Counting on and back (PDF, 42 KB)
Additive Strategies progression, 2nd step
The purpose of the activity
In this activity, the learners develop their understanding of addition and subtraction by counting on in ones to solve problems. They learn that you do not have to start counting from one to solve problems.
The teaching points

Counting tells how many objects there are in a set. The last word in the counting sequence tells how many objects are in the set.

When counting on, start from the largest number. For example, count on from 18 rather than 4 to solve 4 + 18 = ?

Addition and subtraction are closely related. Addition names the whole in terms of its parts. Subtraction names the missing part when the whole and one of the parts is known.

Addition problems have a variety of types or structures. It is important that the learners solve problems of all types to strengthen their understanding of addition.

Result unknown. In this type of problem, two numbers are given, and you have to find the result (for example, 4 + 6 = ?).

Change unknown. In this type of problem, an initial and a final number are given, and you have to find the number in between (for example, 5 + ? = 10).

Start unknown. In this type of problem, you know what has happened to an unknown number to give a particular answer. You have to find the starting number (for example, ? + 5 = 7).

Subtraction problems have a variety of types or structures. It is important that the learners solve problems of all types to strengthen their understanding of subtraction.

Result unknown. In this type of problem, you know the largest or the starting amount and the amount that has been taken away. You need to find the resulting amount (for example, 20 – 5 = ?).

Change unknown. In this type of problem, the starting amount and the amount that is left are given. You need to find the amount that was taken away (for example, 20 – ? = 15).

Start unknown. In this type of problem, the amount taken away and the resulting amount are known. You have to find the starting amount (for example, ? – 5 = 15).

Discuss with the learners the contexts or situations where they need to use addition and subtraction.
Resources

Sets of digit cards (10 cards labelled from 0 to 9).
The guided teaching and learning sequence
1. Show the learners the set of digit cards. Ask a volunteer to select a card from the set. Record this number on the board (for example, 4).
2. Ask another volunteer to select two cards from the set of digit cards and form a twodigit number. Record this number on the board (for example, 45).
3. Tell the learners that you want them to join or add the two numbers together. Record 4 + 45 = on the board.
4. Ask the learners to first work out the answer and then think about how they would explain what to do to someone who was having difficulty working it out.
“What would you say to someone who was having difficulty working out 4 + 45?”
5. Ask the learners to share their solutions, ensuring they:

start with the largest number (for example, 4 + 45 becomes 45 + 4)

use the counting sequence to count on from the largest number (for example, 46, 47, 48, 49).
6. Give pairs of learners a set of digit cards. Explain that they are to take turns selecting the numbers and solving the problems posed. The learner who selects the cards first places them in front of the other learner and asks that person to join or add the numbers together. The learner who is solving the problem must explain how they found the answer to their partner and record the problem and answer on a sheet of paper.
7. Repeat the exercise a couple of times. As the learners work together, circulate, checking that the learners are beginning with the largest number and that they understand the sequence of numbers used.
8. Ask a volunteer to draw two digits from the set of digit cards and form the largest number possible with these two digits. Doing this provides you with another opportunity to assess the learners’ understanding of numbers. Record the number on the board (for example, 62).
9. Tell the learners that this time they are going to subtract or take amounts away from 62. Ask a volunteer to select a card from the set of digit cards to indicate how much is to be taken away from 62 (for example, 6). As a class, count back 6 from 62 to find the answer to 62 – 6 = ?
10. Check the learners understand that they count:
61, 60, 59, 58, 57, 56. Record
62 – 6 = 56 on the board.
11. Give pairs of learners a set of digit cards. Explain they are to take turns selecting the numbers and solving the problems posed. The learner who selects the cards first places them in front of the other learner and asks that person to subtract the singledigit number from the doubledigit number. The learner who is solving the problem must explain how they found the answer to their partner and record the problem and answer on a sheet of paper.
12. Repeat the exercise a couple of times. As the learners work together, circulate, checking that the learners are correctly counting back to solve the problem.
The rest of this guided learning sequence repeats the above steps with different types of addition and subtraction problems. These could be taken as separate teaching sessions.
13. Write the following addition start unknown problem on the board:
“Sandy is collecting donations for a charity. A person gives her $7. She now has $71. How much did she have to start with?”
14. Discuss with the learners how this problem can be recorded as ? + 7 = 71.
15. Ask the learners to think about how they would solve the problem. Ask for volunteers to share their solutions. For example “I counted back from 71 to get to 64”. Ensure the learners understand this approach means that they have used subtraction (or counting back) to solve the addition problem.
16. Ask the learners to work in pairs to pose problems to one another where the starting amount is unknown, using numbers drawn from the set of digit cards (as in step 6 above). Alternatively, the learners could work as a group to solve the following problems:

? + 6 = 78

? + 9 = 35

? + 2 = 40
17. Write the following addition change unknown problem on the board:
“Sandy is collecting donations for a charity and has $79. A person makes a donation. She now has $85. How much did the person donate?”
18. Discuss with the learners how this problem can be recorded as 79 + ? = 85.
19. Ask the learners to think about how they would solve the problem. Ask for volunteers to share their solutions. For example “I counted on 6 from 79 to get to 85” or “I counted 6 back from 85 to get 79”. Ask the learners to work in pairs to pose problems to one another where the starting amount is unknown, using numbers drawn from the set of digit cards (as in step 6 above). Alternatively, the learners could work as a group to solve the following problems:

36 + ? = 45

12 + ? = 19

75 + ? = 82
Followup activity
Ask the learners to write addition and subtraction problems that involve twodigit and onedigit numbers on strips of paper for others to solve. On the reverse side of the strip of paper, have the learners write the solution and the counting on or back strategy they used to solve the problem.
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