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Understanding Mathematical Connections at the Fifth Grade Level

Tuesday, August 18th, 2015

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

This blog is the final post of a four-part series that explores mathematical connections and offers guidelines, strategies and suggestions for helping teachers elicit this type of thinking from their students.

In the first blog post we defined mathematical connections, examined the basis for making good mathematical connections and defined why the CCSSM, NCTM and Exemplars view them as critical elements of today’s mathematics curriculum. We also reviewed the Exemplars rubric and offered strategies for teachers to try in their classroom to help their students become more proficient in making mathematical connections:

As part of the other blogs in this series, we reviewed solutions from a first grade student and third grade student to observe how they successfully included mathematical connections as well as the other problem-solving criteria of the Exemplars rubric in their work.

Blog 4: Mathematical Connections at the Fifth Grade Level

In today’s post, we’ll look at a fifth grade student’s solution for the task “Seashells for Lydia.” This task is one of a number of Exemplars tasks aligned to the Number and Operations in Base Ten standard 5.NBT.2. It would be given toward the end of the learning time dedicated to this standard.

In addition to demonstrating the Exemplars criteria for Problem Solving, Reasoning and Proof, Communication, Connections and Representation from the assessment rubric, this anchor paper shows evidence that students can reflect on and apply mathematical connections successfully. For many students, mathematical connections begin with the other four criteria of the Exemplars rubric, regardless of their grade.

After reviewing our scoring rationales below, be sure to check out the tips for instructional support. Try these along with the task and the Exemplars assessment rubric in your classroom. How many mathematical connections can your students come up with?

5th Grade Task: Seashells for Lydia

Lydia started collecting seashells when she was five years old. At age seven, Lydia had 12(10)2 seashells. At age nine, Lydia had 24(10)2 seashells. At age eleven, Lydia had 48(10)2 seashells. Lydia wants to collect 75(10)3 seashells. Lydia continues to collect seashells at the same rate. How old will Lydia be when she has 75(10)3 seashells? Show all of your mathematical thinking.

Common Core Alignments

  • Content Standard 5.NBT.2: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
  • Mathematical Practices: MP1, MP3, MP4, MP5, MP6, MP7

Understanding Mathematical Connections at the Third Grade Level

Wednesday, August 5th, 2015

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

This blog is Part 3 of a four-part series that explores mathematical connections and offers guidelines, strategies and suggestions for helping teachers elicit this type of thinking from their students.

In the first blog post we defined mathematical connections, examined the basis for making good mathematical connections and defined why the CCSSM, NCTM and Exemplars view them as critical elements of mathematics curriculum. We also reviewed the Exemplars rubric and offered strategies for teachers to try in their classrooms to help their students become more proficient in making mathematical connections.

As part of the second blog, we reviewed a first grade solution and how this student successfully included mathematical connections as well as the other problem-solving criteria of the Exemplars rubric in his or her work.

Blog 3: Mathematical Connections at the Third Grade Level

In today’s post, we’ll look at a third grade student’s solution for the task “Bracelets to Sell.” This task is one of a number of Exemplars tasks aligned to the Operations and Algebraic Thinking Standard 3.OA.3. It would be given toward the end of the learning time dedicated to this standard.

In addition to demonstrating the Exemplars criteria for Problem Solving, Reasoning and Proof, Communication, Connections and Representation from the assessment rubric, this anchor paper shows evidence that students can reflect on and apply mathematical connections successfully. For many students, mathematical connections begin with the other four criteria of the Exemplars rubric, regardless of their grade.

After reviewing our scoring rationales below, be sure to check out the tips for instructional support. Try these in your classroom along with the sample task and the Exemplars assessment rubric. How many mathematical connections can your students come up with?

3rd Grade Task: Bracelets to Sell

Kathy has thirty-six bracelets to sell in her store. Kathy wants to display the bracelets in rows on a shelf. Kathy wants to have the same number of bracelets in each row. What are four different ways Kathy can display the bracelets in rows on the shelf? Each bracelet costs three dollars. If Kathy sells all the bracelets, how much money will she make? Show all of your mathematical thinking.

 Common Core Alignments

  • Content Standard 3.OA.3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
  • Mathematical Practices: MP1, MP3, MP4, MP5, MP6

Understanding Mathematical Connections at the First Grade Level

Monday, July 20th, 2015

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

Summer Blog Series Overview:

This blog represents Part 2 of a four-part series that explores mathematical connections and offers guidelines, strategies and suggestions for helping teachers elicit this type of thinking from their students.

In the previous blog post we defined mathematical connections, examined the basis for making good mathematical connections and defined why the CCSSM, NCTM and Exemplars view them as critical elements of mathematics curriculum.

We also reviewed the Exemplars rubric and offered the following strategies for teachers to try in their classroom to help their students become more proficient in making mathematical connections:

  1. Develop students’ abilities to use multiple strategies or representations to show their mathematical thinking and support that their answers are correct.
  2. Encourage students to continue their representations.
  3. Explore the rich formal language of mathematics.
  4. Incorporate inquiry into the problem-solving process.
  5. Encourage self- and peer-assessment opportunities in your classroom.

Blog 2: Mathematical Connections at the First Grade Level

In today’s post, we’ll look at a first grade student’s solution for the task, “Pictures on the Wall.” This anchor paper demonstrates the criteria for Problem Solving, Reasoning and Proof, Communication, Connections and Representation from the Exemplars assessment rubric. It also shows a solution that goes beyond arithmetic calculation and provides the evidence that a student can reflect on and apply mathematical connections. The beauty of mathematical connections is that they often begin with the other four rubric criteria. In other words, the Exemplars rubric provides multiple opportunities for a student to connect mathematically!

In this piece of student work, you’ll also notice that the teacher has “scribed” the student’s oral explanation. Scribing allows teachers to fully capture the mathematical reasoning of early writers.

This blog will offer tips for the type of instructional support a teacher may provide during this learning time as well as the type of support students may give each other. Teacher support may range from offering direct instruction to determining if a student independently included mathematical connections in her or his solution. After reading this post, give the task a try in your own classroom along with the Exemplars rubric. You may view other Exemplars tasks here.

First Grade Task: Pictures on the Wall

There are sixteen pictures on a wall. The art teacher wants to take all the pictures off the wall to put up new pictures. The art teacher takes seven pictures off the wall. How many more pictures does the art teacher have to take off the wall? 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, MP3, MP4, MP5, MP6

Understanding Mathematical Connections

Wednesday, June 24th, 2015

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

What is a mathematical connection? Why are mathematical connections important? Why are they considered part of the Exemplars rubric criteria? And how can I encourage my students to become more independent in making mathematical connections?

This blog represents Part 1 of a four-part series that explores mathematical connections and offers guidelines, strategies and suggestions for helping teachers elicit this type of thinking from their students. We find many students enjoy making connections once they learn how to reflect and question effectively. As part of this series, student work will be examined at Grades 1, 3 and 5.

A Brief Introduction to the Exemplars Rubric

The Exemplars assessment rubric allows teachers to examine student work against a set of analytic assessment criteria to determine where the student is performing in relationship to each of these criteria. Teachers use this tool to evaluate their students’ problem-solving abilities.

The Exemplars assessment rubric is designed to identify what is important, define what meets the standard and distinguish between different levels of student performance. The rubric consists of four performance levels — Novice, Apprentice, Practitioner (meets the standard) and Expert — and five assessment categories (Problem Solving, Reasoning and Proof, Communication, Connections and Representation). Our rubric criteria reflect the Common Core Standards for Mathematical Practice and parallel the National Council of Teachers of Mathematics (NCTM) Process Standards.

The Importance of Mathematical Connections

Exemplars refers to connections as “mathematically relevant observations that students make about their problem-solving solutions.” Connections require students to look at their solutions and reflect. What a student notices in her or his solution links to current or prior learning, helps that student discover new learning and relates the solution mathematically to one’s own world. A student is considered proficient in meeting this rubric criterion when “mathematical connections or observations are recognized that link both the mathematics and the situation in the task.”

NCTM defines mathematical connections in Principals and Standards for School Mathematics as the ability to “recognize and use connections among mathematical ideas; understand how mathematical ideas interconnect and build on one another to produce a coherent whole; recognize and apply mathematics in contexts outside of mathematics.” (64)

The Common Core State Standards for Mathematics (CCSSM) support the need for students to make mathematical connections in problem solving. Reference to this can be found in the following Standards for Mathematical Practice:

  • MP3: Construct viable arguments and critique the reasoning of others. “… They justify their conclusions, communicate them to others, and respond to the arguments of others.”
  • MP4: Model with mathematics. “… They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.”
  • MP6: Attend to precision. “Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose … They are careful about specifying the units of measure and labeling axes … They calculate accurately and efficiently express numerical answers with a degree of precision appropriate …”
  • MP7: Look for and make use of structure. “Mathematically proficient students look closely to discern a pattern or structure …”
  • MP8: Look for and express regularity in repeated reasoning. “… They continually evaluate the reasonableness of their intermediate results.”

The CCSSM also state, “The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word ‘understand’ are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations …” (Common Core Standards Initiative, 2015)

When students apply the criteria of the Exemplars rubric, they understand that their solution is more than just stating an answer. Part of that solution is taking time to reflect on their work and make a mathematical connection to share.

What Can Teachers Do to Help Students Make Mathematically Relevant Connections?

When students begin to explore mathematical connections, teachers should take the lead by providing formative assessment tasks that introduce new learning opportunities and provide practice, so they may become independent problem solvers. As part of this process, teachers will want to focus on five key areas to help students develop an understanding of mathematical connections.

(1) Develop students’ abilities to use multiple strategies or representations to show their mathematical thinking and support that their answers are correct. When students demonstrate an additional or new strategy or representation in solving a problem, a mathematical connection is made. The Common Core includes a variety of representations students can apply to solve a problem and justify their thinking. Examples include manipulatives, models, five and ten frames, diagrams, keys, number lines, tally charts, tables, charts, arrays, picture graphs, bar graphs, linear graphs, graphs with coordinates, area/visual models, set models, linear models and line plots. By practicing these different approaches, students will begin to create new strategies and representations that are accurate and appropriate to their grade level. This in turn opens the door for them to use a second or even third representation to show their thinking in a new way or to justify and support that their answer(s) is correct.

Using formative problem-solving tasks to introduce and practice new strategies and representations is part of the problem-solving process. Teachers should provide formal instruction so that students may grow to independently determine and construct strategies or representations that match the task they are given. An example of this can be seen in the primary grades when many teachers introduce representations in the following order: manipulative/model, to diagram (including a key when students are ready), to five/ten frames, to tally charts, to tables, to number lines. This order allows students to move from the most concrete to the more abstract representations.

(2) Encourage students to continue their representations. Mathematical connections may be made when students continue a representation beyond the correct answer. Examples of this can be seen when a table or linear graph is continued from seven days to 14 days or when two more cats are added to a diagram of 10 cats to discover how many total ears a dozen cats would have. Another example includes adding supplemental information to a chart such as a column for decimals in a table that already has a column indicating the fractional data. In this case, the student extends his or her thinking to incorporate other mathematics to solve the task. It is important to note that connections must be relevant to the task at hand. In order to meet the standard, a connection must link the math in the task to the situation in the task.

(3) Explore the rich formal language of mathematics. Mathematical connections may be made as students begin to use the formal language of mathematics and its connection to their representations, calculations and solutions. Mathematical connections can be seen in the following examples: two books is called a pair; 12 papers is a dozen, the pattern is a multiple of 10; 13 is a prime number so 13 balls can’t be equally placed in two buckets; and the triangle formed is isosceles. The input and output on a table can also help students generalize a rule with defined variables. Students will quickly learn that making connections promotes math communication (formal terms and symbols) and that using math communication promotes connections. Again, these connections must link the math in the task to the situation that has been presented.

(4) Incorporate inquiry into the problem-solving process. Asking students to clarify, explain, support a part of their solution to a math partner, the whole class, or a teacher, not only helps develop independent problem solvers but also leads to more math connections. In your discussions, use verbs from Depth of Knowledge 2 (identify, interpret, state important information/cues, compare, relate, make an observation, show) and from Depth of Knowledge 3 (construct, formulate, verify, explain math phenomena, hypothesize, differentiate, revise). By asking students questions that provide them the opportunity to show and share what they know, connections become a natural part of their solutions.

Instead of asking, “Do you see a pattern in your table?” say, “Did you notice anything about the numbers in each column in your table?” Try asking a primary student, “I know you have a cat. Would you like your cat to join the cats in your problem?” “What new numbers are you using?” “I heard you tell Maria that all the numbers in your second column are even. Can you help me understand why they are all even numbers?” Every time a student provides you with a correct answer to your or another student’s inquiry, stop and say, “Thank you for explaining/showing/sharing your thinking. You just made a mathematical connection about your problem.” If you hear a student make a mathematical connection outside of class, stop and comment, “You just made a math connection!” Some examples of these student connections may include, “Look, we are lined up as girl, boy, girl, boy, girl, boy for lunch. That is a pattern,” “In four more days it is my birthday,” “Art class is in 15 minutes because we always go to art at 10 o’clock,” “We can have an equal number of kids at each table because four times six equals 24,” “My dad says we have to drive 45 miles per hour because that is the speed limit, so I think I can write each student as ‘per student’” or “I think I can state all the decimals on my table as fractions.”

(5) Encourage self- and peer-assessment opportunities in your classroom. Encourage students to self-assess their problem-solving solutions either independently, with a math partner or with the support of their teacher. The more opportunity students have to use the criteria of the Exemplars assessment rubric to evaluate their work, the more independent they become in forming their solutions, which will include making mathematically relevant connections.

Exploring Authentic Examples of Mathematical Connections

In the next blog post of this series, we’ll look at a problem-solving task and student solution from Grade 1 to observe how mathematical connections have been effectively incorporated. We’ll also explore the type of support a teacher may provide during this learning time as well as the type of support students may give each other. (Solutions from Grades 3 and 5 will follow in subsequent posts of this series.)

 

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

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