1 Example. How many different ways can you group the following expression: 8 + 2 + 9 + 1 ? Below are three possibilities. For each example, order stays the same, but groupings change. Can you think of any more?
(+ (+ 8 (+ 2 9)) 1)
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(+ (+ 8 2) (+ 9 1))
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(+ (+ (+ 8 2) 9) 1)
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2 Your turn! Draw as many equivalent Circles of Evaluation as you can in the boxes below for the expression 98 + 3 + 7 + 26 + 4. Then, answer the questions at the bottom of the page. For each equivalent expression, change the groupings but not the order. To get you going, we’ve completed one sample Circle of Evaluation and started a second one.
(+ (+ (+ (+ 98 3) 7) 26) 4)
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(+ (+ ) (+ (+ ) ))
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3 Which Circle (above) seems like it would be the most difficult to solve in your head? Put a star by that one. (You may award more than one star if it feels right.) Then, in the space below, explain what makes that Circle challenging to evaluate.
4 Which Circle of Evaluation seems like it would be the easiest to solve in your head? Put a check mark by that one. (You may award more than one star if it feels right.) Then, in the space below, explain what makes that one easier to evaluate.
These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). 
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