Exponents Numbers & Operations in Base 10

Exponents Mistake from Middle School

cathy campbell

What makes this mistake so common?

Thanks to Cathy Campbell for the student work!

4 replies on “Exponents Mistake from Middle School”

I’m curios what this is assessing. Were we trying to see 49+343=392 and ummmmm. Whatever 8^4+8^3 is?

Anyway, this appears to be an extension mistake of the multiplying with same base rule.

Great point above. If this is a calculator assessment, then shame on the student for not using the available tools. If it is a test of exponent rules, then why muddy the issue with calculations like 8^4? Either way, the mistake is rooted in kids memorizing rules with little or no understanding of why the rules work. Even with some of my better students, it boils down to a mental coin toss about when to add exponents versus when to multiply them despite my best efforts at trying to uncover why the rules work the way they do. I always have this (egotistical, I admit) dream that if I had a batch of kids from Algebra I onward that we would avoid many of these mistakes.

Thanks for your comments here. You’ve got me thinking! And yes, the students could have used technology if they chose to.

The outcome (standard) that I was assessing was: Demonstrate an understanding of operations on
powers with integral bases (excluding base 0) and whole number exponents. This outcome includes the power laws.

I had other questions where the students provided some proofs and more lengthy written responses to ‘demonstrate an understanding’. I chose to use the same base to see if students could distinguish when to use the power laws and when to follow the order of operations.

Do you think it was misleading using the same bases?

When you say ‘muddy the waters’ with this calculation, what exactly to you mean?

I agree with all who reflect that this is a mis-application of the Power Laws… I also find that an over-reliance on algorithms is a symptom of a math curriculum that is built around the 3 R’s: Rules, Responses, and Repetition.

For many of us, this worked; we were able to put the right rules in the right places and eventually came to ‘see’ what was really happening behind the rules… But that simply didn’t happen for many students, including the ones shown.

That’s one of the common-sense things about asking kids to develop and explain personal strategies- they learn to look for the sense behind the processes they use.

In the above examples, a student looking for a strategy might find the common bases and factor them… ie: 7^2 + 7^3 = 7*7 + 7*7*7 = 7^2(1+7) = 49*8 = 392 … Estimate 50*8=400

Looking at the second one, he might see (-8)^3 * (-9), leading to a positive answer (+4608). Estimate -500 * -9 = +4500

Patterns are the basis of math. Questions like these are great for exploring patterns, not so good for finding out if kids can apply them.

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