# Lesson 3: Functions and Formulas/Square Roots

Students will understand what a square root is and that squares and square roots are inverse operations and can be used to manipulate equations as long as "whatever is done to one side of the equation is done to the other."

**OBJECTIVE**

Students will understand what a square root is and that squares and square roots are inverse operations and can be used to manipulate equations as long as "whatever is done to one side of the equation is done to the other."

**MATERIALS**

- Student calculators (for Worksheet 3; can also be done without calculators)
- Worksheet 3 (PDF) — Content Connections: Square Roots
- Bonus Worksheet 3 (PDF) — Content Connections: Probability
- Take-Home Activity 3 (PDF) — Content Connections: Variables
- Resource: Answer Key (PDF)
- Resource: Mini-Poster (PDF)

Click for *whiteboard-ready printables.*

**DIRECTIONS**

Time required: 20-30 minutes, depending on class review needs; additional time for worksheets

Getting Started

1. Ask the class to define what a square is. After students mention that a square has four equal sides and four right angles, draw a square on the board.

2. Label one of the sides "4 feet." Ask how the area of the square can be determined (by multiplying 4 · 4 or, more generally, squaring one of the sides) and write *A* = *s²* where* A* represents area and *s *represents the length of a side. If necessary, explain how the superscript "2" is used to note exponents.

3. Ask the class how to find the length of a side of a square if we know its area, for example when A = 25. Depending on the level of class support needed, indicate that the problem can be solved by finding out what number times itself equals 25.

4. Introduce the radical sign notation, e.g., that 5 = √25.

5. Go back to the formula for area, *A* = *s*². Then write 25 = *s*².

6. Ask how we can find out what s equals. Indicate that we can take the square root of both sides. Referring to the scale example, remind students that what we do to one side of an equation we must do to the other. Provide other examples as needed using perfect squares for the area.

7. Introduce an example where the area is not a perfect square, for example, *A* = 29. Using a calculator, show that √21 is approximately equal to 5.385.

8. To show how squares are inverse operations of square roots, ask what

√ 9² is equal to. Break down the problem for students as needed, showing step by step how 9² = 81 and √ 81 = 9. To generalize, this means that √*x*² = *x*. Provide other examples with perfect squares as needed.

9. Finally, provide an example of an equation involving taking the square root of both sides of an equation. Write *x²* + 6 = 15. Ask students to solve and explain. Show the solution on the board, first subtracting 6 from both sides, leaving *x*² = 9. Indicate that the equation is still in balance and that it will stay in balance if we take the square root of both sides, leaving *x *= 3. Remind students that whatever is done to one side of an equation must be done to the other.

10. Distribute Worksheet 3: The Case of the Screeching Tires (PDF) and calculators. Read the introduction and review the key facts. Depending upon student support needed, it might be necessary to compute the first vehicle's speed as a class and/or review the calculator's square root function.

11. Ask students to complete the worksheet. Review answers as a class. Worksheets Answer Key (PDF)

12. Students will further develop skills at working with equations in Bonus Worksheet 3: The Case of the Tardy Transportation (PDF) and Take-Home Activity 3: The Case of the Kid Bargain Hunter (PDF). If students require additional support, review worksheets as a class. Worksheets Answer Key (PDF)