### Lesson Plan

# Exponents and Variables in Expressions

Students will write and evaluate algebraic expressions that represent real-world situations.

Grades

6–8

### Objectives

**Students will:**

- Evaluate numerical expressions involving whole-number exponents.
- Write algebraic expressions to represent relationships between terms.
- Write expressions involving whole-number exponents to represent problem situations.
- Write algebraic expressions to represent problem situations.
- Evaluate algebraic expressions that represent problem situations at given values of the variables.

See standards below.

### Materials

- Ski Time: Exponents in Algebraic Expressions printable
- Answer Key for Adventures in Expressions and Equations

### Lesson Directions

### INTRODUCTION TO NEW MATERIAL

**Step 1:** As an introduction to exponents, ask students to write the following multiplication expression as a repeated addition expression: 4 x 7. (7 + 7 + 7 + 7)

Ask:

**Why would someone use a multiplication expression instead of a repeated addition expression?**

It is more efficient to write.

Tell students that using exponents is a way to more efficiently write a repeated multiplication expression. The *base* is the number that is multiplied over and over. The *exponent* is the number of times the base is multiplied together. For instance, in the expression 6^{5}, 6 is the base, and 5 is the exponent. 6^{5} = 6 x 6 x 6 x 6 x 6 = 7,776.

**Step 2: **Ask students to show that multiplication is commutative using the expression 4 x 7 and repeated addition (4 x 7 = 7 + 7 + 7 + 7 = 28 = 4 + 4 + 4 + 4 + 4 + 4 + 4 = 7 x 4). Ask students to use the expression 6^{5} to decide whether the base and the exponent in an exponential expression are commutative (No: 6^{5} = 6 x 6 x 6 x 6 x 6 = 7,776; 5^{6} = 5 x 5 x 5 x 5 x 5 x 5 = 15,625).

**Step 3:** If your students would benefit from practice evaluating exponential expressions before moving on to discussing problem situations, provide them with some practice expressions to evaluate (examples: 3^{8}, 8^{3}, 11^{2}, 9^{4}, 4^{8}).

**Step 4:** Tell students to imagine a skier ascending a ski slope on a chairlift. The skier drops a glove. Ask:

**Does the glove fall to the ground below at a constant rate?**

No.

Tell students that velocity of the glove is equal to the product of the acceleration of gravity *g* and the time *t* in seconds that the glove has been falling. That means that the longer it falls, the faster it travels. Ask:

**What expression describes the velocity of the glove?**

*g* x *t*

**The acceleration of gravity on Earth is 9.8 m/s**^{2}. What expression describes the velocity of the glove since it is falling in Earth’s atmosphere?

9.8*t*

**Find the velocity of the glove after 1, 2, and 3 seconds.**

9.8 m/s, 19.6 m/s, and 29.4 m/s

**Step 5: **Tell students that the distance an object falls is half the product of the acceleration of gravity *g* and the square of the time *t* in seconds that the object falls. Ask:

**What expression describes the distance the glove has fallen?**

0.5*g* x *t*^{2}, or 0.5 x 9.8 x *t*^{2} on Earth

**Ask students to find the distance the glove has fallen after 1, 2, and 3 seconds.**

4.9 m, 19.6 m, and 44.1 m

**Step 6: **The skier reaches the top of the chairlift to begin her descent down the slope. It takes the skier 165 seconds to race down the mountain at 20 meters per second. Ask:

**If the formula for distance is distance = speed • time, how long is the mountain?**

3,300 meters

**Step 7:** Skiers #2, #3, and #4 race down the same course. Write an equation that represents how fast the skier races given the time it takes the skier to get down the same course (speed = 3,300 ÷ time). Model for students how to find the speed of the remaining three skiers if it takes Skier #2 three minutes and 7 seconds, Skier #3 three minutes and 12 seconds, and Skier #4 two minutes and 40 seconds to race down the mountain (17.647 meters per second, 17.1875 meters per second, 20.625 meters per second).

**
Step 8:** Tell students that at 8 a.m., there are four lone skiers on the slopes. Each hour, the number of skiers on the slopes quadruples. Ask:

**How many skiers are on the slopes at 9 a.m.?**

16

**What expressions can you write to show this?**

4 x 4 or 4^{2}

**What about at 10 a.m.?**

64

**What expressions can you write to show this?**

4 x 4 x 4 or 4^{3}

**What about at 11 a.m.?**

256

**What expressions can you write to show this?**

4 x 4 x 4 x 4 or 1,024

Have students work individually to write expressions for the number of skiers at noon, 1 p.m., and 2 p.m. Have them determine the number of skiers on the slopes at those various times. Then ask:

**What expression could you write to describe the number of skiers on the slopes***x*hours after they open at 8 a.m.?

4^{x+1}

### GUIDED PRACTICE

**Step 9:** Split your class into partners and post the following questions. For each question, the partners should:

- Write an expression to represent the word problem.
- Evaluate the expression; if it is an algebraic expression, choose a value for the variable.

**A.** You are on a bus driving to Tampa, Florida, for a training trip with your tennis team. Your average speed is 65 miles per hour.** **After a certain number of hours, how far have you traveled?

65*h*; sample chosen variable value: 65*h* when *h* = 7 would be 455 miles

**B.** You are kneading dough to make bread. Each time you knead, the dough doubles in size. If you knead the dough 3 times, how many times bigger is the dough when you are finished compared to when you started?

2^{3} or 8 times bigger

**C.** Your coach is 13.5 years older than you are. When you are *y* years old, how old is your coach?

13.5 + *y*; sample chosen variable value: 13.5 + *y* when *y* = 14 would be 27.5 miles

**D.** You baked 4 dozen cookies for the trip. If there are *n* players on your team, how many cookies does each person get?

48/*n*

**How many cookies are left over?**

48 – 48/*n*; sample chosen variable value: 48/*n* when *n* = 11 would be 4.3636…, so 4 cookies per person, and then 4 would be left over since 48 – 44 = 4

**Step 10:** If partners finish early, hand them two number cubes. They can roll one number cube to generate a base and one number cube to generate an exponent. Have them each evaluate the value of their expressions. Whichever partner has the larger value wins that round.

**Step 11:** **Checking for Understanding:** Review answers as a class and respond to any questions.

### INDEPENDENT PRACTICE

**Step 12:** Assign the Ski Time: Exponents in Algebraic Expressions printable for classwork or homework.

**Step 13:** **Checking for Understanding**: Review answers to the printable with the class, making sure students explain their mathematical thinking. Address any misconceptions that may arise.

### Standards

**Grade 6**: Exponential Expressions, Algebraic Expressions (CCSS 6.EE.A.1, 6.EE.A.2)**Grade 6–8**: Making Sense of Problems, Reasoning Abstractly and Quantitatively, Attending to Precision, Making Sense of Structure (CCSS Practice MP1, 2, 6, 7)