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Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Fifth Grade Level

Thursday, September 4th, 2014

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 6 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress toward being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Practices is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with assessment rationales that demonstrate the alignment between the Exemplars assessment rubric and the CCSS for Mathematical Content and Mathematical Practice can be insightful for educators. Anchor papers and assessment rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Practices are embedded in the assessment rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in hand while reviewing a piece of student work.

Blog 6: Observations at the Grade 5 Level

The final anchor paper and set of rationales we’ll review in this series is taken from a fifth grade student’s solution for the task, “Newspaper Layout.” This task is one of a number of Exemplars tasks aligned to the Number and Operations–Fraction Standard 5.NF.6.

“Newspaper Layout” would be used toward the end of the learning time allocated to this standard. This particular task provides provides fifth graders with an opportunity to apply different strategies to determine how much the mathematics department pays for each part of the layout and the total cost of the advertisement. The task requires students to bring prior conceptual understanding of area and multiplying with money to their solution. In assessing this task, teachers will be able to determine if their students can apply these concepts and multiply mixed numbers.

Students have a variety of strategies to consider in forming their solutions. Some examples include creating a diagram of the newspaper layout, using grid/graph paper to correctly scale the newspaper area layout, applying the formula for area and money calculations or using a table to record the necessary data to support two correct answers. Students may also demonstrate their conceptual understanding of decimals.

5th Grade Task: Newspaper Layout

The newspaper staff is designing a layout to advertise the mathematics department’s “I Love Math” celebration. The newspaper staff will charge the mathematics department for the advertising by finding the number of square inches for each part of the layout. Below is a diagram of the layout. The newspaper staff charges fifty cents per square inch. How much does the mathematics department pay for each part of the advertisement? What is the total cost of the advertisement?  Show all of your mathematical thinking.

Common Core Alignments:

  • Content Standard 5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP8

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Fourth Grade Level

Thursday, August 28th, 2014

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 5 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress toward being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Practices is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with assessment rationales that demonstrate the alignment between the Exemplars assessment rubric and the CCSS for Mathematical Content and Mathematical Practice can be insightful for educators. Anchor papers and assessment rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Practices are embedded in the assessment rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in hand while reviewing a piece of student work.

Blog 5: Observations at the Fourth Grade Level

The fifth anchor paper and set of rationales we’ll review in this series is taken from a fourth grade student’s solution for the task “Sharing Muffins.” This task is one of a number of Exemplars tasks aligned to the Numbers and Operations–Fractions Standard 4.NF.3c.

“Sharing Muffins” would be used toward the end of the learning time allocated to this standard. This task provides fourth graders with an opportunity to apply different strategies to determine the number of muffins needed for each of nine friends to have one and one-third muffins. In solving this task, there are a variety of strategies for students to consider. Some examples include using actual muffins to model one and one-third muffins per friend or diagramming the muffins using a table, tally chart or number line. In their solutions, students may replace each mixed number with an equivalent fraction. Addition, subtraction and multiplication of fractions may also be used.

Fourth Grade Task: Sharing Muffins

Nine friends are going to equally share some muffins. Each muffin is the same size. Each friend gets one and one-third muffins. How many muffins did the nine friends equally share? Show all your mathematical thinking.

Common Core Task Alignments

  •  Content Standard 4.NF.3c: Add and subtract mixed numbers with like denominators e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP7, MP8

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Third Grade Level

Wednesday, August 13th, 2014

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 4 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress toward being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Practices is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with assessment rationales that demonstrate the alignment between the Exemplars assessment rubric and the CCSS for Mathematical Content and Mathematical Practice can be insightful for educators. Anchor papers and assessment rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Practices are embedded in the assessment rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in hand while reviewing a piece of student work.

Blog 4: Observations at the Third Grade Level

The fourth anchor paper and set of assessment rationales we’ll review in this series is taken from a third grade student’s solution for the task, “Henry’s Lego Structure.” This task is one of a number of Exemplars tasks aligned to the Operations and Algebraic Thinking Standard 3.OA.8.

“Henry’s Lego Structure” would be used toward the end of the learning time allocated to this standard. This particular task provides third graders with an opportunity to apply different strategies to determine how many Legos are needed to build a three-level structure and if “Henry” has enough Legos to build a fourth level. Students need to bring an understanding of the terms twice, three times and pattern to the task as well as the correct calculation. When assessing this task, teachers can observe which forms of calculation a student chooses to use and if s/he can solve a two-step problem.

There are a variety of strategies for students to consider in forming their solutions. Some examples include using actual Legos to model the structure, diagramming the structure, creating a table, tally chart or using a number line.

Third Grade Task: Henry’s Lego Structure

Henry wants to build a structure with his new Lego set. The Lego set contains five hundred Legos. The structure will be three levels high. The first level is made of twenty-seven Legos. Henry uses twice as many Legos for the second level as for the first level. Henry uses three times as many Legos for the third level as for the second level. How many Legos does Henry use to build his structure with three levels? If this pattern continues, does Henry have enough Legos in his new set to build a fourth level on his structure? Show all of your mathematical thinking.

 Common Core Alignments:

  • Content Standard 3.OA.8: Solve two-step problems using the four operations.
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP7, MP8

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Second Grade Level

Friday, August 1st, 2014

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 3 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress toward being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Practices is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with assessment rationales that demonstrate the alignment between the Exemplars assessment rubric and the CCSS for Mathematical Content and Mathematical Practice can be insightful for educators. Anchor papers and assessment rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Practices are embedded in the assessment rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in hand while reviewing a piece of student work.

Blog 3: Observations at the Second Grade Level

The third anchor paper and set of assessment rationales we’ll review in this series is taken from a second grade student’s solution for the task, “A New Hamster Toy.” This is one of a number of Exemplars tasks aligned to the Measurement and Data Standard 2.MD.8.

“A New Hamster Toy” would be used toward the end of the learning time allocated to this standard. This task provides second grade students with an opportunity to apply different strategies to determine if there is enough money to buy a hamster toy for $2.25. The task does not provide the symbolic notation for $2.25, $0.05, or 5¢. Students need to bring this understanding to their solutions, which provides the teacher with an opportunity to assess if they can correctly notate money. This task also provides students with the opportunity to use comparison and to solve a problem that includes two steps. Students need to determine the popcorn bag sales for one day, determine the total sales for five days and compare that total to $2.25.

When forming their solutions, students have a variety of strategies to consider. Some examples include using actual money to model the bag sales and total bag sales, diagramming the bags and/or money earned, creating a table to indicate popcorn sales for one or five days, using a printed number line, creating a number line or a tally chart.

Second Grade Task: A New Hamster Toy

Some students want to earn two dollars and twenty-five cents to buy a toy for their class hamster. The students decide to sell small bags of popcorn at snack time for five cents each. The students sell ten bags every day for five days. Do the students earn enough money to buy a toy for their class hamster? Show all your mathematical thinking.

Common Core Alignments

  • Content Standard 2.MD.8: Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately.
  •  Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP7, MP8

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the First Grade Level

Monday, July 21st, 2014

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 2 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress toward being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Practices is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with assessment rationales that demonstrate the alignment between the Exemplars assessment rubric and the CCSS for Mathematical Content and Mathematical Practice can be insightful for educators. Anchor papers and assessment rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Practices are embedded in the assessment rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in hand while reviewing a piece of student work.

Blog 2: Observations at the First Grade Level

The second anchor paper and set of assessment rationales we’ll review in this series is taken from a first grade student’s solution for the task, “A Birdbath.” In this piece, you’ll notice that the teacher has “scribed” the student’s oral explanation. This practice was also used with the Kindergarten task that was published in the first blog. Scribing allows teachers to fully capture the mathematical reasoning of early writers.

“A Birdbath” is one of a number of Exemplars tasks aligned to the Operations and Algebraic Thinking Standard 1.OA.6. This task would be used toward the end of the learning time allocated to this standard. “A Birdbath” provides first grade students with an opportunity to apply different strategies to find the sum of addends six and 14 by decomposing six into five and one and decomposing 14 into 10 and four, or by finding the sum of six and four and adding that sum to 10. The student can use counters, ten frames, a Rekenrek, number lines or a tally chart to support her/his numerical thinking.

First Grade Task: A Bird Bath

Leah counts the birds that came to her birdbath. In the morning, Leah counts six birds that came to her birdbath. In the afternoon, Leah counts fourteen birds that came to her birdbath. Leah says nineteen birds came to her birdbath. Is Leah correct? Show all your mathematical thinking.

Common Core Alignments

  • Content Standard 1.OA.6: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6, MP7

Supporting the Standards for Mathematical Practice With Exemplars Performance Tasks and Rubric at the Kindergarten Level

Thursday, July 10th, 2014

Written By: Deborah Armitage, M.Ed., Exemplars Math Consultant

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Common Core through problem solving. This blog represents Part 1 of a six-part series that features a problem-solving task linked to a CCSS for Mathematical Content and a student’s solution in grades K–5. Evidence of all eight CCSS for Mathematical Practice will be exhibited by the end of the series.

The Exemplars Standards-Based Math Rubric allows teachers to examine student work against a set of analytic criteria that consists of the following categories: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. There are four performance/achievement levels: Novice, Apprentice, Practitioner (meets the standard) and Expert. The Novice and Apprentice levels support a student’s progress towards being able to apply the criteria of a Practitioner and Expert. It is at these higher levels of achievement where support for the Mathematical Practices is found.

Exemplars problem-solving tasks provide students with an opportunity to apply their conceptual understanding of standards, mathematical processes and skills. Observing student anchor papers with assessment rationales that demonstrate the alignment between the Exemplars assessment rubric and the CCSS for Mathematical Content and Mathematical Practice can be insightful for educators. Anchor papers and assessment rationales provide examples of what to look for in your own students’ work. Examples of Exemplars rubric criteria and the Mathematical Practices are embedded in the assessment rationales at the bottom of the page. The full version of our rubric may be accessed here. It is often helpful to have this in-hand while reviewing a piece of student work.

Blog 1: Observations at the Kindergarten Level

The first anchor paper and set of assessment rationales we’ll review in this series is taken from a Kindergarten student’s solution for the task, “Boots.” You will notice that the teacher has “scribed” the student’s oral explanation. This method allows teachers to fully capture the mathematical reasoning of early writers.

“Boots” is one of a number of Exemplars tasks aligned to the Counting and Cardinality Standard K.CC.5. This task would be used toward the end of the learning time allocated to this standard. Prior to “Boots” being given, students have already completed a number of tasks with questions that state, “How many ears?”, “How many shoes?”, “How many balloons?”, etc. “Boots” gives students an opportunity to bring a stronger understanding of the concept how many to their solution.

Kindergarten Task: Boots

Five students wear boots to go outside for recess. When the students come in from recess they must put all boots on a rubber mat to dry. The teacher counts seven boots on the mat. The teacher thinks some boots are missing. Is the teacher correct? Show and tell how you know.

Common Core Task Alignments

  • Content Standard K.CC.5: Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle or, as many as 10 things in a scattered configurations: given a number from 1-20, count out that many objects.
  • Mathematical Practices: MP1, MP2, MP3, MP4, MP5, MP6

Bridge the Gap Between Common Core, Your Classroom and the Real World

Tuesday, March 25th, 2014

Written By: Elaine Watson, Ed.D., Exemplars Math Consultant

To most nonscientists, mathematics is counting and calculating with numbers. That is not at all what a scientist means by the word. To a scientist, counting and calculating are part of arithmetic and arithmetic is just one very, very small part of mathematics. Mathematics, the scientist says, is about order, about patterns and structure, and about logical relationships.

By, Keith Devlin, Life by the Numbers

The word “scientist” above could be replaced by the word “doctor, lawyer, engineer, accountant, CEO, military officer, government worker, homeowner, citizen …” In other words, anyone who uses numbers to make decisions needs to look beyond the calculations and be able to discern what the numbers are telling them.

Math textbooks have developed “word problems” in response to the question so often asked by students as they learn to follow algorithms and solve equations in order to find the correct answer: “When am I ever going to use this in real life?” The question is often answered by the jaded teacher, who has heard it from each new generation of students in this way: “You’re going to use it on the test!” This answer seals the students’ belief that what they learn in math class is not applicable to the real world, but merely a set of exercises that need to be done in order to pass the course.

Past mathematics standards documents have focused on the hard content, the factual and procedural content students should learn, which is of course important. The focus on the soft content, the habits of mind and thought processes that are practiced by students when solving a problem, has traditionally either been relegated to the end of the standards document as an afterthought or omitted altogether.

The Common Core State Standards in Mathematics (CCSSM) recognize that the soft content, the practices students used to approach and solve a mathematical task, are as important as the hard standards. Soft does not mean unimportant. In the same way that a computer (hardware) cannot function effectively without appropriate software, CCSSM Content Standards cannot be accessed and used without students using the supporting Practice Standards.

The Practice Standards have to be learned, and practiced, alongside the Content Standards, but because of the “soft” nature of Practice Standards, they are harder to pin down. Phil Daro, one of the three authors of the CCSSM, describes the Practice Standards as “the content of a student’s mathematical character.”

It is important to remember that it is the students who practice the Practice Standards. Teachers should model the practices in their instruction, but more importantly, teachers should explicitly plan lessons that include teacher pedagogical moves, student activities and tasks that will elicit the Practice Standards in students.

The tasks created by Exemplars are excellent examples of rich problem-solving that naturally elicit the Practice Standards. Below we will look at the Grade 2 task “Barnyard Buddies” and discuss how it meets each of the eight Mathematical Practice Standards as well as content standard 2.OA.A.1.

Barnyard Buddies

A farmer has 8 cows and 10 chickens. The farmer counts all the cow and chicken legs. How many legs are there altogether? Show all your mathematical thinking.

CCSSMP.1 Make sense of problems and persevere in solving them.

There is no hint in this task as to how to go about solving the task. It is not a generic type of problem with which the student has had previous experience. The student must make sense of the task before being able to develop an approach for solving it. Some approaches may be more efficient than other approaches.

CCSSMP.2 Reason abstractly and quantitatively.

In order to solve the problem, students will need to use an approach in order to organize their thinking and keep track of the quantities involved.  One approach is to draw 4-legged animals and 2-legged animals and count.  Another approach is to create a table. Both of these approaches have created an abstraction (mathematical model) of the situation. The student work below shows how two students modeled the problem.

Student 1 created abstractions of the chickens (square with 2 legs) and cows (circles with 4 legs).

Student 2 simply drew the legs without the bodies, which was a step toward greater abstraction. She or he then went on to use an even more abstract approach by noticing that there was a pattern and deciding to use a table. This student work is also a good illustration of Practice Standard 8: Look for and express regularity in repeated reasoning.

CCSSMP.3 Construct viable arguments and critique the reasoning of others.

This task will elicit a lot of different ideas as to how to approach it. Students will need to persuade others as to why their approach will work the best. In order for students to exhibit this practice standard, a classroom culture needs to be developed where student discussion of their work is the norm. The teacher’s role is to encourage the discussion and question and guide as needed.

CCSSMP.4 Model with mathematics.

In order to solve this task, students will need to go through the steps of the Modeling Cycle. They formulate an approach, compute, and then check their answer to see if they have correctly counted all 8 cow’s legs and all 10 chicken’s legs. If their answer makes sense, they report it out. If it doesn’t make sense, they need to go back through the cycle, determining where they went wrong. Were their pictures correct? Did they have the right number of each type of animal and the correct number of legs on each type of animal? If they used a table, did they skip count correctly by 2 and by 4? Did they add correctly? The cycle continues until they are satisfied that their result is a viable answer for the problem.

CCSSMP.5 Use appropriate tools strategically.

Tools are not necessarily physical, like a ruler or a calculator. On this problem, the student’s drawing or table can be considered a tool, since it helps make sense of and solve the problem.

CCSSMP.6 Attend to precision.

Precision is needed in the drawings or table, in the counting, and in the addition. Students also need to be precise in labeling their answer. If a student answers with only a number without the label “legs,” they are not attending to precision.

CCSSMP.7 Look for and make use of structure.

The student needs to visualize the structure of the situation. In this case the structure involves a given number of animals with 4 legs and a given number of animals with 2 legs. That structure will inform how the student approaches and solves the problem. If the student notices that 4 consists of 2 copies of 2, this will help in counting, since he or she should be proficient at counting by 2s.

CCSSMP.8 Look for and express regularity in repeated reasoning.

The student is repeatedly adding 2 or adding 4 for a given number of times. The student can count by 2s while pointing to each chicken. For the cows, students can either count by 4s, or they can count by 2s when pointing to the cows and touching each of the two pairs of legs on every cow.

Support for Common Core Content Standards

In addition to eliciting the Common Core Practice Standards, Exemplars tasks are also aligned Common Core Standards for Mathematical Content.

To solve “Barnyard Buddies,” students need to model the situation by using some type of drawing to represent the 10 chickens and the 8 cows as well as the number of legs on each animal.  Creating such a representation is an early form of algebraic thinking. After developing the pictorial model, students then need to count the total number of legs. Most students will skip count by either 2 or 4. Some students may organize their counting by making groups of 10 (2 cows and 1 chicken or 5 chickens).  Whichever approach students use for counting, they are recognizing a numerical pattern, which is also an underpinning of algebraic thinking.  This type of thought process is best matched by the Common Core Domain Operations and Algebraic Thinking.  Within this Domain, “Barnyard Buddies” aligns with the cluster, Represent and solve problems involving addition and subtraction. The specific content standard addressed is 2.OA.1.

2.OA.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Download a copy of the “Barnyard Buddies” task complete with anchor papers and scoring rationales to try with your students!

 

A Common Core “Must Read” Paper

Friday, November 30th, 2012

By: Ross Brewer, Ph.D., Exemplars President

Jay McTighe and Grant Wiggins have written a “must read” paper – “From Common Core Standards to Curriculum: Five Big Ideas,” in which they offer key ideas to guide the work of transforming the Common Core Standards to a functioning curriculum in a school or district. The paper highlights some of the important misconceptions that readers bring to the Common Core and focuses on important processes that will lead schools and districts to creating an effective curriculum that actually embraces the Common Core Standards.

You may download a copy of this important paper here.

Using Exemplars for School Improvement

Tuesday, October 30th, 2012

By Tammy Krejcarek, Assistant Principal, Virginia

Several years ago, I was hired as a math specialist to help a school entering school improvement status. Our math scores were the lowest in the county, and our disparity gap was over 30 points. One of the first changes we made was to implement Exemplars. I began by training my teachers on how to lead students in math talk while sharing their various strategies for solving the problems. I found the anchor papers were a great place to start when introducing Exemplars to students, as they promote that rich discussion you are looking for. The built-in rubrics teach students how to self-evaluate their progress from one Exemplars task to another. The rubrics also offer teachers a tool for providing timely and meaningful feedback to students. One of the key benefits of Exemplars is that students don’t have to get to the correct answer in order to be successful or to stretch their thinking. The dynamics of students sharing and discussing their thought processes with one another is what’s so invaluable — it is NOT always about the answer; it’s about the process. Exemplars allows students to explore strategies without the pressure of getting a “bad” grade.

We started doing Exemplars school wide every Friday during math time. Depending on the focus, we worked as a whole group, in smaller groups, as partners and sometimes individually. Our students quickly learned from one another how to represent their thinking with pictures AND words, as well as how to create tables and diagrams. These strategies helped students organize information into a format they could understand without being overwhelmed. Because Exemplars tasks are based on real-world situations, they provided relevant and engaging context from which the students could make meaning. The organization of the tasks into strands and concepts made it easy for teachers to correlate Exemplars with their mathematics lessons. The tiered levels allowed for differentiated instruction to ensure the success of ALL students.

That first year we implemented the program, our math scores increased by over 30 points to well above passing while decreasing the disparity gap to within 10 points. That was five years ago, and Exemplars is still thriving. Based on this success, most schools in the district are now implementing Exemplars into their mathematics program.

This year, I am at a new district that is excited to begin using Exemplars in their buildings. With the increasing rigorous demand of high-stakes testing, Exemplars is a “must-have” component to any mathematics program. I have been in education for over 18 years and have seen programs come and go. Exemplars is one of the few initiatives that has proved effective time and time again!

Exemplars New Math Samples, K-8

Friday, September 7th, 2012

Our real-world performance tasks are differentiated and aligned to the Common Core Content Standards as well as to the Standards for Mathematical Practice. The anchor papers have been assessed using the Exemplars rubric.

New Samples:

Please share these with your colleagues. We suggest trying our problem-solving tasks in your school and discussing the student work as a team to see how your students approach them.

Have a great start to the school year!

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