Questions: Addition Strategy: Counting On

Question 1

A student solves 7 + 3 by counting: '7, 8, 9' and writes 9 as her answer. What error did she make?

• A. She started at the wrong number; she should have started at 3
• B. She counted 7 as her first step rather than holding it, so she only moved 2 steps instead of 3
• C. The counting-on strategy doesn't work when the first number is larger than the second
• D. She should have counted backward from 10 instead

Answer: (B)

Counting on requires holding the first number in mind and counting ON from it — not saying it aloud as a step. The student said '7' as step 1, then '8, 9' for two more steps, counting only 2 steps total instead of 3. The correct approach: hold 7, then count '8' (step 1), '9' (step 2), '10' (step 3). The answer is 10, not 9. This off-by-one error is the most common mistake with counting on.

Question 2

To solve 3 + 8 most efficiently using the counting-on strategy, where should you start?

• A. Start at 3 and count on 8 steps: 4, 5, 6, 7, 8, 9, 10, 11
• B. Start at 8 and count on 3 steps: 9, 10, 11
• C. Count all numbers from 1 to 11
• D. Start at 3 and count on 3 steps, since both numbers are in the problem

Answer: (B)

Because addition is commutative (3 + 8 = 8 + 3), you can always start from the larger number. Starting at 8 and counting on just 3 steps (9, 10, 11) reaches the answer in 3 counts. Starting at 3 and counting on 8 steps reaches the same answer in 8 counts — correct, but more than twice as slow. Always starting from the larger number makes counting on as efficient as possible.

Question 3

When using counting-on to solve 5 + 4, you should say '5' aloud as the first counting step.

• T. True
• F. False

Answer: False

The key rule of counting on is to hold the first number in your mind without counting it, then count on from there. Saying '5' as the first step means you start at 4 and end at 8 instead of 9. The correct method: hold 5, then say '6' (step 1), '7' (step 2), '8' (step 3), '9' (step 4). The last number said — 9 — is the answer.

Question 4

The counting-on strategy is more efficient than counting all because it skips the steps from 1 up to the starting number.

• T. True
• F. False

Answer: True

Counting all starts from 1 every time, requiring as many steps as the total sum. For 7 + 3, counting all means counting 10 numbers (1 through 10). Counting on skips the first 7 steps by holding 7 in mind and counting only the 3 remaining steps (8, 9, 10). This efficiency advantage grows with larger starting numbers.

Question 5

Why should you always start the counting-on strategy from the larger of the two numbers, even if the problem writes the smaller number first?

Model answer: Because addition is commutative — the order of the numbers doesn't change the sum. Starting from the larger number reduces the counting steps needed to just the smaller number's worth, making the strategy faster. For 2 + 9, starting at 9 requires only 2 counting steps (10, 11) instead of 9 steps if you started at 2.

The commutative property makes this swap valid: 2 + 9 = 9 + 2 = 11 regardless of order. Recognizing this and always using the larger number as the starting point is what separates an efficient counter-on from one who takes the long route. This habit also builds toward mental math fluency by training students to find the most efficient path through every addition problem.