Oh man, this is going to be tough for kids. Good mistake.

What makes this so hard? Or am I over-estimating its difficulty?

Thanks Matt!


  • Why are the (-) signs in so high in the brackets? I find that confusing!

    • Joshua Zucker

      I suspect it comes from a text that wants to distinguish the unary negation operator from the binary subtraction operator in some clear way. Of course, there’s also the third use of the – sign which is as the names for negative numbers…

  • My guess is that the rule of negative times negative = positive is at play here in the mind of this student. Then we look at the exponent after that. I think that this is actually a promising mistake because it should be easy to get at the heart of. Talk about order of operations and ask the following two questions which might give insight into the student’s thinking.

    (x)^5 * (x)^2 =

    (-3)^6 * (-3)^2 =

    • Joshua Zucker

      It would be great if this was an unthinking use of “negative times negative = positive” — I hope that’s the case! Then we can focus in on the order of operations and so forth, and ask whether (-3)^5 is positive or negative, and similarly with (-3)^2.

      I also like the recommendation of having them expand (-3)(-3)(-3)(-3)(-3) * (-3)(-3) and think about that for a while.

  • To help this student work through their thinking, I’d have them expand the expression and see what they notice. There’s definitely a lot of “rules” in this situation which may be causing cognitive conflicts.

  • It’s actually not a surprising mistake (in my opinion). Given that the student changed the sign to positive – multiplying the negative number by itself – I wouldn’t be surprised if the base turned to +9, not +3. What I have done in the past when teaching multiplying powers is to expand the multiplication of 2^2 and 2^3 for example – demonstrating that if you multiply the bases as well as add the exponents, you are increasing the product – well, exponentially!

  • Audrey McLaren

    It’s an order of operations mistake at its simplest – they multiplied before doing the exponent. But of course I’d have them expand, or ask them is this a bunch of -3’s or 3’s being multiplied?