# Hands-on ActivityJudgement with Jellybeans

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### Quick Look

Time Required: 1 hour

Expendable Cost/Group: US \$2.00

The cost is from purchasing individual packets of Jelly Belly jelly beans for students to take home, so it is not necessary. Outside of the expendable cost per group, there exists a purchase of the clear, round, sealed-bottom, plastic tube of the same diameter and varying heights for the students to use to collect data. You may consider contacting a company, such as Cleartec Packaging to request a free sample. Additionally, a one-pound, resealable bag of Jelly Belly jelly beans was purchased for each partner pair for data collection purposes only.

Group Size: 2

Activity Dependency: None

Subject Areas: Algebra, Number and Operations

### Summary

Students collect data and apply mathematical modeling, specifically linear approximation, to predict what will happen in a specific situation. In this activity, students collect data to determine the number of Jelly Belly jelly beans it takes to fill up the respective tube. Students plot their data, graph a linear approximation model, and write an equation representing their model. Students use their linear model to predict the number of Jelly Belly jelly beans that are in a similar cylindrical tube with a given height. Students discuss the accuracy of their results and limitations in their model and data collection process. They then apply their predictions to make suggestions to Jelly Belly for potential packaging of jelly beans based on quantity instead of net weight.
This engineering curriculum aligns to Next Generation Science Standards (NGSS).

### Engineering Connection

Engineers deal with real-world problems, and often in real-world problems, numbers and data do not follow a perfect model like they often do in a mathematics classroom. To accommodate for this, engineers use mathematical modeling to investigate the relationship between variables, which allows them to make an accurate prediction of situational outcomes. Engineers follow the engineering design process while they work, and throughout this activity, the process of collecting data, creating a model, testing the model, and making necessary amendments to the model post-testing are applicable to the engineering design process. In this activity, students more specifically explore how packaging engineers apply mathematical modeling to help determine optimal packaging for jelly beans and even offer a suggestion of linear approximation model to Jelly Belly’s logistics team to switch their packaging system from a net weight model to a model based on quantity and cost-effective packaging.

### Learning Objectives

After this activity, students should be able to:

• Investigate relationships between quantities by using points on scatter plots.
• Model an approximately linear situation.
• Apply lines of fit to make and evaluate predictions.

### Educational Standards Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards. All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org). In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.

###### NGSS: Next Generation Science Standards - Science
• Analyze data using tools, technologies, and/or models (e.g., computational, mathematical) in order to make valid and reliable scientific claims or determine an optimal design solution. (Grades 9 - 12) More Details

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• Use mathematical representations of phenomena to describe explanations. (Grades 9 - 12) More Details

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###### Common Core State Standards - Math
• Model with mathematics. (Grades K - 12) More Details

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• Use appropriate tools strategically. (Grades K - 12) More Details

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• Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (Grades 9 - 12) More Details

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• Summarize, represent, and interpret data on two categorical and quantitative variables (Grades 9 - 12) More Details

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###### International Technology and Engineering Educators Association - Technology
• Students will develop an understanding of the characteristics and scope of technology. (Grades K - 12) More Details

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###### Michigan - Math
• Model with mathematics. (Grades K - 12) More Details

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• Use appropriate tools strategically. (Grades K - 12) More Details

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• Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (Grades 9 - 12) More Details

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• Summarize, represent, and interpret data on two categorical and quantitative variables (Grades 9 - 12) More Details

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Suggest an alignment not listed above

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### Materials List

For the teacher’s role:

• cylindrical tube with 2.54 cm diameter and 30.28 cm height (see “To share with the entire class” for details)
• 16 oz re-sealable bag of Jelly Belly jelly beans (see “Each group needs” for details)
• MATLAB license (if applicable) or Desmos.com (free) to display student-data
• computer and projector
• Document camera for student presentations (not necessary, but provides nice visual display of student work)
• 1 oz bag of jelly beans (not necessary, but provides a nice reward for students who successfully complete the challenge) -  one per student; case of 30 available at https://www.jellybelly.com/jelly-belly-20-flavor-assorted-jelly-beans-1-oz-bags-30-count-case/p/72571

Each group needs:

To share with the entire class:

### More Curriculum Like This

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Mathematical Modeling- Linear Approximations

Students investigate the idea of linear approximation. Students apply mathematical modeling, specifically linear approximation, to an already collected data set to make a prediction.

High School Lesson

Students learn about linear programming (also called linear optimization) to solve engineering design problems. They apply this information to solve two practice engineering design problems related to optimizing materials and cost by graphing inequalities, determining coordinates and equations from ...

### Pre-Req Knowledge

An understanding of linear functions - graphing, writing equations, and slope. An understanding of representing data as a table and scatter plot. An understanding of how to use a ruler. An understanding of modeling an approximately linear situation with a line of fit. See the Mathematical Modeling - Linear Approximations lesson.

### Introduction/Motivation

Who has ever opened a bag of candy and thought “Darn, this package is only ¾ full! What a waste of materials!” Sometimes, that extra space is essential for items such as Doritos, as the air provides a type of cushion for the chips during the process of getting the chips from the manufacturers to the consumers. However, for items that are not easily crushed, like jelly beans, why would the company want to waste extra money on materials?

This is why Jelly Belly, the company famous for their jelly beans, wants to employ you as a packaging engineer to remediate this problem. Packaging engineers have an important role on any company’s supply chain team, with one of their main priorities being able to ensure that the packaging of a product is cost-effective. For Jelly Belly, they do not like spending money on the wasted material that the current packaging provides, and instead want to transition to a packaging model where the jelly beans will be packed into cylindrical tubes, with only the height of the tube varying. They want the tubes to remain the same diameter to give a sense of unity in their products, and want the tubes to be filled, so that there is no wasted material.

Though Jelly Belly has the technology and manufacturing equipment to make the new containers, they need you to create a linear approximation model so they can have a precise prediction of the quantity of jelly beans it will take their machines to fill up a tube with a specific height. To demonstrate the effectiveness of your linear regression model, Jelly Belly has provided you with a 12-inch tube (30.48 cm) that is filled with a known quantity of jelly beans [show students pre-filled 12-inch tube]. Jelly Belly wants you to create your linear approximation models with actual data, so they have also provided nine other tubes of different lengths, as well as bags of jelly beans to experiment with. To be successful, Jelly Belly is requiring that your model can accurately predict the number of jelly beans within a plus/minus range of three jelly beans. If successful, Jelly Belly will honor your work and pay you (with jelly beans of course)! Do you accept their job offer?

### Procedure

Background

This activity is geared towards the basic understanding of using a line of fit to model an approximately linear situation, and does not take into account the calculation of error to find the best line of fit. If a group does finish earlier, challenge them to modify their linear approximation model to create a more accurate prediction and explain the importance of modifications for engineers and the design process.

MATLAB (or Desmos) is used to show students a graph and regression line for the entire class’s data. For more information on how to create this plot, see https://www.mathworks.com/help/matlab/ref/scatter.html?s_tid=doc_ta.

Before the Activity

• Instruct students using the Mathematical Modeling - Linear Approximations lesson plan.
• Purchase enough 16 ounce bags of Jelly Belly jelly beans to account for your class being grouped into pairs, as well as one for your class model that the students are trying to predict.
• Fill a 30.48 cm (12-inch) tube with jelly beans. Carefully count the jelly beans as you add them and record the total number of jelly beans in the tube. Place the pre-filled tube in a secured location away from the students.
• Have enough rulers for your class to account for one ruler per pair of students.
• Print enough copies of the Judgement with Jelly Beans Activity Handout for each student in the class.
• Have the nine tubes of varying length clearly labeled, numbered, and centrally located for the students. Recommended lengths (in cm): 2.54, 5.08, 7.62, 10.16, 12.7, 17.78, 20.32, 22.86, 25.4.

With the Students

1. Read the activity introduction to the students to introduce the activity and provide motivation for the students, as well as getting them in the engineering mindset.
2. Divide the class into groups of two (if comfortable, allow students to select a partner to work with).
3. Delegate one student to distribute one 16-ounce, resealable bag of jelly beans to each group.
4. Delegate one student to distribute one ruler to each group.
5. Delegate one student to distribute one Judgement with Jelly Beans Activity Handout to each student.
6. Explain to the students that there are nine tubes, all marked differently, that all groups must share. Set a norm that a group may only possess one tube at a time and when completed, exchange it with another tube at the centrally located table.
7. Go through the process and expectations for the students that are laid out on the Judgement with Jelly Beans Activity Handout. Make sure they know, as engineers, they are to collect and record data, create a scatter plot of that data, add in a line of fit to their scatter plot, write a linear equation for their line of fit, and use their line of fit to predict the number of jelly beans in the 30.48 cm tube.
8. Ask for any last clarifying questions that the students may have. If none, let the students work collaboratively in their groups.
9. Monitor student work, progress, and ideas. Make sure they are measuring correctly as you monitor. A common mistake you may see is to mix up the height and quantity of jelly beans as their independent and dependent variables in their table/graph/equation (height should be independent).
10. As you monitor, collect all student data into MATLAB (or Desmos) with x being height (in cm) and y being number of jelly beans (this will be used at the end of the activity).
11. Once all students have collected their data, created their scatter plot, graphed their line of fit, wrote an equation for their line of fit, and made a prediction using their model, have each group give a quick one-minute discussion of their outcomes (only have a few groups if there’s not ample time). It is nice to use a document camera to display each group’s scatter plot with line of fit, as well as have one of the group members write their equation with prediction on the board for all groups to see. Express that an important part of the engineering design process is to communicate your ideas with other engineers (student peers). It may be useful to track the group and their jelly bean prediction on the board as groups present.
12. Once all groups have shared their materials and predictions, reveal the actual quantity in the 30.48 cm tube. Determine which group had the most accurate prediction, and which groups were deemed “successful” by Jelly Belly’s standards of being within plus/minus three jelly beans.
13. Demonstrate for the students using MATLAB (or Desmos) by projecting on the screen how you collected the entire class’ data and put all of it onto one scatter plot. To do this, complete the following steps in the livescript:
• x=[.......]
• y=[.......]
• scatter(x,y)
• xlabel(‘Tube Height in cm’)
• ylabel(‘Number of Jelly Beans’)
• Title(‘Judgement with Jelly Beans’)
• Hold on
• In the graph settings, select ‘tool’ followed by ‘basic settings’ and check the equation box to see the linear regression model.

Then give a brief discussion of how some lines of fit are better than others because of the total error between the line of fit and the data points, and how there is a best line of fit. Elaborate on the fact that because data sets are often large, engineers use tools such as MATLAB (or Desmos) to quickly create scatter plots and best lines of fit to make their predictions.

1. Have a class discussion on the comparisons of the different group predictions, the MATLAB (or Desmos) prediction, and the actual quantity of jelly beans. Ask why there are these differences and how we could possibly deal with that problem.
2. Have a final discussion on the importance of the engineering design process, specifically collecting data, modeling that data, and using the model to make a prediction.
3. If groups were deemed “successful,” make sure to give them one of the 1-ounce bags of jelly beans as their earned reward.

### Vocabulary/Definitions

consumer: A person who purchases goods and services for personal use.

cylindrical: Having straight parallel sides and a circular or oval cross-section; in the shape or form of a cylinder.

diameter: A straight line passing from side to side through the center of a body or figure, especially a circle or sphere.

linear approximation: Linear approximation attempts to model the relationship between two variables by fitting a linear equation to observed data. One variable is considered to be an explanatory variable (independent variable), and the other is considered to be a scalar response (dependent variable).

manufacturer: A person or company that makes goods for sale.

### Assessment

Pre-Activity Assessment

Entrance Ticket: Have students begin by individually taking five minutes to work on a one-question Linear Approximation Entrance Ticket checking their understanding of the concepts from the Mathematical Modeling - Linear Approximations lesson.

Activity Embedded Assessment

Group Presentations and Observation of Student Progress: As the groups are working on this activity, the teacher should be monitoring student progress to make sure groups are moving forward with the activity. Listening for group collaboration, group questions, and looking at the mathematical accuracy should all be part of the feedback to the teacher for the students’ understanding. Each pair’s presentation of their final data, graph, and model provides a chance for the teacher to hear how well those two particular students understood the overall learning objectives from this activity.

Post-Activity Assessment

Exit Ticket: Have students complete the Linear Approximation Exit Ticket. Tell students that this will provide them an opportunity to show the growth in their understanding from completing this activity. Allow the students five minutes of individual time to complete the one-question exit ticket. As a follow-up, a grade could be assessed to the exit ticket if desired, but more importantly, students could be given back both their entrance and exit ticket for a time of reflection after the activity.

### Investigating Questions

Provided in steps 14-15 of the “Procedure - With Students”

### Safety Issues

• Make sure there are no allergies to dyes used in the Jelly Belly jelly beans.

### Troubleshooting Tips

Students will commonly switch the independent variable, height of the tube, and the dependent variable, quantity of jelly beans. Be aware of this as mixing the variables would cause students to have skewed predictions when trying to use a given height as their input.

### Activity Extensions

If a group finishes early, pose the question “Is there a ‘best’ line of fit to model approximately linear situations? If so, how do you find it?”

### Activity Scaling

• For higher grades or more advanced algebra classes, students could learn how to calculate total-squared-error and apply the Least Square Method to find the best line of fit.

### Other Related Information

As an alternative to MATLAB or Desmos, you can also display student data using MS Excel. For more information on how to do this, see: https://support.microsoft.com/en-us/topic/present-your-data-in-a-scatter-chart-or-a-line-chart-4570a80f-599a-4d6b-a155-104a9018b86e and https://support.microsoft.com/en-us/office/add-a-trend-or-moving-average-line-to-a-chart-fa59f86c-5852-4b68-a6d4-901a745842ad

### Contributors

William Harnica; Leyf Starling

### Supporting Program

RET Program, College of Engineering, Michigan State University

### Acknowledgements

This material is based upon work supported by the National Science Foundation under grant no. 1609339—a Research Experience for Teachers program at Michigan State University. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.