Clarifying Our Terms

Today’s lesson comes after a re-teaching lesson (Let’s simplify matters) when students struggle with combining like terms. In context and with physical models students are able to understand it and correctly combine like terms (zombies, to change or not to change), but they have trouble transferring it to the abstract. This lesson has them taking a closer look at the different types of terms in a purely mathematical sense.

Students start by simplifying three expressions:

3x + 2 + 4 + x                      4m + c + 3c + 6m               4x^2 + 3 + 2x + 3x^2 + 3x

While students are simplifying these I circulate to help them bridge from wherever they are to the correct model. I expect several different difficulties. If any of my students combine the variable with the constant terms I remind them of my disorganized wallet (Let’s simplify matters). Some students may ask me if this is like the wallet.

As they finish their expressions I have students present all their solutions (right or wrong) and let the class consider, critique, and correct each other (MP3). I might suggest that they try out some values for the variables to test out their reasoning to be sure it is equal to the original expression. 

• Some students will get stuck with 3x + x, because there is no coefficient on the x. 
• Some students will mistake the number of exes with the number of terms and say 3x + x is two exes. 
• Some students may be combining all terms or all variable terms regardless of exponents. 
• Some students may make a list without plus signs.

I also may help guide the conversation by asking students each step of the way

• “why do you think we can combine these terms?”
• "can we combine these terms? why/why not?"
• “what tells you they are like terms?”
• “why can we not combine these terms?”
• “what could these variables represent?"
• “are they in the correct standard order?”

However, if the confusion is more widespread I know they need more time and experience with the algebra tiles and maybe some context problems.

Now that we have simplified expressions I want to show them why it is helpful to simplify. I cover everything except the original expression for number one and I assign a value to x (x=2) and ask them to figure the final single right answer in their head.

3x + 2 + 4 + x                                   

Then we try it with the simplified expression with the same value and maybe even larger ones and I ask which one was simpler to do?

4x + 6  (maybe for x=5 or 10 as well)

Then we do the same for problems 2 and 3 for which they may not be able to do the math in their heads using the unsimplified expressions.

 4m + c + 3c + 6m          then 4c + 10m

 4x^2 + 3 + 2x + 3x^2 + 3x      then 7x^2 + 5x + 3  (maybe for x=10)

This should help them see the difference between what it means to solve and what it means to simplify. It will also reinforce that the process of simplifying does not change the value of the expression. They will also like that it makes math easier for them.