You may have heard that "addition is commutative, so 𝑎 + 𝑏 can always be written as 𝑏 + 𝑎."
We know, for example, 1 + 2 can be transformed to 2 + 1.
Suppose another student tells you that 1 + 2 × 3 can be rewritten as 2 + 1 × 3.
This is obviously wrong, but why isn’t that how the commutative property works? Take a moment to think: What’s the problem?
1 Draw the Circles of Evaluation to figure it out!
| 1 + 2 × 3 | 2 + 1 × 3 |
|---|---|
2 What do these Circles of Evaluation show us about why we can’t use the commutative property to rewrite 1 + 2 × 3 as 2 + 1 × 3?
3 Draw the Circles of Evaluation to decide whether or not these expressions will evaluate to the same thing.
| 5 + 21 × 36 | 21 × 36 + 5 |
|---|---|
4 Will 5 + 21 × 36 and 21 × 36 + 5 evaluate to the same thing? How do you know from looking at the Circles of Evaluation?
These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). 
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