nine”: “seven” (one), “eight” (two), “nine” (three). Now the fingers are the answer. Three fingers are up, so Maya needs three more shells.
This is the same as asking what 9 - 6 is. Using counting on to answer a take-away question is the idea of subtraction as a missing part.
Frogs on a log
Read aloud: “8 frogs sit on a log. 3 frogs jump into the pond. How many frogs are still on the log?”
SolutionThe story starts with 8 frogs, so the count starts at 8. Three frogs jump, so say three numbers back, putting up a finger for each: “7” (one finger), “6” (two fingers), “5” (three fingers). Three fingers are up, one for each frog that jumped, so stop. The last number said is 5.
The first finger goes up on “7”, not on “8”, because a finger stands for one frog that has jumped, and when 8 is said no frog has jumped yet: 8 is where the count stands. On a number line this is easy to see. A hop lands on a number, and the first hop from 8 lands on 7. The three hops land on 7, 6 and 5, so the three numbers said are the three places the hops land, and the last one is the answer.
5 frogs are still on the log. Check. Put the 3 frogs back: 5 + 3 = 8.
Birds on a fence
Read aloud: “12 birds sit on a fence. 3 fly away. How many birds are left on the fence?”
SolutionStart at 12, because that is how many birds there are before any fly. Three fly away, so say three numbers back: “11, 10, 9.” Three numbers have been said, so stop. The last one is 9.
9 birds are left. Going past ten needs nothing new: the number before 10 is 9, just as the number before 5 is 4. What the child does need is to know the teen numbers backwards as well as forwards: 12, 11, 10, 9. Check. 9 + 3 = 12.
How many flew away?
Read aloud: “There were 9 geese on the pond. Some flew away. Now there are 6. How many flew away?”
SolutionThis time the start and the end are known, and the number taken away is missing. Counting back still works. Start at 9 and count back until you reach 6, putting up a finger for each number said: “8” (one), “7” (two), “6” (three). You reached 6 after three numbers.
Three numbers were said, so three geese flew away:
Here the fingers are the answer and the last word is only the signal to stop. On the number line (Figure 9) it is counting the hops from 9 back to 6. Check. 6 + 3 = 9. A story like this is one of the problems with the change missing. It is the mirror of Maya’s shells: counting on from 6 to 9 finds the same 3, so either move answers it.
Bigger numbers, same moves
Read aloud: “There are 31 children in the hall. 2 go home. How many are still in the hall?”
SolutionThis goes past twenty to show that the same moves work for bigger numbers. Start at 31 and say two numbers back: “30, 29.” Two numbers, so stop.
29 children are still in the hall. All the child needs is to count backwards past 30, where the tens change. Check. 29 + 2 = 31.
Where counting back gets slow
Read aloud: “What is 14 take away 11?”
SolutionCounting back works, but look how long it is. Start at 14 and say eleven numbers back:
“13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3.”
That is eleven words and eleven fingers, more fingers than one child has. The answer is 3:
Figure 10 shows the eleven hops, and every one of them is a chance to lose count. The number of words in counting back is always the number taken away, so it is quick when that number is small and slow when it is big. Here 11 and 14 are close together, and asking “11 and how many more make 14?” is much shorter: count on from 11, “12, 13, 14”, three numbers. That way of thinking is its own lesson, subtraction as a missing part. Check. 3 + 11 = 14.
Boundaries
The first number must say how many. Counting on from eight works because “eight” is the last word of a count of eight things, so it stands for the whole group. Counting back from seven works for the same reason: “seven” is how many there are before anything goes. The number we start from has to be the size of a group, not just a label on something.
Every step is exactly one. Each word said after the first number adds one object, or takes one away, no more and no less. That is why the second number has to be counted out one at a time, and why the fingers must match it exactly.
Adding may start from either part; taking away always starts from the whole. Starting from the bigger number works in adding because the parts can be swapped. That is not true of taking away. 9 - 2 is 7, but “2 take away 9” has no answer among the numbers a child uses now (0, 1, 2 and on). 7 - 2 starts at 7 and says two numbers; starting at 2 and counting back seven would be 2 - 7, a different question, because subtraction has an order (see the properties of addition).
Zero means saying nothing more. For 6 + 0 no words come after “six” and no fingers go up, so the total is 6. For 9 - 0 no numbers are said backwards, so the answer is where you started, 9.
Counting back stops at zero. With whole numbers, the counting words backwards run out at 0. For 3 - 5 the child says “2, 1, 0” and still has two more to take away, with no number left to say. That matches taking away: you cannot take away more than you have. Much later, negative numbers carry the line on past zero, but not as a count of things left. Counting on has no such end: the words forwards never run out.
Always correct, but quick only when the number counted is small. 9 - 1 costs one word and 14 - 11 costs eleven; 8 + 2 costs two words and 2 + 8 from the two would cost eight. In adding, starting from the bigger part fixes this. In taking away, when the number taken away is close to the start, counting up from it (subtraction as a missing part) is shorter. When the start is a teen number and several are taken away, bridging back through ten makes two jumps instead of many single steps.
More than two parts. The same reason lets a child start from the biggest of three parts: for 2 + 9 + 1, start at nine and count on three, “ten, eleven, twelve”. The order of the parts does not matter however many there are. Choosing which parts to put together first is adding three numbers.
It answers the number sentence, whatever the story. “Mia has 9 shells and Theo has 2. How many more does Mia have?” is also 9 - 2, and counting back two from 9 gives 7, even though nothing in the story is taken away. The count answers 9 - 2; the story’s picture is comparing two groups.
Where this goes next
Counting on and counting back are a bridge from counting to knowing. A child who has counted on from eight by two many times starts simply to know that 8 + 2 = 10, and after enough rounds of “9, then 8, 7” simply knows 9 - 2 = 7. Those known facts are what adding within ten and subtracting within ten build. When the number to count is too big to count comfortably, doubles, near doubles and making ten take over, and they are shorter because they use facts instead of counts.
Counting on to a known total, as Maya did, finds a missing part, and it grows into counting up to find a difference and the “how many more” questions of comparing. For bigger amounts, counting in ones gives way to counting in bigger steps. Stopping at ten on the way down is bridging back through ten, as in 13 - 5 = 13 - 3 - 2. A single jump of ten forwards or backwards, “53, 43, 33”, is ten more and ten less, and hops of any size drawn on a blank line make the open number line, which carries counting on and counting back into subtracting two-digit numbers. The permission to turn a sum around is named in order does not change a sum, part of the properties of addition.
Exercises
Count on from the cup
Use Figure 11. Each cup has a number on it telling how many counters are hiding underneath. Say the number on the cup, then count on the counters beside it. How many counters are there in all?
- Cup A.
- Cup B.
- Cup C.
Count on
Say the first number, then count on. Use your fingers to keep track.
- 7 + 2 = \square
- 9 + 1 = \square
- 10 + 3 = \square
- 15 + 2 = \square
- 8 + 4 = \square
- 6 + 0 = \square
Start with the bigger number
- 2 + 8 = \square
- 1 + 16 = \square
- 3 + 9 = \square
- Ivy has 3 marbles. Her brother gives her 8 more. How many marbles does Ivy have now?
Hops and missing parts
- Put your finger on 11 on a number line. Hop forward three times. Where do you land?
- Omar has 7 toy cars. How many more cars does he need to have 10?
Read the number line
Use Figure 12. Write the number sentence each number line shows.
- Line A.
- Line B.
- Line C.
Count back
Start at the first number and count back. Use your fingers to keep track.
- 8 - 2
- 6 - 3
- 10 - 1
- 13 - 3
- 12 - 4
- 7 - 0
- 20 - 3
Stories to count back
- “There are 10 apples in a bowl. You eat 2. How many apples are left?”
- “11 children wait for the slide. 3 go down and run off to play. How many are still waiting?”
- “Sam is on step 14 of the stairs. He walks down 4 steps. Which step is he on now?”
- “There were 8 birds on a branch. Some flew away. Now there are 5. How many flew away?”
Explain
- Zoe works out 6 + 3. She says “six, seven, eight” and puts up a finger for each word. She stops at eight. Is she right? What should she say?
- To find 2 + 9, Ben starts at 9. His sister says he has to start at 2, because 2 comes first. Who is right? Show it with cubes.
- Kai works out 8 + 3 by saying “nine, ten, eleven”. How does he know he does not have to count the eight?
- Ravi counts back to find 9 - 3, with a finger for each number he says. Is the first number he says 9 or 8? How do you know? Show it with hops on a number line.
- Would you count back to find 12 - 10? Why or why not? What is the answer?
- Count back to find 3 - 5. What happens?
- Show with 7 counters why counting back two from 7 lands on the same number as taking 2 counters away.
Answers
- 8. The cup says 6, so say “six”, then count the two counters: “seven, eight”. So 6 + 2 = 8.
- 12. Say “nine”, then count the three counters: “ten, eleven, twelve”. So 9 + 3 = 12.
- 17. Say “fourteen”, then count the three counters: “fifteen, sixteen, seventeen”. So 14 + 3 = 17.
- 9. “Seven ... eight, nine”, two fingers. 7 + 2 = 9.
- 10. One more than nine is the next number: “nine ... ten”. 9 + 1 = 10.
- 13. “Ten ... eleven, twelve, thirteen”, three fingers. 10 + 3 = 13.
- 17. “Fifteen ... sixteen, seventeen”. 15 + 2 = 17.
- 12. “Eight ... nine, ten, eleven, twelve”, four fingers. 8 + 4 = 12.
- 6. Say “six” and count on nothing: no words, no fingers. 6 + 0 = 6.
- 10. Start at the bigger number, eight, and count on two: “nine, ten”. 2 + 8 = 10.
- 17. Start at sixteen and count on one: “seventeen”. 1 + 16 = 17.
- 12. Start at nine and count on three: “ten, eleven, twelve”. 3 + 9 = 12.
- 11 marbles. The story says 3 + 8. Start at eight and count on three: “nine, ten, eleven”. 3 + 8 = 11. The story’s order does not change how many marbles there are at the end.
- 14. The hops land on 12, 13 and 14. 11 + 3 = 14.
- 3 more cars. Say “seven” and count on until you say “ten”: “eight, nine, ten”, three fingers. 7 + 3 = 10.
- 9 - 3 = 6. The hops start at 9, and three hops back land on 6.
- 11 - 2 = 9. Two hops back from 11: 10, then 9.
- 15 - 4 = 11. Four hops back from 15: 14, 13, 12, 11.
- 6. Say “7, 6”, so 8 - 2 = 6.
- 3. Say “5, 4, 3”, so 6 - 3 = 3.
- 9. One number back is the number just before, so 10 - 1 = 9.
- 10. Say “12, 11, 10”, so 13 - 3 = 10.
- 8. Say “11, 10, 9, 8”, so 12 - 4 = 8.
- 7. Taking away 0 means saying no numbers, so you stay at the start: 7 - 0 = 7.
- 17. Say “19, 18, 17”, so 20 - 3 = 17.
- 8 apples. Say “9, 8”, so 10 - 2 = 8.
- 8 children. Say “10, 9, 8”, so 11 - 3 = 8.
- Step 10. Each step down is one number back: “13, 12, 11, 10”, so 14 - 4 = 10.
- 3 birds. Count back from 8 until you reach 5: “7, 6, 5” is three numbers, so 8 - 3 = 5.
- No, the total is 9. “Six” is for the six she already has, so her first finger should go up with “seven”: “six ... seven, eight, nine”. Three fingers, and the last word is nine, so 6 + 3 = 9.
- Ben is right. Build two dark cubes and nine light cubes as one train. Read one way it is 2 + 9; turn it around and it is 9 + 2, the same eleven cubes. The order does not change the total, so he may start at 9 and count on “ten, eleven”.
- He already knows there are eight, so counting them would only end on “eight” again. Saying “eight” stands for all of them, and he only needs to count the three new ones, one word each.
- 8. Ravi stands on 9 before anything is taken away, and each finger is one thing taken away. On the number line the first hop from 9 lands on 8, so “8” is the first number said. Three hops land on 8, 7 and 6, so 9 - 3 = 6. Check: 6 + 3 = 9.
- Probably not. Taking away 10 means saying ten numbers back, from 11 all the way down to 2, which is a long count. Since 10 and 12 are close, count up from 10 instead: “11, 12” is two numbers. So 12 - 10 = 2. (Seeing 12 as ten and two gives it at once as well.)
- “2, 1, 0” uses three fingers, and two more are still to be taken away, but there is no whole number below 0. You cannot take 5 away from 3 things, so 3 - 5 has no whole-number answer. (For the adult: negative numbers give it one, -2.)
- Put 7 counters in a row and number them 1 to 7. Take the last one away: counter 7 goes, and the last one left is counter 6, the first number said back. Take another away: the last one left is counter 5, the second number said back. So taking 2 away leaves 5, and counting back two from 7, “6, 5”, also ends on 5: 7 - 2 = 5.