According to one student at Tokyo International School (TIS), “Our teachers want us to really understand the mathematics we learn.” That’s why Exemplars plays a vital part of the school’s approach to teaching problem solving. Watch the video below to see Exemplars in action in TIS’s first-grade class — and then read on to learn how our materials enhance the school’s approach to assessment.

Written By: Josef Kaufhold, TIS primary school teacher

Tokyo International School is a pre-K through 8th-grade school located in central Tokyo, Japan. Our 350 students represent more than 40 nationalities, making TIS a truly international school. For TIS, the Exemplars Library represents a step forward in assessing mathematical thinking.

As an International Baccalaureate Primary Years Program school, we are committed to finding resources that support students’ development of knowledge, concepts, skills, and attitudes. Exemplars tasks pose students with authentic problems that require them to demonstrate their understanding, knowledge, and skills. Through iconic representations, students visualize their conceptual understandings as they develop solutions.

Because the expectation that they show their mathematical thinking yields such detailed responses, we can effectively assess student understanding and thoroughly report on Common Core outcomes to our students and their families. Using Exemplars, we are able to assess students’ growth when comparing rubric scores for pre- and post-assessments. Alongside the rubrics, the anchor papers provide specific qualitative indicators of student performance.

Exemplars problems engage curiosity and require commitment to solve. Students learn to cooperate and communicate effectively when working on these problems, which helps to develop positive attitudes towards mathematics. Through practice, students develop a strong, transferable grasp of the problem-solving process. Our staff has welcomed Exemplars into our balanced repertoire of tools used for effectively teaching math.

This blog is a reflection written by Dr. Courtney Baker, Ph.D., an Assistant Professor in Math Education Leadership at George Mason University, on her use of Exemplars with her elementary teacher candidates. The findings outlined in this piece would be beneficial to other professional development courses in a school or district setting.

Exploring Exemplars

Promoting discourse from rich tasks that move mathematical thinking forward challenges elementary teacher candidates, as their past experiences in working with both mathematics and children is often limited.

In teaching practice, the ability to come up with alternative approaches that vary from the traditional algorithm is necessary in order to anticipate student responses and to plan appropriate instructional opportunities. For new teacher candidates, however, this can represent a significant hurdle. Last semester, I incorporated resources from Exemplars into my practice to support my students with developing the background knowledge required to successfully anticipate student responses.

Upon my initial exploration of the Exemplars Library I was intrigued. How could I use these resources to not only introduce how children might authentically approach a particular problem, but also to elicit discussions centered on mathematics content? To meet these objectives, I needed to find a rich task that promoted a deep understanding of the pedagogical and content knowledge required to teach elementary mathematics. After looking through the expansive Library, I realized that multiple tasks met my needs as an instructor. I decided on a fifth-grade task titled “Crab Walk Relay Race.”

Doing the Math

I handed the pre-service teachers in my class hard copies of the corresponding overhead and created small groups to “do the math” collaboratively. I asked them to think about possible approaches using multiple representations (concrete, pictorial, and abstract) as well as possible student misconceptions. I wanted the teacher candidates to anticipate how a student might approach the task. I also wanted them to think critically about the manipulatives that would prove beneficial. My hope was for the teacher candidates to apply the knowledge gained from both their readings and fieldwork as they explored the Exemplars task. What happened in class, exceeded my expectations.

What started out as a 25-30 minute activity became a 60+ minute experience. Grounded in authenticity, the task promoted conversations that connected the teacher candidates’ background experiences with problem solving. Specifically, many of my students shared connections to either coaching sports teams or creating similar environments with children.

My class explained their strategies to one another as they tested their hypotheses. While they searched for an efficient strategy, they constantly compared their thinking to ensure that they were creating the most equitable teams. This continual communication allowed teacher candidates to share their thinking while using mathematically rich vocabulary.

Connecting to the Curriculum

After each group anticipated how students might approach the task, I provided them with the corresponding Exemplars Preliminary Planning Sheet. This resource proved to be an amazing support in the exploration of the mathematics content. Due to their limited experience with the standards, these pre-service teachers previously struggled to find connections between rich tasks and state standards. However, with the support of this document, my students were able to clearly articulate which mathematics standards the task was aligned with.

Analyzing Anchor Papers as Student Work Samples

At the heart of my activity was the analysis of student work samples. I took the levels (Novice, Apprentice, Practitioner and Expert) off of each anchor paper, and provided all samples to each group for evaluation. Initially, I thought that without the labels the teacher candidates would easily be able to sort and classify the student work samples. While I knew the discussion would center on determining what exactly each student knew, what I did not expect was the rich discussion that followed.

Teacher candidates were amazed at the variety of ways a student could approach the problem (many of which they had not anticipated). While some papers were easy to score, others were more challenging. Additionally, the anchor papers provided guidance to the depth of knowledge required to be identified as a successful problem solver. Their discussions centered on questions such as:

How do we define the difference between a student who is an Apprentice and one who is a Practitioner?

What questions could we ask the student to help them explain their thinking to us?

To what extent must a student display their knowledge to be identified as an Expert problem solver?

Is it possible for a student to identify a correct solution, but not be an Expert problem solver?

These questions arose organically from the use of this Exemplars task, and pushed my students’ understanding of problem solving.

Supporting Teacher Development

Preparing individuals to effectively teach elementary mathematics is a challenge, as it is impossible to provide each teacher candidate with all the knowledge and resources required to effectively teach elementary mathematics. Looking at how students might authentically approach a problem is essential for teachers to further their practice. Incorporating the Exemplars task into my teaching allowed teacher candidates to critically reflect on their practice and predict how students might approach problems through the analysis of these materials.

Sharing this Exemplars task with my students made for a great discussion that furthered their understanding of teaching elementary mathematics, and highlighted the complexities of problem solving. It also provided teacher candidates with a quality resource that they can potentially access as they enter into classrooms of their own.

Written By: Danielle Descarfino, Fifth Grade Teacher at P.S. 94 in Brooklyn

Getting Started

From the beginning of the school year, I used Exemplars problem-solving tasks regularly to create routines that have helped my fifth grade students grow and succeed. Following the Problem-Solving Procedure is a central part of this.

Although each task is different, the procedure helps kids internalize a framework for approaching a problem. I provided each student with his or her own color copy (in a sheet protector for safe keeping.) Each time we begin an Exemplars task, the students take out their Problem-Solving Procedures and refer to it. I also have a poster-sized version prominently displayed in the classroom, which I hold up and point to while guiding and facilitating tasks.

Building Background Knowledge

My class is made up of English Language Learners and former English Language Learners, so I anticipate that reading and understanding the problem may be especially challenging for them. We read the problem together, I ask questions to activate their background knowledge, and I often provide pictures that help them visualize the problem.

For example, we recently completed “A New Aquarium,” a 5.MD.C.5a task involving volume. We had been working on this math concept for only a few days and this was our first volume Exemplars task. Before reading the problem, I displayed a photo of an aquarium on the Smart Board and discussed the following questions with the class:

What is an aquarium?

What type of solid is this aquarium?

How could you figure out how much space this aquarium takes up? What steps would you take?

Although many students initially were not familiar with the word “aquarium,” after this discussion, they understood that an aquarium is a fish tank and a rectangular prism, which meant that we would be calculating its volume to find out how much space it takes up. Using visual aids and background questions to ensure that students understand the situation in the problem has been very helpful when completing Exemplars with English Language Learners.

Differentiation

We always utilize the differentiated Exemplars tasks. Students are aware of which problem-solving group they are in and know where to sit when it is time for an Exemplars task. One group gets the More Accessible Version; they are guided through the problem as they work with the Special Education teacher at a kidney-shaped table. The other two groups receive the Grade Level and More Challenging versions and sit with their groups in desk clusters, like a team of problem solvers.

For the Grade Level and More Challenging groups, we discuss background information, read the problem out loud, annotate it, and write our “I have to find …” statements. Then the students go on to work with their groups to complete the task while the teacher takes on the role of a facilitator, conferring with groups. Students share ideas, address misconceptions, and explain their mathematical reasoning to one another as they solve.

Motivating Students

I love hanging Exemplars tasks on bulletin boards. I think it’s useful for students to look at the page and see all of the different ways their classmates organize and express their mathematical thinking through equations, representations, and writing.

From day one, I have made it clear that it is expected that their finished work clearly communicate their problem-solving steps to the reader. Not only should the students make an effort to write neatly, but they should also organize their problem-solving steps on the page in a way that makes sense. Sometimes if a student is not showing all of their steps or it is unclear, I’ll say, “I am confused. When I look at your paper, I don’t understand the steps you took to solve the problem.” When the students have the understanding that a goal is to communicate their math thinking to a reader, it helps them create a higher-quality finished product.

Another great way to motivate students is through mathematical connections. I have given a strong emphasis to connections, as I initially noticed that once students solve the problem, they feel like they are done! This is not the case, because noticing mathematical connections, patterns, and alternate strategies really helps students understand mathematics on a deeper level and practice critical thinking skills.

To help them stretch their thinking, I discourage students from writing “boring” connections, like “This number is greater than that number” or “John ate the least amount of pizza.” Instead, I encourage them to use mathematical language, create a second representation, show steps to solving with alternate strategies, convert fractions/decimals/percents, or extend the problem by adding to the story in the original problem. Once they get the hang of it, they start being more creative, going above and beyond to make more complex math connections. During the volume unit, I taught students how to use grid paper to make scaled models of rectangular prisms. When completing these tasks, many students decided to build models to represent the rectangular prisms in the task and attach them to make 3-D Exemplars. They looked great, and the students loved making them!

Peer Assessing

At the beginning of the year, I explained each portion of the Exemplars rubric to the students. The rubrics are very student-friendly and I find that they inspire students to want to become Experts.

Each time I assess Exemplars, I use the rubric along with a sticky note full of feedback. The sticky note always contains one “Glow,” something the student did well, and one “Grow,” something the student could improve upon. At the beginning of the year, I let the students know that when they become more comfortable with Exemplars, they would learn how to peer assess. After a few months, I told the students that they were ready to peer assess one another’s work. They were so excited! This made them feel proud that they had reached a new level of expertise in problem solving and feel empowered that they were now trusted to assess a classmate’s work.

To peer assess, they do exactly as the teacher has done all year: complete the student rubric and use a sticky note to write “Glow and Grow” feedback. An example of this can be seen below. From the start, I was so impressed at how well the students were able to assess one another’s work with Exemplars. I found that regularly providing students with written feedback and referring to the rubric when expressing expectations is a great way to model peer-assessment. Furthermore, the experience of assessing Exemplars helps students get new ideas from their classmates and become more aware of how their own work will be graded.

Task: A New Aquarium

(More Accessible Version)

Joseph has a new rectangular aquarium. The aquarium has a length of four feet, a width of two feet, and a height of two feet. What is the volume of Joseph’s new aquarium? An aquarium holds one inch in length of fish for each twelve square inches of the area of the base of the aquarium. Joseph can buy fish in two different sizes—about three inches in length or about five inches in length. About how many three-inch fish can Joseph put in the new aquarium? About how many five-inch fish can Joseph put in the new aquarium? Show all your mathematical thinking.

Danielle’s Biography

Danielle Descarfino is a fifth grade teacher at P.S. 94 in Sunset Park, Brooklyn. She graduated from Fordham University with a Masters of Science in Teaching English to Speakers of Other Languages. Danielle grew up in Tappan, New York, and currently lives in Brooklyn. She was inspired to become a teacher after spending time as a volunteer teaching English at an orphanage and community center in Salvador, Brazil.

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!

As you begin preparing your staff to focus on the new math TEKS this year, rubrics should play a key role in terms of helping your teachers and students achieve success with the new standards.

What are rubrics?

A rubric is a guide used for assessing student work. It consists of criteria that describe what is being assessed as well as different levels of performance.

Rubrics do three things:

The criteria in a rubric tell us what is considered important enough to assess.

The levels of performance in a rubric allow us to determine work that meets the standard and that which does not.

The levels of performance in a rubric also allow us to distinguish between different levels of student achievement among the set criteria.

Why should teachers use them?

The assessment shifts in the new math TEKS pose challenges for many students. The use of rubrics allow teachers to more easily identify these areas and address them.

For Consistency. Rubrics help teachers consistently assess students from problem to problem and with other teachers through a common lens. As a result, both teachers and students have a much better sense of where students stand with regard to meeting the standards.

To Guide Instruction. Because rubrics focus on different dimensions of performance, teachers gain important, more precise information about how they need to adjust their teaching and learning activities to close the gap between a student’s performance and meeting the standard.

To Guide Feedback. Similarly, the criteria of the rubric guides teachers in the kind of feedback they offer students in order to help them improve performance. Here are four guiding questions that teachers can use as part of this process:

What do we know the student knows?

What are they ready to learn?

What do they need to practice?

What do they need to be retaught?

How do students benefit?

Rubrics provide students with important information about what is expected and what kind of work meets the standard. Rubrics allow students to self-assess as they work on and complete problems. Meeting the standard becomes a process in which students can understand where they have been, where they are now and where they need to go. A rubric is a guide for this journey rather than a blind walk through an assessment maze.

Important research shows that teaching students to be strong self-assessors and peer-assessors are among the most effective educational interventions that teachers can take. If students know what is expected and how to assess their effort as they complete their work, they will perform at much higher levels than students who do not have this knowledge. Similarly, if students assess one another’s work they learn from each other as they describe and discuss their solutions. Research indicates that lower performing students benefit the most from these strategies.

Rubrics to Support the New Math TEKS.

Exemplars assessment rubric criteria reflect the TEKS Mathematical Process Standards and parallel the NCTM Process Standards. Exemplars rubric consists of four performance levels (Novice, Apprentice, Practitioner (meets standard) and Expert) and five assessment categories (Problem Solving, Reasoning and Proof, Communication, Connections and Representation).

Students can begin to self-assess in Kindergarten. At Exemplars, we encourage younger students to start by using a simple thumbs up, thumbs sideways, thumbs down assessment as seen in the video at the bottom of the page.

Our most popular student rubric is the Exemplars Jigsaw Rubric. This rubric has visual and verbal descriptions of each criterion in the Exemplars Standard Rubric along with the four levels of performance. Using this rubric, students are able to:

Self-monitor.

Self-correct.

Use feedback to guide their learning process.

How to introduce rubrics into the classroom.

In order for students to fully understand the rubric that is being used to assess their performance, they need to be introduced to the general concept first. Teachers often begin this process by developing rubrics with students that do not address a specific content area. Instead, they create rubrics around classroom management, playground behavior, homework, lunchroom behavior, following criteria with a substitute teacher, etc. For specific tips and examples, click here.

After building a number of rubrics with students, a teacher can introduce the Exemplars assessment rubric. To do this effectively, we suggest that teachers discuss the various criteria and levels of performance with their class. Once this has been done, a piece of student work can be put on an overhead. Then, using our assessment rubric, ask students to assess it. Let them discuss any difference in opinion so they may better understand each criterion and the four performance levels. Going through this process helps students develop a solid understanding of what an assessment guide is and allows them to focus on the set criteria and performance levels.

Deidre Greer, professor at Columbus State University, works with students at a Title I elementary school in Georgia. Greer states that in her experience,

The Exemplars tasks have proven to be engaging for our Title I students. Use of the student-scoring rubric helps students understand exactly what is expected of them as they solve problems. This knowledge then carries over to other mathematics tasks.

At Exemplars, we believe that rubrics are an effective tool for teachers and students alike. In order to be successful with the learning expectations set forth by the new math TEKS, it is important for students to understand what is required of them and for teachers to be on the same “assessment” page. Rubrics can help.

To learn more about Exemplars rubrics and to view additional samples, click here.

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.

Written By Exemplars Math Consultants: Deborah Armitage, M.Ed. and Dinah Chancellor, M.Ed.

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Math TEKS through problem solving. This blog represents Part 4 of a six-part series that features a problem-solving task linked to a Math TEKS Unit of Study and a student’s solution in grades K-5. Evidence of all seven Mathematical Process Standards 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 Processes 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 Math TEKS 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 Processes 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 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 Adding and Subtracting Fractions Unit.

“Sharing Muffins” could be used toward the end of the learning time allocated to this Unit. 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.

Math TEKS Alignments:

Adding and Subtracting Fractions Unit

The Adding and Subtracting Fractions Unit involves using a variety of methods to join or separate parts referring to the same whole. Methods may include replacing mixed numbers with equivalent fractions; using properties of operations and the relationship between addition and subtraction; and using visual models of fractions. Questions to answer may include:

Why must we use the same “whole” when adding or subtracting fractional parts?

How can a number line represent adding or subtracting fractions?

How can benchmark fractions help to determine whether a sum or difference makes sense?

The standards covered in this Unit include:

4.3 Number & operations. The student applies mathematical process standards to represent and generate fractions to solve problems. The student is expected to:

4.3A represent a fraction a/b as a sum of fractions 1/b, where a and b are whole numbers and b > 0, including when a > b.

4.3B decompose a fraction in more than one way into a sum of fractions with the same denominator using concrete and pictorial models and recording results with symbolic representations such as 7/8 = 5/8 + 2/8; 7/8 = 3/8 + 4/8; 2 7/8 = 1 + 1 + 7/8; 2 7/8 = 8/8 + 8/8 + 7/8;

4.3E represent and solve addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations;

4.3F evaluate the reasonableness of sums and differences of fractions using benchmark fractions 0, 1/4, 1/2, 3/4, and 1, referring to the same whole.

Mathematical Process Standards

The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:

4.1A apply mathematics to problems arising in everyday life, society, and the workplace;

4.1B use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

4.1C select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

4.1E create and use representations to organize, record, and communicate mathematical ideas;

4.1F analyze mathematical relationships to connect and communicate mathematical ideas; and

4.1G display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Written By Exemplars Math Consultants: Deborah Armitage, M.Ed. and Dinah Chancellor, M.Ed.

Summer Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Math TEKS through problem solving. This blog represents Part 4 of a six-part series that features a problem-solving task linked to a Math TEKS Unit of Study and a student’s solution in grades K-5. Evidence of all seven Mathematical Process Standards 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 Processes 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 Math TEKS 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 Processes 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 Algebraic Reasoning Unit.

“Henry’s Lego Structure” could be used toward the end of the learning time allocated to this Unit. 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.

Math TEKS Alignments:

Exemplars Algebraic Reasoning Unit

The Algebraic Reasoning Unit involves analyzing numerical patterns and the relationships between addition and subtraction; multiplication and division in order to answer questions such as:

What models can be used to represent addition and subtraction situations?

What models can be used to represent multiplication and division situations?

Given a multiplication or division situation, how can you generate an equation in which the unknown is either a missing factor or a missing product?

Given an equation with a missing factor or a missing product, how can you create a situation to match it?

How can a table of real-world number pairs help to generate multiplication or division equations to represent the relationships in the table?

The standards covered in this Unit include:

3.5 Algebraic Reasoning:

3.5A represent one- and two-step problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, such as strip diagrams and number lines, and equations.

3.5B represent and solve one- and two-step multiplication and division problems within 100 using arrays, strip diagrams, and equations.

3.5C describe a multiplication expression as a comparison such as 3 x 24 represents 3 times as much as 24.

3.5D determine the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is either a missing factor or product such as the value 4 makes 3 x [ ] = 12 a true equation.

3.5E represent real-world relationships using number pairs in a table and verbal descriptions such as 1 insect has 6 legs, 2 insects have 12 legs, and so forth.

Mathematical Process Standards

3.1A apply mathematics to problems arising in everyday life, society, and the workplace;

3.1B use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

3.1C select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

3.1D communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

3.1E create and use representations to organize, record, and communicate mathematical ideas;

3.1F analyze mathematical relationships to connect and communicate mathematical ideas; and

3.1G display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Written By Exemplars Math Consultants: Deborah Armitage, M.Ed. and Dinah Chancellor, M.Ed.

Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Math TEKS through problem solving. This blog represents Part 3 of a six-part series that features a problem-solving task linked to a Math TEKS Unit of Study and a student’s solution in grades K-5. Evidence of all seven Mathematical Process Standards 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 Processes 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 Math TEKS 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 Processes 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 Problem Solving with Money Unit.

“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 provide 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.

Math TEKS Alignments

Exemplars Problem Solving with Money Unit

The Problem Solving with Money Unit involves knowing the relationship between U.S. coins, how to use skip counting and other methods to find the value of a collection of coins, and answer questions such as:

If you have saved a given amount of money and you know a toy costs a certain amount, how can you figure out how much more you need to save to buy the toy?

If you have a collection of coins worth 30 cents and put a quarter in your bank every week, how much money will you have in three months?

The standards covered in this Unit include:

2.5 Number & Operations:

2.5A determine the value of a collection of coins up to one dollar

2.5B use the cent symbol, dollar sign, and the decimal point to name the value of a collection of coins.

Mathematical Process Standards

2.1A apply mathematics to problems arising in everyday life, society, and the workplace;

2.1B use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

2.1C select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

2.1D communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;

2.1E create and use representations to organize, record, and communicate mathematical ideas;

2.1F analyze mathematical relationships to connect and communicate mathematical ideas; and

2.1G display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Written By Exemplars Math Consultants: Deborah Armitage, M.Ed. and Dinah Chancellor, M.Ed.

Blog Series Overview:

Exemplars performance-based material is a supplemental resource that provides teachers with an effective way to implement the Math TEKS through problem solving. This blog represents Part 2 of a six-part series that features a problem-solving task linked to a Unit of Study for the Math TEKS and a student’s solution in grades K–5. Evidence of all seven Mathematical Process Standards 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 Processes 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 Math TEKS 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 Processes 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 tasks aligned to the Strategies for Addition and Subtraction Unit designed by Exemplars for the new Math TEKS. This task could be used toward the end of the learning time allocated to this Unit. “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 Birdbath

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 of your mathematical thinking.

Math TEKS Alignment:

Exemplars Strategies for Addition & Subtraction Unit

The Strategies for Addition and Subtraction Unit involves understanding the processes of addition and subtraction in order to solve problems and answer questions such as:

If we know all of the parts, how can we find the whole?

If we know the whole and one of the parts, how can we find the missing part?

Given an equation, can you create an addition or subtraction situation to match it? How can you prove it matches the equation?

The standards covered in this Unit include:

1.3 Number & Operations:

1.3A use concrete and pictorial models to determine the sum of a multiple of 10 and a one-digit number in problems up to 99.

1.3D apply basic fact strategies to add and subtract within 20, including making 10 and decomposing a number leading to a 10.

1.3E explain strategies used to solve addition and subtraction problems up to 20 using spoken words, objects, pictorial models, and number sentences.

1.3F generate and solve problem situations when given a number sentence involving addition or subtraction of numbers within 20.

1.5 Algebraic Reasoning:

1.5D represent word problems involving addition and subtraction of whole numbers up to 20 using concrete and pictorial models and number sentences.

1.5G apply properties of operations to add and subtract two or three numbers such as if 2 + 3 =5 is known, then 3 + 2 = 5.

Mathematical Process Standards:

1.1A Apply mathematics to problems arising in everyday life, society, and the workplace;

1.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution;

1.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems;

1.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

1.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.