Fishtank Learning

8th Grade Math
Unit 1: Exponents and Scientific Notation
Lesson 2

Objective

Evaluate numerical and algebraic expressions with exponents using the order of operations.

Warm Up

Both of the numerical expressions below are equal to 12.

$$\left(2^{4}\right) \div2$$$$$$

$$\frac{36}{3^{2}} + \frac{2}{3} (16-4)$$

Write at least two more expressions that equal 12. Use at least one exponent in each expression.

Anchor Problems - Problem 1

Two expressions are shown below.

Expression A: $${(2^2+3^2)}$$

Expression B: $${(2+3)^2}$$

Are the two expressions equivalent? Justify your response.

Purpose

In this problem, students consider two similar-looking expressions, differing only in the placement of the parentheses. It reaches back to their work in 6th and 7th grades, where they spent a lot of time looking at the structure of expressions, understanding what is happening with the numbers and operations. This skill will be important throughout the unit as students encounter more complex exponential expressions.

Facilitation Guidance

Launch this problem by asking students what they observe about the two expressions.
- What do you notice?
- What is the same and different about each expression?
- What bases do you see?
- How would you read each expression out loud?
Supporting Access for Students

A common way to launch many of the problems in this unit will be to invite students to take a close look at the content and share what they observe. In doing this, students not only get practice observing structure, but they are also able to hear what their fellow peers have observed that they themselves may not have seen.
As review content, students may be able to work on this independently or in pairs. As you circulate, keep an eye out for strategies that students use to determine if there is equivalence. This may include using the order of operations to evaluate each expression or drawing a picture to represent the expressions as areas.
In the debrief, have students share their conclusions and justifications with a focus on the justifications. Additional questions to discuss as a class are:
- Are the parentheses in Expression A necessary? Are they necessary in Expression B?
- Write the parentheses in a new position to create Expression C. Is your new expression equivalent to either A or B?

Anchor Problems - Problem 2

Evaluate the following expression:

$${{2-4(-2-1)^2}\over{\left ({1\over2} \right )^2}}$$

Purpose

The purpose of this problem is to have students work with a complicated-looking expression, using the order of operations to simplify it (or evaluate it) to a single value. In working with this problem, they practice applying their understanding of exponents and also observe how a complex-looking expression can be written into subsequent equivalent, more simple expressions, a theme that will come up throughout the unit.

Facilitation Guidance

Launch the problem by having students observe the structure of the expression. In Anchor Problem 1, students did this as a class; in this problem, you could have students share with a partner. Doing this type of launch in partners helps to ensure every student has the chance to comment on what they notice, and in a form of communication that works best for them (annotating the expression, drawing, writing, etc.) CRMT Culturally Responsive Math Teaching . In particular, tune in to student observations around the exponents and the bases in the problem.
When finding the value of numerical expressions, there will be one value that the expression will be equivalent to, yet there is more than one way to arrive at that value. This can be a valuable distinction for students and help them see that there is not just one right solution pathway CRMT Culturally Responsive Math Teaching .
- Can there be more than one answer to this expression?
- Can there be more than one way to approach solving this problem?
- What convention do we follow when operating with numerical expressions?
Give students time to find the value of the expression by writing a series of equivalent, more simple expressions until they end up with a single value. They can do this either independently or in pairs. Some questions to ask as you circulate and look at student work include:
- What is the value of the base of the exponent in the numerator?
- What operations do you see going on in the numerator?
- Will the value of your answer be positive or negative?
In the debrief, ask students to share the first step that they took. If all students took the same first step, ask if there was a different action they could have done first. Also ask if there is an action they could not do first (getting at the idea they cannot subtract 2−4 because the 4 is involved in multiplication). Walk through any computations as needed.

Anchor Problems - Problem 3

Evaluate the expression below for $${x=2}$$ and $${y=-3}$$.

$${{xy}^2 + xy}$$

Purpose

This problem reviews evaluating an algebraic expression involving exponents and negative values. It specifically involves evaluating the square of a negative number, ensuring students pay close attention to the value of the base and how to represent it in the expression using parentheses.

Facilitation Guidance

Launch the problem by asking students how this expression is different from the previous expressions in the lesson. Welcome all ideas, but be sure to cover the idea that this is an algebraic expression whereas the others are numeric expressions.
Similar to Anchor Problem 1, as review content, students may be able to work on this independently or in pairs. A main detail to keep an eye out for in student work is how students treat the −3 in the expression when they substitute it in for y, and if they recognize the need for parentheses.
Supporting Access for Students

If students struggle with accurately substituting −3 into the expression, then share some of the following expressions and have them consider what each one is communicating and which one matches the intent of the algebraic expression:
- $$2-3 ^{2}+2-3$$
- $$2\times-3 ^{2}+2\times-3$$
- $$2(-3) ^{2}+2(-3)$$
- $$(2\times-3) ^{2}+2(-3)$$
In the debrief, be sure to convey the importance of the parentheses around the −3 to accurately represent it as a base of the exponent MP.6 Attend to precision. .

Problem Set

View Problem Set

Target Task

Find the value of the expression below when $${x=-1}$$ and $${y=2}$$.

$${3x + y^2 - (x^y+y)}$$

Fishtank Learning

8th Grade MathUnit 1: Exponents and Scientific NotationLesson 2

Objective

Warm Up

Anchor Problems - Problem 1

Anchor Problems - Problem 2

Anchor Problems - Problem 3

Problem Set

Target Task

8th Grade Math
Unit 1: Exponents and Scientific Notation
Lesson 2