Unit Activity Unit: Rules of Exponents and Polynomials

This activity will help you meet these educational goals:

Mathematical Practices—You will make sense of problems and solve them, reason abstractly and quantitatively, use mathematics to model real-world situations, and look for and express regularity in repeated reasoning.

Introduction

In this unit, you used different methods to create and solve linear equations. You modeled scenarios using linear inequalities and solved them with the help of graphs. In this activity, you will analyze real-world situations by modeling them as equations and looking at their graphs. Directions and Analysis

Task 1: Tiles

Paulina is remodeling her bathroom. The tile she has chosen is shown below. There are squares and trapezoids in the tile. The side length of each square in the tile is x centimeters. The height and the length of one of the bases of each trapezoid is x centimeters. The other length is 2x centimeters.

a. Write a simplified equation to solve for x in terms of AT, the area of the tile. If necessary, use rational coefficients instead of root symbols.

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b. If the tile is a square with a length of b centimeters, what would AT be in terms of b?

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c. Using the result from part b, rewrite the equation for x in terms of b and explain how you simplify it. What does this equation really represent?

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d. Using the equation from part c, solve for x if the side length of the tile is 6 centimeters.

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e. After seeing her remodeled bathroom, Paulina’s friend Sabrina also wants to remodel her bathroom. The tile she has chosen is shown below.

If each of the smaller squares have sides c centimeters long, write an expression for the area of the whole tile, AT, and find the area of the tile if the length of the squares is 2.5 centimeters.

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f. Once she completes a wall, Sabrina notices that the number of squares along each side of the wall is equal to the number of square centimeters in each tile’s area. Write an equation for the number of squares on the wall, SW, in terms of c. Then, solve for the number of squares on the wall.

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g. Write an equation for the area of the wall, Aw. Then solve for the area of the wall.

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Task 2: E = mc2

E=mc2 is one of the most recognizable equations in the world. In this task, you’ll attempt to understand it better. Let’s start with what the letters mean. The letter E stands for energy, the letter m stands for mass, and the letter c stands for the speed of light. The meaning of this equation is that you can completely convert mass into energy.

a. Rewrite the equation to solve for the speed of light, c. Use rational exponents instead of roots.

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b. Using the properties of exponents, apply the rational exponent to the numerator and the denominator, and then rationalize the denominator.

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c. Rewrite this equation without writing it as a fraction. (Fractional exponents are OK, though.)

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d. Using whichever equation you prefer (part a, b, or c), find the speed of light in meters per second if a mass of 3 kilograms converts to 2.7 × 10∧17 joules (J = kg · m²/s²)

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