# Ducks in a Row!

## Objective

SWBAT simplify multiple step variable expressions using distributive property and combining like terms.

#### Big Idea

Students recognize that they need to follow correct order of operations while simplifying expressions.

## Intro & Rationale

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.

## Warm up

15 minutes

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.

## Mini lesson

7 minutes

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

## White Boards

32 minutes

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)