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Case Vignettes for

Community-based Math Modeling

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The Big Picture

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The Modeling Cycle

(Zavala & Aguirre, 2021)

Return to Table of Contents

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CRMT² Framework To Advance Equity

Community-Based Modeling Cases

Math Modeling Tasks

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Inclusive Playground Task - 3rd & 4th Grade

Inspiration for the Task

Building Mathematical Models

Reporting Out and Taking Action

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Equity Move

Distribute intellectual authority by creating

space for many students to communicate their ideas and experiences in multiple ways

Community-based Mathematical Modeling

To start, 3rd and 4th graders considered accessible playgrounds in the media. They planned a weekend field trip to a local inclusive playground to learn more about how everyone can play together. Kids in both grade levels leveraged school-wide data for their decision-making. Data included their peers’ interests and opinions, as well as an inventory of their current playground to assess for accessibility.

Investigation the Context & The Modeling Problem

During a teacher brainstorm of community issues, one teacher noted that at their school there were a number of students who could not access the playground during recess. This was often a topic of conversation with the students in her class. The task was created: how can we design an inclusive playground?

Students shared their area and budget designs in vertical 3rd-4th grade teams and created a presentation to share with the local community to advocate for an inclusive playground.

Students in 3rd grade considered concepts of area. Utilizing large, scaled map of the playground space, students considered what they would put in the new, inclusive playground. 3rd graders built models by:

• First starting with objects they knew would be smaller, then looking for equipment with larger area.
• Finding area by counting squares.
• With larger area, kids figured out it was much harder to count every square. They began finding area by using concepts within the distributive property: partial products and an area model.

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Students in 4th grade built models through budgeting both money and area. 4th graders built models by:

• Using multiplication and addition when buying more than one of an item.
• Using an aerial picture of the playground, estimating area to then figure out the price of accessible material (given in square feet) to recover base and replace the mulch.

Representation in Our Library Task - 2nd & 3rd Grade

Inspiration for the Task

Building Mathematical Models

Reporting Out and Taking Action

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Community-based Mathematical Modeling

To start, students investigated a Numberless graphic of nationwide data of racial representation in libraries. Students determined that it was not fair, recognizing the invisibility of certain racial groups. Students were asked:

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How fair is our classroom library book collection? Create a plan to help figure out if we have enough books with main characters from different racial and cultural backgrounds. Then, use your data and a budget to purchase new books for our library.

Investigation the Context & The Modeling Problem

Students in a local school noticed that they were having a hard time finding books that had main characters that mirrored their own racial and cultural backgrounds, prompting them to ask for more books. Teachers also noticed the disconnect between books on shelves and the racial identities of students in their classrooms.

As students analyzed their libraries they determined different ways to sort and count books using the race of main characters, including making decisions on categories for race. As students sorted and counted between 500 and 1,200 books, they used a variety of organizational systems to keep track of the data. Some mathematical idea of the task included:

• Counting in various ways, including grouping and skip counting
• Counting smaller groups and then adding those together to find the larger amount
• Using post-its and/or large paper to keep track of their counting and numerical data.
• Making class decisions about a sample of the library or the whole library to analyze.
• Interrogating: what is fair? What would a fair graph look like?

Then, students were tasked with creating a graphical representation. Using the data they collected, student groups made decisions about how to spend money on new books, within a given budget, to create a more fair library.

Students wrote letters to their principal explaining why they needed better representation of race in their library and why they had chosen the specific books with their budget. They also created presentations to share with others, ordered books to make classroom libraries more fair, and wrote letters to Scholastic.

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Lots of people can’t see themself in book’s. We found that most books in the classroom library were white and animals.

Future students could talk to teachers, principals, adults, and students to diversify other libraries. Scholastic should make more diverse books.

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Pacific islander had no books so we felt very disappointed and sad that some races had no books about there self.

It is very important to see how we are shown in books. We can read about ourselves and see ourselves in different ways. We want to be able to relate to a story.

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I know its an animal, but it is in Spanish and not all students speak English. (Eli) likes when you read in Spanish!

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I think this girl is a little like me. She looks like me. I know Yasmin!

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The Coin Harvest- 4th grade

A school wide service learning project focused on a coin drive to fight hunger. The issue of food insecurity is situated in a local context but is a global issue. This modeling task took place in the fall and was called the “Coin Harvest”.

Inspiration for the Task

Learning about the Context

Students could spend the money any way they chose, but had to justify why theirs was the “best” meal. They also had to estimate how many baskets could be created on their plan.Students worked in small groups to create grocery shopping lists. They made choices on how much of each item to buy to have enough food for a family of four. Students of all mathematical ability levels were able to participate due to the open-ended nature of the task. Students coached their partners with computations and checked each other’s work. Groups used strategies such as rounding decimals to whole numbers and repeated addition or subtraction to estimate how many meals could be purchased.

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Building Mathematical Models

Reporting Out and Taking Action

Decisions about Important Quantities

The Modeling Problem

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Equity Move

Distribute intellectual authority by creating

space for many students to communicate their ideas and experiences in multiple ways

Ms. R wanted students to better understand what “1 in 5 children face food insecurity” might look like in their community. Students analyzed population data and created and extended number patterns. Many students used 100s charts, coloring multiples of 5 to determine that 20 out of 100 children might be hungry. Students estimated this was about one classroom of kids. Students then used ten hundreds charts to show 1,000 and counted multiples of 20 to see that 200 out of 1,000 children, or about one grade level, might not have enough food. Students used similar strategies to see how many children could be affected out of 10,000 and 100,000, concluding that of the approximately 200,000 children in their county, 40,000 may experience food insecurity.

Students posted a number line in the school lobby to track how much money had been raised, and each day, students excitedly announced the new totals. Every time someone dropped coins in the class jar, the entire class would erupt in wild cheers. Students used the data to predict how much money they would raise. Initially, the class wanted to donate enough money for ten families but after they calculated the cost of a basket, they revised their plan to feed a smaller number of families. Mrs. R. commented on how students used data they collected and the actual cost of meals to inform their revised solutions. The project not only helped build community and empathy in the classroom, but students were empowered by the fact that they had mobilized such a large scale effort within the school. In one month, students raised a total of $1,300.00!

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Learning about the context inspired students to raise money to create food baskets for the community. The modeling problem was, ‘how can you plan the best, well-balanced Thanksgiving meal to feed the most families?

Students decided how many food baskets to make, and what to include in each basket. They also made assumptions about a reasonable price for each basket. To inform these decisions, students gathered data from the USDA food plans to see how much an average family of four spends on healthy meals. http://www.cnpp.usda.gov/USDAFoodPlansCostofFood/reports) (see chart below)

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Safe Water in our Schools - 3rd Grade - 5th Grade

Building Mathematical Models

Connecting to the Context

Reporting Out and Taking Action

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Equity Move

Distribute intellectual authority by creating

space for many students to communicate their ideas and experiences in multiple ways

Community-based Mathematical Modeling

Inspiration for the Task

Teachers brainstormed environmental justice topics. Access to clean water and the Flint Water Crisis were mentioned. In fact, many teachers commented on having the school’s water tested every year as a result of what happened in Flint, MI.

The Modeling Problem

Every year, schools check the drinking water for poisonous chemicals such as lead and copper. In some schools, water fountains are covered. Signs are posted, “do not use.” If our school water fountains were unsafe, each classroom would use a water dispenser with a large jug of water to make sure everyone has enough water to drink.

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In Mrs. G’s 4th grade class, the lesson hit close to home. A week prior, the school’s water had been tested and found to have lead. All the water was shut off and classrooms had to share water dispensers with large 5-gallon jugs. To build more context, Mrs. G created a cross-curricular connection ELA. Students practiced the 3 read strategy as they read scholastic news articles about the Flint Water Crisis and watched a short video on the situation

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Equity Move

Connect to students’ experience

Students were given some information about drinking water needs for different age groups. They were also given measurement information related to how many gallons in the large jug (5) and how many cups in 1 gallon.

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Students created models that used the information they had about drinking water and ages along with their own lived experiences to determine how much water 1 person would need in 1 day. Different assumptions were made about the number of people needing water, how much water a person might get water from the dispenser, and how long they were at school.

Mrs. G invited students to share their work. While students shared out, she recorded their results on a summary table. She then asked students to help other children in similar situations. What would they tell them to do? She asked them to generalize, to name specific quantities that someone would need to know to help their school in this situation. This is a taking action component of the task. Students could then develop an infographic that helps others understand the importance of access to clean water at school and in their communities. In addition, they could connect with school’s administrator about their findings to make sure there is enough water for their needs through the end of the school year.

Bird Feeders - 1st Grade

Inspiration for the Task

Building Mathematical Models

Understanding the Context

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Reporting Out and Taking Action

Decisions about Important Quantities

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The Modeling Problem

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Equity Move

Analyze and Take Action

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Equity Move

Distribute intellectual authority by creating

space for many students to communicate their ideas and experiences in multiple ways

Students had been noting that birds in the community were their friends and liked watching them. They discussed feeding the birds and that sometimes squirrels would eat the bird seed. This first grade teacher decided to create a task where students could design and build bird feeders to take home for their friendly neighborhood birds.

After deciding as a class to use four egg cups for each bird feeder, students finished planning their designs by choosing whether to use two strings or four strings for the feeder. Table groups shared their plans with the class and used their plans to build the feeders on the following day.

When students returned to their groups to investigate whether the egg cartons collected by the teacher would be enough for everyone to make a six egg cup feeder, they quickly realized they would run out. One group explained that if they used six cups there would only be enough for thirteen bird feeders. Another groups suggested revising the class plan to make bird feeders with four egg cups instead. They explained that making 4 egg cup bird feeders would leave enough materials for the teacher to make one too. To support their claim, they drew their plan on the egg carton diagrams, circling groups of four cups to show there would be enough.

Students worked in groups to brainstorm different ways to cut the egg cartons to make the bird feeders. The teacher provided various tools to support their reasoning including images of egg cards of different sizes, and physical realia ( egg cartons). After small group time, the teacher pulled students back together to share ideas. The class decided that everyone would make a six cup bird feeder because this size feeder would provide more room for birds to land.

Students shared what they know about bird feeders, noting “it’s like a house but the bird can live in it” and “birds like seeds”. The teacher displayed pictures of bird feeders made of egg cartons (the kind they would make) and asked students to share what they noticed and wondered. Students’ ideas included: “how many seeds in each one?”, “why are they different sizes?” “do we have the egg cartons here?”

After watching a video that showed how to make an egg carton bird feeder, students identified several important quantities such as the number of pieces of string, the size of each feeder (how many sections of the egg carton are used) and the number of feeders they would make. While the class decided on some of these quantities all together - such as each person in the class will make 1 bird feeder, students worked in small groups to make decisions about other quantities, such as how many pieces of string they would use.

Building on students’ ideas, the teacher displayed several egg cartons she had selected and stated the modeling problem, “do we have enough egg cartons for everyone in the class to make a bird feeder?”.

Making Lanterns - 2nd Grade

Inspiration for the Task

The Modeling Problem

Decisions about Important Quantities

Taking Action:

Making lanterns to light a path!

Building Mathematical Models

Learning about Context

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Equity Move

Distribute intellectual authority by creating

space for many students to communicate their ideas and experiences in multiple ways

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Equity Move

Connect to students’ experience

In honor of Lunar New Year, Ms. F decided to try a modeling task focused on making lanterns out of paper bags and tealights. Ms. F wanted her class to figure out how many lanterns would be needed to light a pathway during the lunar new year parade at her school. She was also curious about student knowledge and experience with measuring.

How many lanterns are needed to light the outside walkway?

Showing lanterns from different cultural traditions (Luminarias in Mexico; Lanterns for Eid Celebrations; Diwali Hindu festival of lights) and local events, students explored different pathway designs. They also read books about Lunar New Year.

Students were empowered to make decisions about important quantities. For example, students decided the distance between their lanterns (must be equal spacing). They also made design decisions to place lanterns: would lanterns be on 1 side of the path or both sides; in pairs, clusters, or individual.

Students created models to show how many lanterns they would need for the pathway. In this model, students created 7 sections with 5 lanterns on each side of pathway for a total of 10 lanterns in each section. This visual model shows 70 lanterns.

Students made the lanterns and put them on the outside walkway for the Lunar New Year parade.

“I think just the openness of having a walkway and not telling them how far apart they need to be was a really great opportunity for them to make decisions.” - Ms.F

Students compared their plans and made decisions about which plan would be best to help Ms. F buy the materials.

Making Picture Frames- 4th Grade

Inspiration for the Task

The Modeling Problem

Decisions about Important Quantities

Taking Action : Sharing Frames with the Community

Building Mathematical Models

Learning about Context

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Equity Move

Distribute intellectual authority by creating

space for many students to communicate their ideas and experiences in multiple ways

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Equity Move

Connect to students’ experience

To honor loved ones from students’ families and communities, Ms. B’s school was planning a Dia de los Muertos (Day of the Dead) altar. Ms. B wanted her class to contribute by making popsicle stick picture frames for the altar, to display photos of special people in their lives. At another school, students were planning a celebration to honor veterans in their school and neighborhood community. Ms. V wanted her class to make “thank you” frames for a school-wide display.

Both teachers needed to know what supplies they would need for the project.

How many packages of popsicle sticks will we need for our class to make frames for the Dia de los Muertos / Honor our Veterans display?

Students explored different frame designs, and different types of popsicle sticks. They designed sample frames and made decisions about the frames they would make as a class.

Teachers provided information about the number of popsicle sticks per package (60 jumbo sticks or 100 multi-colored sticks), and then empowered students to make decisions about other important quantities. For example, students decided how many popsicle sticks they would need for one frame and how many frames each person in the class would make (1 or 2).

Students created models to show how many popsicle sticks they would need for one frame, and for all the frames. Students compared their plans and selected one or two. For example, Ms. V’s class selected a model that used 6 jumbo sticks per frame and because it allowed the class to make more frames (38 total) with few leftover sticks.

My frame is for someone who passed away and it will be very special for my family to see it. - student in Ms. B’s class

Students honored loved ones by placing frames on a school ofrenda, and thanked veterans with frames gifted at a community event.

The frame displays were meaningful for students,their families and communities.

Friendship Bracelet - 4th Grade

Inspiration for the Task

The Modeling Problem

Building Mathematical Models

Reporting Out and Taking Action

Decisions about Important Quantities

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Equity Move

Connect to students’ experience

The students at West Elementary had been on virtual mode all last year and wanted to celebrate coming back to school in person and making new friends. To celebrate this event, the class decided to make bracelets for one another.

Students researched different designs for bracelets made of beads and fuzzy stems. For each design, they noticed how many beads were used, and how many fuzzy stems. In their own designs, students made decisions about how many fuzzy stems and how many beads to use.

Students also decided how many people were making bracelets, and how many bracelets each class member should make.

We are making friendship bracelets. Make a design and create a model that shows how many beads and fuzzy stems you will need for your design.

“Simple and Unique” design took 13 beads and 2 fuzzy stems for each bracelet. They also decided each student would make just 1 bracelet.

Model:

Beads needed=(13 beads per bracelet) x (number of students)

Fuzzy stems =2 times # of students

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Equity Move

Distribute intellectual authority by creating

space for many students to communicate their ideas and experiences in multiple ways

“Fancy Friendship Bracelet” design took 36 beads and 1 fuzzy stem. They decided each student would make just 1 bracelet.

Model:

Beads needed=(36 beads per bracelet) x (number of students)

Fuzzy stems needed =1 times # of students

The class presented their unique designs for the bracelets along with their models for how many beads and pipe cleaners they would need for their small group as well as their whole class. The class voted and they celebrated by making bracelets for one another.

Building a Friendship River - 3rd Grade

Inspiration for the Task

The Modeling Problem

Building Mathematical Models

Reporting Out and Taking Action

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Equity Move

Distribute intellectual authority by creating

space for many students to communicate their ideas and experiences in multiple ways

Community-based Mathematical Modeling

The school had a strong commitment to art-based community building projects. Teachers were planning a “Friendship River” made of river rocks painted by each student and staff member for the school’s courtyard. The school proposed that if everyone painted one rock, they would have enough to fill the space and create the river. Ms. R, a 3rd grade teacher, invited her students to investigate this plan and decide whether this would be enough rocks. Since students recently completed a unit on area, Ms. R thought they could apply their understanding to a real situation at the school.

Ms. Megan [the outdoor learning teacher] claims that we need 500 rocks to fill the Friendship River. Do you agree? Why or why not?

Students visited the river space and shared what they noticed, and what they would need to know to solve the problem. They decided that if they knew how many rocks they needed for one section of the river, they could figure out how many rocks were needed altogether. To help students divide the river into sections, Ms. R prepared a diagram using grid paper. Each square on the diagram represented one square foot. Students placed rocks of various sizes on square foot templates to see how many rocks fit in one square foot. After several trials, they selected one number to represent how many rocks fit “on average.” One group explained, “We picked 12 rocks in a square foot because it is in between 15 (highest) and 9 (lowest value)”,

While groups assumed different numbers of rocks per square foot, they all determined that more than 500 rocks were needed, and that Ms. Megan was wrong. “Ms. Megan said 500 rocks and we estimated 1355!” Students shared their results with the school and proposed a revised river design that was narrower and curvy and required fewer rocks.

Students used multiple methods to estimate how many rocks they would need in all, including skip counting by the number of rocks in one square foot to fill the river, and grouping multiple square feet together to making friendly groups. One group assumed that 11 rocks fit in each square foot, so 9 square feet would need 99 rocks, which is close to 100 - so they counted by 100s to find the total.

Cafeteria Waste - 3rd and 4th Grade

Students were upset by the district-wide shift to disposable trays and plastic utensils in school cafeterias. The district claimed they were saving water by not washing reusable materials, but students were concerned about the waste produced. Students began collecting disposable trays after each meal to visualize the waste. They were shocked by the number of trays collected in just a few days and decided to take action!

Inspiration for the Task

The Modeling Problem

Building Mathematical Models

Reporting Out and Taking Action

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Equity Move

Distribute intellectual authority by creating

space for many students to communicate their ideas and experiences in multiple ways

Community-based Mathematical Modeling

How much waste does the cafeteria produce with disposable trays and utensils? You decide- focus on breakfast or lunch, our school or many schools, one week or one month or even the whole year! Make a poster to show your conclusions. Include the question you asked, the quantities and assumptions you included, your mathematical work, and your solution.

Students formed small groups and each explored their own modeling question related to cafeteria waste. One group built a model to project the number of trays that would be thrown away at breakfast at their school during the month of October. Another group estimated how many trays were discarded across all elementary schools each month. Groups gathered breakfast and lunch count data from their own school, and additional data from the district to inform assumptions about the number of meals served each day.

To visualize this amount, students found that 120 trays made a stack about 1 foot high, so a stack of 1,170 trays would 10 feet tall!

1,170 trays thrown away in 1 month after breakfast!

Students presented their solutions to the school principal and cafeteria staff. They emphasized the environmental impact of the increased waste: “ it is making he landfills bigger.” Cafeteria staff promised to look into reusable trays, and the principal suggested a follow up modeling question focused on how much water is used wash trays. Students were eager to do more -”We are solving this problem.”

Garden Task - 2nd grade

Outside of Amanda’s math class, students were learning about pollinators; there were books and posters about pollinators, wasp’s nests, and art projects of pollinators around the classroom. Amanda’s school has a well-established vegetable garden that is a source of school pride. The garden is run by a farm-to-school-coordinator who visits the class monthly to teach a lesson and the students were always excited for her lessons.

Inspiration for the Task

The Modeling Problem

Building Mathematical Models

Reporting Out and Taking Action

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In a discussion of the size and shape of the new pollinator garden, the class discussed properties of shapes. They decided that the new pollinator garden should have the same area as the vegetable garden. Amanda gave the students differently scaled pieces of labeled grid paper to use to represent the existing garden. The students drew the garden to scale, choosing to use one side to equal one 1 foot or 2 feet. To keep the same area, students agreed to keep the same number of squares but to arrange the squares to design their own garden. Students cut the grid paper into different shapes and arranged the shapes on another paper to show how they wanted to design the new pollinator garden.

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Equity Move

Scaffolding up to maintain high rigor and support

Community-based Mathematical Modeling

The farm-to-school-coordinator created a video asking the students to make a design of a pollinator garden to be planted near the vegetable garden.

To understand the size of the new garden, Amanda

suggested that the class describe the size of the current

garden and students added that they could measure it. Amanda gave her students many measuring tools to choose from. At first, students chose between standard and non-standard measuring tools. After students realized that they were finding wildly different measurements, Amanda lead a discussion about using measuring tools, the importance of showing mathematical work, and being precise. Students explained which tools they found helpful and which were difficult to use. They then choose new tools or decided to continue measuring with their initial tools and measured and recorded the vegetable garden.

The farm-to-school coordinator came to Amanda’s class, and the students reported their results to her. Amanda and the coordinator commented about aspects of the designs that she liked and she concluded the class by discussing the benefits of building a pollinator garden and told the class that their designs would help her.

Animal Shelter Task - 3rd grade

In planning a task in relation to a geometry unit considering area and perimeter, Maya, a third grade teacher developed a modeling task around an animal shelter because she knows her students love animals and many of her students are familiar with the shelter. Maya reaches out to the local animal shelter and a representative agrees to give students a virtual tour of the space.

Inspiration for the Task

Learning More about the Context and Defining the Modeling Problem

Students receive the dimensions of the new wing, but identify other quantities that might be important. The class decides they need to know the dimensions of the various sizes of kennels, which Maya then provides. In groups, students make decisions about walkways, stacking kennels, and what proportions of the various

Building Mathematical Models

Reporting Out and Taking Action

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Equity Move

Distribute intellectual authority by creating

space for many students to communicate their ideas and experiences in multiple ways

Community-based Mathematical Modeling

The shelter representative tells students about the shelter’s potential future project of adding a new wing and concerns. While students take the virtual tour, they write notes about their noticings and wonderings.

Maya asked her students to think about an addition that would help the animal shelter house more dogs. They focused on how the shelter might set up the dog pens and generated a list of questions related to the situation. Students ask a wide variety of questions—both mathematical and nonmathematical—and decided to investigate the following: “The animal shelter’s goal is to comfortably house as many dogs as possible. How many small, medium, and large pens can we fit into the new addition?”

The class reported their models and considered different decisions and strategies taken. Then, groups wrote letters to the animal shelter containing their suggestions.

sizes of kennels to use. Maya provides many tools for her students, such as calculators, blocks, rulers, and graph paper in various sizes.

Maya encouraged her students to make their mathematical work visible through drawings, expressions, and explanations. Some students were invested in their drawings and forgot to record their thinking or the units associated with their drawing so she reminded and helped students organize their work to include descriptions and units. Periodically, Maya would regroup her class to have groups share their different approaches and mathematical ideas.