Counting on and counting back
When a child already knows how many are in a group, how can they find the new amount without counting the whole group again from one?
Where you are 23 in Counting and cardinality
When you reach for it
Once a group has been counted, its number stands for the whole group, so there is no need to count it again from one: say that number once and keep counting the new things. That is counting on. Taking things away one at a time while saying the words backward is counting back. These four days build both with real objects, first with a group hidden under a cup, then past ten, then backward, and finally with a written number standing in for the group, all within twenty.
The child should already know that the last word said tells how many and be able to say the counting words starting from any number. Adding or taking away a single object is one more and one less; this week the child goes on for two, three or four steps.
Day 1: Five under the cup
You need: a paper cup, about 10 counters (buttons or beans work).
Count 5 counters with the child and put them under the upside-down cup. Ask, “How many are under the cup?” The child should say “five” without lifting it. Then set 3 more counters in a row beside the cup. Ask, “How many altogether?”
Many children lift the cup and count from one. Show the shortcut: tap the cup and say “five” in a long, stretched voice, then touch each counter outside, “six, seven, eight.” Lift the cup and count all of them to check that the answer is 8.
Ask, “Why did we say five for the cup?” You want to hear “because there are five under it.” The cup’s word stands for all five counters at once.
Practice. Hide the first number under the cup and set the second beside it.
- 4 under the cup, 2 beside it. Answer: “four, five, six”, so 6.
- 6 under the cup, 3 beside it. Answer: “six, seven, eight, nine”, so 9.
- 3 under the cup, 4 beside it. Answer: “three, four, five, six, seven”, so 7.
- 7 under the cup, 2 beside it. What word does the child say first? Answer: “seven”, for the cup; then “eight, nine”, so 9.
Day 2: Counting on past ten
You need: a small paper bag, a marker, about 20 counters.
Count 12 counters into the bag together and write 12 on the bag. Put 4 counters in a row next to it. Tap the bag, “twelve,” then touch each counter, “thirteen, fourteen, fifteen, sixteen.” Empty the bag and count everything to check.
The teen words are where children slip, so go slowly and let the child lead. Ask, “What word did you say for the bag? What was the last word? So how many counters are there?” The last word, sixteen, is how many.
Practice. Write the number on the bag and put that many inside.
- A bag of 10 and 2 beside it. Answer: “ten, eleven, twelve”, so 12.
- A bag of 15 and 3 beside it. Answer: “fifteen, sixteen, seventeen, eighteen”, so 18.
- A bag of 9 and 3 beside it. Answer: “nine, ten, eleven, twelve”, so 12. The count goes past ten without stopping.
- A bag of 16 and 4 beside it. Answer: “sixteen, seventeen, eighteen, nineteen, twenty”, so 20.
Day 3: Counting back
You need: about 10 counters and a cup.
Lay 8 counters in a row and count them together. Then take the last one away and drop it into the cup, saying “seven.” Take the next, “six.” Take one more, “five.” Ask, “How many are left on the table?” Count the row to check: 5.
Each counter taken away moves the count back one word, so the words run backward. Keep it to a few steps; counting back is harder than counting on, because the backward words are less familiar. For a warm-up, count down from 10 to 0 together like a rocket launch.
Practice. Start with the row, then take counters away one at a time, saying each new count.
- 10 counters, take away 2. Answer: “nine, eight”, so 8 are left.
- 6 counters, take away 3. Answer: “five, four, three”, so 3 are left.
- 4 counters, take away 2. Answer: “three, two”, so 2 are left.
- 3 counters, take away all 3. Answer: “two, one, zero”, so none are left.
Day 4: On or back, without looking
You need: the paper bag, a marker, about 20 counters.
Count 14 counters into the bag, write 14 on it and hold the top nearly closed so the child cannot see inside. Drop in 3 more, one at a time. The child holds up a finger for each one dropped in and says the new count: “fifteen, sixteen, seventeen.” Three fingers are up, so the counting stops at seventeen. Open the bag and count to check.
Now the other way: with 13 in the bag, take 2 out, one at a time, counting back, “twelve, eleven.” The fingers keep track of how many steps to take; the words give the new amount.
Ask, “Did we put some in or take some out? So do we count on or count back?”
Practice. Write the starting number on the bag each time.
- 18 in the bag, 2 put in. Answer: “nineteen, twenty”, so 20.
- 20 in the bag, 3 taken out. Answer: “nineteen, eighteen, seventeen”, so 17.
- 11 in the bag, 3 put in. Answer: “twelve, thirteen, fourteen”, so 14.
- 15 in the bag, 2 taken out. Answer: “fourteen, thirteen”, so 13.
Exercises
- There are 8 counters under the cup. How many counters are there altogether? Watch whether the child starts at “eight” or lifts the cup and counts from one.
- A jar holds 19 marbles. Take out 3, one at a time, and say the count each time. How many marbles are left in the jar?
Answers
- 12 counters. Say “eight” for the cup, then touch each counter outside: “nine, ten, eleven, twelve.” A child who has the idea starts at eight and says four more words.
- 16 marbles. Each marble taken out moves the count back one word: “eighteen, seventeen, sixteen.” A child who has the idea counts back three words without emptying the jar.
Where this goes next
Counting on from a known amount becomes counting on to add and counting back to subtract, where the starting number comes from an addition sentence. Saying a number and carrying on is also how a child keeps counting past twenty in counting to one hundred. Putting some in is the idea of adding by joining, and taking some out is taking away.