Study Guide

Field 222: Multi-Subject: Teachers of Childhood
(Grade 1–Grade 6)
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 and Operations

1. A bag contains a number of plastic disks that are either red, green, or blue. One-quarter of the disks are red and two-thirds of the disks are green. Which of the following is a possible value for the number of disks in the bag that are either red or green?

  1. 18
  2. 36
  3. 42
  4. 88
Answer
Correct Response: D. This question requires the examinee to perform operations on fractions. Add the portions that are red and green: 1/4 + 2/3 = 3/12 + 8/12 = 11/12, so 11/12 of all the disks are either red or green. Since there must be a whole number of disks, the total number of disks must be a multiple of 12, and for every 12 disks, 11 of them are either red or green, so the number of red and green disks must be a multiple of 11. The only multiple of 11 given is 88.

Competency 0002
Ratios and Proportional Relationships and Number Systems

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

The steps for making a cube-shaped box by cutting and folding a rectangular piece of paper are shown in the diagram. If all the small squares shown in step 2 are congruent and the volume of the box produced in step 3 is 216 cubic units, what is the area of the rectangular sheet of paper in step 1?

  1. 72 square units
  2. 144 square units
  3. 432 square units
  4. 864 square units
Answer
Correct Response: C. This question requires the examinee to use cube roots to solve problems. Working backward, if the volume of the cube in step 3 is 216 cubic units, then 216 = e3, where e = the length of one edge of the cube, and e = 3√216 or 6. Each of the squares shown in step 2 has an edge of 6, so each square has an area of 6 × 6, or 36. There are 12 squares that make up the rectangle, so the area of the rectangle is 12 × 36 = 432 square units.

Competency 0003
Algebra, Measurement, Geometry, and Data

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

Which expression can be used to estimate the number of feet in 2 kilometers?




Answer
Correct Response: C. This question requires the examinee to solve problems involving conversions of measurement units. Using an estimate of 5 cm = 2 in, 1000 m = 1 km, 100 cm = 1 m and 1 ft = 12 in, set up a multiplication of ratios so that the units "cancel": 2km/1 x 1000m/1km x 100cm/1m x 2in/5cm x 1ft/12in.

Competency 0004
Instruction in Mathematics

4. A third-grade teacher is preparing to teach the following standard from the New York State P–12 Common Core Learning Standards for Mathematics.

Number & Operations—Fractions (3.NF)

Develop understanding of fractions as numbers.

4. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

Which strategy is likely to be most effective as part of an introductory lesson designed to meet this standard?

  1. teaching that is equivalent to by showing how they represent the same point on a number line

  2. teaching that is equivalent to because according to the rules of fractions

  3. teaching that is equivalent to because 6 is the least common denominator of 2 and 3

  4. teaching that is equivalent to by showing cross multiplication of 1 x 6 = 2 x 3
Answer
Correct Response: A. This question requires the examinee to apply strategies for extending understanding of fractions, equivalence and ordering. The key words here are compare fractions by reasoning about their size. The position of a number on a number line is a representation of its size, so showing that two fractions represent the same point on a number line shows that they are the same size and therefore are equivalent fractions.