A student solves 13 − 8 by counting: '13, 12, 11, 10, 9, 8' and writes the answer as 6. What went wrong?
AThe student used the wrong strategy — they should have drawn a ten frame
BThe student counted the starting number (13) as one of the steps; counting back from 13 means the first step lands on 12, not 13
CThe student subtracted in the wrong direction — you should always count forward
DThe student stopped too early; they should have counted back 9 steps
The classic counting-back error is including the starting number in the count. Counting back 8 from 13 means: 12 (one), 11 (two), 10 (three), 9 (four), 8 (five), 7 (six), 6 (seven), 5 (eight). The answer is 5. Starting the count at '13' means you've only actually subtracted 5, not 6.
Question 2 Multiple Choice
Which of the following correctly shows that subtraction and addition are inverse operations?
A15 − 8 = 7 because 7 + 9 = 16
B15 − 8 = 7 because 8 + 7 = 15
C15 − 8 = 7 because 15 − 7 = 8, and 7 is smaller
D15 − 8 = 7 because 10 − 3 = 7 and 10 is close to 15
The 'think-addition' strategy works because every subtraction fact is secretly an addition fact. 15 − 8 = 7 because 8 and 7 are the two parts that make 15. If you know 8 + 7 = 15, you instantly know both 15 − 8 = 7 and 15 − 7 = 8. This inverse relationship is the key insight of this topic.
Question 3 True / False
The phrase 'subtraction undoes addition' means that if you know 6 + 9 = 15, you can use that fact to immediately find 15 − 9 without counting.
TTrue
FFalse
Answer: True
Yes — this is exactly the think-addition strategy. 6 + 9 = 15 means 15 − 9 = 6 and 15 − 6 = 9. The addition fact and the two subtraction facts are all part of the same fact family. Knowing any one of them gives you the others instantly, which is why connecting subtraction to addition is so powerful.
Question 4 True / False
Counting back is typically the most reliable strategy for subtraction within 20.
TTrue
FFalse
Answer: False
Counting back is error-prone (it's easy to miscount the starting number) and slow when the numbers are far apart. The think-addition strategy, make-ten strategy, and count-up strategy are all often faster and more accurate. For example, 17 − 9 is much easier solved by 'what plus 9 makes 17?' (answer: 8) or by counting up from 9 to 17 (8 steps), rather than counting back 9 steps from 17.
Question 5 Short Answer
Why can subtraction mean both 'take away' and 'how far between two numbers'? Give an example of each interpretation for 13 − 8.
Think about your answer, then reveal below.
Model answer: Take-away: Start with 13 objects, remove 8, and 5 remain. Distance: Start at 8 and count up to 13 — the gap between them is 5 steps. Both give the same answer (5) because subtraction measures the difference between two numbers, whether you think of it as removing or as comparing.
Understanding that subtraction has two interpretations unlocks two different solving strategies. 'Take away' naturally leads to counting back. 'How far between?' naturally leads to counting up (or think-addition). Choosing the right interpretation for a given context makes the calculation easier — for instance, 13 − 12 is much easier to solve by counting up (just 1 step) than by counting back 12 steps.