K-12 Mathematics Benchmarks



Mathematics Benchmarks & Indicators

with Ohio Achievement Test Questions

Grade 7

Includes questions from the released

2008, 2007, 2006 and 2005 Ohio Achievement Tests

and the 2005 Practice Test

Far East Regional Partnership

for Conceptually Based Mathematics

Youngstown State University

Compiled by A. Crabtree, 2006

Revised by A. Crabtree and L. Holovatick, 2007

Revised by A. Crabtree, J. Lucas, and T. Cameron, 2008

A. Select appropriate units to measure angles, circumference, surface area, mass and volume, using:

• U.S. customary units; e.g., degrees, square feet, pounds, and other units as appropriate;

• metric units; e.g., square meters, kilograms and other units as appropriate.

7.1. Select appropriate units for measuring derived measurements; e.g., miles per hour, revolutions per minute.

|Grade 7 – 2008 Test – Problem # 38 |

| | |

|A woman saves a part of each paycheck and hasn’t spent any of her |A. years/dollar |

|savings. She has saved a total of $6,500 in the last 5 years. |B. dollars/year |

| |C. months/dollar |

|Which unit could be used to measure her average rate of savings? |D. cents/dollar |

|Grade 7 – 2007 Test – Problem # 30 |

| | |

|May Li read the first 30 pages of a novel in half an hour. The entire |A. hours per page |

|novel is 250 pages long. She wants to estimate how long it will take |B. minutes per book |

|her to finish the book. |C. pages per minute |

| |D. seconds per page |

|Which rate would be an appropriate measure of her reading speed? | |

|Grade 7 – 2005 OAT – Problem # 13 |

| | |

|At 7:00 a.m., the temperature was 55ºF. The |A. degrees per hour |

|temperature was 72ºF at 3:00 p.m. |B. degrees per minute |

| |C. hours per degree |

|Which is the most appropriate way to |D. minutes per degree |

|describe the average rate of change in | |

|temperature? | |

B. Convert units of length, area, volume, mass and time within the same measurement system.

7.2. Convert units of area and volume within the same measurement system using proportional reasoning and a reference table when appropriate; e.g., square feet to square yards, cubic meters to cubic centimeters.

|Grade 7 – 2006 OAT – Problem # 36 |

| |

|After a game, Coach Larson wants to serve punch to the players on her soccer team. She will mix 1 quart of ginger |

|ale and 1 gallon of fruit punch together. |

| |

|In your Answer Document, find the number of 8-ounce servings she can make. Show how you found the answer. |

|Grade 7 – 2005 OAT – Problem # 17 |

| | |

|Aaron is tiling a floor. The floor measures |A. 30 tiles |

|6 feet by 5 feet. Aaron is using square tiles |B. 90 tiles |

|with sides that measure 3 inches. |C. 270 tiles |

| |D. 480 tiles |

|How many tiles will it take to cover the entire | |

|floor? | |

|Grade 7 – 2005 Practice Test – Problem # 4 |

| | |

|Tasha needs to measure 1.4 liters of liquid |A. 14 mL |

|for an experiment. |B. 140 mL |

| |C. 1,400 mL |

|How many milliliters of liquid will |D. 14,000 mL |

|Tasha need? | |

C. Identify appropriate tools and apply appropriate techniques for measuring angles, perimeter or circumference and area of triangles, quadrilaterals, circles and composite shapes, and surface area and volume of prisms and cylinders.

7.6. Use strategies to develop formulas for finding area of trapezoids and volume of cylinders and prisms.

7.7. Develop strategies to find the area of composite shapes using the areas of triangles, parallelograms, circles and sectors.

|Grade 7 – 2008 Test – Problem # 29 |

| | |

|A tabletop in the shape of a trapezoid is made up of a rectangle and |A. 240 sq cm |

|two congruent right triangles. |B. 5,600 sq cm |

| |C. 8,000 sq cm |

|[pic] |D. 10,400 sq cm |

| | |

|The bases of the trapezoid are 70 cm and 130 cm, and the height is 80 | |

|cm. What is the area of the tabletop? | |

|Grade 7 – 2007 Test – Problem # 43 |

| | |

|A window consists of a semicircle and a rectangle, as shown. |21 square feet |

| |24 ½ square feet |

|[pic] |28 square feet |

|What is the approximate area of the window? |35 ½ square feet |

| | |

|Grade 7 – 2006 OAT – Problem # 44 |

| | |

|A circle is cut out of a square as shown. |A. 22 square feet |

|[pic] |B. 37 square feet |

| |C. 69 square feet |

|What is the approximate area of the shaded |D. 85 square feet |

|portion of the figure? | |

|Grade 7 – 2005 OAT – Problem # 32 |

| | |

|Tina created a display as shown below. |A. 1,152 square inches |

| |B. 1,440 square inches |

|[pic] |C. 2,016 square inches |

| |D. 2,880 square inches |

|What is the total area of her display? | |

|Grade 7 – 2005 Practice Test – Problem # 17 |

| | |

|The base of a cylinder has a radius of |A. circumference |

|5 inches. |B. diameter |

| |C. height |

|What additional measurement is needed |D. weight |

|to find the volume of this cylinder? | |

D. Select a tool and measure accurately to a specified level of precision.

7.3. Estimate a measurement to a greater degree of precision than the tool provides.

|Grade 7 – 2006 OAT – Problem # 30 |

| | |

|An angle is shown on this protractor. |A. 23º |

|[pic] |B. 26º |

| |C. 27º |

|Estimate the measure of the angle to the |D. 37º |

|nearest degree. | |

|Grade 7 – 2005 OAT – Problem # 38 |

| | |

|Dagny measured a paper clip as shown. |A. 4.25 cm |

| |B. 4.5 cm |

|[pic] |C. 4.75 cm |

| |D. 4.95 cm |

|Which measurement is closest to the exact | |

|length of the paper clip? | |

E. Use problem solving techniques and technology as needed to solve problems involving length, weight, perimeter, area, volume, time and temperature.

7.4. Solve problems involving proportional relationships and scale factors; e.g., scale models that require unit conversions within the same measurement system.

|Grade 7 – 2008 OAT – Problem # 43 |

| | |

|Lisa’s father is an architect. He builds a |A. 24 feet |

|cardboard model of each building he |B. 96 feet |

|designs. The scale of his model to the actual |C. 192 feet |

|building is 1 inch = 8 feet. |D. 288 feet |

| | |

|How tall will the actual building be when the | |

|model is 3 feet tall? | |

|Grade 7 – 2007 Test – Problem # 31 |

| |

|An Olympic-size pool is shown. |

| |

|[pic] |

| |

|Troy is building a pool that is proportional to the Olympic-size pool. The width of his pool will be 625 |

|centimeters. |

| |

|In your Answer Document, find the scale factor Troy will use to build his pool. |

| |

|Then show or explain how you found the scale factor. Use the scale factor to find the length of Troy’s pool. Show or|

|explain how you found the length. |

|Grade 7 – 2005 OAT – Problem # 6 |

| |

|A drawing for a house uses the scale of ¼ inch = 1 foot. |

| |

|[pic] |

|The dimensions of a kitchen on the drawing are 2 inches by 4 inches. |

| |

|In your Answer Document, determine the actual dimensions of the kitchen. Show or describe how you found the length |

|and width. Label your answer with appropriate units. |

|Grade 7 – 2005 Practice Test – Problem # 10 |

| |

|During her summer vacation, Rachel mows lawns. Her lawn mower cuts a path 2 feet wide, and she moves forward at an |

|average speed of |

|180 feet per minute. Rachel is mowing a rectangular section of lawn that measures 120 feet long and 80 feet wide. |

| |

|In your Answer Document, make a reasonable estimate for how long it will take her to mow this section of lawn. Draw |

|a sketch of the lawn to explain how you made your estimate. |

| |

|Rachel started mowing the lawn at 3:00 p.m. |

|After 10 minutes of mowing she increased her speed to 200 feet per minute. Approximately what time will Rachel |

|finish mowing the lawn? |

|Show your work or explain your answer. |

F. Analyze and explain what happens to area and perimeter or surface area and volume when the dimensions of an object are changed.

7.9. Describe what happens to the surface area and volume of a three-dimensional object when the measurements of the object are changed; e.g., length of sides are doubled.

|Grade 7 – 2008 Test – Problem # 26 |

| |

|Carson wanted to make a cylindrical pillow for his mother’s birthday. The pillow was to be 15 inches long, with a |

|diameter of 6 inches and would be filled with stuffing. |

| |

|In your Answer Document determine how many cubic inches of stuffing Carson will need to make the pillow. Show your |

|work. |

| |

|Before he started making the pillow, Carson decided he wanted to make it bigger. |

| |

|Compare the amount of stuffing needed when he doubles the length to the amount of stuffing needed when he doubles |

|the diameter. Show work or provide an explanation to support your comparison. |

|Grade 7 – 2006 OAT – Problem # 22 |

| | |

|The Burton Company produces cylindrical barrels. |A. The volume triples. |

| |B. The volume doubles. |

|If the height of a cylindrical barrel is reduced |C. The volume is reduced to one half of the|

|from 12 feet to 4 feet, what will happen to the |original. |

|volume of the barrel? |D. The volume is reduced to one third of |

| |the original. |

|Grade 7 – 2005 OAT – Problem # 44 |

| | |

|A packing company is changing the size of a |A. 5 |

|shipping carton. |B. 6 |

| |C. 9 |

|[pic] |D. 36 |

| | |

|The carton is in the shape of a rectangular | |

|prism. The height of the carton will be | |

|doubled and the width will be tripled. | |

| | |

|By what scale factor will the volume of the | |

|original carton increase? | |

G. Understand and demonstrate the independence of perimeter and area for two-dimensional shapes and of surface area and volume for three-dimensional shapes.

7.8. Understand the difference between surface area and volume, and demonstrate that two objects may have the same surface area, but different volumes or they may have the same volume, but different surface area.

|Grade 7 – 2007 Test – Problem # 28 |

| |

|How do the volumes and surface areas of these rectangular prisms compare? |

|[pic] |

|A. The volumes are the same, but the surface areas are different. |

|B. The volumes and surface areas are the same. |

|C. The volumes are different, but the surface areas are the same. |

|D. The volumes and surface areas are both different. |

|Grade 7 – 2005 OAT – Problem # 26 |

| |

|A company needs to create a box shaped like a rectangular prism. The volume must be 216 cubic inches, but the |

|surface area needs to be as small as possible. One possible box is shown. |

|[pic] |

|In your Answer Document, sketch or describe a different box that has the same volume as Box A, and a surface area |

|less than that of Box A. Show work or provide an explanation to verify that the new box meets the criteria. |

| | |

|OAT – Grade 7 | |

| | |

|Measurement | |

| | |

| | |

|Test | |

|Year | |

|Question # | |

|Answer | |

| | |

|A | |

|2008 | |

|38 | |

|B | |

| | |

| | |

|2007 | |

|30 | |

|C | |

| | |

| | |

|2005 | |

|13 | |

|A | |

| | |

|B | |

|2006 | |

|36 | |

|S.A. | |

| | |

| | |

|2005 | |

|17 | |

|D | |

| | |

| | |

|2005* | |

|4 | |

|C | |

| | |

|C | |

|2008 | |

|29 | |

|C | |

| | |

| | |

|2007 | |

|43 | |

|B | |

| | |

| | |

|2006 | |

|44 | |

|A | |

| | |

| | |

|2005 | |

|32 | |

|B | |

| | |

| | |

|2005* | |

|17 | |

|C | |

| | |

|D | |

|2006 | |

|30 | |

|A | |

| | |

| | |

|2005 | |

|38 | |

|C | |

| | |

|E | |

|2008 | |

|43 | |

|D | |

| | |

| | |

|2007 | |

|31 | |

|S.A. | |

| | |

| | |

|2005 | |

|6 | |

|S.A. | |

| | |

| | |

|2005* | |

|10 | |

|** | |

| | |

|F | |

|2008 | |

|26 | |

|E.R. | |

| | |

| | |

|2006 | |

|22 | |

|D | |

| | |

| | |

|2005 | |

|44 | |

|B | |

| | |

|G | |

|2007 | |

|28 | |

|A | |

| | |

| | |

|2005 | |

|26 | |

|S.A. | |

| | |

| | |

|* Half-Length Practice Test | |

|** Scoring Rubric Not Released | |

|MEA – Benchmark F |

|2008 OAT – Grade 7 – Problem # 26 Scoring Guidelines: |

|Points |Student Response |

|4 |The focus of the task is finding the volume of a three-dimensional cylindrical object and describing changes in the |

| |volume of a three-dimensional object when the measurements of the object are changed. The response provides a |

| |correct strategy to find the number of cubic inches of stuffing Carson will need to make the pillow AND a comparison|

| |of the amount of stuffing needed when Carson doubles the length to the amount of stuffing needed when he doubles the|

| |diameter. The response also shows or explains all calculations. |

| | |

| |Sample Correct Responses: |

| |Work for 6-inch diameter pillow: |

| |V = r²h |

| |V = x3²x15 |

| |V = 424.12 in³ |

| |Carson needs 424.12 in³ of stuffing for the 6-inch diameter pillow |

| |Work for 12-inch diameter pillow (doubles diameter): |

| |V = r²h |

| |V = x6²x15 |

| |V = 1696.46 in³ |

| |Carson needs 1696.46 in³ of stuffing for the 12-inch diameter pillow |

| |Work for the 30-inch long pillow (doubles the length): |

| |V = r²h |

| |V = x3²x 30 |

| |V = 848.23 in³ 1696.46 / 848.23 = 2 |

| | |

| |Carson would only need ½ the amount of stuffing if he doubles the length compared to if he doubles the diameter OR |

| |The amount of stuffing needed for the larger pillow is more than the amount needed for the smaller pillow. |

|3 |The response provides clear evidence of finding the volume of a three dimensional cylindrical object and describing |

| |changes in the volume of a three-dimensional object when the measurements of the object are changed; however, the |

| |solution is incomplete, slightly flawed with a minor calculation error. |

| | |

| |For example, the response may: |

| |Finds all three volumes correctly but makes no comparisons. |

| |Makes a calculation error when finding volume (the same error may be carried out through all three volume |

| |calculations), and makes an accurate comparison based on those errors. |

| |Calculation errors include: |

| |Not properly squaring the radius |

| |Using the diameter instead of the radius but squares correctly. |

| |Using πdh or 2πrh to find volume. |

| |Doubling the diameter. |

|2 |The response provides partial evidence of finding the volume of a three-dimensional cylindrical object and |

| |describing changes in the volume of a three-dimensional object when the measurements of the object are changed; |

| |however, the solution is incomplete or slightly flawed. |

| | |

| |For example, the response may: |

| |Provides a correct strategy to find the number of cubic inches of stuffing Carson will need to make the original |

| |pillow, showing or explaining most calculations. |

| |Provide a correct comparison of the amount of stuffing needed when Carson doubles the length to the amount of |

| |stuffing needed when he doubles the diameter, based on flawed calculations. |

| |Provide a strategy to find the number of cubic inches of stuffing Carson will need to make the pillow AND a |

| |comparison of the amount of stuffing needed when Carson doubles the length to the amount of stuffing needed when he |

| |doubles the diameter; but, contains multiple calculation errors. |

| |State for 12-inch-diameter pillow (doubled the diameter) |

| |V = dh |

| |V = x12x15 |

| |V = 565.49 in³ |

| | |

| |For 30-inch-long pillow (doubles the length): |

| |V = dh |

| |V = x6x30 |

| |V = 565.49 in³ 565.49/ 565.49 = 1 |

| |Carson would need the same amount of stuffing if he doubles the length instead of doubling the diameter. |

|1 |The response provides minimal evidence of finding the volume of a three-dimensional cylindrical object and |

| |describing changes in the volume of a three-dimensional object when the measurements of the object are changed; |

| |however, the solution is incomplete or flawed. |

| | |

| |For example, the response may: |

| |Provide a partially correct strategy to find the number of cubic inches of stuffing Carson will need to make the |

| |pillow with multiple calculation errors and no comparison. |

| |Provide an inadequate comparison of the amount of stuffing needed when Carson doubles the length to the amount of |

| |stuffing needed when he doubles the diameter with multiple calculation errors. |

|0 |The response provides inadequate evidence of finding the volume of a three-dimensional cylindrical object and |

| |describing changes in the volume of a three-dimensional object when the measurements of the object are changed. The |

| |response provides irrelevant information or major flaws in reasoning. |

| | |

| |For example, the response may: |

| |State that doubling the length is definitely going to be much larger than doubling the diameter because the length |

| |is a bigger number to begin with. |

| |Be blank or make unrelated statements. |

| |Copy information from the stem. |

|MEA – Benchmark E |

|2007 OAT – Grade 7 – Problem # 31 Scoring Guidelines: |

|Points |Student Response |

|2 |The focus of this task is solving problems involving proportional relationships and scale factors. The response |

| |provides the scale factor for the two pools and work showing how it was found. It gives the length of Troy's pool, |

| |with supporting work. |

| | |

| |Exemplar Response: |

| |• The scale factor used to build Troy's pool is 1/4. I found it by comparing the width of Troy's pool (625 cm = 6.25|

| |m) to the width of the Olympic-size pool: 6.25 / 25 = 1/4. To find the length of Troy's pool, I divided the length |

| |of the Olympic-size pool by 4: 50 ÷ 4 = 12.5. Troy's pool is 12.5 meters long. |

| |• The width of Troy's pool compared to the width of the larger pool is 625:2500 or 1:4. To find the length of Troy's|

| |pool: 625/2500 = x/5000 , x = 1250 cm. The length of Troy's pool is 1250 centimeters. |

| |• 625 cm = 6.25 m. The scale factor of Troy's pool to the Olympic-size pool is 6.25 / 25 or 1/4. The length and |

| |width of Troy's pool are ¼ the length and width of the larger pool. ¼ of 50 is 12.5, so the length of Troy's pool is|

| |12.5 meters. |

| | |

| |NOTE: A scale factor of 4:1 or 4 is acceptable, provided the response shows or states the relationship correctly. |

|1 |The response shows partial evidence of solving problems involving proportional relationships and scale factors; |

| |however, the solution may be incomplete or slightly flawed. |

| | |

| |For example, the response may: |

| |• Provide only the length of Troy's pool, with supporting work but no scale factor. |

| |• Provide only the scale factor, with supporting work. |

| |• Provide the scale factor and length but show no work. |

| |• Find an incorrect scale factor but use it correctly to arrive at a length for Troy's pool. |

|0 |The response provides inadequate evidence of solving problems involving proportional relationships and scale |

| |factors. The response provides major flaws in explanations, or irrelevant information. |

| | |

| |For example, the response may: |

| |• State an incorrect length. |

| |• Restate the information provided in the item. |

| |• Be blank or give irrelevant information. |

|MEA – Benchmark B |

|2006 OAT – Grade 7 – Problem # 36 Scoring Guidelines: |

|Points |Student Response |

|2 |The focus of the task is to provide evidence of converting units within the same measurement system to figure out |

| |the number of 8-ounce servings. The response indicates the correct number of servings/cups AND provides work showing|

| |the correct conversion of units. 1 quart = 32 ounces and 1 gallon = 128 ounces, so Coach Larson has a total of 160 |

| |ounces of punch. At 8 ounces per serving/cup (160 oz ÷ 8 oz = 20), she can make 20 servings/cups. |

| |Sample correct responses: |

| |[pic] |

| |4 servings + 16 servings = 20 servings, there would be 20 servings in all. |

| |1 quart = 32 ounces = 4 cups. 1 gallon = 4 quarts. 4 x 4 = 16. 16 + 4 = 20. She can make 20 cups (servings). |

| | |

| |NOTE: Other conversion methods are acceptable (e.g., gallons to cups, quarts to cups). |

|1 |The response indicates partial evidence of how to convert units of measure; however, the solution is incomplete or |

| |slightly flawed. |

| | |

| |For example, the response may: |

| |• Make one correct conversion and one incorrect conversion. |

| |• Indicate that one quart is 32 ounces and one gallon is the same as 128 ounces, and state that Coach Larson has 160|

| |ounces of punch. |

| |• State the correct answer without providing work (e.g., 20 cups can be made with this recipe). |

| |• Use an inaccurate conversion but show an appropriate process. |

|0 |The response indicates inadequate evidence of how to convert units of measure. The response may have major flaws or |

| |be completely incorrect. |

| | |

| |For example, the response may: |

| |• State that a gallon is larger than a quart. |

| |• Copy information that is stated in the stem. |

|MEA – Benchmark E |

|2005 OAT – Grade 7 – Problem # 6 Scoring Guidelines: |

|Points |Student Response |

|2 |Sample Correct Responses: |

| |• ¼ Inch = 1 foot, therefore 1 inch = 4 feet, so 4 × 2 = 8, and 4 × 4 = 16. Dimension of kitchen is 8 feet by 16 |

| |feet. |

| |• ¼ inch = 1 foot; ½ inch = 2 feet, ¾ inch = 3 feet, 1 inch = 4 feet. So length = 4 × 4 = 16 feet and width = 2 × 4 |

| |= 8 feet |

| |• ¼ inch = 1 foot; ¼ inch = 12 inches; 1 inch = 48 inches; length = 48 × 4 = 192 inches; width = 48 × 2 = 96 inches.|

| | |

| |• (1/4) /1 = 2/x so x = 8 feet = width AND (1/4) /1 = 4/y so y = 16 feet = length |

| | |

| |The focus of the task is solving problems involving scale factors. The response includes the correct dimensions of |

| |the kitchen AND includes work or an explanation that shows how to find the length and width. Answers are labeled |

| |with the appropriate units. |

|1 |The response provides partial evidence of solving problems involving scale factors; however, the solution is |

| |incomplete or slightly flawed. |

| |For example, the response may: |

| |• Provide a calculation error with correct set up. |

| |• Provide an incorrect set up, but correct follow through. |

| |• Provide the correct answer with insufficient or no supporting work or explanation. |

|0 |The response provides inadequate evidence of solving problems involving scale factors. The response will provide |

| |major flaws in reasoning or irrelevant information. |

|MEA – Benchmark G |

|2005 OAT – Grade 7 – Problem # 26 Scoring Guidelines: |

|Points |Student Response |

|2 |Sample Correct Responses: |

| |The box is a rectangular prism with length 4, width 6 and height 9. Volume 4×6×9=216 cubic in. Surface area: 2(4)(9)|

| |+ 2(4)(6) + 2(9)(6)=228 sq in. This is less than the surface area of Box A, which is: 2(3)(9) + 2(3)(8) + 2(9)(8) = |

| |246 sq in. |

| |The box is a cube with height, length and width all equal to 6 inches. The volume of this cube will be 216 cubic |

| |inches (6×6×6) and the surface area is 2(6×6)+2(6×6)+2(6×6)=216 square inches. This is less than the surface area of|

| |Box A, which is: 2(3)(9) + 2(3)(8) + 2(9)(8) = 246 sq in. |

| | |

| |The focus of the task is to understand the difference between surface area and volume and that objects may have the |

| |same volume but different surface areas. The response includes an appropriate sketch or description of a different |

| |box that has the same volume, 216 cubic inches, but a smaller surface area than Box A. The response shows work or |

| |provides an adequate explanation to verify that the box has the same volume but a smaller surface area. A sketch and|

| |dimensions may or may not be provided. |

|1 |The response provides partial evidence of understanding the difference between surface area and volume and that |

| |objects may have the same volume but different surface areas; however, the solution may be incomplete or slightly |

| |flawed. |

| |For example, the response may: |

| |• Include an appropriate drawing or description of a different box with dimensions that result in the same volume |

| |with greater surface area or without stating surface area. |

| |• Provide an adequate explanation but minor calculation errors contribute to a volume that is not equal to 216 or a |

| |surface area that is not less than 246. |

|0 |The response provides inadequate evidence of understanding the difference between surface area and volume and that |

| |objects may have the same volume but different surface areas. The response provides major flaws in reasoning or |

| |irrelevant information. |

| |For example, the response may: |

| |• Draw a box with length 200, height 8 and width 8. 200 + 8 + 8 = 216. |

| |• Draw the box from the problem. |

| |• Draw a figure that is not three-dimensional. |

| |• Be blank or make unrelated statements. |

| |• Recopy information from the stem. |

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Measurement Standard

Measurement Standard

♦

Answer Key

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