You want to add 28 + 37 in your head. Which decomposition strategy takes best advantage of the flexibility numbers offer?
AThere is only one right way to decompose: always split each number into tens and ones
BDecompose by tens and ones: 20 + 8 + 30 + 7, then add tens first (50) and ones next (15) to get 65
CYou cannot decompose both numbers at once — only one number can be broken apart at a time
DDecomposing makes addition harder, so keep both numbers whole
The power of decomposition is flexibility — you can split numbers however helps most. Decomposing both by tens and ones (20 + 8 + 30 + 7) lets you add the easier tens together first (50), then the ones (15), then combine (65). Option A is the misconception: tens-and-ones is the most common split but not the only valid one.
Question 2 Multiple Choice
Which of the following are BOTH valid decompositions of 46?
A4 + 6 (using the digits) and 40 + 6
B40 + 6 and 30 + 16
C40 + 6 only — the others change the number
D46 cannot be decomposed except as 40 + 6
Both 40 + 6 and 30 + 16 equal 46 — they are both valid decompositions. The key insight is that decomposing does not change the total; you are simply choosing where to 'cut' the number. Note that 4 + 6 = 10, not 46 — writing the digits side by side is not the same as splitting the number.
Question 3 True / False
There is mainly one correct way to decompose a two-digit number — the standard tens-and-ones split.
TTrue
FFalse
Answer: False
A two-digit number can be decomposed in many valid ways. The number 37, for example, can be written as 30 + 7, or 20 + 17, or 25 + 12, or 36 + 1 — all are true equations. The tens-and-ones split is the most common starting point, but the 'best' decomposition is whichever one makes a particular calculation easiest.
Question 4 True / False
Decomposing 37 as 30 + 7 and as 20 + 17 both give the same total.
TTrue
FFalse
Answer: True
Both 30 + 7 = 37 and 20 + 17 = 37. Decomposing never changes the value of a number — it only changes how you look at it. This is why decomposition is a safe mental-math tool: you can split a number any way you like without affecting the final answer.
Question 5 Short Answer
Why is it useful to decompose a number in different ways rather than always using the standard tens-and-ones split?
Think about your answer, then reveal below.
Model answer: Different decompositions make different calculations easier. By choosing a split that fits the problem — such as splitting a number to create a round number — you can simplify mental arithmetic. Flexibility with how you see a number is the foundation of number sense.
The tens-and-ones split is reliable, but not always the fastest route. For instance, adding 38 + 25 is easier if you decompose 38 as 40 − 2 and then compute 40 + 25 − 2 = 63 mentally. The ability to choose the right decomposition for the situation is exactly what number sense means.