Assessment/Accountability
Massachusetts Comprehensive Assessment System

Grade 8

Mathematics

Content and Skills to be Assessed by MCAS

January 1998

Massachusetts Department of Elementary and Secondary Education

Number Sense

Number and Number Relationships

Curriculum Framework Learning Standards

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

• represent and use equivalent forms of numbers, including integers, fractions, decimals, percents, exponents, and scientific notation.
• apply ratios, proportions, and percents.
• investigate and describe the relationships among fractions, decimals, and percents.
• represent numerical relationships in one- and two-dimensional graphs.

Assessment Expectations

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

• apply number sense to estimate and choose appropriate forms of numbers for various purposes. [see Question 1]
• identify and use appropriate representations of numbers, e.g., number lines. [see Question 2]
• recognize and use equivalent forms of integers, fractions, decimals, percents, exponents, and scientific notation. [see Question 4]
• use ratio, proportion, and percent to analyze a variety of problem-solving situations. [see Questions 3, 4]

For details and samples of student learning related to these learning standards, see page 40 of the Massachusetts Mathematics Curriculum Framework.

Sample Questions

Multiple-choice Questions:

1. Which is the most reasonable estimate of the thickness of a strand of human hair?

*A. 3 x 10^-3 cm

B. 3 x 10^-30 cm

C. 3 x 10⁰ cm

D. 3 x 10³ cm

2. Use the number line below to answer the question.

Which point represents the number (^­2)⁴?

A. Point A

B. Point B

C. Point C

*D. Point D

Short-answer Question:

3. Ms. O'Reilly invests $500 in the stock market. After ten years, if her investment is worth 300% of its original cost, how much will it then be worth?

Correct response: $1,500

Open-response Question:

4. Last week, Michael found that the stereo system he wants to buy costs exactly the same at three different stores. Today, he sees these ads for the three stores:

a. Suppose that the stereo that Michael wants costs exactly $300. Write the fraction in simplest form that tells the discount Michael would get if he bought the stereo at Your Town Electronics. Show or explain how you found your answer.

b. Now suppose that the stereo Michael wants costs more than $300.

• If possible, tell which store sells the stereo for the most money and which store sells the stereo for the least money. Justify your answers.
• If it is not possible to tell which store sells the stereo for the most money and which store sells it for the least, explain in detail why not.

Number Sense

Number Systems and Number Theory

Curriculum Framework Learning Standards

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

• explain the need for numbers other than whole numbers.
• know and use order relations for whole numbers, fractions, decimals, integers, and rational numbers.
• use operations involving fractions, decimals, integers, and rational numbers.
• demonstrate how basic operations are related to one another.
• create and apply number theory concepts, including prime numbers, factors, and multiples.

Assessment Expectations

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

• determine the effect of operations on different types of numbers, e.g., the sum of two negative numbers is less than either addend. [see Question 9]
• use relationships among operations, e.g., use the fact that multiplying a number by 1/3 is the same as dividing it by 3. [see Question 7]
• solve problems involving the ordering of whole numbers, fractions, decimals, and integers, e.g., decide which item would be next largest within a set of items that are graduated by fractions of an inch. [see Question 5]
• factor numbers into component parts, e.g., prime factorizations. [see Question 8]
• analyze relationships among numbers, e.g., relationships involving factors, multiples, and divisibility. [see Questions 6, 8]

For details and samples of student learning related to these learning standards, see page 41 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Questions:

5. Juan and Kimberley need 2/3 yard of ribbon for a school project. Ribbon is sold only in multiples of 1/8 yard. How much ribbon should they buy so that they will have enough, but as little as possible left over?

A. 5/8 yard

B. 7/8 yard

C. 1/2 yard

*D. 3/4 yard

6.

Colin promised to water his neighbor's three plants from May 31 through June 30. One plant needs water every 2 days, one every 3 days, and one every 4 days. If he waters all three plants on May 31, how many days in June will he not have to water any plants?

A. 24

*B. 10

C. 8

D. 12

Short-answer Questions:

7. Fernando chose a number. He multiplied it by 18, added 22 to the product, and got an answer of 274. What is the number he originally chose?

Correct response: 14

8.

Jessica rolled three number cubes, each with sides labeled 1 through 6. She told Amy that the product of the numbers she rolled was 36. Amy said that she must have rolled a 6, a 6, and a 1, but Jessica said she rolled a different combination. Give one other possible combination that Jessica might have rolled.

Correct responses: 2, 3, 6
3, 3, 4
either combination in any order

Open-response Question:

9. Suni believes that the product of any two positive numbers is always greater than either of the two factors.

a. Give an example of a product of two positive numbers in which the product is greater than either of the two factors.

b. Give an example of a product of two positive numbers in which the product is greater than one of the factors and less than the other.

c. Write a note to Suni explaining when his belief is true and when it is false.

Number Sense

Computation and Estimation

Curriculum Framework Learning Standards

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

• compute with whole numbers, fractions, decimals, integers, and rational numbers.
• develop analyze, and explain procedures for computing, estimating, and solving proportions.
• select and use an appropriate method for computing from among mental arithmetic, paper-and-pencil, calculator, and computer methods.
• use computation, estimation, and proportions to solve problems.
• estimate to check the reasonableness of results of computations and problems involving rational numbers.

Assessment Expectations

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

• compute with whole numbers, fractions, decimals, integers, rational numbers, and exponents. [see Questions 10, 12, 13, 14, 15, 16, 17]
• use efficient strategies to estimate sums, differences, products, and quotients, as well as square roots of whole numbers, e.g.,
  ______
  v 17. [see Question 11]
• apply the correct order of operations. [see Question 16]
• use estimation to check the reasonableness of solutions to problems.

For details and samples of student learning related to these learning standards, see page 42 of the Massachusetts Mathematics Curriculum Framework.

Note: The Mathematics Curriculum Framework substrand, Computation and Estimation, has been divided into two reporting categories for grade 8: Computation and Estimation and Ratio, Proportion, Percent.

• On this page, the assessment expectations pertain to computations and estimations that do not involve ratio, proportion, or percent.
• The assessment expectations pertaining to ratio, proportion, and percent can be found on page 71.

Multiple-choice Questions:

10. When Ms. Yoshimora arrived in Boston from Japan, 1 yen was worth 0.0086 U.S. dollars. If she exchanged 10,000 yen for U.S. dollars, what dollar amount would she receive?

A. $86,000

B. $8,600

C. $860

*D. $86

11. The diameter of the sun is approximately 1,392,000 kilometers, while the diameter of Earth is approximately 12,760 kilometers. About how many times greater is the diameter of the sun than the diameter of Earth?

A. about 90 times greater

*B. about 110 times greater

C. about 900 times greater

D. about 1,100 times greater

Short-answer Questions:

12. Compute: 3/4 ­ 1 7/8 =

Correct response: ^­1 1/8

13. Compute: 0.4 ÷ 1/3 =

Correct response: 1.2 or equivalent

14. Compute: 0.09 ÷ 1.5 =

Correct response: 0.06

15. Compute: (^­2)(^­5)(^­1) =

Correct response: ^­10

16. Compute: 4 + 2 * 3 ­ 3² + (^­6) =

Correct response: ^­5

Open-response Question:

17. Webster Office Supply Company agrees to keep Mr. Orlando's photocopier in working condition for one year if he chooses one of these two payment plans:

Plan 1: Mr. Orlando pays $300. For the first 16,000 copies made, he pays nothing extra. For each additional copy, he pays 2.2¢.

Plan 2: Mr. Orlando pays $500. He pays nothing extra for copies, no matter how many are made.

a. Suppose Mr. Orlando makes 20,000 copies in one year. Which plan will cost him less? Explain how you found your answer.

b. How many copies would Mr. Orlando need to make so that he would pay the same amount for Plan 1 as for Plan 2? Explain how you found your answer.

Number Sense

Ratio, Proportion, Percent

Curriculum Framework Learning Standards

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

• compute with whole numbers, fractions, decimals, integers, and rational numbers.
• develop, analyze, and explain procedures for computing, estimating, and solving proportions.
• select and use an appropriate method for computing from among mental arithmetic, paper-and-pencil, calculator, and computer methods.
• use computation, estimation, and proportions to solve problems.
• estimate to check the reasonableness of results of computations and problems involving rational numbers.

Assessment Expectations

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

• compute ratios, proportions, and percents and apply appropriately, e.g., in enlarging pictures, using scale models, finding sale prices after a series of discounts. [see Questions 18, 19, 20, 21, 22, 23]

For details and samples of student learning related to these learning standards, see page 42 of the Massachusetts Mathematics Curriculum Framework.

Note: The Mathematics Curriculum Framework substrand, Computation and Estimation, has been divided into two reporting categories for grade 8: Computation and Estimation and Ratio, Proportion, Percent.

• On this page, the assessment expectations pertain to computations and estimations involving ratio, proportion, and percent.
• The assessment expectations pertaining to other computations and estimations can be found on page 67.

Multiple-choice Questions:

18. Use the advertisement below to answer the question.

Mr. Howard bought a bike with an originally-marked price of $400. What was the price of the bike at 12:15 p.m.?

*A. $262.44

B. $240.00

C. $291.60

D. $280.00

19. In recent years, the ratio of new vehicles to used vehicles sold by car dealerships has been about 1:2. If a total of about 30,000,000 vehicles is sold in a year, about how many are used vehicles?

A. 10,000,000

B. 15,000,000

*C. 20,000,000

D. 25,000,000

Short-answer Questions:

20. Compute: 3.5% of 470 =

Correct response: 16.45

21. Compute: 63 is what percent of 350?

Correct response: 18%

22. Compute: 75 is 30% of what number?

Correct response: 25081

Open-response Question:

23. Anton has a picture, shown at the right. He wants to reduce the picture so that it will fit into a column in the school yearbook. Each column is 3 inches wide.

8 1/2 inches

11 inches

a. Anton needs to reduce the size of his picture on a photocopier. What percent of the original size should the reduction be so that the new width is 3 inches? Explain how you found your answer. Show your work.

b. When the width of Anton's picture is reduced to 3 inches, how tall will the picture be? Explain how you found your answer. Show your work.

Patterns, Relations, and Functions
Patterns and Functions

Curriculum Framework Learning Standards

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

• describe, extend, analyze, and create a wide variety of patterns.
• describe and represent relationships with models, tables, graphs, and rules, using sentences and algebraic expressions.
• analyze functional relationships to explain how a change in one quantity results in a change in another.
• use patterns and functions to represent and solve problems.

Assessment Expectations

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

• extend and analyze the following types of patterns:
  * numeric patterns, e.g., sequences, input-
  output tables, Pascal's triangle;
  * geometric patterns, e.g., triangular numbers, patterns of dots; and
  * patterns in real-world or mathematical situations, e.g., periods of a pendulum,
  patterns of digits in repeating decimals. [see Questions 26, 27, 28, 29]
• determine the rules or general terms for patterns of the types described above. [see Question 29]
• describe and represent real-world or mathematical relationships by using verbal descriptions of rules, or charts and tables, e.g., input-output tables, graphs. [see Question 24]
• describe how a change in one variable affects another variable in a functional relationship that is presented with words or pictures. [see Question 25]

For details and samples of student learning related to these learning standards, see page 60 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Questions:

24. Which of the following graphs most likely depicts the rate of speed of a car traveling 1 hour on a country road, 1 hour on a high-speed highway, and then 1 hour again on a country road?

*A.

B.

C.

D.

25. Michelle is making "trains" of equilateral triangles with 1-inch sides as shown below.

How does the perimeter of the entire train change each time she adds one more triangle to the end of the train?

*A. It increases by 1 inch.

B. It increases by 2 inches.

C. It increases by 3 inches.

D. It depends on how many triangles are already in the train.

Short-answer Questions:

26. Find the next two numbers in this pattern:

1, 1, 2, 3, 5, 8, _____, _____

Correct response: 13, 21

27. Find the missing number in this pattern:

60, 50, 41, _____, 26, 20

Correct response: 33

28. Find the next number in this pattern:

^­6.2, ^­2.6, 1.0, 4.6, _____

Correct response: 8.2

Open-response Question:

29. Use the pattern below to answer the question.

a. How many dots are in the 5th term in the pattern?

b. How many dots are in the 20th term in the pattern?

c. Write a rule that tells how to find the number of dots in the nth term of this pattern, where n stands for any positive whole number.

Patterns, Relations, and Functions

Algebra

Curriculum Framework Learning Standards

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

• understand and apply the concepts of variable, expression, and equation.
• represent situations and number patterns with tables, graphs, verbal rules, and equations and explore the interrelationships of these representations.
• analyze tables and graphs to identify properties and relationships.
• demonstrate an ability to solve linear equations, using concrete, informal, and formal methods.
• describe the strategies used to explore inequalities and nonlinear equations.
• apply algebraic methods to solve a variety of real-world and theoretical problems.
• construct expressions or equations that model problems.
• explore and describe a variety of ways to solve equations, including hands-on activities, trial and error, and numerical analysis.
• know and apply algebraic procedures for solving equations and inequalities.

Assessment Expectations

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

• evaluate algebraic expressions for given values of variables (substitution). [see Question 33]
• solve linear equations with one variable. [see Question 32]
• graph on a coordinate plane. [see Question 34]
• describe and represent relationships with algebraic expressions and equations. [see Questions 30, 34]
• describe and represent relationships with inequalities. [see Question 31]
• describe how a change in one variable affects another variable in a functional relationship represented by an equation.

For details and samples of student learning related to these learning standards, see page 61 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Questions:

30. Which equation states the rule for the relationship shown in this table?

A. y = x ­ 2

*B. y = (x ­ 2)²

C. y = x² ­ 4

D. y = x² ­ x

31. Mr. Garvey has a budget of $50 to spend on algebra tiles. Each set of tiles costs $4.95. The shipping and handling charge for orders of $50 or less is $5.50. Which inequality shows the possible numbers of sets, n, that he can order?

*A. $4.95n + $5.50 ¾ $50

B. ($4.95 + $5.50)n ¾ $50

C. $4.95n ¾ $50 + $5.50

D. $5.50n ¾ $50 + $4.95

Short-answer Questions:

32. What does x equal in this equation?

2/3 x + 5 = ^­3

Correct response: x = ^­12

33. The volume of a square pyramid is given by the expression:

s²h)
_________
3

where s is the length of a side of the square base and h is its height. Find the volume in cubic feet of a pyramid that is 5 feet tall and has a square base whose sides each measure 4 feet.

Correct response: 26.67 ft³ or equivalent

Open-response Question:

34. When Jeff was 3 years old, his sister Erin was 4 times as old as he was.

a. How old was Erin when Jeff was 5?

b. Write an equation showing the relationship between Jeff's and Erin's ages. Be sure to tell what the variables in your equation represent.

c. Draw a graph on a coordinate plane showing the relationship between Jeff's and Erin's ages. Be sure to properly label your graph.

Geometry and Measurement

Geometry

Curriculum Framework Learning Standards

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

• identify, describe, compare, and classify geometric figures.
• explore and describe the properties of points, lines, and planes.
• visualize and draw geometric figures.
• explore and describe transformations of geometric figures.
• represent and solve problems, using geometric models.
• apply geometric properties and relationships.
• develop and explain the concept of ¼.
• develop and explain the concept of the Pythagorean theorem.

Assessment Expectations

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

• identify, describe, compare, and classify geometric figures, e.g., explain how a square is a parallelogram. [see Question 35]
• use the properties of geometric figures, e.g., find the measure of an angle in a regular polygon, recognize the fact that all circles are similar. [see Question 38]
• identify congruent and similar figures and use their properties.
• demonstrate understanding of spatial relationships, e.g., divide and separate shapes, use tesselations, determine two-dimensional patterns (nets) that can be folded into three-dimensional shapes. [see Question 39]
• identify properties of parallel, perpendicular, and intersecting lines and their resulting angles. [see Question 37]
• identify and use transformations, e.g., translations, rotations, reflections of objects and of figures on the coordinate plane. [see Question 36]
• explain the concepts of ¼ and the Pythagorean theorem.

For details and samples of student learning related to these learning standards, see page 75 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Questions:

35. A rhombus is to a square as a parallelogram is to

*A. a rectangle.

B. a trapezoid.

C. an equilateral triangle.

D. an equilateral quadrilateral.

36. Use the figure below to answer the question.

If the triangle shown above is reflected across the y-axis, what will be the coordinates of the image of point C?

A. (^­4, 2)

B. (^­4, ^­2)

*C. (4, ^­2)

D. (4, 2)

Short-answer Questions:

37. In the figure below, lines m and n are parallel.

Name two pairs of congruent angles.

Correct responses: angle 4 and angle 8; angle 1 and angle 5; angle 4 and angle 2; and other correct pairs.

38. Triangle ABC below is an isosceles triangle in which AB = BC and angle ABC measures 100°.

What is the measure of angle BAC?

Correct response: 40°

Open-response Question:

39. A stack of blocks is shown below. Next to it is a drawing showing how the stack looks when viewed from the front at eye level to the stack.

a. Draw how the stack might look viewed from the back at eye level to it.

b. Draw how the stack might look viewed from the top directly above it.

c. Draw how the stack might look viewed from the left at eye level to it.

d. Draw how the stack might look viewed from the right at eye level to it.

Geometry and Measurement

Measurement

Curriculum Framework Learning Standards

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

• select appropriate units and tools to measure to the degree of accuracy required in a particular situation.
• describe the meaning of perimeter, area, volume, angle measure, capacity, density, weight, and mass.
• develop and describe the concepts of rates and other derived and indirect measurements.
• develop and apply formulas and procedures for determining measures to solve problems.

Assessment Expectations

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

• use appropriate tools to measure with reasonable accuracy and apply those measurements.
• estimate length, capacity, weight, and mass. [see Question 41]
• use concepts of accuracy, precision, and error of measurement, e.g., determine reasonable units to use in measuring a wall for the purpose of buying baseboard or, by contrast, for the purpose of determining the length of baseboard to cut for installation. [see Question 40]
• convert units of length, capacity, weight, and time within either the metric system or the customary system. [see Question 43]
• solve problems involving rates. [see Question 42]

For details and samples of student learning related to these learning standards, see page 76 of the Massachusetts Mathematics Curriculum Framework.

Note: The Mathematics Curriculum Framework substrand, Measurement, has been divided into two reporting categories for grade 8: Measurement and Geometric Measurement.

• On this page, the assessment expectations pertain to measurement of length, capacity, weight, time, and rates.
• The assessment expectations pertaining to geometric measurement can be found on page 90.

Multiple-choice Questions:

40. Use the chart below to answer the question.

The lengths of Andrew's and Nicole's jumps have been rounded to the nearest foot. Given this information, which statement best describes the greatest possible difference in the lengths of Andrew's jump and Nicole's jump?

A. One jump could be up to 1/4 ft longer than the other.

B. One jump could be up to1/2 ft longer than the other.

*C. One jump could be up to 1 ft longer than the other.

D. One jump could be up to 2 ft longer than the other.

41. Which of the following is most likely to be the weight of a man who is about 1.8 meters tall?

*A. 80 kg

B. 25 kg

C. 160 kg

D. 240 kg

42. When Whitney's family started their 200-mile trip at 7:00 a.m., the car's odometer read 25,523 miles. At 9:30 a.m., the odometer read 25,652 miles. Based only on this information, which is the best estimate of when they will arrive at their destination?

A. at 10:30 a.m.

*B. at 11:00 a.m.

C. at 11:30 a.m.

D. at noon

Open-response Question:

43. You may use the unit equivalents on the reference sheet provided to answer the question.

Uri wants to use this recipe for punch, but he wants to make only 10 servings.

Uri has only the following measuring tools:

• a 1-cup measuring cup
• a 1/2-cup measuring cup
• a 1/4-cup measuring cup
• a set of measuring spoons that includes 1 tablespoon, 1 teaspoon, 1/2 teaspoon, and 1/4 teaspoon

Rewrite the recipe for Uri so that it will make 10 servings. The new recipe should include only measurements that can be made accurately using Uri's measuring tools. Be sure the new recipe has exactly the same proportions of ingredients as the original. Show how you found each new measurement.

Geometry and Measurement

Geometric Measurement

Curriculum Framework Learning Standards

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

• select appropriate units and tools to measure to the degree of accuracy required in a particular situation.
• describe the meaning of perimeter, area, volume, angle measure, capacity, density, weight, and mass.
• develop and describe the concepts of rates and other derived and indirect measurements.
• develop and apply formulas and procedures for determining measures to solve problems.

Assessment Expectations

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

• estimate measures of angles.
• estimate areas of irregular figures using grids or rulers.
• find perimeter, circumference, area, surface area, and volume. [see Question 44]
• apply the concepts and formulas for perimeter, circumference, area, surface area, and volume to solve problems, e.g., determine the volume of any cube when the lengths of its edges are doubled. [see Questions 45, 47]
• solve problems involving indirect measurements, e.g., problems requiring use of the Pythagorean theorem and similar triangles. [see Question 46]

For details and samples of student learning related to these learning standards, see page 76 of the Massachusetts Mathematics Curriculum Framework.

Note: The Mathematics Curriculum Framework substrand, Measurement, has been divided into two reporting categories for grade 8: Measurement and Geometric Measurement.

• On this page, the assessment expectations pertain to geometric measurement.
• The assessment expectations pertaining to measurements of length, capacity, weight, time, and rates can be found on page 86.

Multiple-choice Questions:

44. Use the figure below to answer the question.

Which statement is true about the two triangles?

*A. They have equal areas but different perimeters.

B. They have equal areas and perimeters.

C. They have equal perimeters but different areas.

D. They have different areas and different perimeters.

45. The area of a rectangle is 36 square centimeters. The perimeter is 30 centimeters. What are the dimensions of the rectangle?

A. 2 cm by 18 cm

*B. 3 cm by 12 cm

C. 4 cm by 9 cm

D. 6 cm by 6 cm

Short-answer Question:

46. Allison is building a bird house. She needs to be sure that the angle between two walls is a right angle. She marks off an 8-inch segment along one wall and a 6-inch segment along the other wall, as shown. Then she measures the distance between the endpoints.

If the angle is a right angle, what will be the distance between the endpoints?

Correct response: 10 in.

Open-response Question:

47. The fish tank in Jeremy's classroom is in the shape of a rectangular prism. It contains the greatest number of fish recommended for the size of the tank. Jeremy bought a tank for home that is one-half as tall, one-half as long, and one-half as wide as the one at school. He thinks his tank will hold one-half as many fish as are in the tank at school. Is he right?

• If he is right, explain how he correctly came to this conclusion.
• If he is wrong, explain why his conclusion is incorrect. Include an explanation of how he can find the greatest number of fish his tank should hold.

Statistics and Probability

Statistics

Curriculum Framework Learning Standards

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

• collect, organize, and describe data systematically.
• construct, read, and interpret tables, charts, and graphs.
• make inferences and convincing arguments that are based on data analysis.
• evaluate arguments that are based on data analysis.
• develop and explain why statistical methods are powerful aids for decision making.

Assessment Expectations

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

• read and interpret data from bar, line, and circle graphs, pictographs, and scatter plots. [see Question 50]
• draw conclusions and evaluate arguments based on data presented in tables, charts, graphs, and advertisements. [see Question 49]
• design surveys, e.g., how the sample should be chosen, what questions should be asked, how the data should be aggregated and reported.
• construct graphs appropriate to the type of data to be represented. [see Question 48]
• compute and interpret mean, median, mode, and range of sets of data, and draw and justify conclusions based on these statistics. [see Question 51]

For details and samples of student learning related to these learning standards, see page 90 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Questions:

48. Shawn found the following information showing the percentages of adults taking part in various leisure activities during the past 12 months:

Exercise program -- 60%

Camping, hiking, canoeing -- 34%

Playing sports -- 39%

Home improvement and repair -- 48%

Gardening -- 55%

Which type of graph is most appropriate for displaying this information?

*A. a bar graph

B. a line graph

C. a circle graph

D. a scatter plot

49. The Morgan family opened a pretzel stand seven weeks ago. Their weekly sales are shown in the graph to the right. Based on the information in the graph, which is the best prediction of the sales for week 8?

Pretzel Sales

A. $1,225

*B. $1,350

C. $1,450

D. $1,525

Short-answer Question:

50. Each boy in Ms. Garcia's class recorded his height to the nearest inch and his weight to the nearest ten pounds. The class made the scatter plot below showing the results.

Height and Weight of Boys

Which is the most common height for the boys in this group?

Correct response: 5'7"

Open-response Question:

51. In the last seven basketball games, Linn scored the following numbers of points:

8, 10, 14, 12, 2, 11, 12

Cathy is writing a story about Linn's performance and wants to report either her mean score or her median score.

a. Find the mean of Linn's scores. Show how you found the mean.

b. Find the median of her scores. Show how you found the median.

c. Tell whether the mean or median best reflects Linn's record. Explain your answer.

Statistics and Probability

Probability

Curriculum Framework Learning Standards

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

• model situations by devising and carrying out experiments or simulations to determine probabilities.
• construct a sample space to determine probabilities.
• describe the power of using a probability model by comparing experimental results with mathematical expectations.
• make predictions that are based on experimental or theoretical probabilities and determine their reasonableness.
• develop and explain an appreciation for the pervasive use of probability in the real world.

Assessment Expectations

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

• determine the number of possible combinations in different types of situations, e.g., find the number of different ways acts in a talent show can be presented.
• construct the set of all possible outcomes (sample space) and determine theoretical probabilities in given situations. [see Questions 52, 54]
• determine empirical probabilities based on given data. [see Question 53]
• make predictions and draw conclusions based on counting procedures and probability. [see Question 55]

For details and samples of student learning related to these learning standards, see page 91 of the Massachusetts Mathematics Curriculum Framework.

Multiple-choice Questions:

52. Of a group of 640 students, 428 were born in Massachusetts. If a newspaper reporter wants to interview one student selected from the group at random, which is the best estimate of the probability that the student will have been born in Massachusetts?

A. 1/2

*B. 2/3

C. 2/1

D. 3/2

53. Batco Company tested 1,500 Type T batteries selected at random from those manufactured last month. They found that 73 of the batteries lasted less than 100 hours. Which is the best estimate of the probability that a Type T battery will last less than 100 hours?

*A. 1/20

B. 1/200

C. 1/15

D. 1/150

Short-answer Question:

54. Use the spinner on the right to answer the question.

What is the probability of moving forward on the next spin?

Correct response: 5/8

Open-response Question:

55. Danika, Craig, and Tiffany created their own board game. One of the players tosses two coins.

Danika moves one space if the result is 2 heads.

Craig moves one space if the result is 2 tails.

Tiffany moves one space if the result is 1 head and 1 tail.

Does each of the players have an equal chance of winning the game?

• If you think the players have equal chances, find the probability that each of the players will win. Explain how you know your answer is correct.
• If you think the players do not have equal chances, explain in detail why you think they do not.

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