K-12 Mathematics Benchmarks · Web viewWrite and use equivalent forms of equations and inequalities in problem situations; e.g., changing a linear equation to the slope-intercept

FERPCBM

Mathematics Benchmarks & Indicatorswith Ohio Graduation Test Questions

Grades 9 – 10

Includes questions from the released 2008, 2007, 2006, 2005, 2004, and 2003

Ohio Graduation Tests

Far East Regional Partnershipfor Conceptually Based Mathematics

Youngstown State University

Compiled by A. Crabtree, 2006Revised by A. Crabtree and L. Holovatick, 2007Revised by A. Crabtree, J. Lucas, and T. Cameron, 2008

Patterns, Functions and Algebra Standard

A. Generalize and explain patterns and sequences in order to find the next term and the nth term.

9.2. Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations.

OGT 2008 – Problem # 1

The owner of a sporting goods store is following a pattern to arrange baseballs into 7 rows for a wall display. The table shows the number of baseballs in the first four rows of this pattern.

How many baseballs are in the seventh row of this pattern?

A. 17B. 15C. 10D. 4


The first four rows of a number array are shown below.

What number will be at the far right end of row 7?

A. 150

B. 210

C. 280

D. 360


Nancy decided to breed gerbils to sell to a pet store. She started with 2 gerbils and determined that they should quadruple in number every 4 months.

Nancy sold all of the gerbils to the pet store at the end of 1 year.

How many gerbils did Nancy sell to the pet store?

A. 24 gerbils

B. 66 gerbils

C. 128 gerbils

D. 512 gerbils


The first five rows of a number array are shown below.

Row 1 1Row 2 2 3Row 3 4 5 6Row 4 7 8 9 10Row 5 11 12 13 14 15

What is the sum of the numbers in row 8?

A. 175

B. 224

C. 231

D. 260


The depth of a lake is measured at the samelocation and on the same day every year for anumber of years. The table below shows themeasurements.

If the pattern continued, what was the depth ofthe lake in 2002?

A. 30 feetB. 29 feetC. 28 feetD. 25 feet

OGT 2004 – Problem # 1 (OGT Practice Test 2004 – Problem # 1)

The table below contains the results of abiology experiment.

Assuming the pattern shown in the tablecontinues, what is the value of b?

A. 108B. 130C. 162D. 243


Maria is making a quilt. She has a large pieceof fabric that is 0.02 millimeters thick. Thefabric is cut in half and one piece is placed ontop of the other to make a pile. The pile is cutin half, and then one half is placed on top ofthe other to make a higher pile.

Continuing this process, what would thethickness of the pile be after the 4th cut andpiling?

A. 0.0016 millimetersB. 0.08 millimetersC. 0.32 millimetersD. 16 millimeters

OGT 2003 – Problem #29

The first 5 terms of a sequence are given in thetable.

For this sequence, which of these representsthe relationship between n, the number of theterm, and t , its corresponding value?

A. t = n2 + 1B. t = 2n + 1C. t = 3n – 1D. t = 2n2 – 1

Patterns, Functions and Algebra OGT Problems

B. Identify and classify functions as linear or nonlinear, and contrast their properties using tables, graphs or equations.

9.1. Define function with ordered pairs in which each domain element is assigned exactly one range element.

9.3. Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations.

10.1. Define function formally and with f(x) notation.

10.2. Describe and compare characteristics of the following families of functions; e.g., general shape, possible number of roots, domain and range.


Which graph represents a linear function?

Prepared by Aimee Crabtree for FERPCBM, Youngstown, OhioSummer 2006 (Rev. 2008)Benchmarks with OGT Problems Page 7 of 45



Which graph represents a linear function?




The maximum heart rate is the highest number of beats per minute recommended for a person while exercising. The rate is dependent upon the age of the person as shown below. The relationship is linear.

In your Answer Document, copy and complete the table.

Write an equation that can be used to find the maximum heart rate for any age. Show your work or provide an explanation for how you determined your equation.




Sheila collects the following statistics about baseball games played by the local Youth League team:

Which statistic shows a non-linear rate of increase over time?

A. number of winsB. average number of spectators per gameC. average number of sodas sold per gameD. average number of hot dogs sold per game

C. Translate information from one representation (words, table, graph or equation) to another representation of a relation or function.

9.2. Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations.


Michael paid $6.00 for a ticket to a football game. Soft drinks at the game cost $0.75. Michael bought x drinks at the game.

Which equation represents the total amount (y) he spent?

A. y = (6 + 0.75)xB. y = 6x + 0.75C. y = 6 – 0.75xD. y = 6 + 0.75x




The table shows values for a function.

Which equation represents this function?

A. y = x2

B. y = x2 + 1C. y = (x - 1)2

D. y = (x + 1)2




Jackie has a job selling cars. Her monthly salary is $2,000 plus a commission of 1% of her total sales up to $100,000 and 1.5% of additional sales above $100,000.

Which graph represents this relationship?




The maximum heart rate is the highest number of beats per minute recommended for a person while exercising. The rate is dependent upon the age of the person as shown below. The relationship is linear.

In your Answer Document, copy and complete the table.

Write an equation that can be used to find the maximum heart rate for any age. Show your work or provide an explanation for how you determined your equation.




Which graph represents the equation y = - x 2 + 3?


The table below shows values for x and y.

Which of these equations represents therelationship between x and y?




Which of these represents the graph of the equation


To solve a math problem, Penny is graphing the equations y = x2 and y = x2 + 1. To graph the equations, she created the tables shown below.

In your Answer Document, copy the tables above and find the y -values for each of the given x -values.

Use the grid provided to graph each equation using the pairs of x - and y -values.

Based on the graphs you have completed, analyze how the graphs differ and write a hypothesis to describe how adding a number to x2 changes the graph of x2.



D. Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.

9.7. Use formulas to solve problems involving exponential growth and decay.

9.11. Add, subtract, multiply and divide monomials and polynomials (division of polynomials by monomials only).

9.12. Simplify rational expressions by eliminating common factors and applying properties of integer exponents.

10.3. Solve equations and formulas for a specified variable; e.g., express the base of a triangle in terms of the area and height.

10.4. Use algebraic representations and functions to describe and generalize geometric properties and relationships.

10.5. Solve simple linear and nonlinear equations and inequalities having square roots as coefficients and solutions.

10.6. Solve equations and inequalities having rational expressions as coefficients and solutions.


A band pays for the use of a location for a concert. The band charges $25 per ticket. If n represents the number of tickets sold and c represents the cost of the location, which inequality below describes how many tickets need to be sold to make a profit?

A. 25c < nB. 25c > nC. 25n < cD. 25n > c


Mrs. Foyle told Yolanda that her test had 38 problems worth a total of 100 points. Each test problem is worth either 5 points or 2 points. Yolanda wanted to determine how many 2-point and how many 5-point questions are on the test.

In your Answer Document, determine how many questions of each point-value are on the test. Show your work or provide an explanation to support your answer.




A local art museum charges admission to groups according to the following rates.

Groups of fewer than 50 people are charged a rate of $4.00/person. Groups of 50 people or more are charged a reduced rate of $3.50/person.

How much money will a group of 49 people save in admission costs if it can recruit one additional member?

A. $ 0.50

B. $ 21.00

C. $ 175.00

D. $ 196.00


For every lawn that she mows, Jane charges$8 per hour for every hour that she works. Foreach lawn that he mows, Bob charges a fixed feeof $20 and an additional $5 for every hour thathe works.

What is the fewest number of hours that bothcould work so that Jane’s total pay for a lawn willbe greater than Bob’s?

A. 1 hourB. 5 hoursC. 6 hoursD. 7 hours

OGT 2004 – Problem # 12Julie does not want to spend more than $300 on ice skating. Her skates will cost $42, her lessons will cost a total of $56, and the practice time will cost $7.50 per hour.

Which inequality should Julie use to determine the maximum number of hours, h, she can practice without spending more than $300?




At the beginning of the summer, Katie openeda new checking account with a $60 deposit.Each week during the summer, she earned $95as a part-time lifeguard. Katie deposited 10%of these earnings into her account. Which ofthese equations represents the amount ofmoney, m, Katie will have deposited in herchecking account after w weeks?

A. m = 15.5wB. m = w + 69.5C. m = 95w + 60D. m = 9.5w + 60


To raise money, a school club is operating adunking booth at the local fair. The club willpay $120 to rent the booth. Each customer willpay $1.50 to throw a ball at the dunkingplatform on which a person sits. Whichinequality could be used to find the number ofcustomers, c, required for the club to make anet profit of at least $350?

A. 1.5c – 120 350B. 1.5c + 120 350C. 1.5c – 120 350D. 1.5c + 120 350



E. Analyze and compare functions and their graphs using attributes, such as rates of change, intercepts and zeros.

9.4. Demonstrate the relationship among zeros of a function, roots of equations, and solutions of equations graphically and in words.

9.5. Describe and compare characteristics of the following families of functions: linear, quadratic and exponential functions; e.g., general shape, number of roots, domain, range, rate of change, maximum or minimum.


The graph of the function f(x) is shown below.

Which of the following is NOT a zero of f(x)?

A. –4B. –3C. 2D. 6




Jamie is buying peanuts and walnuts. He wants to spend exactly $30. The graph below shows all possible ways that he can split the $30 between the peanuts and the walnuts.

What does the point N on the graph represent?

A. spending all $30 on walnuts

B. spending all $30 on peanuts

C. spending $20 on peanuts and $10 on walnuts

D. spending half of the $30 on peanuts and half of the $30 on walnuts


The graph, as shown, represents the amount of money Sarah can earn at her part-time job.

Which of the following equations best represents the relationship between Sarah’s pay and the hours she works?

A. y = 4x

B. y = 6.5x

C. y = 4x + 10

D. y = 6.5x + 10




Jessica participated in a walk-a-thon to raise money for a local charity. She began walking at a rate of 3 miles per hour. The graph represents her distance walked as a function of time.

If Jessica had walked at an average rate of 2 miles per hour, which of the following accurately illustrates how this graph would appear, using the same scale?




Ted and Bob each must type a 1,500-word research paper. The graph below represents their normal typing rates.

Based on the information in the graph, which of these is a valid conclusion?

A. Bob can type his research paper in half the time it takes Ted to type his paper.

B. Ted can type his research paper in half the time it takes Bob to type his paper.

C. Ted will take 4 minutes longer than Bob to type his research paper.D. Bob will take 4 minutes longer than Ted to type his research paper.




Travis went on a long trip. The graph below represents the relationship between distance and time.

During what interval was Travis’ average rate of travel the fastest?

A. 0 to 6B. 6 to 8C. 8 to 11D. 11 to 16


Which pair of equations represents lines that are parallel and perpendicular,

respectively, to the graph of ?




To solve a math problem, Penny is graphing the equations y = x2 and y = x2 + 1. To graph the equations, she created the tables shown below.

In your Answer Document, copy the tables above and find the y -values for each of the given x -values.

Use the grid provided to graph each equation using the pairs of x - and y -values.

Based on the graphs you have completed, analyze how the graphs differ and write a hypothesis to describe how adding a number to x2 changes the graph of x2.


The area of the rectangle illustrated below is 3 square inches.

Marisa used the equation (x – 2)(x – 4) = 3 to determine the values for x to be 1 and 5.

In your Answer Document, show whether Marisa’s solutions are correct for the problem situation. Support your answers by showing work or providing an explanation.



F. Solve and graph linear equations and inequalities.

9.6. Write and use equivalent forms of equations and inequalities in problem situations; e.g., changing a linear equation to the slope-intercept form.

9.8. Find linear equations that represent lines that pass through a given set of ordered pairs, and find linear equations that represent lines parallel or perpendicular to a given line through a specific point.

10.10. Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions.


A sports club is planning a cookout. The food service charges $1.75 per person plus a flat fee of $200. The club can only spend $500 for food service.

In your Answer Document, determine the maximum number of people the club will be able to serve. Use a table, graph, equation or inequality to support your answer.

OGT 2007 – Problem # 24 Marcy and Troy disagreed about the answer to a problem. Marcy said that the equation they were working on had more than one solution. If Marcy is correct, on which of these equations could they have been working?

A. 2x + 4 = 3x + 4

B. 2x + 4 = 3x + 5

C. 2x + 4 = 2(x + 2)

D. 2x + 4 = 2(x + 3)


A producer has $2 million budgeted for costs related to filming a movie on location. She estimates the costs to be:

c = $108,000n + $175,000

where c = costs in dollars and n = number of days.

What is the greatest number of days she can film on location and remain within the budget?

A. 11 days

B. 16 days

C. 18 days

D. 20 days




Neal is selecting a new health club. The one helikes has monthly dues of $24 and a start-up feeof $400. He has determined that the equation

can be used to find y, the total costof being a member at the club based on thenumber of months, x.

After how many months will Neal have spentexactly $1,000 at this health club?

A. 16B. 25C. 41D. 58


Which equation is equivalent to3(2x – 5) = 4(x + 3)?

A. 2x = –27B. 2x = 27C. 10x = –27D. 10x = –3


Pippi calculates her total earnings for themonth with the equation

E = 15m + 5b,where E is the total number of dollars she earns,m is the number of lawns she mows, andb is the number of hours she baby-sits.

If Pippi mows 6 lawns, how many hours mustshe baby-sit to earn a total of $200?

A. 20B. 22C. 40D. 45


A telephone manufacturing company hasdetermined that the cost of producing acertain type of telephone can be found byusing the equation y = 42x + 2,000, where y isthe production cost and x is the number oftelephones produced. The companyaccountant calculates an average dailyproduction cost of $8,426. Approximately howmany telephones does the company producedaily?

A. 153B. 248C. 6,426D. 355,892




The formula for converting temperature on the Celsius scale, C, to the

Fahrenheit scale, F, is . Which graph represents this equation?

G. Solve quadratic equations with real roots by graphing, formula and factoring.

9.10. Solve quadratic equations with real roots by factoring, graphing, using the quadratic formula and with technology.

10.8. Graph the quadratic relationship that defines circles.

10.10. Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions.


Which equation is equivalent to

x2 – 100x + 2,400 = 0?

A. (x – 40)(x – 60) = 0

B. (x – 30)(x – 80) = 0

C. (x + 20)(x – 120) = 0

D. (x + 20)(x + 120) = 0




The area of the rectangle illustrated below is 3 square inches.

Marisa used the equation (x – 2)(x – 4) = 3 to determine the values for x to be 1 and 5.

In your Answer Document, show whether Marisa’s solutions are correct for the problem situation. Support your answers by showing work or providing an explanation.

H. Solve systems of linear equations involving two variables graphically and symbolically.

9.9. Solve and interpret the meaning of 2 by 2 systems of linear equations graphically, by substitution and by elimination, with and without technology.

10.7. Solve systems of linear inequalities.

10.11. Solve real-world problems that can be modeled, using systems of linear equations and inequalities.


Roni and Kelsey bought the same types of flowers from a florist. Roni bought 5 roses and 2 carnations and was charged $17.85 before tax. Kelsey purchased 1 rose and 6 carnations and was charged $12.25 before tax.

How much did the florist charge for 1 carnationbefore tax?

A. $1.12B. $1.55C. $2.15D. $2.80




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

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

For Option B, he prices items at $250 and gives $2.00 off for every day the item 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 find the day on which the two options yield the same price.


Mrs. Foyle told Yolanda that her test had 38 problems worth a total of 100 points. Each test problem is worth either 5 points or 2 points. Yolanda wanted to determine how many 2-point and how many 5-point questions are on the test.

In your Answer Document, determine how many questions of each point-value are on the test. Show your work or provide an explanation to support your answer.


Sara ordered 2 slices of pizza and a 12-ouncecola and paid $3.00. Sydney ordered 3 slices ofpizza and 2 12-ounce colas for $4.75.How much does a slice of pizza cost?

A. $0.50B. $1.00C. $1.25D. $2.50


A system of equations is shown below.

3x + 2y = 192xy = 1

What is the solution to the system of equations?

A. x = 1, y = 1B. x = 3, y = 5C. x = 7, y = –1D. x = 19, y = 1




Cameron had $500 in savings on January 1. Quinn had $800 in savings on January 1. Cameron deposits $20 per week into his savings account. Quinn withdraws $15 per week from his savings account.

In your Answer Document, write two equations: one for the amount of money in Cameron’s savings x weeks after January 1st, and one for the amount of money in Quinn’s savings x weeks after January 1st.

Determine the number of weeks until Cameron ill have more money in his savings account than Quinn. Show your work or provide an explanation for your answer.

I. Model and solve problem situations involving direct and inverse variation.

9.13. Model and solve problems involving direct and inverse variation using proportional reasoning.

9.14. Describe the relationship between slope and the graph of a direct variation and inverse variation.


For his business, Gil has determined that the time it takes to finish a job varies inversely with the number of workers. This can be expressed as:

where T = time, k is a constant, and w = number of workers.

Gil’s records show that 18 workers can finish a job in 6 days.

How many days will it take 12 workers to do the same job?

A. 4B. 9C. 12D. 36




Under ideal conditions, there is an inverserelationship between the pressure (P) and thevolume (V) of a gas. The table shows therelationship of the experimental pressuresrecorded for four different volumes of the gas.

Which equation shows the relationship betweenthe pressure and the volume of this gas?


Density is a physical property of matter which measures the mass of an object per unit of volume.

As most stars grow older, the mass remains the same while the radius becomes smaller. What happens to the star’s density as the star ages?

A. The density increases.B. The density decreases.C. The density stays the same.D. The density increases and decreases periodically over time.



J. Describe and interpret rates of change from graphical and numerical data.

9.15. Describe how a change in the value of a constant in a linear or quadratic equation affects the related graphs.

10.9. Recognize and explain that the slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals.

10.12. Describe the relationship between slope of a line through the origin and the tangent function of the angle created by the line and the positive x-axis.


A construction company buys $38,400 worth of new equipment, the value of which decreases linearly so that its value after 8 years is $2,200. By what amount is the value of the equipment decreasing per year?

A. $ 275

B. $4,525

C. $4,800

D. $5,075




Jessica participated in a walk-a-thon to raise money for a local charity. She began walking at a rate of 3 miles per hour. The graph represents her distance walked as a function of time.

If Jessica had walked at an average rate of 2 miles per hour, which of the following accurately illustrates how this graph would appear, using the same scale?




Last spring, the employees of an Akron tire company contributed to a local charity. The graphic represents the total dollar contributions as of several dates in April.

In your Answer Document, determine on what date the charity could have expected to reach its goal of $1,000, if the pattern of donations remains about the same. Show your work or provide an explanation for your answer.


Patterns, Functions, and Algebra Standard

♦Answer Key

Patterns, Functions and Algebra - Answer Key OGT Problems

Patterns, Functions, and Algebra

TestYear Question # Answer

A

2008 1 D

2007 2 C

2007 18 C

2006 15 D

2005 27 C

2004 1 D

2004 38 C

2003 29 A

B

2008 38 B

2006 6 C

2006 28 S.A.

2005 7 C

C

2008 7 D

2008 41 D

2007 44 C

2006 28 S.A.

2005 8 B

2004 9 C

2004 27 C

2003 20 E.R.**

D

2008 35 D

2006 10 S.A.

2006 36 B

2005 26 D

2004 12 D

2003 3 D

2003 14 A

E2008 31 A

2007 7 A

Patterns, Functions, and Algebra

TestYear Question # Answer2006 24 B

2006 41 B

2005 2 A

2004 7 B

2003 11 D

2003 20 E.R.**

2003 35 S.A.**

F

2008 11 S.A.

2007 24 C

2006 1 B

2005 4 B

2004 3 B

2004 14 B

2003 23 A

2003 31 C

G2006 32 A

2003 35 S.A.**

H

2008 17 B

2007 38 S.A.

2006 10 S.A.

2005 38 C

2004 18 B

2004 32 S.A.

I2008 21 B

2005 14 C

2003 26 A

J2007 30 B

2006 41 B

2005 24 S.A.** Scoring Rubric Not Released



PFA – Benchmark F2008 OGT – Problem # 11 Scoring Guidelines:

Sample Responsec = 1.75n + 200500 = 1.75n + 200n = 300 ÷ 1.75n = 171.4

The maximum number is 171 people OR n < 172.

OR

They should be able to serve about 170 people for $500.

Points Description2 The focus of the item is to use a table, graph, equation or inequality to solve a

word problem. The response determines the maximum number of people (171 or 171.4) the club will be able to serve at a cookout for less than $500 using one of these techniques.

1 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 a correct equation or inequality but the maximum number of people is missing or incorrect.-Contain an incomplete or partially correct linear graph, but the maximum number of people is missing or incorrect.-Contain an incomplete or partially correct table, but the maximum number of people is missing or incorrect.-Contains the correct answer (171 or 171.4) with incomplete or missing work.

0 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:-Be blank or give irrelevant information.



PFA – Benchmark H2007 OGT – Problem # 38 Scoring Guidelines:

Sample ResponseOption A price after d days: p = 225 − dOption B price after d days: p = 250 − 2d225 − d = 250 − 2dd = 25Equal after day 25. Take discount at end of day

OR

Option A price: p = 225 − ( d-1)Option B price: p = 250 − 2(d −1)225 – (d –1) = 250 – 2(d – 1)225 – d + 1 = 250 – 2d + 2d = 26Equal on the 26th day of the sale with full price on day one.

Points Description2 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.)


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 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.



PFA – Benchmark D and H2006 OGT – Problem # 10 Scoring Guidelines:

Sample Response Use a system of equations:

x + y = 38 2x + 5y = 100 ∴x = 30 2-point questions

y = 8 5-point questions OR Use a table to guess-and-check:

There are 8 5-point questions and 30 2-point questions.

Points Description2 The focus of the item is to determine the number of 2-point and 5-point

questions on the test. The response indicates that there are 30 two-point and 8 five-point questions with supporting work or explanation.


For example, the response may:

Provide the correct answer with missing or incomplete work. Provide an incorrect answer based on slightly flawed equations. Provide correct equations or demonstrate a valid process but contain a

calculation error in determining the solution.

0 The response indicates inadequate understanding of the task. For example, the response may: Contain one answer (8 or 30) with no correct work. Include unrelated statements or work.



PFA – Benchmark B & C2006 OGT – Problem # 28 Scoring Guidelines:

Sample Response

a R10 210 15 205 20 200 25 195 30 190 35 185 40 180 45 175 50 170

R decreases by 1 for each additional year in age, so the slope is –1.

OR

210 = –1 (10) + b, so the y-intercept is 220 .

Sample Equations:R= -A + 220 or R = 220 – A or A = 220 – R or R– 210 = -1(A-10) or R – 210 = -A + 10 or R = -A + 220

Points Description2 The focus of the item is to complete a table and to determine an equation that

models the data in the table. The response contains a completed table and an equation for the relationship shown in the table.


For example, the response may: Contain a completed table, but the equation is incomplete or missing. Contain a correct equation, but the table is incorrect, incomplete, or

missing. Identify the correct slope or y-intercept of the equation with or without

the completed table.

0 The response indicates inadequate understanding of the task.

For example, the response may: Contain a partially completed or incorrectly completed table with no

additional work. Include unrelated statement or work.

PFA – Benchmark J



2005 OGT – Problem # 24 Scoring Guidelines:Sample Response

Find the average amount received every 5 days.700 ÷ 4 = $175/5 days700 + 175 = 875 on April 25.875 + 175 = 1050 on April 30

The tire company should reach their goal on the 28th or 29th of April because, by the 30th, they will have exceeded the goal.ORFind the average amount received every day.

700 ÷ 20 = $35/day$35 x 28 = $980 on April 28.$35 x 29 = $1015 on April 29.

OR$1,000 ÷ $35 228.57 days

The tire company should reach its goal during the day on the 29th.ORThis response contains a description of how the diagram was used to estimate when the tire company would reach their goal on the (range of 28th to 30th) of April.

Points Description2 The focus of this item is to determine the approximate date the company will

meet the $1,000 goal based on the current rate of giving. The response indicates a date from the 28th to the 30th of April with supporting work or detailed explanation based on the information provided in the item.

1 The response shows a partial understanding of the solution process or key elements of the task. The response contains gaps or flaws in determining the solution.For example the response may:Indicate the correct date but show no work or the reasoning may be flawed.ORIndicates a date outside the acceptable range resulting from an error in computation. An appropriate solution process is shown or explained.ORContain a correct method for determining the date, but the date is incorrect or missing.

0 The response fails to provide evidence of minimal understanding of the task.For example the response may:Recopy information provided in the question with no work toward a solution.ORState a date outside the acceptable range with no supporting work or explanation.ORBe blank or the student writes “I do not know” or includes unrelated



statements or work.



PFA – Benchmark H2004 OGT – Problem # 32 Scoring Guidelines:

Sample ResponseC = 500 + 20x or equivalent equationQ = 800 – 15x or equivalent equation

500 + 20x > 800 15x35 x > 300x > 8.57

There will be more money in Cameron’s account in the 9th week or after the 8th week or about 8 ½ weeks or between 8 and 9 weeks or the first week in March (assumes four weeks per month).

Alternate solution processes for determining the number of weeks until Cameron will have more money in his account than Quinn include making a table (see example below) or graphing the equations.

Points Description2 The task includes two components—writing equations that represent the

amount of money in each account for a variable number of weeks and determining the number of weeks until the amount of money in Cameron’s account is greater than that in Quinn’s account. The response contains accurate equations for finding the amount in each account for any number of weeks and indicates Cameron will have more money in his savings account than Quinn in week 9 (or other accurate response such as those provided in sample responses) with clear, accurate work or explanation.

1 The response provides evidence of a partial answer and/or solution process. The response has an error(s) or does not carry out all parts of the task.

For example, the response may:Provide correct equations for each account with an incorrect answer. Supporting work or explanation is incorrect or missing for the number of weeks until Cameron will have more money in his account than Quinn.OR



Find the correct number of weeks with supporting work but does not provide two correct equations.ORProvide one correct equation and one flawed equation, but then determine an accurate number of weeks based upon the equations.ORProvide one correct equation with no additional work.ORIndicate the 9th week without any supporting work or equations.

0 The response indicates inadequate or no understanding of the taskand/or incorrect use of the key elements or information; e.g., writing an equation to represent a problem situation and finding a value that when used in both equations meets a specified criteria. The response does not meet the criteria required to earn one point.For example, the response may:State an incorrect number of weeks with no supporting work orexplanationORGive two incorrect equations and no other work or explanation.ORRecopy information provided in the item with no work.ORShow no apparent understanding or relationship to the key components in the task or a possible solution process; e.g., finds the sum of the dollar amounts in the item.ORBe blank, or the student writes, “I do not know” or includes unrelated statements or work.


Documents

K-12 Mathematics Benchmarks · Web viewWrite and use equivalent forms of equations and inequalities in problem situations; e.g., changing a linear equation to the slope-intercept