# Study Guide

## Field 242: Multi-Subject: Secondary Teachers

(Grade 7–Grade 12)

Part Two: Mathematics

### Sample Selected-Response Questions

The following reference material will be available to you during the test:

**Competency 0001**

Number and Quantity

1. Grapes are 80% water by weight and raisins are 15% water by weight. If 100 pounds of grapes are dried to make raisins, which weight is closest to the weight of the raisins?

- 24 pounds
- 32 pounds
- 75 pounds
- 85 pounds

- Answer
**Correct Response: A.**This question requires the examinee to use proportional relationships to solve a multistep ratio and percent problem. If grapes are 80% water by weight, then 20% of the weight is not water, so 100 pounds of grapes contain 20 pounds of non-water matter. These 20 pounds must make up the non-water matter in the raisins. Since raisins are 15% water by weight, the non-water matter is 85% of the weight, so 20 = 0.85 (total weight) and total weight of the raisins is 20 ÷ 0.85 or about 24 pounds.

**Competency 0002**

Algebra and Functions

2. **Use the graph below to answer the question that follows.**

The graph shows a function of the form *y* = *f*(*x*) = for constant *c* and *x* > 0. What is the inverse function of *y* = *f*(*x*) for *x* > 0?

*y*=*y*= 2*x*^{2}*y*=*y*=

- Answer
**Correct Response: C.**This question requires the examinee to build new functions from existing functions. Since the graph contains the point (1, 2), substituting 1 for*x*and 2 for*y*calculates*c*= 2. Then*y*= . Inverse functions reverse the roles of*x*and*y*, so an equation for the inverse is*x*= . Squaring both sides,*x*^{2}= →*x*^{2}*y*= 4 →*y*= .

**Competency 0003**

Geometry and Statistics

3. **Use the diagram below to answer the question that follows.**

The fuselage of an airplane is modeled as a right circular cylinder with a diameter of 20 feet and a length of 200 feet. A platform is built into the cylinder to serve as a floor for cargo, as shown in the cross-section diagram. How much floor space, in square feet, is provided if the platform is positioned 4 feet above the bottom of the fuselage?

- 2400 ft
^{2} - 3200 ft
^{2} - 4800 ft
^{2} - 9600 ft
^{2}

- Answer
**Correct Response: B.**This question requires the examinee to analyze the shape of a two-dimensional cross section of a three-dimensional object. The floor in question is a rectangle of length 200 feet, whose width must be calculated. Using the diagram below,*AB*is the width of the floor and also a chord of the circular cross section whose diameter is 20 feet and radius is 10 feet. A radius drawn perpendicular to a chord bisects it, so (*AD*)^{2}+ 6^{2}= 10^{2},*AD*= 8, and The area of the floor is 16 × 200 = 3200 square feet.*AB*= 16.