# Study Guide

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Field 232: Multi-Subject: Teachers of Middle Childhood

(Grade 5–Grade 9)

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 Systems

1. A student approximates as . Which of the following is closest to the error of this approximation?

- 0.000831
- 0.000832
- 0.795831
- 0.795832

- Enter to expand or collapse answer. Answer expanded
**Correct Response: B.**This question requires the examinee to apply knowledge of numbers that are not rational and find rational approximations of irrational numbers. Using a calculator, = 4.795832, rounded to six decimal places. Converting to thousandths, 4 = 4.795. Subtracting 4.795832 4.795 finds an error of 0.000832.

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**Competency 0002**

Algebra and Functions

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

Altitude Above Sea Level (miles) Percentage of Oxygen Available for Breathing 0 100% 10 11%

The table shows the percentage of oxygen available for breathing at two different altitudes above sea level. One model for determining the percentage of oxygen available for breathing as a function of the altitude above sea level is *P*(*h*) = 100*e*^{0.22h}, where *P*(*h*) is the percentage of oxygen available for breathing and *h* is the altitude in miles above sea level. If a linear function is also used to model the data, how do the predictions made for the percentage of oxygen available for breathing by the two models compare at an altitude of 5 miles above sea level?

- A linear model predicts a value of about 1.1 percentage points less than the exponential model.
- A linear model predicts a value of about 8.9 percentage points less than the exponential model.
- A linear model predicts a value of about 22 percentage points greater than the exponential model.
- A linear model predicts a value of about 33 percentage points greater than the exponential model.

- Enter to expand or collapse answer. Answer expanded
**Correct Response: C.**This question requires the examinee to construct and compare linear and exponential models. To construct a linear model, use slope = = 8.9 and intercept (0, 100), so*P*(*h*) = 8.9*h*+ 100 and + 100 = 55.5%. Using the given exponential model,*P*(5) = 8.9(5)*P*(5) = 100*e*^{0.22(5)}= 33.3%, so the linear model predicts a value about 22 percentage points greater than the exponential model.

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**Competency 0003**

Measurement, Geometry, Statistics, and Probability

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

two triangles that share a vertex triangle a b c has its longest side a c laying horizontally with a being the bottom left vertex c being the bottom right vertex and b being the top vertex the second triangle a e d is more vertical with a being the bottom left vertex d being the bottom right vertex and e being the top vertex side a e is a continuation of side a b from triangle a b c vertex d is located between a and c on triangle a b c side a b is congruent to side a d and angle a b c is congruent ro angle a d e.

Given:

ABADandABCADEA student wants to use a triangle angle-side relationship and the given information to prove that Δ

AEDis congruent to ΔACB. Some of the statements made by the student are shown below.

ABAD.ABCADE.- Δ
AEDΔACB.

Which of the following is the missing statement for line 3?

*AE**AC*.-
*EAC**CAE*. *BC**DE*.-
*BCA**DEA*.

- Enter to expand or collapse answer. Answer expanded
**Correct Response: B.**This question requires the examinee to analyze the proof of a theorem about triangles. Given the pair of congruent angles and the pair of congruent sides, the choices are to use either angle-side-angle or side-angle-side to prove the required triangles congruent. Since the triangles overlap, they share angle*A*, and*EAC*is congruent to*CAE*by the reflexive property, making the triangles congruent by angle-side-angle.