A Program of The Actuarial Foundation. Aligned with Common Core State and NCTM Standards.

What is an actuary? An actuary is an expert in statistics who works with businesses, governments, and organizations to help them plan for the future. Actuarial science is the discipline that applies math and statistical methods to assess risk.

4-8

DURATION
30

COLLECTION
Solving the Unknown with Algebra

# Lesson 3: Functions and Formulas/Square Roots

Students will simplify expressions that include squares and square roots. They will use squares and square roots as inverse operations to manipulate equations. They will also approximate rational numbers.

STANDARDS (CCSS and NCTM):

• Grade 6: Solving equations, expressions with exponents, perfect squares (CCSS 6.EE.A.1 & B.5)
• Grade 7: Solving multi-step mathematical problems (CCSS 7.EE.B.3)
• Grade 8: Approximating irrational numbers, evaluating square roots (CCSS 8.NS.A.1 & 2, 8.EE.A.2)
• Grades 6–8: (CCSS MP1, 4, 5, 6 & 7); NCTM Algebra

OBJECTIVE
Students will be able to

• Find perfect squares of numbers and use square notation.
• Solve for a variable in an equation that includes the square of that variable.
• Define rational and irrational.
• Approximate irrational numbers.
• Use squares and square roots as inverse operations.
• Use square root symbols to express numbers or solve equations.

TIME REQUIRED: 30–40 minutes, depending on class review needs; additional time for worksheets

MATERIALS

DIRECTIONS

INTRODUCTION TO NEW MATERIAL

1. Ask the class to define a square. Students should mention that a square has four equal sides and four right angles.

2. Distribute square tiles to students. Tell them to use the tiles to make a 6-by-6 square (filled in). Ask: How many tiles did you use to make the square? (36) Tell students to also make a 3-by-3 square (filled in) with the tiles. Ask: How many tiles did this square require? (9) Do you notice anything about the side lengths and the area of (the number of tiles required to model) the square? (6 x 6 = 36 and 3 x 3 = 9)

3. Continue to bring students from concrete to abstract by displaying a drawing of a square. Label one of the sides “4 feet.” Ask: How can you find the area of the square? (by multiplying 4 • 4). Remind students that they can use superscript numbers to reflect exponents. Ask: Is there a way you can rewrite this using exponents? (42; ensure that students are able to write this notation as well as express it in words as “four squared” and “four to the second power”) Note that if students need additional practice on how to rewrite expanded expressions using exponents, this might be a good time. Tell students that the generic formula for the area of a square is A = s2 where A represents area and s represents the length of a side.

4. Have students make a square with 25 tiles. Ask: What is the length of each side? (5) Have students make a square with 16 tiles. Ask: What is the length of each side? (4) Then ask students if they can determine the length of a side of a square with an area of 49 without using tiles. Depending on the level of class support needed, provide them with the hint that this problem can be solved by finding out what number times itself equals 49. (7)

5. Introduce the radical sign (square root) notation: √, e.g., 5 = √25. For a quick check of understanding, display the following, and ask students to simplify: √16; √9; √36; √49; √100.

6. Revisit the formula for area, A = s2. Then write 25 = s2. Ask: How can you find out what s is? Demonstrate that you can take the square root of both sides, which will isolate s, since squares and square roots are inverse operations: √25 = √s2; 5 = s. Remind students that what they do to one side of an equation, they must do to the other side. Provide other examples as needed using perfect squares.

7. To more explicitly 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 √x2 =x. Provide other examples with perfect squares as needed.

8. Introduce an example where the area is not a perfect square—for example, A = 29. Ask students to arrange 29 tiles in the shape of a filled-in square. (They will not be able to complete this task.) Using a calculator, show that √29 is approximately equal to 5.385. Have students look at the decimal expansion of √29 on their calculators (0.385164807134504). Ask: Do you see any repeating digit(s) within the decimal expansion? (No.) Tell students that there are no repeating digits in the decimal expansion because √29 is an irrational number. Display the following definitions:
• rational number: a number expressible in the form a/b or –a/b for some fraction a/b. The rational numbers include the integers.
• irrational number: a number that cannot be expressed in the form a/b or –a/b (a number that is not rational).
9. Have students estimate the value of √54. Ask: Which two integers must √54 be between? Why? (7 and 8 because 72 is 49 and 82 is 64. 54 is between 49 and 64.) Tell students to find the exact value of √54 using their calculators. (7.34846922834953…) Tell students to round √54 to the nearest tenth. (7.3) Which two tenths on the number line is √54 between? (7.3 and 7.4) How might you approximate √54 even more precisely? (provide a rounded number with more decimal places) Round √54 to the nearest hundredth. (7.35) Round √54 to the nearest thousandth. (7.348) Ask: So far, which of these approximations is the most precise? (7.348) Ask: Which two thousandths on the number line is √54 between? (7.348 and 7.349) Show that 7.348 is a rational number. (You can write 7.348 as 7348/1000.)

10. Finally, provide an example of an equation involving taking the square root of each side of an equation. Display the equation x2 + 6 = 15. Ask students to solve the equation and explain their solution strategy. Show the solution, first by subtracting 6 from each side, leaving x2 = 9. Indicate that the equation is still in balance and that it will stay in balance if you take the square root of each side, leaving x = 3. Remind students that whatever is done to one side of an equation must be done to the other side. After demonstrating how to solve for a variable, use substitution to verify that the solution is accurate: 32 + 6 = 15; 9 + 6 = 15; 15 = 15.

GUIDED PRACTICE:

In pairs, have students solve for the following variables. After pairs are done, have volunteers present solution strategies to the class.

1. h2 – 18 = 103
2. 4 + m2 = 68
3. (k + 2)2 = 100
4. g + 27 = 122
5. 152f = 169

INDEPENDENT PRACTICE:

11. 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.

13. 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)

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