# Test Information Guide

## Field 47: Mathematics (Middle School)

Sample Multiple-Choice Questions

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

FormulasCalculators will NOT be provided for the Mathematics (Middle School) test.

### Number Sense and Operations

**Objective 0002**

Understand principles and operations related to integers, fractions, decimals, percents, ratios, and proportions.

1. **Use the sequence of steps below to answer the question that follows.**

Start layout, first row four point two six times two point two equals open parenthesis four hundred and twenty six times one over one hundred close parenthesis times open parenthesis twenty two times one over ten close parenthesis. Second row, equals four hundred and twenty six times open parenthesis one over one hundred times twenty two close parenthesis times one over ten. Third row, equals four hundred and twenty six times open parenthesis twenty two times one over one hundred close parenthesis times one over ten. Fourth row, equals open parenthesis four hundred and twenty six times twenty two close parenthesis times open parenthesis one over one hundred times one over ten close parenthesis. Fifth row open parenthesis four hundred and twenty six times twenty two close parenthesis times open parenthesis one over one thousand close parenthesis end layout.

The sequence of steps above could be used to answer which of the following questions?

- How can a decimal be converted to a fraction reduced to lowest terms?
- Where should the decimal point be placed in the product of two decimals?
- Why is it necessary to invert and multiply when solving problems involving fractions?
- How is scientific notation used to multiply decimals?

- Answer
**Correct Response: B.**

Correct Response: B.

**Objective 0003**

Understand and solve problems involving integers, fractions, decimals, percents, ratios, and proportions.

2. Which of the following situations represents 2 ÷ ?

- A 2-acre lot needs to be plowed. If four workers split the plowing evenly, how many acres will each person plow?
- One side of a one-fourth-square-foot rectangle is 2 feet. How long is the other side?
- Alix sawed off one fourth of a 2-yard log. How many yards did Alix saw off?
- Terry picked 2 pounds of berries and put them into one-quarter-pound containers. How many containers of berries did Terry fill?

- Answer
**Correct Response: D.**

Correct Response: D.

### Patterns, Relations, and Algebra

**Objective 0007**

Understand properties of functions and relations.

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

Function machine with an Input (x) and an Output (y). There are two main rectangular tanks with a smaller rectangular passage between them that is centered vertically in relation to the two main rectangles. The input is at the top left of the rectangle on the left and has a v shaped opening and an arrow pointing down. The left rectangle is labelled Square value. The smaller middle passage has an arrow pointing to the right. The right rectangle is labelled Subtract from two. The output is on the right side of the right rectangle at the same level of the smaller middle passage and has an arrow pointing to the right.

Which of the following graphs represents the function described in the above diagram?

- Answer
**Correct Response: B.**

Correct Response: B.

**Objective 0008**

Understand properties and applications of linear relations and functions.

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

Graph showing several points with times until next eruption t in minutes on the y axis and length of eruption l in minutes on the x axis. The points start at about open parenthesis one comma forty seven close parenthesis and have a general pretty steady increase through open parenthesis three comma sixty eight close parenthesis up to the highest point of open parenthesis five comma ninety two close parenthesis

The graph shown above represents the relationship between the length of a geyser's eruption, ℓ, and the time until the next eruption, *t*. Which of the following linear equations best models the data?

*t*+ 12ℓ = 33- 2
*t*– 3ℓ = –66 - 12ℓ –
*t*= –33 - 3ℓ + 2
*t*= 66

- Answer
**Correct Response: C.**

Correct Response: C.

**Objective 0009**

Understand properties and applications of quadratic relations and functions.

5. In solving the quadratic equation *x*^{2} + 14*x* – 4 = –30 by completing the square, the first step is to add 4 to both sides of the equation. The second step is to:

- add 49 to both sides of the equation.
- factor
*x*from the binomial*x*^{2}+ 14*x*. - factor the number 2 from 14, –4, and 30.
- take the square root of both sides of the equation.

- Answer
**Correct Response: A.**

Correct Response: A.

### Geometry and Measurement

**Objective 0012**

Understand the principles of Euclidean geometry and use them to prove theorems.

6. **Use the geometric proof below to answer the question that follows.**

For the proof, there is a diagram of a circle with center point, P. Point R is on the circle at 12:00, and point S is on the circle at about 8:30. Radius PR and radius PS are drawn. A line is drawn between points R and P. Point Q is drawn on line RS, and a line is drawn between point Q and center point P.

Given: Circle P;

Line PQ is a median of triangle PRS.
Prove: Triangle RPQ is congruent to triangle SPQ.

Statements and Reasons.

Statement 1, Circle P; line PQ is a median of triangle PRS.
Reason, Given.

Statement 2, Line RQ is congruent to line QS.
Reason, Definition of median.

Statement 3, Line PR and line PS are radii of P.
Reason, Definition of radius.

Statement 4, Line PR is congruent to line PS.
Reason, All the radii of a circle are congruent.

Statement 5, Question mark.
Reason, Question mark.

Statement 6, Triangle RPQ is congruent to triangle SPQ.
Reason, SSS postulate.

Which of the following statements and reasons would be most appropriate in step 5 of this proof?

- Answer
**Correct Response: D.**

Correct Response: D.

**Objective 0014**

Solve problems involving three-dimensional shapes.

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

Square corners, 5 units on a side, are removed from a 15-unit-by-25-unit rectangular piece of cardboard. The sides of the cardboard are then folded to form an open box. Which of the following is the volume, in cubic units, of the box?

- 375
- 625
- 1000
- 1125

- Answer
**Correct Response: A.**

Correct Response: A.

### Data Analysis, Statistics, and Probability

**Objective 0016**

Understand descriptive statistics and the methods used in collecting, organizing, reporting, and analyzing data.

8. Which of the following situations best represents a random sampling?

- Ask every tenth person coming out of a health spa how many times a week they exercise to determine how often people in the town exercise.
- Survey students in advanced biology classes to determine the average amount of time students in a certain school study each week.
- Find the heights of all boys in a senior gym class to determine the average height of all boys in the school.
- Count the number of chocolate chips in every fifth cookie to determine the average number of chocolate chips per cookie in a bag of cookies.

- Answer
**Correct Response: D.**

Correct Response: D.

**Objective 0017**

Understand the fundamental principles of probability.

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

Rectangle laid out on a coordinate plane in the first quadrant the corners of the rectangle are the origin, open parenthesis zero comma eight close parenthesis open parenthesis four comma eight close parenthesis open parenthesis four comma zero close parenthesis

A point is randomly selected within the rectangle shown in the diagram above. What is the probability that the *y*-coordinate of the point is less than or equal to 2?

- Answer
**Correct Response: C.**

Correct Response: C.

### Trigonometry, Calculus, and Discrete Mathematics

**Objective 0019**

Understand the conceptual basis of calculus.

10. The graph of the function f(*x*) = 2*x*^{2} – 6*x* + 4 has its minimum value at the point (, –). Which of the following statements must be true?

- The derivative of f(
*x*) is zero when*x*is equal to . - The value of f(
*x*) approaches a limit of zero as*x*approaches . - The graph of f(
*x*) shifts from a concave downward shape to a concave upward shape. - The slope of the line tangent to f(
*x*) at*x*= is equal to –.

- Answer
**Correct Response: A.**

Correct Response: A.