# Study Guide

## Field 202: Middle Grades (5–8) Mathematics

Sample Multiple-Choice Questions

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

Each multiple-choice question has four answer choices. Read each question and its answer choices carefully and choose the ONE best answer.

During the test you should try to answer all questions. Even if you are unsure of an answer, it is better to guess than not to answer a question at all. You will NOT be penalized for choosing an incorrect response.

**Objective 0001**

Understand calculus.

1. The graph of a parabola intersects the *x*-axis when *x* = 4 and *x* = –1. If the *y*-intercept is a positive number, which of the following equations could represent this graph?

*y*= 2(*x*– 4)(*x*+ 1)*y*= –2(*x*– 4)(*x*+ 1)*y*= 2(*x*+ 4)(*x*– 1)*y*= –2(*x*+ 4)(*x*– 1)

- Answer and Rationale
**Correct Response: B.**

A polynomial function factored in the form*y*=*a*(*bx*±*c*)(*dx*±*e*) will have*x*-intercepts when*bx*±*c*= 0,*dx*±*e*= 0, or*a*= 0. The graph of the polynomial will open upward if*a*> 0 and downward if*a*< 0. For the polynomial to have the given*x*-intercepts of*x*= 4 and*x*= –1, (*x*– 4) and (*x*+ 1) must be binomial factors. The*y*-intercept is a positive number, so the graph of the polynomial will open downward; thus,*a*< 0. The function*y*= –2(*x*– 4)(*x*+ 1) satisfies these conditions.

**Objective 0001**

Understand calculus.

2. Which of the following equations should be solved to find the *x*-coordinates of the maximum or minimum points of the function *y* = (*x*^{2} – 3)^{2}?

- 2(
*x*^{2}– 3) = 0 - 2
*x*(*x*^{2}– 3) = 0 - 4(
*x*^{2}– 3) = 0 - 4
*x*(*x*^{2}– 3) = 0

- Answer and Rationale
**Correct Response: D.**

The maximum or minimum values of any continuous function*y*are found by calculating the derivative of the function, setting the derivative equal to zero, and solving this equation for*x*. The maxima and minima of function*y*will occur at these*x*values. The product rule can be used to find the derivative of*y*. If*y*= (*x*^{2}– 3)^{2}= (*x*^{2}– 3)(*x*^{2}– 3), then = (*x*^{2}– 3)2*x*+ 2*x*(*x*^{2}– 3) = 4*x*(*x*^{2}– 3). The minimum or maximum values of function*y*can be found by finding the*x*-values that solve 4*x*(*x*^{2}– 3) = 0. Note: the chain rule may also be used to find the derivative of function*y*in this case.

**Objective 0002**

Understand college statistics.

3. A student's median score on three tests is 87. The mean of the student's scores is 89, and the range is 6. Which of the following scores could be one of the student's other two test scores?

- 85
- 86
- 90
- 93

- Answer and Rationale
**Correct Response: D.**

The median is found by ordering all observations numerically and then locating the middle observation, and the mean is found by summing the values of all observations and dividing the sum by the number of observations. In this case the order of the observations is:*a*, 87,*b*. The mean is 89; therefore , which yields*a*+*b*= 180. The range is the difference between the maximum and minimum values; that is,*b*–*a*= 6. Using elimination, solve the system*a*+*b*= 180 and*b*–*a*= 6 to obtain*a*= 87 and*b*= 93, so 93 is one of the student's other two test scores.

**Objective 0003**

Understand ratios and proportional relationships.

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

The number line shown best models which of the following fraction computation problems?

- Answer and Rationale
**Correct Response: A.**

The number line shows of the fraction shaded. This indicates multiplication. The correct response is .

**Objective 0003**

Understand ratios and proportional relationships.

5. Property taxes in a particular community are calculated on an amount equal to 70% of the market value of the property. In this community, the tax rate for this year is $30 per $1,000 of taxable value. If a homeowner's property tax bill is $5,250, what is the market value of the home?

- $204,750
- $227,500
- $235,000
- $250,000

- Answer and Rationale
**Correct Response: D.**

The town calculates the property tax by charging $30 for each $1000 of taxable value; that is, The property tax on this home is $5250. After substituting $5250 in for*property tax*= (*taxable value*).*property tax*, solve for*taxable value*to find*taxable value*= $5250. The*taxable value*is equal to 70% of the*market value*; that is,*taxable value*= 0.70(*market value*). After substituting $5250 in for*taxable value*, solve for*market value*to find*market value*= $250,000. The market value of this home is $250,000.

**Objective 0004**

Understand the structure and properties of the real number system.

6. **Use the Venn diagram below to answer the question that follows.**

The Venn diagram shown represents the prime factorization of two integers *A* and *B*. Based on the diagram, which of the following expressions is the quotient of the least common multiple divided by the greatest common factor of *A* and *B*?

*x*^{3}•*z**x*•*y*^{2}•*z**x*•*y*^{2}•*z*^{3}*x*^{3}•*y*^{2}•*z*^{3}

- Answer and Rationale
**Correct Response: C.**

The least common multiple (LCM) of*A*and*B*is the smallest positive integer that is divisible by both*A*and*B*; it can be found by finding the prime factorizations of*A*and*B*and multiplying together the prime numbers with the highest exponents from both*A*and*B*. The prime factorizations of*A*and*B*are*x*^{4}*y*^{2}*z*^{4}and*x*^{3}*z*^{4}, respectively. The prime numbers with the highest exponents are*x*^{4},*y*^{2},*z*^{4}, so the LCM is*x*^{4}*y*^{2}*z*^{4}. The greatest common factor (GCF) of*A*and*B*is the greatest factor that divides both*A*and*B*; this is the intersection of*A*and*B*in the Venn diagram,*x*^{3}*z*. Therefore, .

**Objective 0005**

Understand expressions and equations.

7. A textbook recommends that factoring polynomials in the form *x*^{2} + *bx* + *c* be taught before factoring polynomials in the form *ax*^{2} + *bx* + *c*. A teacher decides to skip *x*^{2} + *bx* + *c* and move directly to factoring polynomials in the form *ax*^{2} + *bx* + *c*. Which of the following statements describes the best reason for doing this?

- The curriculum minimizes factoring polynomials in the form
*x*^{2}+*bx*+*c*because expressions like*ax*^{2}+*bx*+*c*are more essential to algebra. - The students will benefit from the rigor of practicing the more difficult factoring techniques involved in factoring polynomials in the form
*ax*^{2}+*bx*+*c*. - The process for factoring polynomials in the form
*ax*^{2}+*bx*+*c*is equivalent to the process for factoring polynomials in the form*x*^{2}+*bx*+*c*. - The teacher wants to give the students more opportunities to ask questions, which factoring polynomials in the form
*ax*^{2}+*bx*+*c*will do.

- Answer and Rationale
**Correct Response: C.**

The process for factoring polynomials in the form*x*^{2}+*bx*+*c*is equivalent to factoring polynomials in the form when*ax*^{2}+*bx*+*c**a*= 1. By factoring*ax*^{2}+*bx*+*c*first, students are less likely to misinterpret the process of factoring when*a*= 1 as a separate factoring procedure from factoring when*a*≠ 1. Skipping*x*^{2}+*bx*+*c*and moving to*ax*^{2}+*bx*+*c*allows students to deduce that*a*= 1 when such examples arise.

**Objective 0006**

Understand relations and functions.

8. The points (*b*, *c*) and (1, *a*) are on the graph of a directly proportional relationship between variables y and *x*. Which of the following equations expresses a valid relationship between (*b*, *c*) and (1, *a*)?

- Answer and Rationale
**Correct Response: C.**

When two variables*x*and*y*vary directly, there is a number*m*such that*y*=*mx*, with*m*representing the slope of the line containing the variables. The slope,*m*, between any two points on the line, (*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}) is a constant, , which for the points (*b*,*c*) and (1,*a*) is calculated as . Substituting the constant*m*and the values*x*= 1 and*y*=*a*in the equation*y*=*mx*gives .

**Objective 0006**

Understand relations and functions.

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

The graph above represents the function *f*(*x*). Which of the following graphs represents the function *f*(*x* – 2) + 4.

- Answer and Rationale
**Correct Response: A.**

The transformation*g*(*x*) →*g*(*x*–*h*) denotes a horizontal shift*h*units to the right in the graph of*g*(*x*), and*g*(*x*) →*g*(*x*) +*k*denotes a vertical shift*k*units up. Therefore,*f*(*x*) →*f*(*x*– 2) + 4 denotes a horizontal shift 2 units to the right and a vertical shift 4 units up for*f*(*x*). Graph A represents this transformation.

**Objective 0007**

Understand linear functions and relations.

10. A tutor charges a flat fee per session and then an hourly rate. For a 2-hour session, the charge is $100, and for a 6-hour session, the charge is $268. At this same rate, what is the charge for a 7-hour session?

- $294
- $310
- $315
- $350

- Answer and Rationale
**Correct Response: B.**

The situation can be modeled by the equation*charge*=*flatfee*+*hour*(*rate*). For a 2-hour session, the equation is 100 =*flatfee*+ 2(*rate*), and for a 6-hour session, the equation is 268 =*flatfee*+ 6(*rate*). This yields the system of equations , which can be solved using elimination: . Thus,*r*= 42 and*f*= 16. The model the tutor uses is*charge*= 16 + 42(*hour*). To determine how much the tutor charges for a 7-hour session, substitute*hour*= 7 into the model, and find*charge*= 310. The tutor charges $310 for a 7-hour session.

**Objective 0008**

Understand nonlinear functions.

11. Which of the following statements describes the roots of the equation *y* = *x*^{2} + 4*x* + 4?

- There are 2 distinct real roots.
- There is 1 real double root.
- There are 2 distinct imaginary roots.
- There is 1 imaginary and 1 real root.

- Answer and Rationale
**Correct Response: B.**

The discriminant,*b*^{2}– 4*ac*, of a quadratic function in the form*y*=*ax*^{2}+*bx*+*c*can be used to determine the number of roots of the function. In*y*=*x*^{2}+ 4*x*+ 4,*a*= 1,*b*= 4, and*c*= 4. Substitute these values into*b*^{2}– 4*ac*to yield (4)^{2}– 4(1)(4) = 0. When the discriminant is 0, there is one real root. In this case,*y*= (*x*+ 2)(*x*+ 2), so there is one double root,*x*= –2.

**Objective 0009**

Understand the principles of two- and three-dimensional geometry.

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

A triangle is drawn in the interior of a circle whose radius measures 2 cm, as shown in the diagram. The central angle measures 60 degrees. The triangle is approximately what percentage of the circle?

- 14%
- 16%
- 28%
- 32%

- Answer and Rationale
**Correct Response: A.**

The percent that the triangle is of the circle can be represented by . The total area of a circle with radius 2 cm is Area_{circle}= π*r*^{2}= 4π cm^{2}. The area of the triangle can be found by first determining the height*h*of the triangle and then using the area formula for triangles. Because the central angle is 60° and two sides of the triangle are radii, the triangle is equilateral. The perpendicular segment from any vertex to the opposite side will create a 30-60-90 special right triangle with shortest leg measuring 1 cm and hypotenuse measuring 2 cm. Using special right triangle properties, or the Pythagorean formula, the height*h*of the triangle is cm. Thus, Area_{triangle}= , so . The triangle is approximately 14% of the circle.

**Objective 0009**

Understand the principles of two- and three-dimensional geometry.

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

The wooden block shown has the shape of a right square pyramid. The volume of the block is 4 cubic inches and its height is 3 inches. What is the length of edge *AC*?

- inches
- inches
- inches
- inches

- Answer and Rationale
**Correct Response: C.**

The volume of a square pyramid is*V*= , where*s*is the side length of the base and*h*is the height of the pyramid. Substituting 4 and 3 into the formula for*V*and*h*, respectively,4 = . Solve for*s*to obtain*s*= 2 inches. If*ED*=*DC*= 2, then by the Pythagorean theorem (*EC*)^{2}= 2^{2}+ 2^{2}so*EC*= 2 inches and*FC*= . Triangle*AFC*can be solved for*AC*using*FC*= inches and*AF*= 3 inches and substituting into the Pythagorean theorem: (*AC*)^{2}= (*FC*)^{2}+ (*AF*)^{2}, and (*AC*)^{2}= ()^{2}+ (3)^{2}, so*AC*= . The length of edge*AC*is inches.

**Objective 0010**

Understand the principles of coordinate and transformational geometries.

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

Triangle *ABC* is transformed using a translation 6 units down followed by a dilation with scale factor of 2 centered at the origin. Which of the following coordinates are the coordinates of the image point *B*?

- (–8, –6)
- (–8, 6)
- (4, –6)
- (4, 1)

- Answer and Rationale
**Correct Response: C.**

Vertical translations of*k*units on the point (*x*,*y*) can be described by (*x*,*y*) → (*x*,*y*±*k*). Dilations undergoing a scale factor of*j*and centered at the origin can be described by (*x*,*y*) → (*jx*,*jy*).*B*is at (2, 3) and is transformed 6 units down, so the initial image of*B*is located at (2, –3). Then, this image is dilated by a scale factor of 2 centered at the origin, so the final image of point*B*is located at (4, –6).

**Objective 0011**

Understand the principles and techniques of probability.

15. A parent and a child are playing a game. Each rolls a standard six-sided die to determine who has the first turn in the game. What is the probability that the child rolls a higher number than the parent?

- Answer and Rationale
**Correct Response: D.**

Three outcomes are possible: the parent rolls a number greater than the child, the child rolls a number greater than the parent, or the parent and the child roll the same number; this can be described by*P*(*child roll*=*parent roll*) +*P*(*child roll*>*parent roll*) +*P*(*parent roll*>*child roll*) = 1. The probability of any given number being rolled is , so the probability the parent and*child roll*the same number is . It is equally likely that the parent or child rolls the higher value. Thus, the probability of all outcomes can be re-expressed as + 2*P*(*child roll*>*parent roll*) = 1, and can be solved to find*P*(*child roll*>*parent roll*) = . The probability the child rolls higher than the parent is .

**Objective 0012**

Understand the principles and techniques of statistics.

16. **Use the boxplot below to answer the question that follows. **

After a recent test, a teacher created a boxplot to display the scores of the 20 students in the class. Based on the information in the boxplot, which of the following measures has the highest value for the set of test scores?

- mean
- median
- range
- standard deviation

- Answer and Rationale
**Correct Response: A.**

The portion of the graph representing the upper 50% of the data (82–100) is considerably larger than the portion of the graph represented by the lower 50% of the data (78–82). This indicates that the distribution of scores is skewed towards the high end, so there are some very high values. These high values will raise the mean, whereas the median is virtually unaffected by a few such values. Therefore, the mean is probably higher than the median.

**Objective 0012**

Understand the principles and techniques of statistics.

17. Students are directed to make twelve displays: a boxplot, a stem plot, and a histogram, one for each of four different sets of data. This exercise particularly reinforces application skills because it helps students:

- remember how to construct each type of display through repeated practice.
- select the most appropriate type of data display for different types of data sets.
- recognize how different displays may present misleading graphs and statistics.
- understand how the strength of a correlation is shown by different displays.

- Answer and Rationale
**Correct Response: B.**

Boxplots, stem plots, and histograms are different methods of displaying quantitative data. Boxplots are generally used for comparing the overall distribution of several groups and for displaying outliers. Histograms are appropriate for viewing the distribution of large sets of data because they can be created efficiently, while stem plots are less efficient for large sets of data because they must show individual values. This exercise allows students to compare and contrast the effectiveness of each type of display when used with different types of data sets.

**Objective 0013**

Apply knowledge of foundations of research-based disciplinary literacy instruction and assessment.

18. As compared with the narrative texts (stories) that middle school students encounter in an English language arts class, the story problems students encounter in mathematics class are often more difficult to comprehend, especially if the students are not native speakers of English. According to research, which of the following statements best explains a research-based rationale for this phenomenon?

- Story problems are more likely to require students to make text-to-self and text-to-world connections in order to determine a story's key idea or theme.
- Story problems are typically short and about random topics, so they often lack contextual clues to help students understand unfamiliar vocabulary and situations.
- Students are more likely to encounter complex syntax and semantic features not used in everyday spoken language when reading story problems than when reading literary texts.
- Students often do not realize that they have to apply the same skills and strategies when analyzing story problems as they do when analyzing literary texts.

- Answer and Rationale
**Correct Response: B.**

All students use a variety of comprehension and vocabulary strategies when they are reading. These strategies are of particular importance for English language learners who rely on them to understand the content presented in discipline-specific texts. According to current research, one such vocabulary strategy is the use of contextual clues in the text (e.g., descriptions, examples, definitions, synonyms) to aid in determining the meaning of an unknown word or concept. Math story problems are typically short and about random topics, activities, or situations that may be wholly unfamiliar to students, particularly students for whom English is a new language. The text of story problems often does not include sufficient context to provide students with clues to understanding such unfamiliar vocabulary and situations.

**Objective 0014**

Apply knowledge of academic-language and vocabulary development to support students' disciplinary literacy development in the mathematics classroom.

19. To promote students' vocabulary and concept development related to quadrilaterals, a middle school mathematics teacher guides students in comparing diagrams of six common four-sided figures. First, students identify and discuss the features of each figure. Then, they draw their own pictures of the quadrilaterals. Finally, the teacher leads students in constructing a semantic features analysis chart. A copy of their completed chart is shown below.

Quadrilaterals Term Four Sides All Sides of Equal Length All Angles Equal Opposite Sides of Equal Length Opposite Sides Parallel Two Sets of Parallel Sides Rectangle + – + + + + Square + + + + + + Rhombus + + – + + + Parallelogram + – – + + + Trapezoid + – – – – – Scalene Quadrilateral + – – – – –

In this mathematics lesson, the teacher addresses the diverse strengths and needs of all students primarily by:

- simplifying target academic vocabulary into simpler, everyday words.
- breaking up a complex, multi-layered concept into smaller, more accessible parts.
- using multiple language modalities and viewing to reinforce target content.
- differentiating instruction for students who fail to meet objectives following initial instruction.

- Answer and Rationale
**Correct Response: C.**

By using a variety of modalities in this lesson (i.e., viewing diagrams of the quadrilaterals; discussing the features of each quadrilateral; drawing pictures of the quadrilaterals; and using listening, speaking, reading, and writing to construct a semantic features analysis chart), the teacher helps promote the students' understanding of the characteristics of quadrilaterals by addressing a range of learning styles and responding to the diverse strengths and needs of all students. Finally, the semantic features analysis chart the class created helps reinforce the new vocabulary. In the far left column of the chart is a list of terms to be compared, while across the top is a list of properties that the objects might share. The activity helps students compare features of objects that are in the same category by providing a visual prompt of their similarities and differences. It helps them see how similar words are related and connected and is a good visual tool to use with diverse learners for this purpose.

**Objective 0016**

Apply knowledge of the development of writing, listening, and speaking skills to support students' disciplinary literacy development in the mathematics classroom.

20. A middle school mathematics teacher wants to incorporate more writing into the mathematics curriculum. Research suggests that which of the following types of writing activities is most effective in developing students' understanding of mathematics concepts and skills?

- asking students routinely to write down the steps they followed to solve each mathematics problem
- providing students with regular brief written feedback on their mathematics work that emphasizes positive reinforcement with minimal elaboration
- asking students routinely to provide descriptive feedback to their peers on the peers' mathematics work
- providing students with regular modeling and guided practice in how to describe mathematics ideas in writing and through graphic organizers

- Answer and Rationale
**Correct Response: D.**

The process of writing requires active engagement on the part of students as they must take in, organize, and process information. Current research shows that requiring and encouraging students to write across the curriculum helps to develop an understanding of the content-area concepts and skills they are learning and expands their literacy skills. Since students may be unfamiliar with this type of informational writing, research shows that it is beneficial for the teacher to provide instructional writing support (e.g., teacher modeling, guided practice, use of graphic organizers) until students internalize the process and are able to write appropriate math responses independently.