Assessment/Accountability
Massachusetts Comprehensive Assessment System

Grade 10

Mathematics

Content and Skills to be Assessed by MCAS

January 1998

Massachusetts Department of Elementary and Secondary Education

Number Sense

Discrete Mathematics

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• represent problem situations, using discrete structures such as finite graphs, matrices, sequences, and recurrence relations.
• represent and analyze finite graphs, using matrices.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• identify, extend, and use a variety of sequences, e.g., applications of the Fibonacci sequence to patterns in nature. [see Question 1]
• create, manipulate, and interpret matrices that represent real-life applications, e.g., structure matrices to represent and process inventory data. [see Question 2]
• create and apply recurrence relations, e.g., find each term in a sequence based on the term or terms before it. [see Question 1]
• use finite graphs to represent and interpret data, e.g., networks, traceable paths, tree diagrams, Venn diagrams, other pictorial representations.

For details and samples of student learning related to these learning standards, see page 44 of the Massachusetts Mathematics Curriculum Framework.

Sample Questions

Short-answer Question:

1.

The first two terms of a sequence, T₁ and T₂, are shown in the table above as 3 and 5.

Using the rule: T_n = T_{n ­ 2} + T_{n ­ 1} , where n >= 3, complete the table.

Correct response: 8, 13, 21, 34

Open-response Question:

2. A sports equipment retailer has a store in Lowell (L) and another in Amherst (A). The stores specialize in two types of bicycles, sports bikes and touring bikes. The inventory data at the beginning of the month for each of the stores is contained in matrix T, and the value of each bike is given in matrix V.

a. Using the information in matrices T and V, determine the total retail value of the bikes at the Lowell store.

b. Write an expression in matrix form that can be used to calculate the total value of the bikes at each store.

c. Bike sales for one month at each store are shown in matrix S.

Write an expression using the matrices S, T, and V that can be used to calculate the total retail value of the bikes at each store at the end of the month.

d. Determine the total retail value of the bikes at the end of the month for the store in Amherst.

Number Sense

Mathematical Structure

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• compare and contrast the real number system and its subsystems with regard to structural characteristics.
• demonstrate the logic of algebraic procedures and their interrelationship with geometric ideas and concepts.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• apply properties of real numbers, e.g., inverses, closure, identity, betweenness/density, properties of operations, order of operations. [see Question 3]
• identify and apply concepts of subsets of real numbers, e.g., integers, rational, irrational numbers. [see Question 4]
• identify and apply properties of other finite systems, e.g., modular systems, 2 x 2 matrices, defined operations. [see Question 5]
• use algebraic procedures as they relate to geometric concepts, e.g., create a geometric model for the product of two polynomials. [see Question 6]

For details and samples of student learning related to these learning standards, see page 45 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Questions:

3. How many real numbers are between 1.71 and 1.72?

A. none

B. 100

C. 1000

*D. more than 1000

4. Which is the largest?

A. 2v5

B. v24

C. v3 + v5

*D. 3v3

Short-answer Question:

5. The operation (+) is defined as:

a (+) b = (2a + b)/2, where a and b are real numbers.

Use this definition to find the value of (p (+) q) (+) r, where p = 3, q = 4, and r = 6.

Correct response: 8

Open-response Question:

6. Use the diagram below to answer the question.

In the figures below, algebra tiles are arranged in a rectangular array to represent each of the quadratic expressions.

Show how to use algebra tiles to represent each expression below in a rectangular array. If an expression cannot be represented, explain why not.

a. x² + 5x + 6

b. x² + 3x + 5

Number Sense

Estimation

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• use estimation strategies to judge the reasonableness of results of computation and problem solving involving real numbers.
• use estimation when making graphs.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• use estimation strategies to determine the reasonableness of results of computations and problems. [see Questions 8, 9]
• use estimation in making or interpreting graphs. [see Question 7]

For details and samples of student learning related to these learning standards, see page 46 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Question:

7.

Which set of percents would best fit the pie graph shown above?

A. 54%, 8%, 30%, 8%

B. 51%, 17%, 15%, 17%

*C. 47%, 22%, 9%, 22%

D. 42%, 14%, 30%, 14%

Short-answer Question:

8. Alicia is a newspaper reporter who wants to estimate the number of people who recently attended a concert at the Civic Center, shown in the scale drawing below. She knows Section A contains 6 rows with 25 seats per row.

If the concert was sold out, estimate the number of people who attended.

Correct response: an estimate between 5,000 and 7,000

Open-response Question:

9. Superburger Fast Foods, Inc., claims that 7% of all Americans eat at their restaurants each year. Evaluate their claim, given that there are 2,000 Superburger restaurants in the United States and the population of the country is 250 million. Explain your reasoning and be sure to justify your answer mathematically.

Patterns, Relations, and Functions

Algebra

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• formulate problems that involve variable quantities with expressions, equations, and inequalities.
• use tables and graphs as tools to interpret expressions, equations, and inequalities.
• simplify algebraic expressions to solve equations and inequalities.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• use variables to formulate and solve problems, e.g., cost formulas, geometric formulas, matrix representations for problems needed to solve systems of equations. [see Questions 10, 12]
• simplify algebraic expressions to solve equations and inequalities, e.g., use equivalent expressions in the process of solving equations.
• solve equations and inequalities. [see Question 12]
• manipulate formulas, equations, and expressions using algebraic procedures. [see Question 11]
• extrapolate and interpolate information from tables and graphs representing equations and inequalities, e.g., applications to spreadsheets.

For details and samples of student learning related to these learning standards, see page 62 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Question:

10.

represents which system of equations?

A. 2x + 4y = 10
3x + 5y = 15

B. 5x = 10
9y = 15

*C. 2x + 3y = 10
4x + 5y = 15

D. 6x = 10
20y = 15

Short-answer Question:

11. The formula F = 9/5 C + 32 converts temperatures from Celsius to Fahrenheit.

Solve the equation for C in terms of F to convert temperatures from Fahrenheit to Celsius.

Correct response: C =5/9(F ­ 32)

Open-response Question:

12. The sophomore class is planning a car wash to raise money for a class trip. They will charge $5 for a wash and $7 for an interior cleaning. The class has a goal of raising at least $1,000 at the car wash.

a. If the class washes 100 cars, what is the fewest number of interior cleanings necessary to reach the goal of $1,000?

b. Using W for the number of cars washed and I for the number of interior cleanings, write an inequality that shows how many washes and interior cleanings are needed to meet or exceed the goal of $1,000.

c. Graph the inequality from part b.

d. The students know that the total number of washes plus the number of interior cleanings that can be done in one day is approximately 250. Explain how your graph can be used to determine the various combinations of washes and interior cleanings that can be done in one day to raise $1,000 or more for the class trip.

Patterns, Relations, and Functions

Functions

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• model real-world phenomena with a variety of functions.
• represent and analyze relationships, using tables, verbal rules, equations, and graphs.
• translate among tabular, symbolic, and graphical representations of functions.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• model real-life situations with graphs on the coordinate plane, e.g., graphs of linear, quadratic, and exponential functions, scatter plots, graphs representing direct and inverse variation.
• identify and extend patterns and translate them into algebraic functions, e.g., number patterns, geometric patterns, patterns including real-world applications. [see Questions 13, 14, 15]
• demonstrate an understanding of the connections between verbal descriptions of functional relationships and input-output tables, equations and inequalities, and graphs on number lines or coordinate planes. [see Question 15]
• use proportional reasoning to solve problems. [see Questions 14, 15]

For details and samples of student learning related to these learning standards, see page 63 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Question:

13. Toothpicks are arranged as follows:

If the pattern is continued, how many toothpicks would be used to make Figure 10?

A. 30

*B. 33

C. 36

D. 39

Open-response Questions:

14. In a recent newspaper article, veterinarians reported that there are several approaches that people use to relate the age of a cat to the age of a human being. Two of these approaches are listed below:

• Approach 1: Each year of a cat's life is equivalent to 7 human years.
• Approach 2: The first year of a cat's life is equivalent to 15 human years, the second year is equivalent to an additional 10 human years, and each year of life past age 2 is equal to an additional 5 human years.

a. Make a chart similar to the one below that shows the equivalent human age for a cat in its first 5 years of life. Be sure to complete your chart.

b. Let c represent a cat's age and let h represent its equivalent human age. For each of the two approaches above, write a mathematical relationship between a cat's age and its equivalent human age.

c. For what cat's age do both Approach 1 and Approach 2 give the same equivalent human age? Show your work.

15. From 1980 to 1990, the population of Massachusetts grew from 5.74 million to 6.02 million.

a. What is the percent of increase in population over this 10-year period?

b. If the population increases by the same percent in the 10-year period from 1990 to 2000, what will the population be in the year 2000?

c. Assume the population continues to increase at the same rate as it did from 1980 to 1990. Write an expression that would represent the population of Massachusetts in the year 2050.

Patterns, Relations, and Functions

Trigonometry

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• apply trigonometry to problem situations involving right triangles.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• use sine, cosine, and tangent trigonometric ratios to solve problems involving right triangles. [see Question 16]

For details and samples of student learning related to this learning standard, see page 64 of the Massachusetts Mathematics Curriculum Framework.

Open-response Question:

16. Use the trigonometric table provided on the reference sheet to answer the question.

One end of a ramp is secured to a dock while the other end moves along a track on a float anchored in a harbor. At low tide the dock is 10 feet above the water.

a. A safe angle of incline, marked as (theta) in the diagram, has been determined to be no more than 30°. Write an equation that can be used to calculate the minimum length of the ramp, R, to meet this safety guideline.

b. Suppose the ramp is 24 ft. in length. To the nearest degree, what is the angle of incline of the ramp at low tide? Show your work and explain your reasoning.

Geometry and Measurement

Geometry and Spatial Sense

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• interpret and draw three-dimensional objects.
• represent problem situations with geometric models and apply properties of figures.
• classify figures in terms of congruence and similarity and apply these relationships.
• deduce properties of, and relationships between, figures from given assumptions.
• develop and defend conclusions.
• formulate counter examples.
• construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• interpret, visualize, and draw three- dimensional objects, e.g., draw the front view of a structure, determine the cross- sectional shape of an object, identify or draw the two-dimensional net of a three-dimensional object. [see Question 18]
• represent and solve problems with the use of geometric models.
• use logical processes (e.g., deduction, induction, counter-examples) to develop, justify, or prove mathematical conclusions or assertions. [see Question 19]
• relate properties of geometric figures to shapes, e.g., parallel and perpendicular sides, exterior and interior angle relationships, the Pythagorean theorem. [see Question 17]
• identify and apply properties and concepts of congruence, similarity, and transformations, e.g., use of proportional reasoning in problems involving one-, two-, and three-dimensional measures.

For details and samples of student learning related to these learning standards, see page 78 of the Massachusetts Mathematics Curriculum Framework.

Note: The Mathematics Curriculum Framework substrand, Geometry and Spatial Sense, has been divided into two reporting categories for grade 10: Geometry and Spatial Sense and Measurement.

• On this page, the assessment expectations pertain to the relationships of geometric figures to shapes.
• The assessment expectations pertaining to the relationships of geometric figures to measurement can be found on page 123.

Multiple-choice Question:

17. The perimeter of an isosceles triangle is 30 cm. Which of the following cannot be the length of the base?

A. 1 cm

B. 5 cm

C. 10 cm

*D. 15 cm

Short-answer Question:

18.

Draw two possible top views of the structure shown above.

Correct responses:

Open-response Question:

19. The bisectors of the angles of a rectangle meet at four points as shown in Figure 1 below.

a. List two properties of quadrilateral PQRS. Justify each property.

b. In the parallelogram in Figure 2, sketch the angle bisectors and label their four points of intersection. Are the properties you listed for quadrilateral PQRS also true for the figure joining the four points of intersection in Figure 2? Explain your reasoning.

Geometry and Measurement

Measurement

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• represent problem situations with geometric models and apply properties of figures.
• classify figures in terms of congruence and similarity and apply these relationships.
• deduce properties of, and relationships between, figures from given assumptions.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• relate properties of geometric figures to measurement, e.g., perimeter, area, volume, symmetry, angle measures, arc length, the Pythagorean theorem, circumference. [see Question 22]
• solve problems related to geometric measurement, e.g., compare volumes related to surface area of three-dimensional figures, compare rates of flow related to volume and time, estimate cost based on area or volume. [see Question 21]
• identify and apply concepts of similarity to measurement in two- and three-dimensional figures. [see Question 20]

For details and samples of student learning related to these learning standards, see page 78 of the Massachusetts Mathematics Curriculum Framework.

Note: The Mathematics Curriculum Framework substrand, Geometry and Spatial Sense, has been divided into two reporting categories for grade 10: Geometry and Spatial Sense and Measurement.

• On this page, the assessment expectations pertain to the relationships of geometric figures to shapes.
• The assessment expectations pertaining to the relationships of geometric figures to measurement can be found on page 120.

Short-answer Questions:

20. When a hot air balloon is inflated to a diameter of 12 feet, its volume is about 900 cubic feet. What is its approximate volume when it is inflated to a diameter of 18 feet?

Correct response: an estimate between 3,000 and 3,100 cubic feet

21. Solar cells convert the energy of sunlight directly into electrical energy. For each square centimeter of solar cell in direct overhead sunlight, about 0.01 watt of electrical power is available. A solar cell in the shape of a regular hexagon is required to deliver 15 watts. Determine the minimum length of side s to the nearest centimeter.

Correct response: 24 cm

Open-response Question:

22. A running track is laid out in the form of a rectangle of dimensions 2r by 3r with a semicircle on each end, as shown below. The running lanes are 1 meter wide.

a. What is the value of r if lane 1 is 400 meters (measured on its inside border)?

b. For a 400-meter race, the runner in lane 1 would start and end at the finish line. Find the starting positions for the runners in lanes 2 and 3 if they must stay in their lanes throughout the race and also run 400 meters to the finish line.

c. Members of an exercise club are allowed to walk or jog in lane 4 when the track is not in use for track events. How many laps in lane 4 are equivalent to a mile [1 mile ~ 1610 meters]?

Geometry and Measurement

Geometry from an Algebraic Perspective

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• translate between synthetic and coordinate representations.
• deduce properties of figures, using transformations and coordinates.
• identify congruent and similar figures, using transformations.
• develop and explain geometric interpretations and applications of slope.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• draw, given the coordinates of the vertices, geometric figures on a coordinate plane and determine lengths and mid-points of segments on the plane.
• deduce and identify the properties of figures using transformations and/or coordinates, e.g., transformation symmetries of a design. [see Question 25]
• apply and interpret transformations of figures in the coordinate plane, e.g., translations, reflections, rotations, scale factors or size changes, compositions of transformations. [see Questions 23, 25]
• develop and explain geometric interpretations and applications of slope. [see Question 24]

For details and samples of student learning related to these learning standards, see page 79 of the Massachusetts Mathematics Curriculum Framework.

Short-answer Questions:

23.

On the diagram to the right, draw the image of arrow W, reflected over the y-axis and rotated 180° about the origin counterclockwise.

Correct response:

24. Use the graph below to answer the question.

Tree Growth

a. Estimate the slope of the line of best fit shown in the scatter plot in the graph above.

b. In terms of the units shown in the graph, write an interpretation of the slope of the line.

Correct responses: a. an estimate between .15 and .25 b. Slope represents the growth in the diameter in inches per year.

Open-response Question:

25.yx

a. Copy scalene triangle ABC onto the x and y axes shown above, with AB on the x-axis and C on the y-axis. Reflect the triangle over the x-axis.

b. The quadrilateral formed by triangle ABC and its reflection image is a kite. List four geometric properties of a kite.

Statistics and Probability

Statistics

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• construct, draw inferences, and reason with charts, tables, and graphs that summarize data from real-world situations.
• use sampling to recognize and describe its role in statistical claims.
• design a statistical experiment to study a problem, conduct the experiment, and interpret and communicate the outcomes.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• construct and interpret data displays, e.g., tables, matrices, frequency distributions, stem-and-leaf plots, circle graphs, line plots, box plots, spreadsheets. [see Questions 26, 27]
• analyze data sets to identify and select appropriate measures of central tendency.
• fit a line to a scatter plot using estimation, determine the equation of the line of best fit, and use the line to make predictions.
• analyze and interpret data presented in graphs. [see Questions 26, 27]
• determine whether a given sample is biased.
• design an experiment/survey involving sampling and analyze data from the experiment/survey.

For details and samples of student learning related to these learning standards, see page 94 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Question:

26. In the scatter plots below, each dot represents a particular product's sales based on the amount of money spent on advertising. Which graph shows the greatest positive correlation between the dollars spent on advertising and the resulting sales?

A.

*B.

C.

D.

Open-response Question:

27. Use the advertisement pictured below to answer the question.

Reprinted with permission from Data Analysis and Statistics across the Curriculum, copyright 1992 by the National Council of Teachers of Mathematics.

a. Identify one way in which the graph of "Average Monthly Income" presents data in a misleading manner. Explain how it is misleading.

b. Identify one way in which the graph is not misleading.

c. Draw a different graph of this data that corrects the misleading feature that you identified in part a.

Statistics and Probability

Probability

Curriculum Framework Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

• use simulations to estimate probabilities.
• determine the likelihood of outcomes, using theoretical probabilities.

Assessment Expectations

On the Mathematics section of MCAS, students will be expected to:

• apply basic counting techniques, e.g., permutations, combinations, the Fundamental Counting Principle. [see Question 30]
• use tree diagrams, organized lists, geometric models, and/or tables to determine possible outcomes, and estimate probability using relative frequency data. [see Question 29]
• determine experimental probability based on data from simulations. [see Question 28]
• calculate theoretical probability.

For details and samples of student learning related to these learning standards, see page 95 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Question:

28. Simon is conducting a probability experiment. He randomly selects a tag from a set of tags that are numbered from 1 to 100 and then returns the tag to the set. He is trying to draw a tag that matches his favorite number, 21. He has not matched his number after 99 draws. What are the chances he will match his number on the 100th draw?

*A. 1 out of 100

B. 99 out of 100

C. 1 out of 1

D. 1 out of 2

Short-answer Questions:

29.

Points A, B, C, and D are spaced at equal intervals along line l, shown above. If another point P is placed randomly on the line segment AD, what is the probability that P is closer to B than to D?

Correct response: 2/3

30. The telephone area codes in the United States were originally assigned using the following rules:

Rule 1: The first digit can be any whole number from 2 through 9.

Rule 2: The second digit is either 0 or 1.

Rule 3: The third digit can be any whole number from 1 through 9.

As the need for area codes increased, the decision was made that additional digits beyond 0 or 1 could be used in Rule 2. Each additional digit in Rule 2 can form how many new area codes?

Correct response: 72

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