Student Work
The problems in this section come from CMP classrooms. The work can be downloaded for teachers to examine during planning time, to use in their classroom, for use in a collaborative meeting, or for use in professional development activities.
Student Work from various CMP Units is provided to stimulate conversations about the mathematical teaching and learning embedded in individual Problems. To add more examples of student work on CMP problems from your classrooms, please email connectedmath@msu.edu.
In the Connected Mathematics Project (CMP) curriculum, Connected Mathematics4, student work as a context for student leaning is central to the success of students’ mathematical understanding. The examples of student work in the materials provide a glimpse into the ways that CMP students use to construct answers. They help students view themselves as doers, creators, and communicators of important mathematics.
In Connected Mathematics4 examples of student work occur in several places:
- Embedded in a problem in the Student Book to show various strategies that students use to solve the problem. These examples may confirm current students’ mathematical identities, or deepen, expand, and refine their mathematical understandings. Most of the examples come from CMP classrooms over the past 35 years.
- Embedded in the Launch-Explore-Summarize phase of the problem in the Teacher Guide.
- Embedded in the Applications-Connections- Extensions in the student book. Some of the work may also occur in the teacher guide for the problem.
For more information about current research on student work carried out by the CMP Staff, visit the Student Work in Curriculum Materials Research Project .
Grade 6
Click Here to View the Student Work Example
Relates to:
CMP4 Bits of Rational Problem 3.3 Sharing Trail Mix: Dividing a Fraction by a Whole Number
These student strategies from CMP3 Let’s Be Rational Problem 3.3 are a teacher’s documentation of student thinking over many years using CMP. When given an opportunity to think about how operations work with various numbers in a problem-solving setting, students will generate both non-standard and standard ways of calculating.
Many people have learned to divide fractions by learning an algorithm that may or may not make sense them. In CMP, students are asked to make sense of fraction division on their way to developing an algorithm
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 7 Comparing and Scaling Problem 3.1 Using Percents: What Is the Tax?
These student strategies from CMP3 Decimal Ops Problem 4.1 are a CMP teacher’s documentation of student thinking. During the Summarize, she recorded the strategies her students used compute the tax on various items. These charts are then posted on the wall as a record of the student strategies.
Many people have learned to compute percents by learning an algorithm that may or may not make to sense them. In CMP, students are asked to make sense of various strategies to understand how and when percents are used to calculate amounts.
CMP3 Decimal Ops Problem 4.1: What is the Tax on This Item? is a chance for students to develop understanding and skill in the use of percents to calculate taxes.
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 6 Variables and Patterns Problem 1.4 Chincoteague Island to Norfolk: Stories, Tables, and Graphs
Note: In the CMP4 version, the story varies from previous versions. Also, the values for time and distance are slightly different.
Problem 1.3 in CMP3 Variables and Patterns challenges students to create a table and graph representation that match a description of the data expressed in words. There is no one single correct answer. The challenge in the problem requires students to portray a written description on a table and graph.
The examples of student work provide a glimpse into the ways that CMP students use to construct answers. In small groups, students created table and graph representations of the story in CMP3 Variables and Patterns, Problem 1.3: From Lewes to Chincoteague Island: Stories, Tables, Graphs. Then staying with their group, the students used sticky notes to provide feedback as they did a gallery walk around the classroom. The gallery walk provided a chance for students to critique the reasoning of others while also allowing them to compare and contrast the work to their own.
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 6 Variables and Patterns Problem 2.3: What’s the Story: Interpreting Graphs, What If… ? Situation A
Note: In CMP4 some of the graphs have changed. Also, in CMP4, the Material Kit cards and Learning Aid offer a matching opportunity.
Three CMP teachers offer strategies for how to engage students in thinking about CMP3 Problem 2.4 in Variables and Patterns. The problem requires students to identify the variables, decide how they are related in “stories”, and then choose the graph that best represents the relationship. Sample student work is shown.
Teachers use different strategies when engaging students in thinking about the graphical representation of variables in CMP3 Variables and Patterns. Here we provide examples of teachers’ thinking and student work. Three different ways to implement the problem are suggested by experienced CMP teachers.
Grade 7
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 7 Shapes and Designs Problem 3.2
CMP3 Shapes and Designs Problem 2.2 asks students to make an important generalization about the angle sum property for any polygon. Students are introduced to the strategies of three students and asked to make sense of the strategies for finding the angle sums of polygons.
This student work can be used with students as an extension of the thinking already presented in Problem 2.2. This strategy can deepen students understanding of polygons and their shapes. It is interesting to note that the different examples of student thinking presented in Problem 2.2 may have prompted one student to explore the possibilities of other strategies.
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 7 Shapes and Designs Problem 3.1 Back to the Bees: Tiling a Plane Experiment
Note: Now for the experiment in CMP4, we limit the set of shapes used from the Shape Set.
The photos show students working during the Explore of CMP3 Shapes and Designs Problem 2.3. Students are creating posters as they tested to see which regular polygons do and do not tessellate. Using the results, the students and teacher discussed how angle measures in the polygons can explain the tiling properties.
Students in the photos were exploring these questions from the teacher:
Which regular polygons will tile a flat surface with no gaps or overlaps?
Which polygons will not do so?
Why do some shapes tile and others do not?
The student work photos provide a glimpse into the CMP classroom during the Explore phase of the lesson. In particular, the photos show the active role of the student in the process constructing their knowledge around well-defined challenges.
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 7 Comparing and Scaling Problem 3.1 Using Percents: What Is the Tax?
These student strategies from CMP3 Decimal Ops Problem 4.1 are a CMP teacher’s documentation of student thinking. During the Summarize, the teacher recorded the strategies her students used compute the tax on various items. These Anchor Charts are then posted on the wall as a record of the student strategies.
Many people have learned to compute percents by learning an algorithm that may or may not make to sense them. In CMP, students are asked to make sense of various strategies to understand how and when percents are used to calculate amounts.
CMP3 Decimal Ops Problem 4.1: What is the Tax on This Item? is a chance for students to develop understanding and skill in the use of percents to calculate taxes.
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 7 Comparing and Scaling Problem 1.1 Mixing Juice: Choosing a Comparing Strategy
Note: In CMP4, some of the strategies shown here have been incorporated into the What If…? situations in the student book.
The “Orange Juice” Problem in the Grade 7 Unit: CMP3 Comparing and Scaling has become a classic CMP problem. Iterations of the problem have been in all three versions of CMP. While working with the problem, students generate many interesting strategies for determining the “most orangey” tasting juice from four different recipes.
The CMP3 Problem 1.2: Mixing Juice occurs early in the unit and provides insights into students’ knowledge about rates and ratios. This group of 22 pieces of student work shows multiple strategies utilized by students to solve the problem. Students use ratio and rates, both part-to-part and part-to-whole, and a variety of number representations including fractions, decimals, and percents.
The student work from this problem has been used in many presentations and several articles about teaching and learning proportional reasoning concepts.
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 7 Comparing and Scaling Problem 2.1 Sharing Pizza: Comparing Rate Strategies
Students use many strategies to decide if the amount of pizza a camper would eat differs by which table the camper selects. This student work represents some of strategies that students use. The examples may help teachers anticipate the strategies that their own students may use to answer the question of whether a camper gets the same amount of pizza at each table.
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 7 Moving Straight Ahead Problem 2.1 Henri and Emile’s Race: Developing Strategies to Find a Solution
Students use many strategies when trying to decide how long the race should be for Henri and Emile in the CMP3 Moving Straight Ahead, Problem 2.1: Henri and Emile’s Race. Here we provide examples of student work produced by students in various CMP classrooms. The strategies include students guessing and checking, using a table, making a graph, using visual diagrams, and using the distance between the two boys to reason about the question.
The work does not show students’ answers to the question. Some of the work is in students’ writing and other work is re-created from actual CMP students. The examples may help teachers anticipate the strategies that their own students may use to answer the question of how long to make Henri and Emile’s Race.
Grade 8
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 8 Growing, Growing, Growing Problem 1.3 Killer Plant Strikes Lake Victoria: y-Intercepts Other Than 1
The examples of student work provide a glimpse into the ways that CMP students use to construct answers. Students use various representations to show the growth of a water plant. The examples may help teachers anticipate how their own students may answer the question of when the lake’s surface will be covered by the water plants.
Click Here to View the Student Work Example
Relates to:
CMP4 Grade 8 Mars, Gravity, and Painted Cubes Investigation 2 Mathematical Reflection
In CMP3, Frogs, Fleas, and Painted Cubes was designated as an Algebra I unit. Many districts taught this unit to all 8th Graders. This student work represents a strategy employed by a teacher to help students reflect and summarize what they knew about the distributive property. This strategy proved to be a beneficial formative assessment that helped both the teacher and the students determine and refine mathematical understandings.
The students in this classroom were concerned about taking Check Up 2 after Investigation 2 in the CMP3 Grade 8 unit: Frogs, Fleas, and Painted Cubes. The teacher told them that the Check Up was primarily about the quadratic expressions and the distributive property. The students asked if they could review the ideas before taking the Check Up the next day.
The teacher gave the following instructions to students.
- Get into groups of two or three.
- On a piece of chart paper, write what you know about the forms of quadratic expressions and the distributive property that we have been discussing.
- After charts are posted in the room, you will choose classmates to explain their paper. You can pick people who have different ideas or ways of showing the thinking.
- You can ask questions and take notes on any of the ideas that you find helpful and that add to what you already know.
The teacher has found this strategy helpful as a way to summarize and refine learning.
Grades 6, 7, & 8
Click Here to View the Student Work Example
Examples from CMP3 Grade 7
The examples are from CMP3 Grade 7 Units. However, this strategy could be used in any grade.
Relates to:
CMP4 Mathematical Reflections for each Unit
At the end of each CMP3 Unit, a teacher in Michigan has students summarize the Investigations on posters. Throughout the year students improved their ability to document and articulate their mathematical thinking.
Comments from the Teacher
As we end each Unit, my students are so excited and start to ask me, "When are we going to do the Unit Posters? It's our favorite thing to do." The day that students draw an Investigation card is my favorite. As the students pick a card, I hear, "Oh, I hope I get Investigation 2, I loved doing the Wumps." (Stretching and Shrinking Unit) Or, "I hope I get Investigation 3 because I loved pouches and coins." (Moving Straight Ahead Unit) These are my favorite days!