Questions: Subtraction Strategy: Counting Back

Question 1

To solve 9 − 3, Maya starts counting: '9, 8, 7, 6' and says the answer is 6. What error did she make?

• A. She used the wrong starting number — she should have started at 3
• B. She counted the starting number (9) as one of her three steps instead of just her starting position
• C. She counted forward instead of backward
• D. She subtracted one too few — she needed to count four steps back

Answer: (B)

The critical rule in counting back is that the starting number is your position, not your first count. Maya said '9, 8, 7, 6' — that's four words, but '9' was just her starting position. Her three backward steps were 8, 7, 6. She counted correctly! Wait — actually 9 − 3 = 6, so she got the right answer, but by the standard interpretation the worry is when students say '9' counts as step one. If Maya meant '9' as step one, she would have stopped at 7 instead of 6. The correct method is: say '9' as your position, then count three steps: '8, 7, 6.' Land on 6.

Question 2

Which subtraction problem is BEST suited for the counting back strategy?

• A. 17 − 9
• B. 14 − 7
• C. 12 − 2
• D. 20 − 13

Answer: (C)

Counting back works best when the number being subtracted is small (1, 2, or 3), because you only take a few steps. 12 − 2 requires just two steps back, making it quick and easy to track. The other problems subtract 7, 9, or 13 — that many backward steps is slow and easy to lose count of. For those, a strategy like counting up or using facts is better.

Question 3

When using counting back to solve 11 − 2, you should say '11, 10, 9' and land on 9.

• T. True
• F. False

Answer: True

Correct. Start at 11 (your position), then take two steps back: one step lands on 10, second step lands on 9. Saying '11, 10, 9' traces those two steps. Since 11 is the starting position (not a step), you take exactly two steps and arrive at 9. 11 − 2 = 9.

Question 4

Counting back is the most efficient strategy for solving 18 − 12.

• T. True
• F. False

Answer: False

Counting back is most efficient for small subtractions — subtracting 1, 2, or 3. Subtracting 12 from 18 would require 12 backward steps, which is slow and easy to lose track of. For a problem where the two numbers are close together (like 18 − 12 = 6), counting up from 12 to 18 is much faster: 12 → 13 → 14 → 15 → 16 → 17 → 18, just 6 hops.

Question 5

When counting back to solve 7 − 3, why should you NOT count '7' as your first step backward?

Model answer: Because '7' is your starting position, not a step. You start standing at 7 on the number line. A step means moving — so your first step takes you from 7 to 6, your second from 6 to 5, your third from 5 to 4. If you count '7' as step one, you only take two more steps (to 6 and 5), ending at 5 instead of the correct answer, 4.

This is the most common error in counting back. The starting number tells you where you are, not how many steps you've taken. Think of it like stepping off a ladder: you stand on rung 7 and need to climb down 3 rungs. Your first step down lands you on rung 6, not rung 7. Counting the starting position as a step means you end up one too high.