# Why do we add here, but not here…

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For some reason there were 6 kids in after school with me…all working on different topics and courses. Yikes. I had to be on my toes that day.

One student in particular was working with polynomial operations.  Why do we add the exponents on these… but not these…?

Well.  I paused. Why do we?  That expert blind spot…I know that’s what we do, but why?  We aren’t actually adding exponents but that’s what appears to be happening. Help.

Hmm. Well.  Uhm.  Flashback to my daughter’s 2nd or 3rd grade math.

Sort of like place value.  The math doesn’t work because of the rule. Someone made up the rule as a short cut because that’s what happens when we do the math…

And this is what I began with…

Too often students see polynomial expressions as some obscure thing. Have we failed them by not encouraging them to see them as numbers?

Please share how you address this same question, why do we add exponents here, but not here?

And saying that’s the rule is not acceptable. 😊

### 4 responses »

1. Great work! I actually had this question come up within the last two weeks as well. I just asked the kids what 3 times 3 equals, and they said 9. Then, I asked them how we could rewrite it based on what we know about exponents. They said 3 squared.

Next, I asked what 3+3 is, and they said 6. Then, I asked why it wouldn’t work to rewrite that sum as 3 squared. They said because 3 squared has to equal 9. It seemed to help a bit.

2. Yes, I have had this problem in my algebra class. I had the opportunity to attend a workshop, about exponential functions.
We started with “problem strings”. You take a higher level concept, like exponent rules, and start with previous knowledge, like multiplying numbers and the student works their way to “seeing” the rule, not me feeding them the rule.
I have since started using “problem strings” in my class and it has changed the way I teach.

3. I have run into the exact same issue. Try focusing on what operation is actually happening… the exponents are being added because the terms are being MULTIPLIED (adding exponents is just the consequence of the multiplication, a short-cut instead of writing out all the bases individually). When the terms are being ADDED, it’s like adding several items: 2 x-squared + 3 x-squared = 5 x-squared, same as 2 pizzas + 3 pizzas = 5 pizzas.

• I agree. I don’t teach them the “shortcut” or rule, but scaffold so they see the structure and make it their thinking. All work with olynomials, I begin with place value examples to connect to their prior knowledge, but too often it’s presented as a single lesson and here’s the rule use it without any conceptual development