2007 Math Released Item Packet

8/19/2019 2007 Math Released Item Packet

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2007 Mathematics 196

CAPT Mathematics

2007 Administration

Released Items andScored Student Responses


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Table of Contents

CAPT Mathematics Framework............................................................... 198

Items Found in This Packet ..................................................................... 204CAPT Mathematics Open-Ended Items and

Scored Student Responses .............................................................. 205

Scoring Rubric for Mathematics Open-Ended Items ............................... 206

Cargo Ship........................................................................................ 207

Martha’s Sales .................................................................................. 217

Hang Gliding ..................................................................................... 227

Two Silos .......................................................................................... 237

Kendra’s Travels............................................................................... 247

Population of New London County ................................................... 257

Chocolate Candy .............................................................................. 267

Organism Lengths ............................................................................ 277

CAPT Mathematics Grid-In Items ............................................................ 287

Soup Cans ........................................................................................ 288

Stopping Distance............................................................................. 289Coffee Special .................................................................................. 290

Entertainment Center ........................................................................ 291

Mary’s Number Cubes ...................................................................... 292

Bloodhound....................................................................................... 293

Joseph’s Final Grade........................................................................ 294

Picnic Food ....................................................................................... 295


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CAPT Mathematics Framework

The CAPT is designed to be directly aligned with the 2005 Connecticut Mathematics Framework thatwas developed by an advisory committee of Connecticut educators. The framework that is presented onthe following pages integrates the skills, competencies, and understandings delineated in the NCTM

Principles and Standards for School Mathematics (National Council of Teachers of Mathematics, 2000),2005 NAEP Mathematics Framework (National Assessment of Educational Progress, 2005), the Trendsin International Mathematics Science Study (TIMMS, 2003), and CSDE Goals 2000 MathematicsCurriculum.

The Mathematics test assesses how well students compute and estimate, solve problems, andcommunicate their understanding. The content standards assessed are Algebraic Reasoning: Patterns andFunctions; Numerical and Proportional Reasoning; Geometry and Measurement; and Working withData: Probability and Statistics. The test focuses on how well students apply important mathematicsconcepts and skills to solve problems that are relevant to everyday experiences.

The CAPT Mathematics test is based on the view that mathematical understanding is best assessed bydoing mathematics, and that doing mathematics means using and discovering knowledge in the courseof solving real-world problems. This means that, instead of assessing long division skills directly,students apply division skills to solve everyday problems. For example, students might be asked to findthe price per pound of 2.38 pounds of ground beef that has a total cost of $6.20, compare this price withthe unit prices of other possible choices and, finally, justify their purchase decision.

Thus the CAPT Mathematics test assesses knowledge, skills, and applications reasonable to expect of allstudents by the end of 10th grade. Students may use a calculator with which they are familiar andcomfortable for the entire Mathematics test.


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CAPT Mathematics Process Strands

Student understanding of mathematical content is assessed through students “doing” mathematics.CAPT items require students to demonstrate their abilities in the following areas:

Problem Solving and Reasoning

• formulate problems from situations and given data;• develop and apply a variety of strategies to solve multi-step and non-routine problems;• make and evaluate conjectures and arguments; and• verify, validate, and interpret results and claims and generalize solutions.

Communicating• model situations using written, concrete, pictorial, graphical, and algebraic representations;• express mathematical ideas and arguments with clarity and coherence; and• use mathematical language and notation to represent ideas, describe relationships, and model

situations.

Computing and Estimating

• select and use appropriate methods for computing, including mental mathematics, estimation, paper-and-pencil, and calculator methods; and

• use estimation to assess the reasonableness of results.

Mathematical problem solving and reasoning relate to thequestions:

• “Does the student apply an appropriate strategy?”• “Does the student use the correct information?”

Mathematical communication relates to the questions:• “Does the student explain what he/she did?”• “Does the student communicate an understanding of

the problem?”

Mathematical computing and estimating relate to thequestion:

• “Does the student perform the mathematical procedures correctly?”


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CAPT Mathematics Content Strands

All CAPT mathematics items are written in a real world context and require students to solve aproblem. In addition, all open-ended CAPT items will require students to show their work OR explaintheir reasoning, thereby communicating their understanding of the relevant mathematics.

All CAPT items will be devised to measure one or more of the following Expected Performances:

ALGEBRAIC REASONING: PATTERNS AND FUNCTIONS

Patterns and functional relationships can be represented and analyzed using a variety ofstrategies, tools and technologies.

How do patterns and functions help us describe data and physical phenomena and

solve a variety of problems?

Students

should…

Performance

StandardsExpected Performances

1.1 Understand

and describe

patterns andfunctional

relationships.

a. Describerelationships and

makegeneralizationsabout patterns andfunctions.

(1) Identify, describe, create and generalize numeric,geometric and statistical patterns with tables, graphs, words

and symbolic rules.(2) Make and justify predictions based on patterns.(3) Identify the characteristics of functions and relations,including domain and range.(4) Describe and compare properties and classes of linear,quadratic and exponential functions.

1.2 Represent

and analyze

quantitative

relationships in a

variety of ways.

a. Represent andanalyze linear andnon-linearfunctions andrelations

symbolically andwith tables andgraphs.

(1) Represent functions and relations on the coordinate plane.(2) Identify an appropriate symbolic representation for afunction or relation displayed graphically or verbally.(3) Recognize and explain the meaning of the slope and x-

and y-intercepts as they relate to a context, graph, table orequation.(4) Evaluate and interpret the graphs of linear, exponentialand polynomial functions.

1.3 Use

operations,

properties and

algebraic symbols

to determine

equivalence and

solve problems.

a. Manipulateequations,inequalities andfunctions to solve problems.

(1) Model and solve problems with linear, quadratic andabsolute value equations and linear inequalities.(2) Determine equivalent representations of an algebraicequation or inequality to simplify and solve problems.(3) Solve systems of two linear equations using algebraic orgraphical methods.


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NUMERICAL AND PROPORTIONAL REASONING:

Quantitative relationships can be expressed numerically in multiple ways in order to makeconnections and simplify calculations using a variety of strategies, tools and technologies.

How are quantitative relationships represented by numbers?

Students

should…

Performance

StandardsExpected Performances

a. Extend theunderstanding ofnumber to includeintegers, rationalnumbers and realnumbers.

(1) Compare, locate, label and order real numbers onnumber lines, scales, coordinate grids and measurementtools.(2) Select and use an appropriate form of number (integer,fraction, decimal, ratio, percent, exponential, scientificnotation, irrational) to solve practical problems involvingorder, magnitude, measures, labels, locations and scales.

2.1 Understand

that a variety of

numerical

representations

can be used to

describe

quantitative

relationships. b. Interpret andrepresent largesets of numberswith the aid oftechnologies.

(1) Use technological tools such as spreadsheets, probes,computer algebra systems and graphing utilities to

organize and analyze large amounts of numerical

information.*

a. Developstrategies forcomputation andestimation using properties ofnumber systemsto solve problems.

(1) Select and use appropriate methods for computing tosolve problems in a variety of contexts.(2) Solve problems involving scientific notation andabsolute value.(3) Develop and use a variety of strategies to estimatevalues of formulas, functions and roots; to recognize thelimitations of estimation; and to judge the implications ofthe results.

2.2 Use numbers

and their

properties to

compute flexibly

and fluently, and

to reasonably

estimate measures

and quantities.

b. Solve proportionalreasoning problems.

(1) Use dimensional analysis to determine equivalentrates.(2) Solve problems using direct and inverse variation.

*Concepts in italics will NOT be tested on CAPT, but should be included in Core instruction.


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GEOMETRY AND MEASUREMENT

Shapes and structures can be analyzed, visualized, measured and transformed using a varietyof strategies, tools and technologies.

How do geometric relationships and measurements help us to solve problems and

make sense of our world?

Students

Should…

Performance

Standards

Expected Performances

a. Investigaterelationshipsamong plane andsolid geometricfigures usinggeometric models,constructions andtools.

(1) Use models and constructions to make, test andsummarize conjectures involving properties of geometricfigures.(2) Use geometric properties to solve problems in two andthree dimensions.(3) Determine and compare properties of classes of polygons.

3.1 Use

properties and

characteristics of

two- and three-

dimensional

shapes and

geometric

theorems to

describe

relationships,

communicate

ideas and solve

problems.

b. Develop andevaluatemathematicalarguments usingreasoning and proof.

(1) Recognize the validity of an argument.(2) Create logical arguments to solve problems anddetermine geometric relationships.

3.2 Use spatial

reasoning,

location and

geometric

relationships to

solve problems.

a. Verifygeometricrelationshipsusing algebra,coordinategeometry andtransformations.

(1) Interpret geometric relationships using algebraicequations and inequalities, and vice versa.(2) Describe how a change in measurement of one or more parts of a polygon or solid may affect its perimeter, area,surface area and volume and make generalizations forsimilar figures.(3) Apply transformations to plane figures to determinecongruence, similarity, symmetry and tessellations.

3.3 Develop and

apply units,

systems,

formulas and

appropriate tools

to estimate and

measure.

a. Solve a varietyof problemsinvolving one-,two- and three-dimensionalmeasurementsusing geometricrelationships andtrigonometricratios.

(1) Select appropriate units, scales, degree of precision, andstrategies to determine length, angle measure, perimeter,circumference and area of plane geometric figures.(2) Use indirect methods including the PythagoreanTheorem, trigonometric ratios* and proportions in similarfigures to solve a variety of measurement problems.(3) Judge the reasonableness of answers to direct andindirect measurement problems.(4) Use two-dimensional representations and formal andinformal methods to solve surface-area and volume problems.

*Concepts in italics will NOT be tested on CAPT, but should be included in Core instruction.


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WORKING WITH DATA: PROBABILITY AND STATISTICS

Data can be analyzed to make informed decisions using a variety of strategies, tools,and technologies.

How can collecting, organizing and displaying data help us analyze information

and make reasonable predictions and informed decisions?

Students

should…

Performance

Standards

Expected Performances

4.1 Collect,

organize and

display data

using

appropriate

statistical and

graphical

methods.

a. Create theappropriate visualor graphicalrepresentation ofreal data.

(1) Collect real data and create meaningful graphicalrepresentations of the data.(2) Develop, use and explain applications and limitations oflinear and nonlinear models and regression in a variety ofcontexts.

4.2 Analyze data

sets to form

hypotheses and

make

predictions.

a. Analyze real-world problemsusing statisticaltechniques.

(1) Estimate an unknown value between data points on agraph (interpolation) and make predictions by extending thegraph (extrapolation).(2) Use data from samples to make inferences about a population and determine whether claims are reasonable orfalse.(3) Determine and use measures of spread and centraltendency to describe and compare sets of data.

4.3 Understand

and apply basic

concepts of

probability.

a. Understand andapply the principles of probability in avariety ofsituations.

(1) Determine outcomes and solve problems involving the probabilities of events.(2) Explore the concepts of conditional probability in real-world contexts.(3) Apply theoretical and experimental probabilitiesappropriately to solve problems and predict experimentalresults.


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Items Found in This Packet

Open-Ended Items

Open-ended items are those for which a student must write a response to a question. For example, theseitems require students to solve mathematical problems by showing their work or explaining their

reasoning and/or procedure(s) they used. Included in this packet are the eight open-ended items thatappeared on the 2007 administration of the Mathematics section of the CAPT.

Open-ended items are scored by trained readers using a four-point holistic scale (0–3). This means thatthe overall quality of a student’s response is considered when making a scoring judgment. The generalscoring rubric for the mathematics open-ended items (see page 207) describes the characteristics of aresponse at each score point. Included with each item is a specific scoring rubric and the classification ofthe item based on the 2005 Connecticut Mathematics Framework . For each score point, there are twoexamples of scored student responses along with a brief explanation of why the response received that particular score.

Grid-In ItemsGrid-in items are those for which a student must arrive at a numeric answer and enter it into a grid.Included in this packet are several grid-in items that appeared on the 2007 administration of theMathematics section of the CAPT.

The grid-in items are scored electronically as either correct or incorrect; however, there may be severalcorrect answers for an item. There are times in mathematics when, because of rounding (38.21 or 38.2)or representing percents (35% or .35), a number of responses are correct and acceptable.


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CAPT Mathematics Open-Ended Items


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Scoring Rubric for Mathematics Open-Ended Items

Each score category contains a range of student responses that reflect the descriptions given below.

Score 3

The student has demonstrated a full and complete understanding of all concepts and processes essentialto this application. The student has addressed the task in a mathematically sound manner. The responsecontains evidence of the student’s competence in problem-solving and reasoning, computing andestimating, and communicating to the full extent that these processes apply to the specified task. Theresponse may, however, contain minor arithmetic errors that do not detract from a demonstration of fullunderstanding. Student work is shown or an explanation is included.

Score 2

The student has demonstrated a reasonable understanding of the essential mathematical concepts and processes in this application. The student’s response contains most of the attributes of an appropriateresponse including a mathematically sound approach and evidence of competence with applicable

mathematical processes, but contains flaws that do not diminish the evidence that the studentcomprehends the essential mathematical ideas addressed in the task. Such flaws include errors attributedto faulty reading, writing, or drawing skills; errors attributed to insufficient, non-mathematicalknowledge; and errors attributed to careless execution of mathematical processes or algorithms.

Score 1

The student has demonstrated a partial understanding of some of the concepts and processes in thisapplication. The student’s response contains some of the attributes of an appropriate response, but lacksconvincing evidence that the student fully comprehends the essential mathematical ideas addressed bythis task. Such deficits include evidence of insufficient mathematical knowledge; errors in fundamentalmathematical procedures; and other omissions or irregularities that bring into question the extent of the

student’s ability to solve problems of this general type.

Score 0

The student has demonstrated merely an acquaintance with the topic. The student’s response isassociated with the task in the item but contains few attributes of an appropriate response. There aresignificant omissions or irregularities that indicate a lack of comprehension in regard to themathematical ideas and procedures necessary to adequately address the specified task. No evidence is present to suggest that the student has the ability to solve problems of this general type.


12/100

2007 Mathematics 207 —RELEASED ITEM—

Cargo Ship Geometry and Measurement

1. Use your ruler to help you answer this question.

The captain of a cargo ship plans to leave Genoa and travel to Naples, Palermo and Cagliari beforereturning to Genoa as shown on the map of Italy below.

The distance from Naples to Palermo is 320 kilometers. The ship’s average speed is 14 nauticalmiles per hour. Estimate the number of hours it will take the cargo ship to complete its round trip,not including stops at ports. Show your work or explain how you found your answer.

Remember to show your work and write your answer in your answer booklet.


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Rubric for Cargo Ship

Score 3

The response contains:• a reasonable value for the number of hours it will take the cargo ship to complete its round trip

resulting from reasonable estimates or measurements and appropriate conversions, with supportingwork or explanation.

The response may contain minor errors that do not detract from a demonstration of full understanding.

Score 2

The response may contain:• an incorrect value due to a significant error in estimating or measuring the round trip, with

appropriate conversions, with supporting work or explanationOR

• an incorrect value due to a significant error converting between units, with reasonable estimates or

measurements for the round trip, with supporting work or explanation.

Score 1

The response may contain:• reasonable estimates or measurements for the round trip with no appropriate conversions

OR

• an estimate between 74 and 86.5 hours, inclusive, with little or no supporting work or explanationOR

• some other demonstration of ability to make appropriate estimations/measurements and/orconversions.

Score 0The student has demonstrated merely an acquaintance with the topic. No evidence is present to suggestthat the student has the ability to solve problems of this general type.


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Scored Student Responses for Cargo Ship

Question 1: Cargo Ship Score 3

The response demonstrates a full and complete understanding of the task. After converting 1 inch to nauticalmiles, the student calculates the distance and time traveled for each leg of the journey and adds those numbersto arrive at a reasonable estimate of the amount of time required for the entire trip.


15/100




The student uses an appropriate process for determining a reasonable estimate of the number of hours traveled. After estimating the scale length of one leg of the journey, the student sets up a ratio and uses it to convert thescale to calculate the number of kilometers traveled. The conversion to hours generates an acceptable response.The response demonstrates a full and complete understanding of the task.


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The response demonstrates a reasonable understanding of the topic. The student estimates the total number ofkilometers traveled for each leg based on a scale of 2 inches equaling 320 kilometers. That scale is a significanterror resulting in a final answer that is outside a reasonable range. Apart from the scaling error, the problem isaddressed using a mathematically sound approach.


17/100




The student estimated that the round trip totaled 6.5 lengths of 320 kilometers. This sound approach was crossedout and replaced with an estimation of time traveled based on 4.5 equal lengths of 320 kilometers, which resultsin an inappropriate estimation.


18/100




The response demonstrates a partial understanding of the task. The student made reasonable estimates of thedistances traveled for each leg and provided a total distance for the trip, but performed no further calculations.


19/100




The response demonstrates a partial understanding of the task. Although it is unclear how the distances for eachleg of the trip were determined, they are proportionate to the actual values. The student then calculates thenumber of hours for the complete trip by dividing the distance by 14 nautical miles per hour.


20/100




The response demonstrates merely an acquaintance with the topic. Estimates are made regarding themeasurements of the distances on the map, but nothing further is done with those measurements.


21/100




This response contains a numerical value that is outside a reasonable range with no supporting calculations or

explanation. No evidence is present to suggest that the student has the ability to solve problems of this generaltype.


22/100


Martha’s Sales Algebraic Reasoning

2. Martha works as a salesperson for Momentum Sales. She earns $1,000 per month plus 15%commission on her sales.

a. Write an equation that expresses T , her income for one month as a function of x, her total salesfor the month.

b. River City Sales, another company in the same town, has offered Martha a job that will pay her$500 per month plus 20% commission on her sales. The benefits and working conditions areequally good at both companies. Explain why Martha should or should not accept the job atRiver City Sales. Support your answer by finding x, the average total sales for the month shewould need to at least equal her present income at Momentum Sales.



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Rubric for Martha’s Sales

Score 3

The response contains:• a correct equation that expresses Martha’s income at Momentum Sales for one month as a function

of her total sales for the monthAND

• an appropriate explanation why Martha should or should not accept the job at River City Sales,supported by the average total sales at which her River City Sales income would equal herMomentum Sales income.


Score 2

The response may contain:• a correct equation and a partially correct or partially complete explanation due to a significant error

or omissionOR

• an incorrect equation with an appropriate explanation, supported by a correct or consistent averagetotal sales at which her River City Sales income would equal her Momentum sales income.

Score 1

The response may contain:• a correct equation with an unacceptable explanation

OR

• an incorrect equation with a partially correct or complete explanation arising from an appropriate process.

Score 0

The student has demonstrated merely an acquaintance with the topic. No evidence is present to suggestthat the student has the ability to solve problems of this general type.


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Scored Student Responses for Martha’s Sales

Question 2: Martha’s Sales Score 3

The response demonstrates a full and complete understanding of the task. The equation for determining Martha’sincome is correct. The student then calculates that the income will be equal under the two wage scenarios at$10,000 in sales and explains the better position for Martha based on that calculation.


25/100




The equation for determining Martha’s income is correct. The student calculates that Martha would make an equalamount of money under the two wage scenarios at $10,000 in sales and provides an explanation of the bestchoice for her that is consistent with that calculation. The response demonstrates a full and completeunderstanding of the task.


26/100




The response demonstrates a reasonable understanding of the task. The equation for determining Martha’sincome is correct. The student then shows the difference in income based on one amount of monthly sales andmakes a recommendation based on that calculation, but does not use when the two amounts will be the same.


27/100




The student provides a correct equation for determining Martha’s income. A computational error results in thestudent mistakenly stating that the income under the two wage scenarios would be equal if sales were $1,000. Anappropriate recommendation is made based on the miscalculation. The response demonstrates a reasonableunderstanding of the task.


28/100




The student provides an acceptable equation. Attempts are made to find the impact of the two wage scenariosbased on $2,500 in sales, but the student does not evidence an understanding of how to make that calculationcorrectly. This process demonstrates a partial understanding of the concepts involved in the task.


29/100




The equation differs from what is expected in that it solves for the amount of sales, rather than the amount ofincome, but it can be used to calculate Martha’s income correctly. The student does not use the equation toresolve the question of which is the better of the two wage scenarios, and the explanation is insufficient. Theresponse demonstrates a partial understanding of the task.


30/100




The response demonstrates merely an acquaintance with the task. The equation provided is not correct. Theexplanation as to the better of the two wage scenarios reflects only the salary amount and does not demonstratean understanding of the impact of differing sales amounts on Martha’s salary.


31/100




The student does not attempt to write an equation. The calculation of the two wage scenarios is incorrect. Thestudent has demonstrated merely an acquaintance with the topic.


32/100


Hang Gliding Algebraic Reasoning

3. Josephine likes to go hang gliding. She took off from a hillside at an elevation of 700 feet. Duringthe first 5 minutes, she went down to 500 feet. Then she rode for another 5 minutes up to a height

of 600 feet. She then descended at a rate of 200 feet every 15 minutes until she landed.

a. Using the grid provided, construct a graph to represent Josephine’s flight. Use height as afunction of time and remember to label the axes.

b. How many minutes did her flight take? Show your work or explain how you found youranswer.



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Rubric for Hang Gliding

Score 3

The response contains:• a correct graph of the data provided and a correct determination of the time of the flight, with

supporting work or explanation.


Score 2

The response may contain:• a correct graph of the data provided and a correct determination of the time of the flight, with

insufficient supporting work or explanationOR

• a correct graph of the data provided and an incorrect determination of the time of the flight due to asignificant error in an appropriate process

OR• a graph containing errors, but demonstrating a reasonable understanding, along with either:

- a correct determination of the time of the flight, with supporting work or explanation or- an incorrect determination of the time of flight that is consistent with the errors, with supporting

work or explanation.

Score 1

The response may contain:• a graph containing errors, only demonstrating limited understanding, and a correct determination of

the time of flight.

Score 0 The student has demonstrated merely an acquaintance with the topic. No evidence is present to suggest

that the student has the ability to solve problems of this general type.


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Scored Student Responses for Hang Gliding

Question 3: Hang Gliding Score 3

The response demonstrates a full and complete understanding. The labeling, scaling, and plotting of the graphare accurate. The correct response of 55 minutes is supported by both the graph and the explanation.


35/100


36/100




The scaling and plotting of the graph are accurate. The accurate calculation of the flight time is consistent with thegraph, but it is not supported by an explanation. The response demonstrates a reasonable understanding of thetask.


37/100




The scaling and the plotting of the graph are accurate. The axes have been reversed. The calculation of the flighttime is accurate and is supported by a sufficient explanation. The student demonstrates a reasonableunderstanding of the task.


38/100




The response demonstrates a partial understanding of the task. The graph does not demonstrate anunderstanding of the topic. The student provides an accurate answer to the question regarding the flight time andsupports the answer with an explanation.


39/100




The response demonstrates a partial understanding of the task. The scaling of the graph is uneven and theplotting ends at 30 minutes of flight time. The response regarding total flight time is incorrect, though theexplanation shows some understanding of a valid process.


40/100




The student uses a bar graph, which is not a correct response to the task presented in the prompt. The responsethat total flight time is 30 minutes is incorrect, and the explanation further demonstrates that the student hadmerely an acquaintance with the task.


41/100




The response demonstrates merely an acquaintance with the task. The plotted points are inappropriate for theflight described in the prompt. This results in the student incorrectly calculating the flight time, and no furtherexplanation is given.


42/100


Two Silos Geometry and Measurement

4. A farmer has two grain silos, both shaped like right circular cylinders, with dimensions shown inthe diagrams below.

a. How much greater is the volume of Silo B than the volume of Silo A? Show your work orexplain how you found your answer.

b. The farmer has the same amount of grain stored in each of the two silos. Silo A is filled to thetop. What is the height, in feet, of the level of the grain in Silo B? Show your work or explainhow you found your answer.



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Rubric for Two Silos

Score 3

The response contains:• a correct value expressing the difference or ratio between the volumes of Silo B and Silo A, with

work or explanation, and a correct value for the height in feet of the level of grain in Silo B, withwork or explanation.


Score 2

The response may contain:• an incorrect value for the difference in volumes due to a significant error, e.g., the use of an incorrect

formula or the use of diameter instead of radius in the correct formula, and a correct or consistentvalue for the height of the grain in Silo B, with work or explanationOR

• a correct value for the difference in volumes, with work or explanation, but an incorrect value for theheight of the grain in Silo B due to a significant error in an appropriate process.

Score 1

The response may contain:• a correct value for the difference in volumes, with work or explanation

OR

• an incorrect value for the difference in volumes due to a significant error, e.g., the use of an incorrectformula or the use of diameter instead of radius in the correct formula, with work or explanation.

Score 0



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Scored Student Responses for Two Silos

Question 4: Two Silos Score 3

The response to part a establishes the size of Silo B relative to Silo A, which is an appropriate response. Forpart b, the student uses that ratio to determine the appropriate height. This response demonstrates a full andcomplete understanding of the task.


45/100




The response demonstrates a full and complete understanding. The correct response to part a is supported bycalculations showing the use of the correct formula. For part b, the student finds the volume of Silo A and solvesfor the height at which Silo B would have the equivalent volume.


46/100




The student uses an incorrect formula for volume (formula for a cone rather than a cylinder) that yields anincorrect response, but correctly states that Silo B is 6.25 times larger than Silo A. That enables the student tocalculate the correct height for part b. This response demonstrates a reasonable understanding of the task.


47/100




The response demonstrates a reasonable understanding of the task. The correct answer in part a is supported bya sufficient explanation of the process. The student subtracts rather than divides when solving for the height inpart b, leading to an incorrect response.


48/100




The correct difference in volumes is provided, with supporting calculations and explanation. The response to partb is incorrect, with an insufficient explanation. The response demonstrates a partial understanding of the task.


49/100




The response demonstrates a partial understanding of the task. The difference in the volumes of the silos isdetermined correctly, with calculation shown. However, the student does not demonstrate the ability to find aprocess for determining the height needed in part b.


50/100




The response demonstrates merely an acquaintance with the topic. An incorrect formula is used for determiningthe differences in volumes, leading to an incorrect result. No attempt is made to solve part b.


51/100




The response offers an incorrect difference in volume, with evidence of the wrong formula being used due to aninappropriate understanding of the process. Part b is not attempted.


52/100


Kendra’s Travels Numerical and Proportional Reasoning

5. Kendra traveled to Europe and Japan on a business trip. In Europe, she exchanged 300 U.S. dollarsfor euros and spent 100 euros. She then went to Japan and exchanged her remaining euros for yen.

She spent 10,000 yen while in Japan.

The exchange rate during the time she traveled is shown below.

1.00 U.S. Dollar = 0.821774 Euro

1.00 U.S. Dollar = 110.565 Japanese Yen

Kendra will exchange her remaining yen for U.S. dollars. How much money, in U.S. dollars, willshe receive? Show your work or explain how you found your answer.



53/100


Rubric for Kendra’s Travels

Score 3

The response contains:• a correct answer ($87.87), with reasonable allowance for rounding differences, with supporting work

or explanation.


Score 2

The response may contain:• an incorrect answer due to a significant error in an appropriate process, with supporting work or

explanationOR

• an appropriate process with correct currency conversions, with supporting work or explanation, butmissing an intermediate spending transaction.

Score 1

The response may contain:• some demonstration of ability to convert between currencies.

Score 0



54/100


Scored Student Responses for Kendra’s Travels

Question 5: Kendra’s Travels Score 3

The response demonstrates a full and complete understanding of the task. The student explains the process andshows calculations. The student converts the dollars to euros and subtracts the number of euros spent. Thestudent then converts the remaining euros back to dollars and then converts them to yen. After subtracting theamount spent in Japan, the student converts the remaining yen back to dollars.


55/100




The student converts the dollars to euros and subtracts the number of euros spent. The student then converts theremaining euros back to dollars and then converts them to yen. After subtracting the amount spent in Japan, thestudent converts the remaining yen back to dollars. The response demonstrates a full and completeunderstanding of the task.


56/100




The response demonstrates a reasonable understanding of the task. Dollars are converted to euros, and thecorrect amount of euros is subtracted before converting the remaining euros back to dollars. The dollars are thenconverted to yen. However, the student does not subtract the 10,000 yen spent before converting back to dollars.


57/100




The response demonstrates a reasonable understanding of the task. The student converts dollars to euros andsubtracts the correct number of euros. In converting the euros back to dollars, the student divides by theconversion factor instead of multiplying. The dollars are then converted to yen, the spent yen subtracted, and theremainder converted back to dollars.


58/100




The student correctly converts dollars to euros and subtracts the 100 euros spent. The remaining euros areconverted back to dollars incorrectly. The dollars are converted to yen and the spent yen are subtracted. Thestudent indicates an intention of converting the remaining yen back to dollars, but does not show the calculation.The response demonstrates a partial understanding of the task.


59/100




The response shows a partial understanding of the task. The student correctly converts the $300 to euros, butthen incorrectly subtracts the amount spent in dollars from the number of euros. The remaining amount is thenincorrectly divided by the conversion factor from euros to dollars. The calculated amount is then converted to yen,the amount spent is subtracted, and the remainder is converted back to dollars.


60/100




The response shows merely an acquaintance with the task. In converting dollars to euros, the student dividesinstead of multiplying. After subtracting the spent euros, the student converts the remaining amount back todollars. However, in converting to yen, the student divides instead of multiplies and then performs no furthercalculations.


61/100




The response shows merely an acquaintance with the task. The student correctly converts dollars to euros, butthen does not use the result of that calculation. Instead, the student subtracts the spent euros from the amount ofdollars and coverts that remainder to euros.


62/100


Population of New London County Working with Data

6. The table below shows the population of New London County, Connecticut, from 1950 to 2000.

Year Population

1950 145,000

1960 186,000

1970 230,000

1980 238,000

1990 255,000

2000 259,000

a. Make a scatter plot of the data. Be sure to label the axes.

b. Make a reasonable prediction for the population in New London County in 2010. Explain howyou found your answer.



63/100


Rubric for Population of New London County

Score 3

The response contains:• a correct graph of the provided data and a reasonable prediction of the population of New London in

2010 with supporting work or explanation.


Score 2

The response may contain:• a correct graph of the provided data and a reasonable prediction but with incomplete work or

explanationOR

• a correct graph of the provided data and an inaccurate prediction due to a significant errorOR

• a graph which may contain errors but demonstrates some understanding of the concept and areasonable prediction with supporting work or explanation.

Score 1

The response may contain:• a correct graph of the data provided and either no prediction or an incorrect prediction resulting from

an invalid approachOR

• a graph which may contain errors but demonstrates a some understanding of the concept and areasonable prediction without supporting work or explanation.

Score 0The student has demonstrated merely an acquaintance with the topic. No evidence is present to suggestthat the student has the ability to solve problems of this general type.


64/100


Scored Student Responses for Population ofNew London County

Question 6: Population of New London County Score 3

The response demonstrates a full and complete understanding of the task. No break in scale symbol is used, butotherwise the scaling is correct and the plotting is correct with the exception of a minor error plotting thepopulation for 1980. The prediction for the population in 2010 is reasonable and is supported by a sufficientexplanation.


65/100




The response demonstrates a full and complete understanding of the task. The graph is appropriately labeled.The scaling is correct and the plotting is correct, with the exception of a minor error plotting the population for1980. The prediction is reasonable and is supported with a sufficient explanation.


66/100




The response demonstrates a reasonable understanding of the task. The graph is appropriately labeled. Thescaling and plotting are correct. The student makes a reasonable prediction, but the explanation does not providea clear rationale for determining the predicted populations.


67/100




The response demonstrates a reasonable understanding of the task. The graph is appropriately labeled. Thescaling and plotting are correct. An appropriate strategy (the drawing of a trend line) for predicting the futurepopulation is employed, but the student makes an error in executing the strategy that leads to a response that isnot reasonable.


68/100




The response demonstrates a partial understanding of the task. The graph is appropriately labeled. No break inscale symbol is used. The plotting is acceptable. The prediction is flawed due to the incorrect execution of thetrend line. In addition, the explanation that the prediction results from the point where the x- and y- axes meet isflawed.


69/100




The response demonstrates a partial understanding of the task. The graph is appropriately labeled. No break inscale symbol is used, and there is a scaling error between 250,000 and 300,000. The student does not arrive at areasonable prediction due to an invalid approach to the task.


70/100




The response demonstrates merely an acquaintance with the task. The graph is not labeled and there aremultiple scaling errors on the y-axis, as well as incorrectly plotted points. The prediction is not within a reasonablerange based on the data, and the student provides no explanation for the prediction.


71/100




The response demonstrates merely an acquaintance with the task. The x- and y- axes are not labeled. Thescaling is not correct due to the use of the population figures for the numbers used at even intervals on the y-axis.The prediction is not within an acceptable range, with an insufficient explanation for the prediction.


72/100


Chocolate Candy Working with Data

7. Below is a picture of a box of chocolate candies.

S

N

S

C

N

S

N

S

S

C

S

N

N

S

C

S

S C

N S

S N

N S

S – Solid Chocolate

C – Creme Filling

N – Nuts

a. If Malik chooses a piece of the candy at random, what is the probability that it is solidchocolate? Show your work or explain how you found your answer.

b. Malik eats a total of 4 pieces of candy, 2 with nuts, 1 solid chocolate, and 1 with cream filling.He then gives the box to Fatima. If Fatima wants a piece of candy with nuts, what is the probability that she will pick one? Show your work or explain how you found your answer.

c. Fatima also eats 4 pieces of candy and then gives the box to Samira. If the probability ofSamira selecting a piece of candy with cream filling is 0.1875, how many cream-filled candiesdid Fatima eat? Show your work or explain how you found your answer.



73/100


Rubric for Chocolate Candy

Score 3

The response contains:• correct answers to Part A (0.5 or equivalent), Part B (0.3 or equivalent), and Part C (0 or None) with

supporting work or explanation.


Score 2

The response may contain:• a missing or incorrect answer to one part of the question, but correct or consistent answers to the

other two parts of the question, with supporting work or explanation.

Score 1

The response may contain:

• a correct answer to only one of the three parts with some work or explanation.

Score 0



74/100


Scored Student Responses for Chocolate Candy

Question 7: Chocolate Candy Score 3

The response demonstrates a full and complete understanding of the task. The student correctly calculates eachprobability and accurately states that Fatima ate no cream-filled chocolates. The student shows calculations andprovides explanations supporting each answer.


75/100




The student correctly determines the probability of choosing a solid candy and the number of cream-filled candieseaten by Fatima. The probability of obtaining a candy with nuts is incorrect; however, the explanation and thecalculations shown reveal that the student made an error in counting which does not detract from theunderstanding of the task. The response demonstrates a full and complete understanding of the task.


76/100




The response demonstrates a reasonable understanding of the mathematical concepts and processes embeddedin the application. The correct probabilities for obtaining solid candies and candies with nuts are calculated andexplained. The student then calculates the number of candies available for choosing, but does not answer thequestion of how many Fatima would have eaten.


77/100




The probabilities of choosing a solid chocolate and a candy with nuts are calculated and explained. The equationused to determine the number of candies eaten by Fatima is not appropriate and results in an incorrect response.The response shows a reasonable understanding of the task.


78/100




The response demonstrates a partial understanding of the topic. The student correctly calculates the probability ofchoosing a solid chocolate. However, in attempting to determine the probability of choosing a candy with nuts, thestudent fails to account for the chocolate candy that has already been eaten. The response to part c misappliesthe calculation of the number of candies available.


79/100




The student correctly calculates the probability of choosing a solid chocolate. However, in attempting to determinethe probability of choosing a candy with nuts, the student fails to account for the chocolate candy that has alreadybeen eaten. The student does not attempt to determine the number of cream-filled candies eaten by Fatima. Theresponse demonstrates a partial understanding.


80/100




The response demonstrates merely an acquaintance with the topic. The student fails to state the probabilities. Noattempt is made to determine the number of cream-filled chocolates eaten by Fatima.


81/100




The response demonstrates merely an acquaintance with the topic. The responses only describe the probabilityof picking any one of the individual pieces of candy. The student does not attempt part c.


82/100


Organism Lengths Numerical and Proportional Reasoning

8. Students in a biology class measured and recorded the lengths of different microscopic organisms.The results are recorded in the table below.

Organism Lengths

Organism Length (millimeters)

A 0.00065

B 4.72 × 10 –4

C8

10,000

D ?

a. A fourth organism (D) was measured and found to be40

1 the length of organism B. What was

the length of organism D? Show your work or explain how you found your answer.

b. Which of the four organisms in the table was longest? Show your work or explain how you

found your answer.



83/100


Rubric for Organism Lengths

Score 3

The response contains:

• The correct length of organism D (1.18 × 10 –5 mm or 0.0000118 mm) for Part A, with supporting

work or explanation, and an indication that organism C is the longest of the four organisms forPart B, with supporting work or explanation.


Score 2

The response may contain:• a value in Part A that is incorrect due to an error and that is between 0 and 1, with work or

explanation, and a consistent answer in Part B (due to the error in Part A) with work or explanationOR

• a correct value in Part A with work or explanation and a correct answer in Part B with insufficient

work or explanation.

Score 1

The response may contain:• a correct value in Part A but an incorrect answer in Part B

OR

• a value in Part A that is incorrect due to more than one error and that is between 0 and 1, and aconsistent answer in Part B (due to the error in Part A). There is some work or explanation in Part Aand/or Part B.OR

• a value in Part A that is incorrect due to an error and that is outside the range of 0 to 1, and a

consistent answer in Part B (due to the error in Part A). There is some work or explanation in Part Aand/or Part B.

The response demonstrates some understanding of scientific notation.

Score 0



84/100


Scored Student Responses for Organism Lengths

Question 8: Organism Lengths Score 3

The response demonstrates a full and complete understanding of the task. The length of the organism iscalculated correctly, with the calculations shown and explained. The student then converts the length of eachorganism into decimal form in order to determine the longest.


85/100




The length of organism d is calculated correctly, with calculations shown. The student then converts the length oforganism c into decimal form and explains why that would be the longest of the organisms. The response, whilebrief, is sufficient to show a full understanding of the task.


86/100




The response demonstrates a reasonable understanding of the task. Part a has one or more transcription errors,and both of the responses are rounded incorrectly. In part b, the student converts the lengths of the organisms todecimal form in order to correctly determine the longest.


87/100




The response demonstrates a reasonable understanding of the task. The student errs in converting the length oforganism b into decimal form, resulting in an incorrect calculation for organism d. However, the remainingorganism lengths are converted correctly by the student, and the longest organism is determined correctly, basedon the answer found in the first section.


88/100




The student correctly determines the length of organism d. However, in attempting to determine the longestorganism, the response does not demonstrate an understanding of how to determine which of the decimals is thelargest number. The student demonstrates a partial understanding of the task.


89/100




The student demonstrates a partial understanding of scientific notation. The length of one organism is convertedinto decimal form. No other correct calculations are shown.


90/100




In this response, the student errs in converting the length of organism b into decimal form and then multiplies by2/5 instead of by .025. This response lacks convincing evidence that the student fully comprehends the essentialmathematical ideas addressed by this task.


91/100




The student converts 1/40 into decimal form as an incorrect means for determining the length of organism d. Thedetermination of the longest organism is not consistent with the calculations shown or the previous response. The

response demonstrates merely an acquaintance with the task.


92/100


CAPT Mathematics Grid-In Items


93/100


CAPT Mathematics Grid-In Item:Soup Cans

Geometry and Measurement

9. Latitia’s Foods is introducing a new line of soups. The soups will be sold in cans that are 4 inchestall and have a diameter of 3 inches. The labels will wrap around the entire outside of each can,excluding the top and bottom.

Determine the area of the label. Round your answer to the nearest tenth of a square inch.

Do your work above and remember to grid your answer in your answer booklet.


94/100


CAPT Mathematics Grid-In Item:Stopping Distance

Algebraic Reasoning

10. The relationship between the distance d , in feet, required to stop a vehicle and s, the speed in miles per hour that the vehicle was traveling, is given by the equation

20.0155sd

f =

where f represents the coefficient of friction between the tires and the road.

It took a car 205 feet to stop. What speed was the car traveling? Use 0.3 f = and round your

answer to the nearest mile per hour.



95/100


CAPT Mathematics Grid-In Item:Coffee Special

Algebraic Reasoning

11. A store sells gourmet coffee at a discount with the purchase of a coffee maker that costs $26.00.The graph below shows the total price in dollars, T , for a coffee maker plus the amount of coffee in pounds, P.

Carl plans to buy a coffee maker and four pounds of coffee. How much does the store charge per pound for the gourmet coffee?



96/100


CAPT Mathematics Grid-In Item:Entertainment CenterGeometry and Measurement

12. José wants to buy a new TV that will fit the opening of his entertainment center. The height of theopening in his entertainment center is 27 inches. Usually, the opening of an entertainment centerhas a width-to-height ratio of 4:3.

What is the diagonal measurement of the opening in José’s entertainment center?



97/100


CAPT Mathematics Grid-In Item:Mary’s Number Cubes

Working with Data

13. Mary rolls two number cubes with sides numbered from 1 to 6.

1

3

2 4

6

5

If she rolls a 3 on one of the cubes, what is the probability that the sum of the numbers facing upon both cubes is greater than or equal to 5? Express your answer as a decimal rounded to thenearest hundredth.



98/100


CAPT Mathematics Grid-In Item:Bloodhound

Numerical and Proportional Reasoning

14. The bloodhound, a type of dog, has 4.0 × 109

scent receptors in its nose. A typical human has1.2 × 10

7 scent receptors. How many times more scent receptors does a bloodhound have than ahuman? Round your answer to the nearest whole number.



99/100


CAPT Mathematics Grid-In Item:Joseph’s Final Grade

Working with Data

15. Joseph’s final averages in science class are shown in the table below. What is the minimum scoreJoseph can get on the final exam in order to receive at least a 90 for his final grade?

Homework Quizzes Tests Final Exam

% Toward Final Grade 20 20 40 20

Average 93 92 85 ?



100/100

CAPT Mathematics Grid-In Item:Picnic Food

Numerical and Proportional Reasoning

16. Each summer, a high school sponsors a picnic for new students, their parents and teachers. Lastyear, 65 pounds of hamburger patties were cooked to serve between 250 and 300 people. Thisyear, the school expects between 325 and 375 people.

Estimate the number of pounds of hamburger patties that should be ordered.


2007 Math Released Item Packet

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