Evaluating Expressions

Students work in partners on the Think About It problems (TAB). Here, they are substituting in a value of n for different expressions. 

For problems a and b, I have students respond chorally, in whisper voices.  I expect the majority of students to get these two problems pretty easily.

For problem c, I tell them that our (imaginary) friend BoBo thinks the value is 124.  I ask for someone to explain what mistake BoBo made.  Students know that 12n means 12 times the number n from previous lessons.  I have the student who explained this to the class call on another student to share the value of 12n, when n =4.

Finally, I frame the lesson by letting students know today we’ll be evaluating more complex expressions using everything we know about expressions and order of operations.

The Intro to New Material (INM) starts with an example:

Evaluate 2x + 8 if x=4

What does 2x mean?      (twice some number x)

What is the 2 called?       (the coefficient)

I model how to solve, without the help of students:

• Re-write the expression 2x + 8.
• Wherever you see x, replace or substitute it with 4 in a parentheses.
  • I ask students to turn and talk for 30 seconds about why I put the 4 in parentheses (so that it is clear we mean 2*4, so that we don’t think it means 24)
  • Finish simplifying by using order of operations: I ask students to list out the order of operations that we follow, and create a quick visual on the  board to represent grouping symbols, exponents, multiplication/division, and subtraction/addition

2x + 8

2(4) + 8

8 + 8

16

For Example 2, we’re given an equation to use to evaluate a situation.  I have a student read the problem out loud as the class annotates the problem.  This includes drawing and labeling a triangle to help us make sense of the problem. 

My students are familiar with the formula to find the area of a triangle, so I quickly have them share out to remind each other of the meaning of each variable.  Then, I ask them what our units should be when we are done (square feet).

We go through the same steps I used for Example 1 – rewrite the equation, substitute in for the variables, using parenthesis, and then evaluate.

This problem is a great one to apply the commutative property of multiplication.  Rather than find half of 5, we can multiply 5 and 6, and then find half (or, find half of 6 and then multiply by 5).

Finally, we look at one more example together that is more complex. 3n² – 4 + 2²  I ask students what is more complex about this problem (the exponents).

We’re still going to follow the same steps.  First, rewrite the expression.  Then, substitute a 3 in every place we see an n.

3(3)² – 4 + 2²

What comes first in order of operations? (grouping symbols).  Do we have any of those? (nope) So what’s next? (exponents)  We evaluate the exponents, which leaves us with:

3(9) – 4 + 4

Now what do we do?  (multiply three by 9)  That leaves us with:

27 – 4 + 4

What’s left? (subtraction and addition, which we perform from left to right, like we’re reading a sentence)

What do you think happens when we subtract 4 and then add 4? (we end up right back to 27) Let’s test it, to be sure.

27 – 4 + 4

23 + 4

27

Students complete two quick problems on their own, and then check their own work against my exemplars, which I put under the document camera.

Finally, we fill in the steps that we’ve used.  Students come up with the words in the blanks. I post the Visual Anchor for the class after they’ve filled in the blanks, so they can reference it once they’ve started to work without my help.

Students work in pairs for 10 minutes on the Partner Practice set.

As they are working, I am circulating and checking:

• Are students rewriting the expression
• Are students using parentheses around the substituted value?
• Are students using the correct order of operations?
• Are students including units, for the formula problems?

I ask partners:

• What did you do first here?
• Where did you put in ___ (value of the variable)?
• What was your first step, when evaluating this expression? What did you do second?

After 10 minutes of work time, I pull a popscicle stick to get a student work sample for the document camera.  The student walks through how (s)he completed the task. During the explanation I ask the same questions that I asked of partners as I circulated.

Students then work independently on the final CFU problem. If students have not seen the formula d = rate * time before, take the time to discuss what each variable stands for.  After a minute of work time, I have students flash me their answers by holding up their papers. 

Students work on their own on the Independent Practice problems. 

As I circulate, I am particularly focused on making sure students are not making the error of representing 6n as 64 when n = 64. 

I’m also focused on the order of operations, especially when there is both a coefficient and an exponent with the variable (like 4n³)

Before students begin work on their exit tickets, I ask them to evaluate my work on problem 8, where h = 5.1. I intentionally make a mistake with my multiplication of 3 and 5.1 (I write the product as 15.1, instead of 15.3). I provide several students with the opportunity to consider my work and explain if it is correct or incorrect.

After our discussion, students work on their Exit Tickets.