a classroom resource built for the new math teks · a classroom resource built for the new math...

A Classroom Resource Built forthe New Math TEKS

K–5 Preview Material

Standards-Based Assessment + Instruction

We Set the Standards!

380+ open-ended and engaging math tasks delivered online

for Texas educators!

New!

For over 20 years, Exemplars has been providing teacher-friendly tools to integrate performance-based instruction and assessment.

Problem Solving for the 21st Century: Built for the TEKS is a supplemental math resource for grades K–5 that features:

• Rich problem-solving supplements organized by Units of Study around the new math TEKS.

• 380+ open-ended tasks that address the big mathematical ideas within each Unit, connecting the TEKS Content Standards to the Mathematical Processes.

• Differentiated material for instruction, exploration and formative assessment.

• Summative assessments that include anchor papers and scoring rationales.

• Standards-based rubrics that provide clear guidelines for assessing student work and support the TEKS Mathematical Process Standards.

• Preliminary Planning Sheets that provide teachers with a useful tool for lesson planning. The Mathematical Concepts, Problem-Solving Strategies, Mathematical Vocabulary/Symbolic Representations, Possible Solutions and Mathematical Connections are provided for each task.

• Online delivery for teachers through the Exemplars Library.

Exemplars authentic tasks engage students and develop theirabilities to reason and communicate mathematically as well as toformulate mathematical connections.

2 exemplars.com

An introduction …

“The Exemplars program is designed to

assess students’ problem-solving and

mathematical-communication skills. It

also supports higher-level thinking and

extension of mathematical reasoning.“

S. DementConverse, TX

exemplars.com

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“I was introduced to Exemplars through

the USI Collaborative for Excellence at the

University of Texas at El Paso ... using

Exemplars has added strength and depth

to our mathematics program.”

V. BrownEl Paso,TX

Our very first order came from Alief ISD

more than two decades ago!

Rubrics are a valuable tool for assessing student work and an important component of Exemplars.Our assessment rubric allows teachers to examine student work against a set of analytic criteria.

4 exemplars.com

Exemplars® Standards-Based Math Rubric

Novice

Problem Solving Reasoning and Proof Communication RepresentationConnections

Apprentice

No strategy is chosen, or a strategy is chosen that will not lead to a solution.

Little or no evidence of engagement in the task is present.

A partially correct strategy is chosen, or a correct strategy for only solving part of the task is chosen.

Evidence of drawing on some previous knowledge is present,showing some relevant engagement in the task.

Arguments are made with no mathematical basis.

No correct reasoning nor justification for reasoning is present.

Arguments are made with some mathematical basis.

Some correct reasoning or justification for reasoning is present.

No awareness of audience or purpose is communicated.

No formal mathematical terms or symbolic notations are evident.

Some awareness of audience or purpose is communicated.

Some communication of an approach is evident through verbal/written accounts and explanations.

An attempt is made to use formal math language. One formal math term or one symbolic notation is evident.

No connections are made or connections are mathematically or contextually irrelevant.

A mathematical connection is attempted but is partially incorrect or lacks contextual relevance.

No attempt is made to construct a mathematicalrepresentation.

An attempt is made to construct amathematical representation to record and communi-cate problem solving but is not accurate.

COPYRIGHT ©2001, REVISED 2014 BY EXEMPLARS, INC. ALL RIGHTS RESERVED.

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There are four levels of performance (Novice, Apprentice, Practitioner and Expert). The Practitioner level is proficient and meets the standard. Our rubric criteria strongly support the TEKS Mathematical Process Standards. To learn more, refer to our crosswalk on pages 24 and 25.

Practitioner

Problem Solving Reasoning and Proof Communication RepresentationConnections

Expert

A correct strategy is chosen based on mathematical situationin the task.Planning or monitoring of strategy is evident.Evidence of solidifying prior knowledge and applying it to the problem- solving situation is present.Note: The Practitioner must achieve a correct answer.

An efficient strategy is chosen and progress toward a solution is evaluated.Adjustments in strategy, if necessary, are made along the way, and/or alternative strategies are considered.Evidence of analyzing the situation in math-ematical terms and extending prior knowl-edge is present.Note: The Expert must achieve a correct answer.

Arguments are con-structed with adequate mathematical basis.A systematic approach and/or justification of correct reasoning is present.

Deductive arguments are used to justify decisions and may result in formal proofs.Evidence is used to justify and support decisions made and conclusions reached.

A sense of audience or purpose is communicated.Communication of an approach is evident through a methodical, organized, coherent sequenced and labeled response.Formal math language is used to share and clarify ideas. At least two formal math terms or symbolic notations are evident in any combination.

A sense of audience and purpose is communicated.Communication at the Practitioner level is achieved, and communication of argument is supported by mathematical properties.Formal math language is used to consolidate math thinking and to communicate ideas. At least one of the math terms or symbolic notations is beyond grade level.

A mathematical connection is made. Proper contexts are identified that link both the mathematics and the situa-tion in the task. Some examples may include one or more of the following:• clarification of the mathematical or situational context of the task• exploration of mathematical phenomenon in the context of the broader topic in which the task is situated• noting patterns, structures and regularities

Mathematical connections are used to extend the solution to other mathematics or to a deeper understanding of the mathematics in the task. Some examples may include one or more of the following:• testing and accepting or rejecting of a hypothesis or conjecture• explanation of phenomenon• generalizing and extending the solution to other cases

An appropriate and accurate mathematical representation is constructed and refined to solve problems or portray solutions.

An appropriate mathematical representationis constructedto analyze rela-tionships, extend thinking, and clarify or interpret phenomenon.

Meets the Standard!

Rubrics also present students with important information about what is expected and what kind of work meets the standard. Our student rubrics may be used to develop a learner’s ability to self- and peer-assess.

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Level

NoviceMakes an effortNo or little understanding

ApprenticeOkay, good tryUnclear if student understands

PractitionerExcellentClearStrong understandingMeets the standard.

ExpertWow, awesome!Exceptional understanding!

Problem Solving Reasoning and Proof Communication Connections Representation

I did not understand the problem.

I only understand part of the problem. My

strategy works for part of the problem.

I understand the problem and my

strategy works. My answer is correct.

I understand the problem. My answer is correct. I used a rule,

and/or verified that my strategy is correct.

My math thinking is not correct.

Some of my math thinking is correct.

All of my math thinking is correct.

I showed that I knew more about a math ideathat I used in my plan.Or, I explained my rule.

I used no math language and/or math notation.

I used some math language and/or math notation.

I used math language and/or math notation accurately throughout

my work.

I used a lot of specific math language and/

or notation accurately throughout my work.

I did not notice anything about the

problem or the numbers in my work.

I tried to notice something, but it is not about the math in the

problem.

I noticed something about my math work.

I noticed something in my work, and used that to

extend my answer and/or I showed how this problem is

like another problem.

I did not use a math representation to help solve the problem and explain

my work.

I tried to use a math representation to help solve the problem and

explain my work, but it has mistakes in it.

I made a math representation to help solve the problem and

explain my work, and it is labeled and correct.

I used another math representation to help solve

the problem and explain my work in another way.

COPYRIGHT ©2005, REVISED 2012 BY EXEMPLARS, INC. ALL RIGHTS RESERVED.

Exemplars® Jigsaw Student Rubric

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TASKBen and Sara have four glass jars. Ben and Sara have one hour and fifteen minutes to collect four different types of insects. Ben and Sara will put each type of insect in a glassjar. Ben and Sara will round the number of each type of insect collected to the nearest ten and let any extra insects go. Ben and Sara start collecting insects at one o’clock in the afternoon. They collect 23 ants, 12 butterflies, 10 beetles and 14 grasshoppers. What time do Ben and Sara stop catching insects? How many insects do Ben and Sara let go from each jar? Show all your mathematical thinking.

Instructional Task Sample: Grade 3

Insects in JarsTask Alignments:• Place Value Unit (TEKS covered 3.2A, 3.2B, 3.2C, 3.2D)• Mathematical Process Standards: 3.1A, 3.1B, 3.1E, 3.1G

View all Units for this online

product on pages 26–27!

To assist Texas educators in teaching a focused mathematics curriculum, Exemplars has grouped individual TEKS standards into rich Units of Study. Throughout the Exemplars Library, both instructional tasks and summative assessments are provided to address the big mathematical ideas within each Unit. A summative assessment sample may be found on page 11.

Task-Specific EvidenceThis task requires students to use place value to round whole numbers to the nearest 10. The students are also expected to find the difference between rounded numbers and given numbers.

Alternative Versions of TaskMore Accessible Version:Ben and Sara have four glass jars. Ben and Sara will collect four different types of insects. Ben and Sara will put each type of insect in a glass jar. Ben and Sara will round the number of eachtype of insect collected to the nearest ten and let any extra insects go. They collect 23 ants,12 butterflies, 10 beetles and 14 grasshoppers. How many insects do Ben and Sara let gofrom each jar? Show all your mathematical thinking.

More Challenging Version:Ben and Sara have four glass jars. Ben and Sara have one hour and fifteen minutes to collect four different types of insects. Ben and Sara will put each type of insect in a glass jar. Ben and Sara will round the number of each type of insect collected to the nearest ten and let any extra insects go. Ben and Sara start collecting insects at one o’clock in the afternoon. They collect 123 ants, 212 butterflies, 100 beetles and 314 grasshoppers. What time do Ben and Sara stop catching insects? How many insects do Ben and Sara let go from each jar? Show all your mathematical thinking.

To meet individual student’s needs, each instructional task is differentiated to include a “more accessible” and a “more challenging” version. These problems may be used as opportunities for students to learn new mathematical strategies, language and representations. Instructional tasks may also be used in combination with Exemplars rubric for formative assessment.

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Four or moreinstructional tasks

are offered for each Unit!

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Possible solution representations and examples of mathematical connections are provided for each instructional task.Mathematical solutions are provided for the alternative versions.

Possible Solutions Ben and Sara stop collecting insects at 2:15 p.m. They let 3 ants, 2 butterflies, 0 beetles and 4 grasshoppers go from the glass jars.

Possible ConnectionsBelow are some examples of mathematical connections. Your students may discover some that are not on this list.

• Ben and Sara caught 59 total insects.• A total of 9 insects were let go.• 15 minutes is 1/4 of an hour.• They caught insects for 75 minutes.• They caught the most ants and the least beetles.• They caught an odd number of ants.• They caught an even number of butterflies, beetles and grasshoppers.• They caught a dozen butterflies.• If they caught 1 more ant they would have 2 dozen.• Relate to a similar task and state a math link.• Solve more than one way to verify the answer.• They caught 11 more ants than butterflies.• The total number of insect legs are found, 6 · 40 = 240 insect legs.

Alternative Version SolutionsMore Accessible Version:Ben and Sara let 3 ants, 2 butterflies, and 4 grasshoppers go from the glass jars.More Challenging Version:Ben and Sara stop collecting insects at 2:15 p.m. They let 3 ants, 2 butterflies and 4 grasshoppers go from the glass jars.

12

3

4567

8

910

11 12 12

3

4567

8

910

11 12 12

3

4567

8

910

11 12

InsectsAntsButterfliesBeetlesGrasshoppers

Numbersof InsectsCaught

23121014

InsectsKept in

Jar20101010

InsectsLet Go

3204

23 12 10 14

a b B g

20 10 10 10insects keptinsects caught

is jara is antsb is butterfliesB is beetlesg is grasshopper

Key

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Each instructional task is accompanied by a Preliminary Planning Sheet. Depending on the intended purpose of a task, this resource enables teachers to anticipate which math concepts and skills students might be required to use or what instruction should be given ahead of time.

Preliminary Planning Sheet

Possible Problem-Solving Strategies• Model (manipulatives)• Diagram/Key• Table• Number line• Chart

Possible Mathematical Vocabulary/Symbolic Representation• Model • Fraction (1/4 ) • Greater than (>)/Less than (<)• Diagram/Key • Hour, minute • Equivalent/Equal to• Table • Most/Least • Time notation (2:15 p.m.)• Chart • Odd/Even • Place value (ones, tens...)• Number line • Dozen • Difference• Total/Sum

Possible Connections• Ben and Sara caught 59 total insects.• A total of 9 insects were let go.• 15 minutes is 1/4 of an hour.• They caught insects for 75 minutes.• They caught the most ants and the least beetles.• They caught an odd number of ants.

Underlying Mathematical Concepts• Rounding whole numbers to the nearest 10• Addition/Subtraction• Number sense to 23• Time notation

PreliminaryPlanning Sheets

serve as a teacher’s

“blueprint.”

Place Value UnitTEKS Covered: 3.2A, 3.2B, 3.2C, 3.2DProcess Standards: 3.1A, 3.1B, 3.1E, 3.1G

Insects in Jars

• They caught an even number of butterflies, beetles and grasshoppers.• They caught a dozen butterflies.• If they caught 1 more ant they would have 2 dozen.• Relate to a similar task and state a math link.• Solve more than one way to verify answer.• They caught 11 more ants than butterflies.• Found the total number of insect legs, 6 · 40 = 240 insect legs.

Possible Solutions Ben and Sara stop collecting insects at 2:15 p.m. They let 3 ants, 2 butterflies, 0 beetles and 4 grasshoppers go from the glass jars.

12

3

4567

8

910

11 12 12

3

4567

8

910

11 12 12

3

4567

8

910

11 12

is jara is antsb is butterfliesB is beetlesg is grasshopper

Key

InsectsAntsButterfliesBeetlesGrasshoppers

Numbersof InsectsCaught

23121014

InsectsKept in

Jar20101010

InsectsLet Go

3204

23 12 10 14

a b B g

20 10 10 10insects kept

insects caught

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Summative assessments are provided for each Unit of Study. These tasks are designed to evaluate if a student meets the TEKS standard(s) in a particular Unit. Anchor papers and scoring rationales are included with each problem. These tasks are not differentiated.

Summative Assessment Sample: Grade 3

One Hundred MilesTask Alignments:• Place Value Unit (TEKS covered 3.2A, 3.2B, 3.2C, 3.2D)• Mathematical Process Standards: 3.1A, 3.1B, 3.1E, 3.1G

TASKA team of five bikers are training to ride a combined total of one hundred miles in a race. The bikers practice every Sunday from seven o’clock in the morning to ten thirty in the morning. During practice a biker who rides 20 or more miles when rounded to the nearest ten is ready for the race. This past Sunday the first biker rode 28 miles. The second biker rode 24 miles. The third biker rode 27 miles. The fourth biker rode 12 miles. The fifth biker rode 8 miles. Did the team of five bikers meet the goal of riding a combined total of one hundred miles? How many bikers are ready for the race? Show all your mathematical thinking.

Task-Specific EvidenceThis task requires students to use place value to round whole numbers to the nearest 10. The students are also expected to add numbers to determine a combined total.

Additional task samples can beviewed online at exemplars.com!

Summative assessments include possible solution representations and examples of mathematical connections.

12 exemplars.com

Possible Solutions 3 bikers are ready to race: the 1st, 2nd and 3rd bikers. The determination of whether the team has ridden 100 miles will depend on whether the student adds the actual miles ridden or the rounded numbers.

Possible ConnectionsBelow are some examples of mathematical connections. Your students may discover some that are not on this list.

• Compare the difference in estimated total miles ridden to exact total miles ridden.

• The 5th biker rode 1/3 as far as the 2nd biker.

• Bikers 1 and 2 ride a total of 52 miles. That is more than 1/2 of the goal.

• Biker 4 rides 1/2 the distance of biker 2.

• If each biker rides 20 miles for a total of 100 miles, it is an equal share of miles per biker.

• 1 mile is 5,280 feet.

• Biker 3 rides an odd number of miles. All other bikers ride an even number of miles.

• 12 miles is a dozen.

• Relate to a similar task and state a math link.

• Solve more than one way to verify the answer.

• Add more bikers to recreate the task.

• Change the number of miles ridden to recreate the task.

28 + 24 + 27 + 12 + 8 = 9930 + 20 + 30 + 10 + 10 = 100

Biker12345

Miles282427128

Tens22210

Ones4 584728

RoundedNumber

3020301010

BikerNumber

12345

MilesBiked

282427128

Miles Rounded tothe Nearest Ten

3020301010

TotalMiles

2852799199

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Each summative assessment task is accompanied by a Preliminary Planning Sheet (PPS). In this setting, the PPS allows teachers to foresee what instruction should be done before the summative assessment is given. It may also be used in combination with the Exemplars rubric to support teachers in assessing student work.

Preliminary Planning Sheet

Possible Problem-Solving Strategies• Model (manipulatives)• Diagram/Key• Table• Number line

Possible Connections• Compare the difference in estimated total miles ridden to exact total miles ridden.• The fifth biker rode 1/3 as far as the second biker.• Bikers 1 and 2 ride a total of 52 miles. That is more than 1/2 of the goal.• Biker 4 rides 1/2 the distance of biker 2.• Change the number of miles ridden to recreate the task.• If each biker rides 20 miles for a total of 100 miles, it is an equal share of miles per biker.

Underlying Mathematical Concepts• Rounding whole numbers to the nearest 10• Adding or combining whole numbers• Comparison

Place Value Unit TEKS Covered: 3.2A, 3.2B, 3.2C, 3.2DProcess Standards: 3.1A, 3.1B, 3.1E, 3.1G

One Hundred Miles

PreliminaryPlanning Sheets also accompany

summative assessments.

Possible Mathematical Vocabulary/Symbolic Representation• Model • Time notation (9:00 a.m.) • Per• Diagram/Key • Hours, minutes • Ordinal numbers• Table • Average (1st, first, 2nd,• Number line • Total/Sum second, 3rd, third...)• More/Less than • Percents (25%, 50%...) • Miles • Estimation • Fractions (1/2, 1/4...) • 5,280’ (ft.)• Odd/Even • Place value (ones, tens) • Dozen

• Add more bikers to recreate the task.• 1 mile is 5,280 feet.• Relate to a similar task and state a math link.• Biker 3 rides an odd number of miles. All other bikers ride an even number of miles.• 12 miles is a dozen.• Solve more than one way to verify answer.

Possible Solutions 3 bikers are ready to race: the 1st, 2nd and 3rd bikers. The determination of whether the team has ridden 100 miles will depend on whether the student adds the actual miles ridden or the rounded numbers.

28 + 24 + 27 + 12 + 8 = 9930 + 20 + 30 + 10 + 10 = 100

Biker12345

Miles282427128

Tens22210

Ones4 584728

RoundedNumber

3020301010

BikerNumber

12345

MilesBiked

282427128

Miles Rounded tothe Nearest Ten

3020301010

TotalMiles

2852799199

NOVICE

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Summative assessment tasks include anchor papers and scoring rationales at the four levels of Exemplars rubric. These tools demonstrate what work meets or does not meet the TEKS standard(s) for a Unit and explains why.The Novice level demonstrates little or no evidence of understanding the mathematical concepts of the task.

Scoring rationalesdescribe why each piece

of student work is assessed at a specific performance level for each of the Exemplars

rubric criterion.

P/S R/P Com Con Rep A/Level N N N N N N

Reasoning & ProofNovice

Problem SolvingNovice

CommunicationNovice

ConnectionsNovice

The student’s strategy of adding the numbers 20, 28, 24, 12, and 8 does not work to solve the task. The student’s answer, “no,” is based on an incorrect strategy so is not consideredcorrect.

The student does not demonstrate correct reasoning of the underlying concepts in the task. The student finds the total of numbers that five bikers ride on Sunday and does notaddress the concept of rounding that is needed to solve the second part of the task.

The student does not use any mathematical language to communicate her/his reasoning and proof.

The student does not make a mathematically relevant observation about her/his solution.

Scoring RationalesCriteria and

Perfomance Levels

RepresentationNovice

The student does not use a representation to support her/his reasoning.

The Apprentice level has the broadest range, from a student who is just beginning to demonstrate mathematical understanding to a student who almost meets the Unit standard(s).

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APPRENTICE P/S R/P Com Con Rep A/Level A P P P A A

Reasoning & ProofPractitioner

Problem SolvingApprentice

CommunicationPractitioner

ConnectionsPractitioner

The student’s strategy of using a table to indicate the number of racers, miles rode, and total miles would work to solve the first part of the task but the student indicates 18 miles instead of eight miles for the fifth biker. This leads to a total of 109 miles. The student’s answer, “YES-109 miles,” is not correct. The student rounds 18 instead of eight, which results in an incorrect answer, “4 can race,” for the second part of the task.

The student demonstrates correct reasoning of the underlying concepts in the task. The student finds the total miles that five bikers ride but makes the error of adding 18 miles instead of eight. The student demonstrates understanding of rounding numbers to the nearest 10. Using 18 instead of eight is a careless error and is not considered a lack of reasoning and proof.

The student correctly uses the mathematical terms miles, total from the task. The student also correctly uses the terms table, dozen.

The student makes the mathematically relevant observation, “12 miles is a dozen miles.”


Perfomance Levels

RepresentationApprentice

The student’s table is appropriate but not accurate. The student states 18 “miles rode” by biker five instead of eight miles. The 109 total miles for the five bikers is not correct.

16 exemplars.com

PRACTITIONER Student 1


Problem SolvingPractitioner



The student’s strategy of using a table to indicate the number of racers, miles ridden, and total miles works to solve the first part of the task. The student’s answer, “did not ride 100 miles,” is correct. The student’s strategy of making a table to indicate the miles each biker rides, the rounding of the miles, and if each biker is ready to race is correct. The student’s second answer, “Bikers 1 2 and 3 can race,” is correct.

The student demonstrates correct reasoning of the underlying concepts in the task. The student finds the total miles that five bikers ride and correctly rounds the miles that each biker rides.

The student correctly uses the mathematical terms miles, total from the task. The student also correctly uses the terms table, “estimatation” (estimation), AM, Sunday, Saturday. The words round and add are not considered mathematicallanguage as the rubric does not accept the use of verbs as communication.

The student makes the mathematically relevant observations, “biker 1-fastest racer,” “biker 8-slowest racer,” and, “add the estimatation-it is 100 miles but you can’t use that just 99 miles. 3 round up 28, 27, 8. 2 round down 24, 12,” “Theyonly ride in the AM,” and, “Biker 4 and 5 should practice longer on Sunday or on Saturday.”

RepresentationPractitioner

The student’s first table is appropriate and accurate. The labels and entered data is accurate. The student’s second table is also appropriate and accurate. The labels and entered data is accurate.


Perfomance Levels

P/S R/P Com Con Rep A/Level P P P P P P

The Practitioner level meets the Unit standard(s) and is proficient in understanding the underlying mathematics of the task. A correct solution must be achieved.

The Goal!

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The student’s strategy of using a diagram to indicate the number of racers, miles ridden, and then adding the rounded miles for a total of 100 miles works to solve the first part of the task. The student’s answer, “YES They did ride 100 miles,” is correct. The student’s strategy of using her/his key again to indicate the miles each biker rides, the rounding of the miles, and if each biker is ready to race is correct. The student’s second answer, “onle 3 can race,” is correct.

The student demonstrates correct reasoning of the underlying concepts in the problem. The student finds the total miles that five bikers ride and correctly rounds the miles that each biker rides to determine that three bikers are ready to race.

The student correctly uses the mathematical term miles from the task. The student also correctly uses the terms diagram, key, more.

The student makes the mathematically relevant observation, “28 m – 8 m = 20 m between B1 and B5. The fastest and slowest bikers.”


The student’s diagram is appropriate and accurate. The key and entered data is accurate.


Perfomance Levels


The Goal!The students’ solutions on pages 16–19 meetthe Unit standard(s).

Throughout this resource there is often more than one anchor paper associated with a level of performance. These are intended to demonstrate different strategies a student might use or different misconceptions a student might have (depending on the performance level).

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PRACTITIONER Student 3 (continued on next page)





The student’s strategy of using a table to indicate the number of racers, total miles, and then adding the miles for a total of 99 miles works to solve the first part of the task. The student’s answer, “NO,” is correct. The student’s strategy of using a number line to indicate the miles each biker rides, the rounding to nearest 10, and circling the miles that would round to or past 20 and putting an X by the number of miles that would not, works to solve the second part of the task. The student’s answer, “3 are ready,” is correct.

The student demonstrates correct reasoning of the underlying concepts in the task. The student finds the total miles that five bikers ride and correctly determines that three bikers are ready to race.

The student correctly uses the mathematical terms miles, total, first from the task. The student also correctly uses the term table, number line.

The student makes the mathematically relevant observation, “1st Biker is the fastest.”


Perfomance Levels


The student’s table is appropriate and accurate. All labels are included and the entered data is correct. The student’s number line is appropriate and accurate. All labels are included and the jump for miles and the line for the rounding results are correct. NOTE: Students may choose printed number lines to use as a representation if they independently select one from a variety of number lines left on a math shelf, table, etc.


The Goal!

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Multiple student work samples are provided

at various performance

levels.

The Expert level goes beyond the Unit standard(s) and shows a deeper understanding of the underlying mathematics.

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EXPERT (continued on next page) P/S R/P Com Con Rep A/Level E E E E E E

Reasoning & ProofExpert

Problem SolvingExpert

CommunicationExpert

ConnectionsExpert

The student’s strategy of using a table to indicate the number of bikers, number of miles, total miles, and ready to race works to solve the task. The student’s answer, “NO, 3 bikers-1, 2, and 3,” is correct. The student brings prior knowledge of fractions, elapsed time, and average miles per hour to her/his solution.

The student demonstrates correct reasoning of the underlyingconcepts in the task. The student finds the total miles that five bikers ride and correctly determines that three bikers are ready to race. The student applies correct reasoning of fractions, elapsed time and miles per hour in her/his solution.

The student correctly uses the mathematical terms miles, total, 5th, 1st from the task. The student also correctly uses the terms more, number, AM, hours/hrs, min., “averge” (average), per. The student correctly uses the mathematical notation 1/4, 3 1/2, 7:00, 8:00, 9:00, 10:00, 10:30.

The student makes the mathematically relevant observations, “They need 1 more mile for 100 miles,” “The 5th biker is slowest,” and, “The 1st biker is the fastest.” The student makes the Expert con-nections, “Biker 2 rode almost 1/4 of 100. 1 more miles is 25-1/4 of 100,” “7:00 AM-8:00 AM-1 hour,” “8:00 AM-9:00 AM- 2 hours, 9:00 AM-10:00 AM-3 hours, 10:00 AM-10:30 AM-3 1/2 hours total time riding.” The student finds “210 min total.” The student also determines, “I think the averge is about 6 miles per hour,” by dividing 20 miles by 3 hours and then by 4 hours. It appears that the student is not able to divide the 20 miles by 3 1/2 hours so rounds both down and up and selects six miles per hour.


Perfomance Levels

RepresentationExpert

The student’s table is appropriate and accurate. All labels are included and the entered data is correct. The student uses the data presented on the table to support work with fractions, elapsed time, and miles per hour.

800-450-4050 21

EXPERT

The achievement level of Expert is not a common

occurrence when assessing student

work.

Sign up for a FREE 30-day Trial!Using the trial feature in Exemplars Library you can explore Problem Solving for the 21st Century: Built for the TEKS firsthand. Simply go to exemplarslibrary.com/teks-trial to sign up.

aView Unit definitions.

aTry grade-specific instructional tasks and corresponding summative assessments.

aPrint Preliminary Planning Sheets to help with your lesson preparation.

aExplore how anchor papers and scoring rationales can be effective assessment tools.

aExperience for yourself just how easy Exemplars Library makes it to find and incorporate problem-solving tasks into your classroom that support the new math TEKS Content Standards and Mathematical Processes.

Exemplars Advantages

22 exemplars.com

Try sample tasks with your

students!

For Teachers:• Authentic performance material that supports the new math TEKS.• Differentiated versions to use with students at varying levels.• Standards-based scoring rubrics at four levels of performance.• Preliminary Planning Sheets that serve as a useful teaching resource.• Anchor papers and scoring rationales that provide guidelines for assessment and offer concrete examples of student work at each level of the rubric.• Tools to develop students’ critical thinking and reasoning skills as well as their abilities to self- and peer-assess.• A method of formative assessment.

For Students:• Real-world tasks that captivate student’s interest and develop their critical thinking, reasoning and communication skills.• Student rubrics, anchor papers and scoring rationales that provide a basis for peer- and self-assessment.• Differentiated material to meet the needs of each student.

For Curriculum Leaders and Staff Developers:• Open-ended performance tasks to help meet the new math TEKS.• Material that can be used to introduce teachers to problem solving.• Rubrics and anchor papers that provide consistency for working with teachers on scoring performance-assessment problems.• Material that provides a basis for reviewing student work and tying together assessment results, curriculum and instruction.

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The TEKS Mathematical Process Standards state that students are expected to:

(A) apply mathematics to problems arising in everyday life, society, and the workplace;

(B) use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

(C) select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

Exemplars rubric criteria from the “Practitioner Level” supports TEKS by requiring students to do the following in order to meet the standard:

Problem Solving• Evidence of solidifying prior knowledge and applying it to the problem-solving situation is present.• There is a correct solution.

Reasoning and Proof• A systematic approach is present.

Representation• An appropriate and accurate mathematical representation is constructed and refined to solve problems or portray solutions.

Problem Solving• A correct strategy is chosen based on the mathematical situation in the task. • Planning or monitoring of strategy is evident.• There is a correct solution.

Reasoning and Proof• Justification of correct reasoning is present.

Communication• Communication of an approach is evident through a methodical, organized, coherent, sequenced and labeled response.

Problem Solving• A correct strategy is chosen based on the mathematical situation in the task.• Planning or monitoring of strategy is evident.• Evidence of solidifying prior knowledge and applying it to the problem-solving situation is present.

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Below is a crosswalk showing the correlation between Exemplars assessment rubric and the TEKSMathematical Processes.

Exemplars® Crosswalk for the TEKS Mathematical Processes

Exemplars rubric criteria from the “Practitioner Level” supports TEKS by requiring students to do the following in order to meet the standard:

Reasoning and Proof• Arguments are constructed with adequate mathematical basis.

Communication• Communication of an approach is evident through a methodical, organized, coherent, sequenced and labeled response.• Formal math language is used to share and clarify ideas.



Connections• A mathematical connection is made. Proper contexts are identified that link both the mathematics and the situation in the task. Some examples may include one or more of the following: • clarification of the mathematical or situational context of the task • exploration of mathematical phenomenon in the context of the broader topic in which the task is situated • noting patterns, structures and regularities

Reasoning and Proof• Arguments are constructed with adequate mathematical basis.

Communication• Formal math language is used to share and clarify ideas.

The TEKS Mathematical Process Standards state that students are expected to:

(D) communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

(E) create and use representations to organize, record, and communicate mathematical ideas;

(F) analyze mathematical relationships to connect and communicate mathematical ideas;

(G) display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication;

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Exemplars performance material provides a tool for teachers to elicit the TEKS Mathematical Processes with their students.

Problem Solving for the 21st Century: Built for the TEKS supports Texas educators in teaching a focused mathematics curriculum by grouping the new TEKS into the following Units of Study:

Kindergarten Units • Counting and Cardinality (covers: K.2A, K.2B, K.2C, K.2D) • Comparing Numbers (covers: K.2E, K.2F, K.2G, K.2H) • Composing and Decomposing Numbers (covers: K.2I) • Addition and Subtraction (covers: K.3A, K.3B, K.3C) • Geometry (covers: K.6A, K.6B, K.6C, K.6D, K.6E, K.6F) • Data Analysis (covers: K.8A, K.8B, K.8C)

Grade 1 Units • Number Concepts and Place Value (covers: 1.2A, 1.2B, 1.2C, 1.5A, 1.5B, 1.5C) • Comparing Numbers (covers: 1.2D, 1.2E, 1.2F, 1.2G) • Composing and Decomposing Numbers (covers: 1.3C) • Strategies for Addition and Subtraction (covers: 1.3A, 1.3D, 1.3E, 1.3F, 1.5D, 1.5G) • Equivalence (covers: 1.5E) • Algebraic Reasoning (covers: 1.3B, 1.5F) • Geometry (covers: 1.6A, 1.6B, 1.6C, 1.6D, 1.6F) • Fractions (covers: 1.6G, 1.6H) • Data Analysis (covers: 1.8A, 1.8B, 1.8C)

Grade 2 Units • Number Concepts and Place Value (covers: 2.2B, 2.2C, 2.3A, 2.7B) • Comparing Numbers (covers: 2.2D, 2.2E, 2.2F) • Fractions (covers: 2.3A, 2.3B, 2.3C, 2.3D) • Addition and Subtraction (covers: 2.4A, 2.4B, 2.4C, 2.4D) • Problem Solving With Money (covers: 2.5A, 2.5B) • Multiplication and Division (covers: 2.6A, 2.6B) • Algebraic Reasoning (covers: 2.7A, 2.7C) • Geometry (covers: 2.8A, 2.8B, 2.8C, 2.8D, 2.8E) • Measurement (covers: 2.9A, 2.9B, 2.9C, 2.9D, 2.9E, 2.9F, 2.9G) • Data Analysis (covers: 2.10A, 2.10B, 2.10C, 2.10D)

Exemplars AdvantagesUnits of Study

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Grade 3 Units • Place Value (covers: 3.2A, 3.2B, 3.2C, 3.2D) • Understanding Fractions (covers: 3.3A, 3.3B, 3.3C, 3.3D, 3.3E, 3.7A) • Comparing Fractions (covers: 3.3F, 3.3G, 3.3H) • Addition and Subtraction (covers: 3.4A, 3.4B, 3.4C) • Multiplication (covers: 3.4D, 3.4E, 3.4F, 3.4G) • Division (covers: 3.4H, 3.4I, 3.4J) • Relationship Between Multiplication and Division (covers: 3.4K) • Algebraic Reasoning (covers: 3.5A, 3.5B, 3.5C, 3.5D, 3.5E) • Geometry (covers: 3.6A, 3.6B) • Measurement (covers: 3.6C, 3.6D, 3.7B, 3.7C, 3.7D, 3.7E) • Data Analysis (covers: 3.8A, 3.8B)

Grade 4 Units • Whole Number and Decimal Place Value (covers: 4.2A, 4.2B, 4.2C, 4.2D, 4.2E, 4.2F, 4.2G, 4.2H, 4.3G) • Multiplying and Dividing Whole Numbers (covers: 4.4B, 4.4C, 4.4D, 4.4E, 4.4F, 4.4G, 4.4H) • Comparing Fractions (covers: 4.3C, 4.3D) • Adding and Subtracting Fractions (covers: 4.3A, 4.3B, 4.3E, 4.3F) • Algebraic Reasoning (covers: 4.4A, 4.5A, 4.5B) • Geometry (covers: 4.6A, 4.6B, 4.6C, 4.6D) • Angle Measurement (covers: 4.7A, 4.7B, 4.7C, 4.7D, 4.7E) • Measurement (covers: 4.5C, 4.5D, 4.8A, 4.8B, 4.8C) • Data Analysis (covers: 4.9A, 4.9B)

Grade 5 Units • Decimal Place Value (covers: 5.2A, 5.2B, 5.2C) • Operations With Whole Numbers (covers: 5.3A, 5.3B, 5.3C, 5.4B) • Products and Quotients With Decimals (covers: 5.3D, 5.3E, 5.3F, 5.3G) • Adding and Subtracting Fractions (covers: 5.3H, 5.3K) • Multiplying With Fractions (covers: 5.3I) • Dividing With Fractions (covers: 5.3J, 5.3L) • Algebraic Reasoning (covers: 5.4A, 5.4C, 5.4D, 5.4E, 5.4F) • Geometry (covers: 5.5) • Measurement (covers: 5.4G, 5.4H, 5.6A, 5.6B, 5.7) • Coordinate Plane (covers: 5.8A, 5.8B, 5.8C) • Data Analysis (covers: 5.9A, 5.9B, 5.9C)

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Unit definitions can be viewed online as part

of a FREE 30-day trial.

Professional DevelopmentExemplars professional development workshops and Institutes empower educators by offering effective instructional and assessment techniques for performance-based learning. The focus of our math sessions is on problem solving and using rubrics to assess student work, using results to improve performance and managing the standards-based classroom. Exemplars instructors are knowledgeable about NCTM and the new TEKS standards, performance assessment, portfolios and using assessment information to improve performance.

Workshops, K–12Exemplars offers professional development workshops in the areas of Math (K–12), Science (K–8), Developing Writers (K–4) and Reading, Writing, Research in the Content Areas (5–8). Sessions are hosted by schools and districts. Our workshops are customized to meet the needs and expertise of the participants. Workshops can be for one, two, three or more days.

Summer InstitutesIn addition to workshops, Exemplars offers Institutes that are hosted by us. The summer locations and dates change annually.

If you are looking for a successful approach to unlocking teacher and student potential, contact us for your next school or district training.

Exemplars preview kits are available for:

• Math, Pre K–K• Science, K–8• Developing Writers, K–4• Reading, Writing, Research, 5–8

Visit us online to download samples and to view other grade levels!

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a classroom resource built for the new math teks · a classroom resource built for the new math...

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