Sometimes my students think that the distributive property is an exception to the order of operations. This lesson should help them see it as just an alternative. I don’t want them to suspend their knowledge and use of the order of operations when they are simplifying multistep expressions and gain a mistaken understanding that the order of operations does not apply when simplifying.
Students begin the warmup on the screen when they enter. They solve the following expressions with the given values for the variables:
4(x + 1) + 2x when x=3 5 + 2(3 + 2n) when n=5 3(3e+1) + 2(4 + e) when e=2
As they solve each of the problems above I remind them of the order of operations by asking them why this is the correct thing to do next. In addition to reinforcing the relevance of the order of operations this also helps them develop better arguments by getting into the habit of explaining/justifying their reasoning. (mp3)
Common misconceptions for this section: students sometimes forget that the number outside parentheses or a number next to a variable means to multiply. If I see this mistake I will usually consult the class and ask them what it means. This helps them to view each other as resources rather than relying on me all the time. Also, since we have just been combining like terms they may try to combine the like terms before they distribute. I will not bring up the order of operations at this point, that will come later after we have gone over the warmup, but I will remind them that we are given the value of the variable so we can substitute the value for the variable and solve.
When we go over these I have them come rewrite the expression on the overhead after substituting the value for the variable. This should catch the mistake of replacing 2x with 23 instead of 2 times 3. If the person writes it correctly I would ask them to explain how they knew to multiply. Once the expression has been rewritten I ask how we proceed. At each step I ask how they knew in order to reinforce order of operations.
I put 5(4x + 3) on the overhead and tell them that I'm not going to give them a value for x so we won't be able to solve it, but we will see how far we can get simplifying it. I remind them that normally we would start with the addition inside the parentheses and I ask them why we get stuck here. After they explain how they know 4x and 3 can’t be added without the value of the variable I ask them if we’re going to let that stop us from going further? What other operations are we asked to do here? Can we do the multiplication before the addition inside parentheses? Some may say no, but I would draw the rectangle and remind them they can use the area model if they want to and I would remind them that the distributive property actually says we can. I ask them to show the multiplication and simplify the expression.
I put a second expression on the overhead (one from the warmup) but we don't assign a value to the variable.
4(x + 1) + 2x
I ask them what we did first when we were given a value for the variable and why can we not do that here. I ask what we did afterwards. When they say multiply I ask which one to make sure they mean the 4. I ask them if we can still do that without having done the addition inside the parentheses first. They should say yes, but if they say no I would draw another rectangle and ask again like the last problem. After the distributive property I ask if there is any more multiplication. If they say no, I ask if they remember what 4x and 2x are telling us to do with the number and the variable. I ask why we can't do the multiplication right now and then ask what the final step would be. When they say "add" I ask if we can do any of the adding and we finish with like terms.
I tell students I am not going to give them the value for the variables and ask them to just simplify the expression and get as far as they can. I start with just distributive property and then I add more terms reminding them of the order of operations at each step. The mistake many of them may be making is trying to combine the like terms before completing the distributive property. They also may be distributing to terms outside the parentheses. If students are combining like terms before they multiply I will probably see lines drawn from a term inside the parentheses to a term outside and the coefficients in their final answer will be too small. If they are distributing to terms outside the parentheses those terms will be too big, for example if the term outside is constant, their constant will be too big. If students are adding before distributing, as in 5+2(3+2n) all the numbers will be too big.
5(3 + 2x) 2(x + 4) 4(x + 1) + 2x 5x + 2(3 + 2x) 6(2x + 3) + 10
5 + 2(3 + 2n) 3(e + 1) + 2(4 + e) 4(2 + x) + 2(3 + 4x)
When we go over these I ask right away what we would normally start with. (inside parentheses) I ask since we can’t do that what do we look for next? They may share what they think we should do next in the particular problem, but I clarify that I am asking in general what we would normally do after the operation inside parentheses. It may help to write PEMDAS. I am trying to slow them down and make sense of what is going on in the problem so they can see that the knowledge they already have applies here. With no exponents what would we normally do next? Do we see any multiplication or division? In the more complex expressions they should notice both the multiplication being distributed and the coefficient and the variable. I ask then which one of those we can do and we do the distributive property. In the simpler ones I still ask what would we do last? Can we do the addition? As we go over each one I may ask if we can add the like terms (for example the x and the 4x in the last problem) before we complete the multiplication. It may also be helpful to bring back the problems from the warm up and remind them how it was solved when we did have the value of the variable (the bolded expressions are the same ones from the warmup) and maybe even substitute the value for the variable in the simplified expression to show they are equivalent.
I am trying to relate simplifying variable expressions to the order of solving expressions. It is really important that they see the connection between how to solve an expression when all the values are given and to simplify an expression when we don't. I think if we work on them side by side and continue to go back and refer to the tools we have used before they may begin to see the relationship.