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Study Guide

Field 232: Multi-Subject: Teachers of Middle Childhood
(Grade 5–Grade 9)
Part Two: Mathematics

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Sample Selected-Response Questions

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

Formulas PDF document

Competency 0001
Number Systems

1. A student approximates square root of twenty three as four and one hundred and fifty nine over two hundred. Which of the following is closest to the error of this approximation?

  1. 0.000831
  2. 0.000832
  3. 0.795831
  4. 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, square root of twenty three = 4.795832, rounded to six decimal places. Converting one hundred and fifty nine over two hundred to thousandths, 4 one hundred and fifty nine over two hundred = 4.795. Subtracting 4.795832 – 4.795 finds an error of 0.000832.

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) = 100e–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?

  1. A linear model predicts a value of about 1.1 percentage points less than the exponential model.
  2. A linear model predicts a value of about 8.9 percentage points less than the exponential model.
  3. A linear model predicts a value of about 22 percentage points greater than the exponential model.
  4. 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 = start fraction numerator eleven minus one hudnred denominator ten minus zero end fraction = –8.9 and intercept (0, 100), so P(h) = –8.9h + 100 and P(5) = –8.9(5) + 100 = 55.5%. Using the given exponential model, P(5) = 100e–0.22(5) = 33.3%, so the linear model predicts a value about 22 percentage points greater than the exponential model.

Competency 0003
Measurement, Geometry, Statistics, and Probability

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

Given: AB is congruent to AD and angleABC is congruent to angleADE

A student wants to use a triangle angle-side relationship and the given information to prove that ΔAED is congruent to ΔACB. Some of the statements made by the student are shown below.

  1. AB is congruent to AD.
  2. angleABC is congruent to angleADE.
  3. ΔAED is congruent to ΔACB.

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

  1. AE is congruent to AC.
  2. angleEAC is congruent to angleCAE.
  3. BC is congruent to DE.
  4. angleBCA is congruent to angleDEA.
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 angleEAC is congruent to angleCAE by the reflexive property, making the triangles congruent by angle-side-angle.