K-12 Mathematics Benchmarks



Mathematics Benchmarks & Indicators

with Ohio Achievement Test Questions

Grades 5 – 6

Includes questions from the released

2008, 2007, and 2006 Ohio Achievement Tests

and the 2005 Practice Tests

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

Measurement

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.

5-1. Identify and select appropriate units to measure angles; i.e., degrees.

|Grade 5 – 2007 OAT – Problem # 13 |

| | |

|Han compared the angles on this diagram. |[pic] |

|[pic] | |

|Which angle appears to be greater than 90º? | |

|Grade 5 – 2006 OAT– Problem # 19 |

| | |

|Bob covered a floor with carpet. |A. inches |

| |B. feet |

|Which unit of measure describes how much carpet he used? |C. square feet |

| |D. cubic inches |

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

4-5. Make simple unit conversions within a measurement system; e.g., inches to feet, kilograms to grams, quarts to gallons. (Grade 4)

5-5. Make conversions within the same measurement system while performing computations.

|Grade 5 – 2007 OAT – Problem # 30 |

| | |

|Julia ran a 6-kilometer race on a 400-meter oval track. |A. 0.15 lap |

| |B. 1.5 laps |

|How many laps around the track did Julia run? |C. 15 laps |

| |D. 150 laps |

|Grade 5 – 2006 OAT– Problem # 31 |

| | |

|A pickup truck weighs 3 tons. |A. 600 pounds |

| |B. 2,000 pounds |

|How many pounds does the truck weigh? |C. 3,000 pounds |

| |D. 6,000 pounds |

|Grade 5 – 2006 OAT– Problem # 38 |

| |

|Peter’s goal is to read 5 hours every school week. He reads every evening during the school week and records his |

|time in the chart shown. |

|[pic] |

| |

|In your Answer Document, determine how much time Peter should read on |

|Friday to meet his goal. Show or explain how you found your answer. |

|(2 points) |

|Grade 5 – 2005 Practice Test – Problem # 8 |

| |

|After swim practice, each of the 30 swim team members gets 8 ounces of juice. The coach brought 2 gallons of juice |

|to practice. |

| |

|In your Answer Document, determine whether the coach brought enough juice for each team member to get 8 ounces. Show|

|work to support your answer. |

|Grade 5 – 2005 Practice Test – Problem # 12 |

| | |

|Carla hiked for 2 ½ hours. How many minutes did she hike? |A. 30 minutes |

| |B. 60 minutes |

| |C. 120 minutes |

| |D. 150 minutes |

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.

5-6. Use strategies to develop formulas for determining perimeter and area of triangles, rectangles and parallelograms, and volume of rectangular prisms.

5-7. Use benchmark angles (e.g., 45°, 90°, 120°) to estimate the measure of angles, and use a tool to measure and draw angles.

6-2. Use strategies to develop formulas for finding circumference and area of circles, and to determine the area of sectors; e.g., ½ circle, 2/3 circle, 1/3 circle, ¼ circle.

6-3. Estimate perimeter or circumference and area for circles, triangles and quadrilaterals, and surface area and volume for prisms and cylinders by:

a. estimating lengths using string or links, areas using titles or gird, and volumes using cubes;

b. measuring attributes (diameter, side lengths, or heights) and using established formulas for circles triangles, rectangles, parallelograms and rectangular prisms.

|Grade 5 – 2007 OAT – Problem # 8 |

| | |

|A parallelogram is shown on the grid. |A. 6 + 6 + 4 + 4 |

|[pic] |B. 8 + 8 + 4 + 4 |

| |C. 6 ( 4 |

|Which expression represents the area of this parallelogram? |D. 8 ( 4 |

|Grade 5 – 2006 OAT– Problem # 9 |

| | |

|Angle KJL is shown. |A. 60° |

|[pic] |B. 80° |

|Use your protractor to find the measure of angle KJL. |C. 140° |

| |D. 160° |

|Grade 5 – 2005 Practice Test – Problem # 3 |

| | |

|Carlos wants to know how many small cubes will fit in the box. |A. the area of the box |

|[pic] |B. the length of the box |

| |C. the surface area of the box |

|Which measurement of the box is Carlos finding when he fills it with |D. the volume of the box |

|cubes? | |

|Grade 5 – 2005 Practice Test – Problem # 20 |

| | |

|The dimensions of Mike’s garden are shown. |A. 3 + 5 |

|[pic] |B. 3 x 5 |

|Which expression shows how Mike could find the number of square meters |C. 3 + 5 + 3 + 5 |

|in his garden? |D. 3 x 5 x 3 x 5 |

|Grade 6 – 2007 OAT – Problem # 5 |

| | |

|A circle with a radius of 8 inches is shown. |A. about 5 inches |

| |B. about 16 inches |

|[pic] |C. about 24 inches |

| |D. about 48 inches |

|Which estimate of the circumference of this circle is reasonable? | |

|Grade 6 – 2006 OAT – Problem # 5 |

| | |

|Square floor tiles will be put on the floor of a |A. 95 tiles |

|school hallway. Each tile is 1 foot by 1 foot. |B. 100 tiles |

|The hallway is 85 feet long and 8 feet wide. |C. 200 tiles |

| |D. 700 tiles |

|About how many tiles will be needed to | |

|cover the floor of the hallway? | |

|Grade 6 – 2006 OAT – Problem # 40 |

| | |

|The triangular street sign shown has a base |A. about 60 square inches |

|30 inches long and a height of 26 inches. |B. about 250 square inches |

| |C. about 400 square inches |

|[pic] |D. about 750 square inches |

|Which estimate of the area of the street sign | |

|is reasonable? | |

|Grade 6 – 2005 Practice Test – Problem # 3 |

| | |

|3. Mary knows that the circumference of a |A. 80 square inches |

|circle is about three times its diameter. |B. 120 square inches |

|She made her mother a vase in the shape |C. 250 square inches |

|of a cylinder. |D. 500 square inches |

|[pic] | |

|About how much material does she need | |

|to cover the outside of the vase, not | |

|including the bottom? | |

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

|Grade 5 – 2006 OAT– Problem # 9 |

| | |

|Angle KJL is shown. |A. 60° |

|[pic] |B. 80° |

|Use your protractor to find the measure of angle KJL. |C. 140° |

| |D. 160° |

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

5-6. Write, solve and verify solutions to multi-step problems involving measurement. (Grade 4).

5-2. Identify paths between points on a gird or coordinate plane and compare the lengths of the paths; e.g., shortest path, paths of equal length.

6-4. Determine which measure (perimeter, area, surface area, volume) matches the context for a problem situation; e.g., perimeter is the context for fencing a garden, surface area is the context for painting a room.

|Grade 5 – 2007 OAT – Problem # 46 |

| | |

|A florist sells roses in bunches of 12. He sold four bunches and has 36 |A. 4 + 12 + 36 |

|roses left at the end of the day. |B. 4 ( 12 + 36 |

| |C. 4 + 12 – 36 |

|Which expression represents the number of roses he had at the beginning |D. 4 ( 12 – 36 |

|of the day? | |

|Grade 5 – 2006 OAT– Problem # 4 |

| | |

|Joe is putting a low fence around all four sides of a rectangular |A. 10 sections |

|flower bed. |B. 20 sections |

|The flower bed is 4 feet wide and 6 feet long. |C. 24 sections |

| |D. 40 sections |

|[pic] | |

| | |

|Each section of fencing is 2 feet long. How many sections of fencing | |

|will Joe need? | |

|Grade 5 – 2006 OAT– Problem # 34 |

| | |

|Point J and point K are shown on the grid. |A. 3 units right and |

|[pic] |2 units up |

| | |

|What is the direction from point J to point K along the grid lines? |B. 3 units right and |

| |3 units up |

| | |

| |C. 4 units right and |

| |3 units up |

| | |

| |D. 4 units right and |

| |2 units up |

|Grade 5 – 2006 OAT– Problem # 38 |

| |

|Peter’s goal is to read 5 hours every school week. He reads every evening during the school week and records his |

|time in the chart shown. |

|[pic] |

| |

|In your Answer Document, determine how much time Peter should read on |

|Friday to meet his goal. Show or explain how you found your answer. |

|(2 points) |

|Grade 6 – 2006 OAT – Problem # 2 |

| | |

|Joe wants to paint the outside of the toy |A. height |

|chest shown. |B. perimeter |

|[pic] |C. surface area |

| |D. volume |

|Which measure of the toy chest should he use | |

|to determine how much paint to buy? | |

|Grade 6 – 2005 Practice Test – Problem # 11 |

| | |

|Tim’s bicycle tire has a diameter of two |A. 10 feet |

|feet. Tim knows that the circumference of |B. 30 feet |

|a circle is about three times its diameter. |C. 50 feet |

|About how far would Tim’s bicycle travel |D. 60 feet |

|when the tire has made five revolutions? | |

| | |

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

5-3. Demonstrate and describe the differences between covering the faces (surface area) and filling the interior (volume) of three-dimensional objects.

5-4. Demonstrate understanding of the differences among linear units, square units, and cubic units.

6-1. Understand and describe the difference between surface area and volume.

6-6. Describe what happens to the perimeter and area of a two-dimensional shape when the measurements of the shape are changed; e.g., length of sides are doubled.

|Grade 6 – 2007 OAT – Problem # 43 |

| |

|A square and a rectangle are shown. |

| |

|[pic] |

| |

|Which statement is true about these two figures? |

| |

|A. They have the same area and the same perimeter. |

|B. They have different areas and the same perimeter. |

|C. They have the same area and different perimeters. |

|D. They have different areas and different perimeters. |

|Grade 6 – 2006 OAT – Problem # 31 |

| |

|Marla and her cousins invented a game. Marla used chalk to draw a rectangle that was 15 feet wide and 25 feet long |

|for a space to play her game. |

| |

|Several other children joined in, and Marla realized that the rectangle was now too small. She doubled the length |

|and doubled |

|the width to create a new play area. |

| |

|In your Answer Document, compare the perimeters of the original play space and the new play space. Determine how |

|much greater the new perimeter is than the original perimeter. Show or explain your work. |

| |

|Then, compare the areas of the original and the new play spaces. Determine how much greater the new area is than the|

|original area. Show or explain your work. |

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

4-8. Use geometric models to solve problems in other areas of mathematics, such as number (multiplication/division) and measurement (area, perimeter, border). (Grade 4).

5-3. Demonstrate and describe the differences between covering the faces (surface area) and filling the interior (volume) of three-dimensional objects.

5-4. Demonstrate understanding of the differences among linear units, square units and cubic units.

6-1. Understand and describe the difference between surface area and volume.

6-5. Understand the difference between perimeter and area, and demonstrate that two shapes may have the same perimeter, but different areas of may have the same area, but different perimeters.

|Grade 5 – 2006 OAT– Problem # 25 |

| |

|Justin keeps his toys in a box like the one shown. |

|[pic] |

|In your Answer Document, explain the difference between the volume and the surface area of the box. (2 points) |

|Grade 6 – 2007 OAT – Problem # 34 |

| | |

|A can of tomato soup and its label are shown. |A. The label represents surface area and |

|[pic] |the perimeter is 300 mL. |

| |B. The label represents surface area and |

|Which statement is true? |the volume is 300 mL. |

| |C. The label represents volume and the |

| |surface area is 300 mL. |

| |D. The label represents volume and the |

| |perimeter is 300 mL. |

|Grade 6 – 2006 OAT – Problem # 42 |

| | |

|The square and the rectangle have the |A. same area, same perimeter |

|dimensions shown. |B. same area, different perimeter |

|[pic] |C. different area, same perimeter |

| |D. different area, different perimeter |

|What description is true about the figures? | |

|Grade 6 – 2005 Practice Test – Problem # 17 |

| |

|In your Answer Document, draw two rectangles that have the same area but different perimeters. Label the dimensions.|

|Give the area and the perimeter of both rectangles. |

Mathematical Processes

A. Clarify problem-solving situation and identify potential solution processes; e.g., consider different strategies and approaches to a problem, restate problem from various perspectives.

B. Apply and adapt problem-solving strategies to solve a variety of problems, including unfamiliar and non-routine problem situations.

C. Use more than one strategy to solve a problem, and recognize there are advantages associated with various methods.

D. Recognize whether an estimate or an exact solution is appropriate for a given problem situation.

E. Use deductive thinking to construct informal arguments to support reasoning and to justify solutions to problems.

F. Use inductive thinking to generalize a pattern of observations for particular cases, make conjectures, and provide supporting arguments for conjectures

G. Relate mathematical ideas to one another and to other content areas; e.g., use area models for adding fractions, interpret graphs in reading, science and social studies.

H . Use representations to organize and communicate mathematical thinking and problem solutions.

I. Select, apply, and translate among mathematical representations to solve problems; e.g., representing a number as a fraction, decimal or percent as appropriate for a problem.

J. Communicate mathematical thinking to others and analyze the mathematical thinking and strategies of others.

K. Recognize and use mathematical language and symbols when reading, writing and conversing with others.

Multiple Choice:

|OAT – Grades 5 & 6 |

|Measurement |

| |Grade |Test |No. |Answer |

| | |Year | | |

|A |5 |2007 |13 |D |

| |5 |2006 |19 |C |

|B |5 |2007 |30 |C |

| |5 |2006 |31 |D |

| |5 |2006 |38 |S.A. |

| |5 |2005* |8 |** |

| |5 |2005* |12 |D |

|C |5 |2007 |8 |D |

| |5 |2006 |9 |C |

| |5 |2005* |3 |D |

| |5 |2005* |20 |B |

| |6 |2007 |5 |D |

| |6 |2006 |5 |D |

| |6 |2006 |40 |C |

| |6 |2005* |3 |C |

|D |5 |2006 |9 |C |

|E |5 |2007 |46 |B |

| |5 |2006 |4 |A |

| |5 |2006 |34 |A |

| |5 |2006 |38 |S.A. |

| |6 |2006 |2 |C |

| |6 |2005* |11 |B |

|F |6 |2007 |43 |B |

| |6 |2006 |31 |E.R. |

|G |5 |2006 |25 |S.A. |

| |6 |2007 |34 |B |

| |6 |2006 |42 |C |

| |6 |2005* |17 |** |

* Half-Length Practice Test

** Scoring Rubric Not Released

Short Answer & Extended Response Rubrics:

Grade 5

|MEA – Benchmark G |

|2006 OAT – Grade 5 – Problem # 25 Scoring Guidelines: |

|Points |Student Response |

|2 |The focus of this task is describing the difference between surface area and volume for a three-dimensional object. |

| |The response provides an adequate explanation of the difference between surface area and volume, using appropriate |

| |examples. |

| |Sample response: |

| |• The surface area is the area of the faces (top, bottom and sides) of the toy box and the volume is the amount of |

| |space inside the toy box. |

| |• The surface area is measured with square units but the volume is measured in cubic units. |

| |• The surface area is how much material is used to cover the toy box and the volume is how many toys Justin can fit |

| |into the box. |

|1 |The response provides partial evidence of describing the difference between surface area and volume for a figure; |

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

| |For example, the response may: |

| |• Provide an adequate explanation of surface area as it pertains to the toy box, but the explanation of volume is |

| |incorrect or missing. |

| |• Provide an adequate explanation of volume as it pertains to the toy box, but the explanation of surface area is |

| |incorrect or missing. |

| |• State that the surface area is outside of the box and/or the volume is inside of the box. |

|0 |The response provides inadequate evidence of describing the difference between surface area and volume for a figure.|

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

| |For example, the response may: |

| |• State that volume and surface area are the same. |

| |• Be blank or provide unrelated statements. |

| |• Recopy information from the stem. |

|MEA – Benchmark B & E |

|2006 OAT – Grade 5 – Problem # 38 Scoring Guidelines: |

|Points |Student Response |

|2 |The focus of this task is converting units of time within the same measurement system. The response correctly |

| |determines the time Peter should read on Friday with supporting work or an adequate explanation. Sample response: |

| |• 30 minutes + 1 hour 15 minutes + 1 hour 5 minutes + 40 minutes = 2 hours 90 minutes; 60 minutes = 1 hour so 2 |

| |hours 90 minutes = 3 hours 30 minutes. 5 hours – 3 hours 30 minutes = 1 hour 30 minutes. |

| |• 60 minutes = 1 hour, so 30 minutes + 75 minutes + 65 minutes + 40 minutes = 210 minutes. 5 hours = 300 minutes, so|

| |300 minutes – 210 minutes = 90 minutes. Peter must read for 90 minutes. |

| |• 3 hours 30 minutes = 3.5 hours so 5 hours – 3.5 hours = 1.5 hours. |

|1 |The response provides partial evidence of converting units of time within the same measurement system; however, the |

| |solution may be incomplete or slightly flawed. |

| |For example, the response may: |

| |• Show a minor computational error when converting, adding or subtracting. |

| |• Find only the total time Peter has already read. |

| |• Provide the correct answer without an explanation or work. |

|0 |The response provides inadequate evidence of converting units of time within the same measurement system. The |

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

| |For example, the response may: |

| |• State 100 minutes. |

| |• Be blank or provide unrelated statements. |

| |• Recopy information from the stem. |

Short Answer & Extended Response Rubrics:

Grade 6

|MEA – Benchmark F |

|2006 OAT – Grade 6 – Problem # 31 Scoring Guidelines: |

|Points |Student Response |

|4 |The focus of this task is finding the perimeter and area of a rectangular shape and describing what happens to the |

| |perimeter and area when the measurements of a shape are changed. The response provides a correct calculation of how |

| |much longer the new perimeter is, showing or explaining work, AND a correct calculation of how much larger the new |

| |area is, showing or explaining work. |

| | |

| |Sample Correct Response: |

| |New Length – 2 × 25 = 50 feet. |

| |New Width – 2 × 15 = 30 feet |

| |Original chalk line: 25 + 25 + 15 + 15 = 80 feet |

| |New Chalk line: 50 + 50 + 30 + 30 = 160 feet 160 – 80 = 80. |

| |The new chalk line is 80 feet longer or double than the original line. |

| |Original Area: 25 × 15 = 375 square feet |

| |New Area: 50 × 30 = 1500 square feet 1500 – 375 = 1125. |

| |The new area is 1125 square feet or 4 times larger than the original area. |

|3 |The response provides adequate evidence of finding the perimeter and area and describing what happens to the |

| |perimeter and area when the measurements of a shape are changed. The response provides work with a minor error or |

| |flaw in a calculation or explanation. |

| | |

| |For example, the response may: |

| |• Provide work with a calculation error in the area or perimeter, but base the comparison on that error. |

| |• Provide correct work for the areas and perimeters, but state a comparison for the area or perimeter. |

|2 |The response provides partial evidence of finding the perimeter and area of a rectangular shape and describing what |

| |happens to the perimeter and area when the measurements of a shape are changed. The response provides work with minor|

| |errors or flaws OR vague explanation. |

| | |

| |For example, the response may: |

| |• Show work with calculation errors in the area and perimeter, and fail to state a conclusion. |

| |• Provide the correct answers to how much larger the new perimeter is and how much larger the new area is; but, lacks|

| |an explanation or work. |

| |• Provide correct work for the perimeters or the areas and comparison for the perimeter or the area. |

| |• State, the new perimeter is 80 feet longer than the original one and the new area is 1,125 square feet larger than |

| |the original area. |

|1 |The response provides minimal evidence of finding the perimeter and area of a rectangular shape and describing what |

| |happens to the perimeter and area when the measurements of a shape are changed. The response provides work with |

| |multiple errors or flaws and a vague explanation. |

| | |

| |For example, the response may: |

| |• Provide the perimeter of the original or the new play space. |

| |• Provide the area of the original or the new play space. |

| |• Provide a partially accurate comparison of perimeter and area as it relates to the original and new play areas. |

| |• State only the area and perimeter of the original play space. |

|0 |The response provides inadequate evidence of finding the perimeter and area of a rectangular shape and describing |

| |what happens to the perimeter and area when the measurements of a shape are changed. The response provides major |

| |flaws in explanation or irrelevant information. |

| | |

| |For example, the response may: |

| |• State the area and perimeter is doubled. |

| |• Be blank or state unrelated statements. |

| |• Recopy information from the stem. |

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Measurement Standard

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