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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.)

 

7 Things I’ve Learned on My Journey to Implementing Problem Solving in the Classroom

Tuesday, March 3rd, 2015

Written By: Suzanne Hood, Instructional Coach, Georgia

I’ve always believed in the power of students to use their own childlike curiosity to problem solve. These problem-solving experiences happen for our students naturally, through the math they use in cooking, playing games and playing with toys, among other things. Problem solving is a life-long skill all mathematicians use. The true power of a mathematician is the ability to see math in all situations and solve problems using a toolbox of proven strategies.

While I believe that students are innate problem solvers, I also believe that learned algorithmic thinking corrupts a child’s natural ability to problem solve and discourages perseverance. Although I have met many teachers who share my belief that problem solving should be the focus of the math, many struggle to create this culture in their classroom.

This is becoming more apparent—and the stakes of ignoring problem solving much higher—as we approach testing season. The classrooms that will likely fall behind in this new era are those who insist on teaching math through algorithmic thinking. Conversely, I am convinced that teachers who use problem solving to teach math, supported by materials like Exemplars, will have students who score proficiently on the state assessment and are more prepared for success beyond the classroom.

So how can teachers help their classrooms make this critical transition to problem solving? My personal story of transformation, which began after participating in one of Exemplars’ Summer Institutes, offers a path forward. This was when I realized two important things: first, I needed to work on my own personal proficiency in teaching problem solving. And second, I wasn’t alone; veteran teachers confessed their frustration in teaching problem solving, and many admitted their backgrounds did not include support in how to instruct students through the problem-solving process. Here are seven things I’ve learned on my journey to becoming an educator fully committed to teaching mathematics through a problem-solving approach.

1. Nurture a community of trust.

Based on my experience as a Mathematical Instructional Coach in Georgia, I believe it is essential to nurture relationships and establish a community of trust between teachers, so that discussions are authentic and all voices are included. Trust is a prerequisite for being able to assess the strengths, weaknesses and gaps of teacher readiness in the classroom. Only when teachers feel they are in an environment where they can share their knowledge, their doubts and their pedagogical weaknesses, will they be able to feel comfortable.

2. Establish a baseline of teacher readiness.

Evaluating teacher readiness and needs and getting them on the same page is an important first step. How can you get teacher teams to have collegial conversations when everyone has a totally different math background? Do all teachers even want a problem-solving classroom? Do they know what that means? Asking these questions can be illuminating, albeit tough. As such, using universally agreed-upon protocols such as those from the National School Reform Facility can establish a baseline to work from, encourage collaboration, and support an atmosphere of trust.

3. Assess student work so you can see where the gaps are.

One way to assess teacher acuity and readiness in teaching problem solving is by assessing student work using an Exemplars task. Here’s how it worked for me: At the first Professional Learning session, I asked teachers to bring classroom samples from their most recent classroom Exemplars task. As a community, we agreed to facilitate the discussion with the protocol Atlas – Learning From Student Work. As I observed teachers at the meeting, I noticed that while some teachers were proud to display their samples, others pretended to forget their samples or chose to stick their student work in their tote bag. As we used the Exemplars standards-based rubric to score our samples, it became clear that our understanding of the skills needed to meet the standards did not align. The journey began; teachers began to talk about problem solving.

4. As a team, align your mathematical beliefs towards problem solving.

When we began, we knew we shared some foundational mathematical beliefs. We also knew that we needed to solidify a shared understanding of how a mathematics culture transfers knowledge from the teacher to the student. We used the Math Framework (a document listing all the mathematical beliefs of the faculty) as a tool to target instructional strengths and weaknesses. As a team, we revised the document to build cohesion and a shared understanding of our beliefs. Next, I had the team read a book rooted in Vygotsky’s constructivist theory to increase our group’s understanding of the problem-solving trajectory. Because we had been working hard to build an atmosphere of trust, teachers felt safe sharing their struggles and personal hardships with teaching problem solving. We discovered that we shared similar experiences, and that we all believed our students would be successful at any problem if we just taught them the necessary skill set. The student samples, however, told a different story.

5. Create simple tools to help teachers and students internalize the standards and assess their progress.

At our next meeting, we reviewed Exemplars student work samples and discovered a misconception: we thought we knew how to teach problem solving, but we were actually teaching skills in isolation. Why? Quite simply, it turns out that many teachers lacked background knowledge about the Standards of Problem Solving. To facilitate the understanding of the standards, I created posters with clear icons for each standard. These anchor charts would support teachers and students. It worked. Now, teachers could explain each standard. Each classroom in our building displayed the posters. It was a great reference for both students and teachers. We made a replica of the posters into a small book that students put in folders for their own reference. Students used the folders as portfolios to track their problem-solving progress, and created data notebooks to reflect on their growth and set goals for their next Exemplars task. Using data notebooks empowered kids to self-reflect on their own progress.

6. Hold individual meetings with students to track progress and set goals.

Currently, I am encouraging teachers to hold one-on-one Exemplars conferences with their students. Individual conferences support differentiated instruction, meet students where they are, and set goals for the next problem-solving task. Although this approach makes some teachers uneasy at first, they become more confident over time. Allowing other teachers or coaches to observe and co-teach the process can lead to greater transparency and effect change in teacher practice.

7. You may not get the teacher of the year award, but you’ll still be changing students’ lives.

At the beginning of my career, I thought Oprah would call me to announce my Disney Teacher of the Year Award. While this hasn’t happened yet, I do have countless memories of the sparkle in a child’s eye when he or she announces, “I get it!” I believe I have the responsibility to show up every day prepared to change the lives of children and equip them with the skills to be life-long mathematicians. Exemplars provides the problem-solving tools necessary to guide teaching and build capacity for each child’s mathematical journey.

Why A Focused Mathematics Curriculum Matters and How Exemplars Can Help Texas Educators

Monday, November 3rd, 2014

Written By: Dinah Chancellor, Exemplars Math Consultant

Prior to 2006, many states—including Texas—had a math curriculum that was perceived to be “an inch deep and a mile wide.” Teachers were required to teach a large number of math skills that spiraled from grade to grade and seemed both disconnected and fragmented. When Texas’ own Cathy Seeley became President of the National Council of Teachers of Mathematics (NCTM), she determined that a more focused mathematics curriculum that was built around fewer “big ideas” would give students and teachers the luxury of time—time to plumb the depths of major math concepts, and time to form a foundation of connected mathematical understandings.

Therefore, in 2006 NCTM published the Curriculum Focal Points—A Quest for Coherence. The Texas response to the Curriculum Focal Points was the new state assessment program—STAAR—the State of Texas Assessment of Academic Readiness taken by students in grades 3-8. STAAR focuses on fewer skills at each grade level and it is expected that these skills will be taught at greater depth. When the new math TEKS were written, released in April 2012 and implemented in the fall of 2014, the writing teams focused on fewer skills at each grade level. Teachers are expected to address these skills and understandings by teaching rich lessons in which students make critical connections between foundational big ideas in mathematics. Because of the need to teach a focused mathematics curriculum, it does not make sense to teach each of the new math TEKS in isolation.

To assist Texas educators in achieving this goal, Exemplars latest K–5 product, Problem Solving for the TEKS, groups the individual math TEKS student expectations into rich Units of Study. Four or more instructional tasks/formative assessments and one or more summative assessment is provided to address the big mathematical ideas within each Unit. Tasks are meant to supplement a school or district’s existing curriculum. Teachers may choose to use all or only a few of the instructional tasks/formative assessments in a Unit. The summative assessments include anchor papers that exemplify each of the performance levels in the Exemplars Rubric—Novice, Apprentice, Practitioner (meets the standard) and Expert.

A Look at a Sample Unit

The Place Value Unit represents one of eleven Units in the third grade. The math TEKS covered in this Unit include: 3.2A, 3.2B, 3.2C, 3.2D. These math TEKS were grouped together to provide a cohesive Unit that enables 3rd grade students to understand the Properties of Place Value and to apply this understanding to compare and order whole numbers. See the full list of Units of Study for K–5.

How Might a Teacher Use the Tasks in This Unit?

As the Place Value Unit progresses, a teacher may want to use one of the instructional tasks to teach students the expectations of the Exemplars Assessment Rubric. A lesson using the task “Tables for a Party” may include the following steps:

  •  Whole Group: Read the task together and ask students to underline the question, identify important information in the problem and summarize the task by restating what the question is asking them to do. Example: I need to find out how many tables need to be set up for 34 students with no more than 10 students sitting at each table. On the Exemplars Rubric, this step is scored in the category for Problem Solving—Does the student understand the problem?
  •  Small Group: Ask students to work together, think of a plan, and write it down. Example: I will draw a diagram of tables with students sitting at them. At this point, students will implement their plan to solve the problem.

Example:

(Refer to the task Planning Sheet for additional examples of solution strategies.)

 Students will check their plan to make sure it works and put a box around their answer.

Example:

On the Exemplars Rubric, this step is scored in the category for Problem Solving—Does the student have a plan? Does the student get the correct answer? It is also scored in the category for Reasoning and Proof—Does the student show a systematic implementation of the plan?

Small Group: Ask students to polish their papers—

  • Organize your solution.
  • Explain your plan and how it solved the problem.
  • Create a representation—such as a diagram with a key, use a model (such as manipulatives), use a table, use a number line.
  • Use mathematical vocabulary and/or symbolic representation.
  • Label your solution.
  • Show your answer. Put a box around it. Make sure it answers the question.

On the Exemplars Rubric this step is scored in the category for Communication—Does the student use at least two mathematical vocabulary words, at least two correct symbolic representations or one of each? It is also scored in the category for Representation—Is the representation correct and appropriate to the solution?

 Small Group: Finally, make a connection—

  • Make an observation.
  • Identify and describe a pattern.
  • Make a comparison between this task and other tasks. Explain how the math is similar.
  • Identify a rule.
  • Create a hypothesis or conjecture to test.
  • Solve the problem using a different strategy to prove the original solution is correct.
  • Recreate the problem and show a different solution.

On the Exemplars Rubric, this step is scored in category for Connections—Does the student include a mathematically relevant connection? Making connections requires students to look at their solutions and reflect.

Using Anchor Papers & Scoring Rationales

Anchor papers and assessment rationales are provided with every summative assessment task. These problem-solving tasks are given at the end of a Unit of Study to assess students’ understanding. A summative assessment must represent a student’s total independent solution. One Hundred Miles is the summative assessment for the grade 3 Place Value Unit.

Anchor papers and scoring rationales provide a great way to show both teachers (in professional development sessions) and students the expectations of the Exemplars Rubric; i.e. What a Practitioner (meets the standard) piece of student work looks like. Analyzing Exemplars anchor paper solutions and rationales at the Practitioner and Expert levels help students polish their own work and measure their own progress toward a specific goal. Analyzing the Novice and Apprentice samples can help identify for students where the work falls short of the goal and specifically how the papers could improve.

To view other sample tasks and anchor papers for grades K–5, you can sign up for a free 30-day Trial for Problem Solving for the TEKS.

 

A Problem-Solving Lab to Support the Math Practices

Friday, October 31st, 2014

Written By: Donna Krachenfels & Debra Sander, Teachers from PS 54

The school administrators at PS 54 had a vision to create a math laboratory based on the eight Standards of Mathematical Practice. The idea was to create a setting in which students could focus on multi-step problem solving.

The Exemplars program has given our students many opportunities to build and strengthen their problem-solving skills. Students were also able to strengthen their close reading skills as they reread problems multiple times to identify and think about the relevant information necessary to find a solution. Collaboration allowed students to become confident in their problem-solving skills and increased their abilities to construct viable arguments as they defended their solutions and critiqued the solutions of their classmates. Students were not afraid to take risks as they tried different representations and strategies to solve problems. As a result of the Exemplars math program, our students became more confident and more independent problem solvers.

The math laboratory is in its second year at PS 54. Last year, our data saw increased math scores for the classes that participated in the problem-solving lab. This year, the trend continued and all general education students passed the state math exam.

Special thanks goes to Exemplars professional development consultant Deb Armitage for all of her help and support. She is a true math educator!

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

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