### Lesson Plan

# Applying Cube Roots, Square Roots, and the Pythagorean Theorem

In this Adventures in Expressions and Equations lesson, students will apply knowledge of exponents to solve problems involving a parkour course.

Grades

6–8

Duration

40 MINUTES

### Quick links to lesson materials:

### Objectives

**Students will:**

- Use properties of integer exponents to write equivalent expressions
- Use square root symbols to represent solutions to equations of the form
*x*=^{2}*p* - Use cube root symbols to represent solutions to equations of the form
*x*=^{3}*p* - Evaluate square roots of small perfect squares
- Evaluate cube roots of small perfect cubes
- Use the Pythagorean theorem to find an unknown side length in a triangle

### Materials

- A Complex Course: Exponents, Square Roots, and Cube Roots printable
- Answer Key: Adventures in Expressions and Equations printable

### Set Up

- Make a class set of the A Complex Course: Exponents, Square Roots, and Cube Roots printable.
- Print a copy of the Answer Key: Adventures in Expressions and Equations printable for your use.

### Lesson Directions

### Introduction to New Material

**Step 1:** As an introduction to the properties of integer exponents, provide students with a number of guided computations by posting the following table:

Model the first few rows with the students, but then have them participate to complete the rest of the table.

Ask:* ***What pattern do you notice between the 2-term exponential expressions and the 1-term exponential expressions?** If two exponential expressions with the same base are multiplied, add the exponents together. If two exponential expressions with the same base are divided, subtract the exponents.

If students would benefit from practicing these properties, provide them with a few more 2-term exponential expressions with the same base to practice with.

**Step 2:** Then, show your students the Pythagorean theorem. Tell them that when you know two side lengths of a right triangle, you can calculate the third side.

In a right triangle, the sum of the squares of the leg lengths is equal to the square of the hypotenuse. The legs are the two sides adjacent to the right angle. The hypotenuse is the side opposite of the right angle.

So, for the picture above, *a*^{2} + *b*^{2} = *c*^{2}. This is the Pythagorean theorem.

**Step 3:** Display the following right triangle for students:

Have students substitute the given values within the formula:

*a*^{2} + *b*^{2} = *c*^{2}

8^{2} + 6^{2} = *c*^{2}

64 + 36 = *c*^{2}

100 = *c*^{2}

Once you have reached this point with the class, explain to students that you can use the square root to find the value of *c*.

√100 = √*c*^{2}

To find the square root of 100, find the value that when squared equals 100. 102 = 100, so √100 = 10; *c* = 10.

**Step 4:** Have students practice square roots by asking them to find the square roots of various smaller numbers. Model this for them by displaying:

√49 is 7 because 7^{2} = 49

Then have them find the square roots of 144, 25, 81, 64, 16, 121, 4, 9, 36, and 1.

**Step 5:** Display the following right triangle for students:

Have students substitute the given values within the formula:

*a*^{2} + *b*^{2} = *c*^{2}

8^{2} + 6^{2} = *c*^{2}

64 + 36 = *c*^{2}

100 = *c*^{2}

Once you have reached this point with the class, explain to students that you can use the square root to find the value of *c*.

√100 = √*c*^{2}

To find the square root of 100, find the value that when squared equals 100. 102 = 100, so √100 = 10; *c* = 10.

**Step 6:** Tell students that, in a similar way, the cube root of a number is the inverse of cubing a number. Display:

^{3}√729 is 9 because 9^{3} = 729

Have students practice by asking them to find the cube roots of 1, 343, 125, 8, 64, 216, 512, and 1000.

### Guided Practice

**Step 7:** Provide students with three sample questions:

- 9 x 48 ÷ 2 =
*x*^{3}; What is*x*? - Using the picture below, find the value of
*k*.

3. 2(4 + 9.5 – 1) = y^{2}; What is *y*?

**Step 8:** Ask students to work in pairs to write five questions that require similar skills to answer. Tell students that they might want to use integers in their problems and to have integer solutions. After they are done writing such questions, have pairs swap their questions so that another pair may attempt to answer the questions.

**Step 9:** **Checking for Understanding:** Review answers to the worksheet with the class, making sure students explain their mathematical thinking. Address any misconceptions that may arise.

### Independent Practice

**Step 10:** Assign the A Complex Course: Exponents, Square Roots, and Cube Roots printable for classwork or homework.

**Step 11:** **Checking for Understanding: **Review the answers to the A Complex Course: Exponents, Square Roots, and Cube Roots printable, which are provided on page 2 of the Answer Key: Adventures in Expressions and Equations printable. Make sure students explain their mathematical thinking. Address any misconceptions that may arise.

### Standards

**Grade 8**: Properties of Exponents, Square Roots, Cube Roots, Pythagorean theorem (**CCSS**8.EE.A.1, 8.EE.A.2, 8.G.B.7)**Grade 6–8**: Making Sense of Problems, Reasoning Abstractly and Quantitatively, Modeling, Making Use of Structure (CCSS Practice MP1, 2, 4, 7)