Problem Solving, Numbers, and Algebra



Problem Solving, Numbers, and Algebra

*include perimeter & area problems(review)-rectangles and triangles (whole numbers)

Open Responses

Carrie’s rectangular garden has the dimensions shown below.

[pic]

a. What is the area, in square feet, of Carrie’s garden? Show or explain how you got your answer.

Carrie wants to put a fence along the perimeter of her garden. She will pay $15 per foot of fence that she uses.

b. What is the amount of money, in dollars, that Carrie will pay for the fence? Show or explain how you got your answer.

Roberto has a garden in the shape of a square. The perimeter of his garden is equal to the perimeter of Carrie’s garden.

c. What is the area, in square feet, of Roberto’s garden? Show or explain how you got your answer.

Lucinda earns $20 each week. She spends $5 each week and saves the rest. The table below shows the total amount that she saved at the end of each week for 4 weeks.

Lucinda's Savings at the End of Each Week

|Week |1 |2 |3 |4 |

|Total Amount |$15 |$30 |$45 |$60 |

|Saved | | | | |

Lucinda continues to save at the same rate.

|a. |What will be Lucinda’s total amount saved at the end of 7 weeks? Show or explain how you got your answer. |

|b. |Use numbers, words, or symbols to write an expression that represents Lucinda’s total amount saved at the end of n weeks. |

|c. |How many weeks will it take for Lucinda to save $300? Show or explain how you got your answer. |

Short Answers

Book covers cost $0.15 each, including tax. Ms. Larkin bought 25 of them.What is the total amount of money that she spent on the book covers?

What is the value of the expression below when [pic]?

[pic]

A rectangle has a width of 6 feet, as shown below.

[pic]

The perimeter of the rectangle is 34 feet. What is the length, in feet, of the rectangle?

In 27 years, Julia will be 43 years old. How old is Julia now?

[pic]

Statistics: Graphing Data, Mean, Median, Mode

Open Responses

Paul bowled 6 games today. His scores are listed in the table below.

|Paul’s Bowling |

|Scores |

|Game |Score |

|1 |158 |

|2 |124 |

|3 |110 |

|4 |167 |

|5 |146 |

|6 |165 |

a. What was Paul’s median score for the 6 games? Show or explain how you got your answer.

b. What was Paul’s mean score for the 6 games? Show or explain how you got your answer.

c. Paul will bowl one more game. What is the minimum score Paul must achieve in the next game so that his mean score for all 7 games is at least 150? Show or explain how you got your answer.

|On the grid in your Student Answer Booklet, copy the x-axis, the y-axis, and line m, as shown below. |

[pic]

| |Triangle PQR has vertices located at the following points: |

| |P(1, 6) |

| |Q(4, 2) |

| |R(4, 6) |

a. Plot points P, Q, and R, and draw triangle PQR on your coordinate grid. Be sure to label the vertices of your triangle with the letters P, Q, and R.

b. Draw the reflection of triangle PQR across line m on your coordinate grid. Label this new triangle STU.

c. Write the coordinates of points S, T, and U.

d. Emily works at a fitness center. She recorded the heart rates of some people immediately after they exercised. Her data are shown below.

Heart Rates (beats per minute)

|120 |128 |144 |136 |130 |

|150 |138 |140 |132 |130 |

e.

|a. |Construct a stem-and-leaf plot to show Emily's data. Be sure to include a key. |

|b. |Based on Emily's data, what is the median heart rate? Show or explain how you got |

| |your answer. |

f. Emily measured the heart rates of two more people. When these heart rates were added to the data set, the mode decreased.

|c. |Explain what must be true of the two additional heart rates in order for the mode to decrease. |

|d. |Explain how the two additional heart rates will affect the median heart rate that you found in part (b). |

The graph below shows the relationship between the number of museum tickets bought and the total cost of the tickets.

[pic]

a. What is the greatest number of museum tickets that can be bought for $21?

b. What is the cost of 1 museum ticket? Show or explain how you got your answer.

c. Using numbers, words, or symbols, write a rule that could be used to find the total cost of any number of museum tickets. You may use n to represent the number of museum tickets bought.

d. Calvin bought a one-year museum pass for $45. The pass allows him to visit the museum an unlimited number of times during one year. What is the least number of times Calvin must visit the museum, during one year, in order for his one-year pass to be less expensive than buying a single museum ticket for each visit? Show or explain how you got your answer.

Katie will take a total of 5 mathematics tests. She has taken 4 mathematics tests so far. The scores on her first 4 tests are shown in the table below.

Katie’s Mathematics Test Scores

|Test |Score |

|1 |94 |

|2 |98 |

|3 |86 |

|4 |92 |

|5 |? |

a. What is the median of Katie’s first 4 mathematics test scores? Show or explain how you got your answer.

b. What is the mean of Katie’s first 4 mathematics test scores? Show or explain how you got your answer.

c. What score must Katie get on her 5th test in order to have a mean score of 90 on all 5 of her mathematics tests? Show or explain how you got your answer.

 

Short Answers

The stem-and-leaf plot shown below displays the number of days it rained last year in

11 different cities.

[pic]

What is the mode of this set of data?

[pic]

Understanding Decimals, Estimating, Adding and Subtracting Decimals

Open Responses

The table below shows the distance, in kilometers, Lisa ran each day for 5 days.

Lisa’s Running Distances

|Day |Distance |

| |(in kilometers) |

|Monday |2.05 |

|Tuesday |4.10 |

|Wednesday |3.80 |

|Thursday |1.95 |

|Friday |4.05 |

a. Write the five distances in order from greatest to least.

b. Estimate the average distance, in kilometers, Lisa ran each day for the 5 days. Explain your estimation strategy.

To prepare for a race, Lisa plans to run 100 kilometers over a number of days.

c. Based on your answer to part (b), estimate the number of days it will take Lisa to run 100 kilometers. Explain your estimation strategy.

d. Carla made the table below to show the populations of five different states.

State Populations

|State |Population (in millions) |

|   Massachusetts |  6.35 |

|   New Hampshire |  1.24 |

|   New York |18.98 |

|   Pennsylvania |12.28 |

|   Vermont |  0.61 |

e.

|a. |Based on the data in the table, write the states in order from the greatest population to the least population. |

|b. |Based on the data in the table, estimate how many more people live in Pennsylvania than in Vermont. Show or explain your |

| |strategy. |

|c. |Based on the data in the table, estimate the total number of people who live in all five states. Show or explain your |

| |strategy. |

|d. |Based on your answer to part (c), estimate the percent of the total population of the five states that is from Massachusetts.|

| |Show or explain how you got your answer. |

[pic]

Measurement- Metric and Customary

[pic]

Using Number Patterns, Fractions and Ratios (GCF, LCM, prime and composite numbers)

| |

|Open Responses |

|A store will celebrate the release of a new game system by giving free gifts to some of its customers. On the day the game |

|system is released, every 5th customer will receive a free video game and every 12th customer will receive a free game system. |

a. What is the total number of customers who will receive a free video game before a customer receives the first free game system? Show or explain how you got your answer.

b. Of the first 50 customers, what is the total number of customers who will receive a free gift? Show or explain how you got your answer.

c. If 200 customers visit the game store on this day, what is the total number of customers who will receive both a free video game and a free game system? Show or explain how you got your answer.

A local bakery celebrated its one-year anniversary on Saturday. On that day, every 4th customer received a free cookie. Every 6th customer received a free muffin.

a. Did the 30th customer receive a free cookie, a free muffin, both, or neither? Show or explain how you got your answer.

b. Casey was the first customer to receive both a free cookie and a free muffin. What number customer was Casey? Show or explain how you got your answer.

c. Tom entered the bakery after Casey. He received a free cookie only. What number customer could Tom have been? Show or explain how you got your answer.

d. On that day, the bakery gave away a total of 29 free cookies. What was the total number of free muffins the bakery gave away on that day? Show or explain how you got your answer.

Short Answers

Write 84 as a product of prime numbers.

[pic]

Exploring Ratio, Proportion, and Percent

Open Responses

Use your MCAS ruler to answer question 27.

NOTE to Web site users: The Web release graphic for item #27 is not to scale; please be aware that students will not get accurate measurements when using the Web printout of this item.

Part of a map is shown below. Each point is labeled to indicate the town that it represents. The map has a scale in which 1 inch represents 20 miles.

[pic]

|a. |Using your MCAS ruler, what is the distance, in inches, between Mayfield and Shelton on the map? |

|b. |What is the actual distance, in miles, between the towns of Mayfield and Shelton? Show or explain how you used the scale to |

| |get your answer. |

|c. |What is the actual distance, in miles, between the towns of Hearne and Shelton? Show or explain how you used the scale to get|

| |your answer. |

|d. |The town of Sawyer is located 50 miles from Mayfield. On the full map, what should be the distance, in inches, between Sawyer|

| |and Mayfield? Show or explain how you got your answer. |

Short Answers

Ms. Edwards has visited 19 of the 50 states in the United States. What percent of the states has Ms. Edwards visited?

[pic]

Algebra: Slope as a Constant Rate of Change

[pic]

Applying Fractions, Decimals, and Percent’s to Probability

| |

|Open Responses |

|Ellen is ordering a new mountain bike. She must choose 1 option from each column in the table shown below. |

[pic]

a. Show all the possible ways Ellen can order her new mountain bike. You may use a tree diagram or an organized list.

b. What is the total number of ways Ellen can order a bike with a front suspension? Show or explain how you got your answer.

c. Ellen learned that there are now 3 more color choices than before. What is the total number of additional ways Ellen can order her new mountain bike now that there are 3 more color choices? Show or explain how you got your answer.

Short Answers

Marcus has a bag of 20 table tennis balls. The probability of selecting a yellow table tennis ball, without looking, is[pic]. What is the total number of yellow table tennis balls in the bag?

[pic]

Adding and Subtracting Fractions

Short Answers

What is the value of the expression below when [pic] = 12?

[pic]

Luke spent 2[pic] hours reading on Saturday and 1[pic] hours reading on Sunday. What was the total time, in hours, that Luke spent reading on Saturday and Sunday?

[pic]

Multiplying and Dividing Fractions

[pic]

Geometry: Investigating Patterns (triangles, angles, transformations etc.)

Open Responses

Six geometric terms are given in the box below.

[pic]

David drew the six triangles shown below.

[pic]

|a. |Identify one of the geometric terms listed in the box that can be used to describe triangle A. Explain your reasoning. |

|b. |Which two of the geometric terms listed in the box can be used to describe triangle B? Explain your reasoning. |

David grouped his triangles as shown below.

[pic]

|c. |Using one or more of the geometric terms listed in the box, explain what the two triangles in each group have in common. Be |

| |sure to label your answers Group 1, Group 2, and Group 3. |

 

Copy triangle ABC and line l, shown below, onto the grid in your Student Answer Booklet. Be sure to label points A, B, and C in your drawing.

[pic]

a. Is triangle ABC equilateral, isosceles, or scalene? Explain your reasoning.

b. On the grid in your Student Answer Booklet, draw the reflection of triangle ABC over line l. Label the new triangle DEF.

c. On the grid in your Student Answer Booklet, draw the translation of triangle ABC after it has been moved 7 units right and 3 units up. Label the new triangle GHI.

d. Are triangle DEF and triangle GHI congruent? Explain your reasoning.

Short Answers

Rectangle ABCD is shown below.

[pic]

Write the name of a line segment in rectangle ABCD that is parallel to [pic].

[pic]

Geometry: Understanding Perimeter, Area, and Volume

Open Responses

A triangular prism and a rectangular prism are shown below.

[pic]

a. How many edges does a triangular prism have?

b. How many more faces does a rectangular prism have than a triangular prism has? Show or explain how you got your answer.

c. How many faces does a hexagonal prism have? Show or explain how you got your answer. 

|The diagram below shows the dimensions of a fish tank at a science museum. |

[pic]

a. What is the volume, in cubic feet, of the fish tank? Show your work.

b. The tank is made of glass and does not have a cover. What is the surface area, in square feet, of the outside of the fish tank? Show your work.

| |A new fish tank is being built for the museum. The new tank will have different dimensions than the first tank, but will |

| |have the same volume as the first tank. |

c. What could be the dimensions of the new tank? Show or explain how you got your answer.

 A city park is in the shape of a square, with each side measuring 40 feet.

a. What is the area, in square feet, of the city park? Show or explain how you got your answer.

The city has decided to put a pond in the shape of a circle in the center of the park. The circle will have a radius of 10 feet, as shown in the diagram below. The remaining portion of the park will be a lawn.

City Park

[pic]

a. What is the approximate area, in square feet, of the circle? Show your work. (Use 3.14 for [pic].)

b. A landscaper plans to fertilize the lawn of the park. What is the approximate area, in square feet, of the lawn of the park? Show or explain how you got your answer.

c. One bag of GrowFast fertilizer can fertilize 50 square feet. How many bags of GrowFast will the landscaper need in order to fertilize the lawn of the park? Show or explain how you got your answer.

Short Answers

What is the area, in square feet, of the triangle shown below?

[pic]

A hexagonal prism is shown below.

[pic]

What is the total number of edges in a hexagonal prism?

The shaded figure below represents Peggy’s garden.

[pic]

Based on the dimensions in the figure, what is the perimeter, in feet, of Peggy’s garden?

What is the volume, in cubic inches, of the cube below?

[pic]

[pic]

Algebra: Investigating Integers



[pic]

Algebra: Exploring Equations

At a recycling center in his state, Cody receives $0.05 for every can he returns. In the equation below,

• m represents the total amount of money that Cody receives, and

• c represents the total number of cans he returns.

m = 0.05c

a.  What is the total amount of money that Cody receives for returning 10 cans? Show

     or explain how you got your answer.

Cody received a total of $10 for the cans he returned last week.

b.  What was the total number of cans Cody returned last week? Show or explain how

     you got your answer.

Cody's cousin lives in another state. She receives $0.05 more than Cody for every can

she returns.

c.  Write a new equation to determine the total amount of money, m, Cody's cousin receives

     for returning c cans in this other state. Explain your reasoning.

Short Answers

Using the number line below, what is the distance between point A and point B?

[pic]

What number best represents the location of point A on the number line below?

[pic]

Write a rule that describes the relationship between the input (x) and the output (y) in the input-output table below.

|Input (x) |   2    |   5    |   10    |   11    |

|Output (y) |   5    |   11    |   21    |   23    |

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