OHIO GRADUATION TESTS

MathematicsScoring Guidelines and Samples of

Scored Student Responses

Spring 2007

© 2007 by Ohio Department of Education

Table of Contents Item 4: Item and Scoring Guidelines ............................................................................................1 Item 4: Samples of Scored Student Responses ........................................................................4

Item 10: Item and Scoring Guidelines....................................................................................... 14 Item 10: Samples of Scored Student Responses................................................................... 18

Item 22: Item and Scoring Guidelines....................................................................................... 28 Item 22: Samples of Scored Student Responses................................................................... 32

Item 28: Item and Scoring Guidelines....................................................................................... 47 Item 28: Samples of Scored Student Responses................................................................... 50

Item 34: Item and Scoring Guidelines....................................................................................... 60 Item 34: Samples of Scored Student Responses................................................................... 62

Item 40: Item and Scoring Guidelines....................................................................................... 72 Item 40: Samples of Scored Student Responses................................................................... 75

1

Mathematics

Item 4

Spring 2007

Item and Scoring Guidelines

2

Item 4. Helga earns $6.30 per hour working part-t ime at the grocery

store. She records her start ing and ending t imes each day on her t ime card.

Helga’s Time Card Date In Out

Monday 7/25 10:30 a.m. 2:00 p.m. Tuesday 7/26 12:45 p.m. 3:30 p.m.

Wednesday 7/27 Thursday 7/28 12:00 p.m. 4:45 p.m.

Friday 7/29 11:15 a.m. 3:45 p.m.

In your Answer Document , determine Helga’s total wages for the week. Show your work or provide an explanation for your answer.

Sample Response for Item 4 (Short Answer):

Mon 3.50 Tues 2.75 Thurs 4.75 Fri 4.50 15.50 hours × 6.30 97.65 OR

1

32

32

43

441

42

10 2 1

13 15 154 4 2

= = = hours 1

15 6.30 97.652

× =

Time card – 86

3

Scoring Guidelines for Item 4: Score Point Description 2 points The focus of the item is to determine the hours worked from a time

card and calculate the weekly wages. The response determines Helga worked 15.5 hours and earned total wages of $97.65.

1 point The response provides evidence of a partially correct answer

and/or solution process. The response shows understanding of some key elements of the task but contains gaps or flaws.

For example, the response may:

• Identify the correct number of total hours worked, but fails to find the total wages.

• Determine the total wages based on an incorrect number of hours (e.g., ignoring the fractional parts of hours, rounding off hours, or miscalculating.) However, this number is multiplied by $6.30 to obtain the total wages based on the incorrect number of hours.

• Contain a detailed explanation of the steps required, but has incorrect or missing totals.

• Only state Helga’s total wages are $97.65. 0 points The response indicates inadequate understanding of the task and

the response does not meet the criteria required to earn one point.

For example, the response may: • Include unrelated statements or work.

14

Mathematics

Item 10

Spring 2007

Item and Scoring Guidelines

15

Item

10. The diagram below shows two of the faces of a prism.

El len is constructing this tr iangular prism out of cardboard. El len plans to draw a shape (a net of the prism) on a piece of cardboard that she can cut out and fold to make the prism.

In your Answer Document , sketch a net of the prism that El len could use. Place the dimensions on the net.

For question 10, respond completely in your Answer Document. (2 points)

Sample Response for Item 10 (Short Answer):

[The triangles may be attached to the net at any side. The length of the rectangular portion may be marked as 9 cm.] OR

16

Nets – 1020 Scoring Guidelines for Item 10: Score Point Description 2 points The focus of the item is to sketch a net of a triangular prism with

given dimensions. The response shows a sketch of a net with dimensions labeled appropriately.

1 point The response provides evidence of a partially correct answer

and/or solution process. The response shows understanding of some key elements of the task but contains gaps or flaws.

For example, the response may:

� Contains a correct net but one or more of the dimensions is labeled incorrectly or missing.

� Contains a slightly flawed net (e.g., the student may omit the second triangle, or place the rectangles in an order that does not correspond to the order of the sides of the triangle.)

17

0 points The response indicates inadequate understanding of the task and

the response does not meet the criteria required to earn one point.

For example, the response may:

• Include unrelated statements or work.

28

Mathematics

Item 22

Spring 2007

Item and Scoring Guidelines

29

Item

22. The Ohio Big Tree Program l ists the largest l iving specimen of each kind of tree found in the state. Three measurements are needed to assign a score to each tree:

Circumference (1 inch = 1 point)

Height (1 foot = 1 point)

Average crown spread (1 foot = 14

point)

The score is found by adding the number of points for each measurement.

Four trees and their measurements are given below.

Selected Ohio Big Trees

Big Tree Circumference

(in.) Height

(ft)

Average Crown

Spread (ft) American Basswood

175.0 69.0 68.0

American Beech

222.0 130.0 75.0

Black Oak 249.6 55.0 94.0

Eastern Hemlock 140.0 138.0 52.2

In your Answer Document , calculate the scores for each of the four trees in the table. Arrange the trees in order based on their scores. Show work or provide an explanation to support your answer.

Write a formula for computing the score assigned by the Ohio Big Tree Program.

For question 22, respond completely in your Answer Document. (4 points) Sample Response for Item 22 (Extended Response):

1. (smallest) Basswood Score = 175 + 69 + 684

= 261

30

2. Hemlock Score = 140 + 138 + 52.2

4 = 291.05

3. Oak Score = 249.6 + 55 + 944

= 328.1

4. (largest) Beech Score = 222 + 130 + 754

= 370.75

Score = c + h + s/4 or c + h + ¼ s Scoring Guidelines for Item 22: Score Point Description 4 points The focus of the item is to calculate scores based on tree

measurements, order these scores from largest to smallest (or smallest to largest) and determine a general formula to calculate these scores. The response determines and orders the scores, 1) American Basswood – 261; 2) Eastern Hemlock – 291.05; 3) Black Oak – 328.1; and 4) American Beech 370.75 (order may be reversed). The response includes the general formula, Score = c + h

+ 4s

, to determine the score for any tree in the Big Ohio Tree

Program. 3 points The response clearly addresses the key aspects of the task;

however, it includes minor flaws. For example, the response may:

• Contain an error in one or two of the scores for the four trees, the ranking is correct based on the scores provided and the formula is provided.

• Contain the correct scores for each of the four trees listed in the chart, the ranking is incorrect or missing. The formula is provided.

• Contain the correct scores for each of the four trees listed in the chart, the ranking is correct. The formula is flawed or missing.

• Omit the scores for the four trees; however, ranks the four trees correctly and provides a correct formula.

2 points The response provides evidence of a partially correct answer

and/or solution process. The response may adequately address some of the components of the task but contain gaps or flaws in other components.

For example, the response may:

31

• Contain the correct numerical scores for at least two of the four trees listed in the chart, and the ranking is correct based on the four scores provided. The formula is flawed or missing.

• Contain the correct scores for each of the four trees. The ranking and formula are incorrect or missing.

• Contain the correct formula and at least two correct scores, the ranking is incorrect or missing.

• Contain an incorrect formula but the scores and ranking are correct based on the formula provided.

1 point The response omits significant aspects of the task. There is evidence

of minimal understanding of the concepts involved in the task and/or solution process; however, the response includes significant errors in most of the components of the task.

For example, the response may:

• Provide only the formula. The scores for the trees are missing

and no ranking is provided. • Contain the correct numerical score for at least two of the four

trees. The ranking and formula are incorrect or missing.

• Ignore the 4s

in calculations and formula; however, the ranking

is correct for the values given. • Contain a correct ranking with missing scores and the formula is

missing. 0 points The response indicates inadequate understanding of the task and

the response does not meet the criteria required to earn one point.

For example, the response may:

• Contain the values added without considering the 4s

, and no

ranking or formula is provided. • Contain an incorrect ranking with only incorrect scores or

missing scores. • Include unrelated statements or work.

47

Mathematics

Item 28

Spring 2007

Item and Scoring Guidelines

48

Item 28. Light travels at a speed of 1.86 × 105 miles per second.

A l ight-year is defined to be the distance that l ight travels in one year.

In your Answer Document , determine the number of miles in one l ight-year. Write your answer in correct scientif ic notation. Show all of your work or provide an explanation for your answer.

For question 28, respond completely in your Answer Document. (2 points)

Sample Response for Item 28 (Short Answer): There are 365 × 24 × 60 × 60 = 31,536,000 seconds in one year. There are (1.86 × 105) × 31,536,000 = 5.865696 × 1012 miles in one light-year. Or (1.86 × 105) × (3.15 × 107) = 5.86 × 1012

Students may also use 3651

4 or 7 × 52 for the number of days in the year resulting

in values of 5.8697136 × 1012 or 5.8496256 × 1012. 360 days per year, as used in business mathematics, is also acceptable. Rounded solutions are also acceptable 5.86 × 1012 or 6.0 × 1012. Scientific notation – 892 Scoring Guidelines for Item 28: Score Point Description 2 points The focus of the item is to determine the number of miles in one

light-year and write this number in scientific notation. The response contains the value of ≈5.86 x 1012 with supporting work.

1 point The response provides evidence of a partially correct answer

and/or solution process. The response shows understanding of some key elements of the task but contains gaps or flaws.

For example, the response may:

49

• Contain the correct value written in scientific notation with incomplete or missing work.

• Contain the correct value, but it is not written in correct scientific notation. Supporting work is incomplete or missing.

• Provide a correct process for determining the number of miles in one light-year but contains computation errors.

• Provide evidence of a partially correct process; (e.g., the conversion process includes the number of minutes in a year, but not the number of seconds.) The value found is written in correct scientific notation.

0 points The response indicates inadequate understanding of the task and

the response does not meet the criteria required to earn one point.

For example, the response may: • Include unrelated statements or work.

60

Mathematics

Item 34

Spring 2007

Item and Scoring Guidelines

61

Item

34. The local newspaper publishes a l ist of the f ive most popular radio stations. Lisa’s favorite station is not on the l ist . At school, she asks two of her fr iends what their favorite station is, and they both choose Lisa’s favorite station. Based on this, Lisa decides her favorite station should have been on the l ist .

In your Answer Document , explain why Lisa’s conclusion is not valid by giving two reasons why her sample is biased.

For question 34, respond completely in your Answer Document. (2 points)

Sample Response for Item 34 (Short Answer): Possible reasons: Lisa’s sample is too small—only two people. Lisa only asked her friends—they probably like the same thing she does. She only asked girls—boys may choose something else. She only asked people her own age—different age groups may have different tastes. Scoring Guidelines for Item 34: Score Point Description 2 points The focus of this item is the identification of sources of bias in the

sample used for a survey. The response provides two distinct reasons why Lisa’s sample may be biased.

1 point The response provides evidence of a partially correct answer

and/or solution process. The response shows understanding of some key elements of the task but contains gaps or flaws.

For example, the response may:

• Provide one valid reason why the sample is biased. 0 points The response indicates inadequate understanding of the task, and

the response does not meet the criteria required to earn one point.

For example, the response may: • Include unrelated statements or work.

72

Mathematics

Item 38

Spring 2007

Item and Scoring Guidelines

73

Item

38. Mr. Richards needs to price items that must be sold within 90 days, and he has two options.

In Option A, Mr. Richards prices i tems at $225 and gives $1.00 off the price for every day that the item does not sell .

For Option B, he prices i tems at $250 and gives $2.00 off for every day the i tem does not sell .

In your Answer Document , write an equation for each option that relates the price of the item to the number of days that the item has not sold.

Use the equations to f ind the day on which the two options yield the same price.

For question 38, respond completely in your Answer Document. (2 points)

Sample Response for Item 38 (Short Answer): Option A price after d days: p = 225 − d Option B price after d days: p = 250 − 2d 225 − d = 250 − 2d d = 25 Equal after day 25. Take discount at end of day OR Option A price: p = 225 − ( -1)d Option B price: p = 250 − −2( 1)d

225 – (d –1) = 250 – 2(d – 1)

225 – d + 1 = 250 – 2d + 2

d = 26

Equal on the 26th day of the sale with full price on day one.

74

Scoring Guidelines for Item 38: Score Point Description 2 points The focus of the item is to write a system of linear equations to

model a problem situation. The response includes two correct equations, and a solution. (Day 25 or 26 acceptable with the appropriate equations.)

1 point The response provides evidence of a partially correct answer

and/or solution process. The response shows understanding of some key elements of the task but contains gaps or flaws.

For example, the response may:

• Contain two correct equations but an incorrect solution. • Contain a correct solution with supporting work (e.g. table) with

flawed or missing equations. • Provide an incorrect solution based on slightly flawed equations.

0 points The response indicates inadequate understanding of the task, and

the response does not meet the criteria required to earn one point.

For example, the response may: • Contain one correct equation with no additional work. • Include unrelated statements or work.