# Study Guide

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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:

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**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

- Enter to expand or collapse answer. Answer expanded
**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.

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**Competency 0002**

Algebra and Functions

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

Graph. Curved line starting high near the y axis and decreasing very quickly almost vertically then decreasing more slowly passing through the single point open parenthesis one comma two close parenthesis then slowing the decrease to an almost horizontal decrease.

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*=

- Enter to expand or collapse answer. Answer expanded
**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*= .

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**Competency 0003**

Geometry and Statistics

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

right cylinder with a rectangular cross section length wise inserted inside of it that is four feet up from the bottom.

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}

- Enter to expand or collapse answer. Answer expanded
**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.

circle with center point p radii drawn from p to a on the bottom left side that is ten feet another radii drawn vertically from p to the bottom of the circle the the right of a there is a segment drawn horizontally from a on the left to b on the right of the circle point d is labeled where the lin segment a b intersects the vertical radii from p to the outer circle segment p d is labeled six feet and the rest of that radi from d to the outer part of the circle is labeled four feet the points p d a make a right triangle with its hypotenuse being a p ten feet and the leg p d is six feet.