Hands-on Activity A Sweet Volume:
Designing a Jumbo Chocolate Bar Using Polynomials

Learning Objectives

After this activity, students should be able to:

• Explain an optimization process in engineering design.
• Use the geometric net to build a regular prism and to express the variable prism’s volume formula as a single variable cubic polynomial function.
• Solve the graph of this polynomial to determine the value for which this function reaches a maximum, and with this value find the dimensions of the geometric net for the maximum volume possible.

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

Common Core State Standards - Math

International Technology and Engineering Educators Association - Technology

State Standards

Materials List

Each group needs:

• two 56 cm x 71 cm (~22 in x 28 in) cardstock sheets (or poster board)
• meter stick (or yardstick)
• regular 30 cm ruler
• scissors
• pencils
• scotch tape
• glue stick
• wrapping paper (students can get creative here and bring in their own)
• computer with Google Sheets, PowerPoint, or other drawing and graphing software
• color pencils, pens, markers, or crayons; if students decide to  decorate the bars by themselves

To share with the entire class:

• Students Activity Guide
• color printer

Worksheets and Attachments

Pre-Req Knowledge

Before undertaking this activity, students should have an understanding of regular prisms, volume computational formulas, right triangles, trigonometric ratios, polynomials multiplication, polynomials relative maximum and minimum, functions analysis with a graphing calculator, systems of equations solution by substitution method, and regular prisms construction from geometry nets.

Introduction/Motivation

What is the biggest candy bar you have ever seen? [Expect students to give several answers such as large Hershey Kisses, Toblerone or Hershey’s chocolate bars. You may want to bookmark a few webpages that show images of these chocolate bars and show them to the class.]

A common practice during the holiday season among confectioners is to produce jumbo candy bars— they make great gift ideas! Imagine that you are engineers working as part of the production team in a brand new chocolate factory. Your team received the instructions from CEO to produce a container for a jumbo chocolate bar. The company wants to produce these bars in order to compete with Hershey, Toblerone, and others. In a meeting with the CEO and the marketing team, you receive the specifications for your design: the container must be in the shape of a regular prism with isosceles trapezoids as bases. The angle between the non-parallel sides, or legs, and the longest base of the trapezoids must be 60 degrees. You also have constraints on your materials: you have to design your prism with a surplus of a packing material the company purchased a long time ago, but did not fully use. The CEO wants you optimize the design by using as much of this material as possible, which means that you must cut as large a volume as possible from a 56 cm x 71 cm (~22 in x 28 in) sheet of cardstock.

You will also work with the company’s marketing team to come up with a marketable name and decoration for your jumbo chocolate bar. One final constraint: you have a relatively short deadline, five days, to present your final prototype!

Procedure

Background

Draw two possible geometric nets on a 56 cm x 71 cm (~22 in x 28 in) sheet of cardstock. Once cut and folded, these nets create the required trapezoidal prism with maximum volume. One is the vertical configuration, with the bases of trapezoidal bases parallel to the poster board’s short side (Figure 1a), and the other is the horizontal configuration, with the bases of the trapezoidal bases parallel to the poster board’s long side, (Figure 1b). The first one will produce a regular bar while the second one makes a “fun-size” shaped bar.

times the length l of the prism (Figure 2):

First, focus on the trapezoidal prism’s base. The angle between the trapezoid’s legs and its longest base must be 60 degrees. With this knowledge, the triangle formed by the trapezoid’s leg x, its height h and the longest base B is a special 30-60-90 right triangle (Figure 3). Using the standard relationships between the sides of this special right triangle (Figure 3) and the above geometric nets (Figure 1), it is possible to express the variables in the volume’s formula (1) in terms of the trapezoid’s leg x only. On the geometric nets, the length x represents the cut needed to create a prism with maximum volume.

(a). Vertical Configuration.

From the prism’s net in Figure 4, figuring in a subtraction of 2.5 cm (~1 in) for the prism’s tab (Figure 1), it is easy to see that the 30-60-90 special right triangle relationships between its sides (Figure 3) and the variables in formula (1) may be written in terms of the trapezoid length x and the cardstock sheet dimensions:

From the graph of equation (10), determine the x-value for which the volume of the trapezoidal prism is at its maximum (Figure 5):

Note: The graph in Figure 5 may lead students to think that any x-value between 0 and 10.5 is possible as a length for the trapezoid’s leg. However, this is not true: variable b will be negative for x-values greater than 7 in equations (6) through (9). In other words, 0 < x < 7.

The trapezoidal prism has a maximum volume of 7.783 L (474.97 in³) for x = 10.5486 cm (4.153 in) Using equations (6) through (9), determine the dimensions of the prism.

When constructing the prism, use centimeters with x = 10.5486 cm:

Following the same reasoning as in the vertical configuration, the trapezoidal prism’s volume can again be expressed in terms of the trapezoid’s length x. From the prism net in Figure 6:

Note: Like the vertical configuration, the graph in Figure 7 could make students think that any x-value between 0 and 12.7, may be used as a length for the trapezoid’s leg. However, variable b will be negative for x-values greater than 9. Because all the dimensions of the prism must be positive, only x-values between 0 and 9 produce positive values for all equations (15) through (18).

Before the Activity

• Bookmark the following webpage: https://jerseykids.net/2015/10/26/review-hershey-chocolate-world-with-families-review/
• This page shows a giant Hershey’s chocolate bar. You can also do a Google search of “giant chocolate bars for sale”, and bookmark some of the results.

Use these bookmarks to engage the students. If you want to achieve a greater impact, present a real extra-large candy bar in class! (Because novelty-size candy bars can be expensive, this is an optional suggestion.)

• Have enough 56 cm x 71 cm (~22 in x 28 in) cardstock for all your groups; at least two per team.
• Ensure students have enough building and measuring material (yardsticks, rulers, scissors, glue stick, tape, etc.) to construct the prisms in class.
• Create a couple of small models before the activity to help students be aware of the possible problems they may encounter during the prism design and construction.
• Have enough copies of the Students Activity Guide available.

With the Students

Day 1

1. Distribute the Students Activity Guide.
2. Give the Area and Volume Pre-Quiz.
3. Conduct the Introduction/Motivation.
4. Allow students to select partners with whom to work as well as the chocolate bar configuration they want to optimize: a regular bar (vertical configuration) or a “fun-size” bar (horizontal configuration).
5. Let students begin working on the geometric nets included in the Students Activity Guide (Figure 4 or Figure 6); tell them to discuss how they can solve the problem.
6. If students struggle to set up the trapezoidal prism’s volume function (which may be likely), tell them check the solution of the simpler problem in the Students Activity Guide - Annex A: the rectangular prism with maximum volume that can be cut from a 56 cm x 71 cm (~22 in x 28) in cardstock. Help them to understand the strategy and the substitution method used to express the prism’s volume formula as a function of a single variable x. Remind them of the importance of the single variable function used to solve and graph this problem. Explain that all the conditions and constrictions of the problem translate into a system of equations. Students can obtain an expression for the prism’s volume, as a function of a single variable, by combining these equations (via the substitution method).
7. Provide the following hint: using the 30-60-90 special right triangle relates to the two bases of the trapezoid. Allow students to continue or finish working on and outlining the geometric nets included in the Students Activity Guide.

Day 2

1. Have students continue working in the Students Activity Guide. Once students obtain the prism’s volume function, a third degree polynomial, students may graph this polynomial with a graphing calculator or graphing software (such as Desmos) to locate the value x for which this function has a maximum.
2. Instruct students to find other dimensions, as well as the volume of the prism, using the other formulas.
3. Using the dimensions they find, instruct students to draw the geometric net of the trapezoidal prism on the 56 cm x 71 cm (~22 in x 28 in) cardstock.
4. Tell students to cut, fold, and glue their prism together.  

Note: Decorating the prism is a task that students may work on at home and depends on the time available time in class. Students may opt to find and bring in materials for the exterior design of their prism.

Day 3

1. Allow students additional time to finish their prism designs.
2. Instruct students to begin working on their in-class or video presentations as outlined in the Students Activity Guide.
3. Allow for time to students to present in Day 3 or extend into Day 4 and Day 5.

Vocabulary/Definitions

30-60-90 triangle: A special right triangle that always has angles of 30 degrees, 60 degrees, and 90 degrees. The measure of any of the three sides can be found simply by knowing the measure of at least one side in the triangle. The hypotenuse is equal to twice the length of the shorter leg, which is the side opposite the 30 degree angle.

chocolate: A food preparation in the form of a paste or solid block made from roasted and ground cacao seeds, typically sweetened.

cosine: The ratio of the side adjacent to a specified angle to the hypotenuse.

geometric net: A pattern that you can cut and fold to make a model of a solid shape.

isosceles trapezoid: A trapezoid with congruent legs.

polygon: A plane figure bounded by a finite chain of straight-line segments closing in a loop to form a closed chain or circuit.

polynomial: An expression consisting of variables (or indeterminates) and coefficients (or constants), that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The highest power present determines the polynomial degree.

prism: A solid geometric figure whose two end faces are similar, equal, and parallel rectilinear figures, and whose sides are parallelograms.

regular polygon: A polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).

regular prism : A prism with bases that are regular polygons. When the bases align (one exactly above the other one), the lateral faces are rectangles; when the bases do not align (one directly above the other), the faces are parallelograms.

relative maximum: The value of a function at a certain point in its domain that is greater than or equal to the values at all other points in the immediate vicinity of the point. Also known as local maximum.

relative minimum: The value of a function at a certain point in its domain that is less than or equal to the values at all other points in the immediate vicinity of the point. Also known as local minimum.

sine: The ratio of the side opposite to a specified angle to the hypotenuse.

tangent: The ratio of the side opposite and adjacent to the specified angle.

trapezoid: A quadrilateral with only one pair of parallel sides, called bases. The non-parallel sides are called legs.

trapezoidal prism: A prism with two congruent trapezoidal bases that are connected to each other by four rectangles.

trigonometric ratio: The ratios of the sides of a right triangle that are related with the non-right angles of the triangle. The sides of the triangle are labeled as opposite (in front of the angle) and adjacent (next to the angle) while the hypotenuse is always the side in front of the right angle.

volume: The amount of space occupied by a three-dimensional figure as measured in cubic units.

Assessment

Pre-Activity Assessment

Check in: Verify that students know the formulas to obtain the area of quadrilaterals; in particular, the area of a trapezoid. Check that they can find the volumes of right angle prisms. Give the Area and Volume Pre-Quiz.

Activity Embedded Assessment

Monitor: Monitor (and help) students to see if they are finding the right equations and doing algebra correctly to obtain the proper expression for volume function.

Post-Activity Assessment

Model: The first evaluation in this activity is to obtain the correct dimensions of the prism. The second one is to build a precise model; good models should not have more than a 0.5 cm error from the optimized prism’s dimensions. The teacher can decide on the point deduction for the final grade. The teacher should emphasize that significant errors (more than 0.5 cm) in the student calculations and in the cut-and-fold process will result in a score penalty as engineering design requires precision.

Poster: The teacher should decide the best way to evaluate student learning. Encourage students to summarize their results in a poster. Check for the following poster guidelines in the Students Activity Guide:

• activity title and students’ names
• project objective and a brief description of all the work completed
• selected net prism
• description of the math process, equations used, and polynomial function found
• polynomial function graph with the maximum point and its coordinates displayed
• dimensions of the optimized prism (in inches and in centimeters) and volume
• pictures of the construction process
• picture of the finished model
• references

Depending on the available class time, students may present their posters in-class and explain their work, results as well as present their model. Alternatively, they can record their findings and submit their presentation to the teacher as a video presentation.

Troubleshooting Tips

Feel free to contact the author by email: mramirez2@galenaparkisd.com or by phone: (832) 386-2888 to ask any questions related to this activity.

Activity Extensions

Surface Area
Ask your students the following question: As part of your presentation to the company’s CEO, you have to provide an estimation of the amount of wrapping material needed to cover your jumbo chocolate bar. Because you were required to optimize the available materials, prepare to answer this following question: Is this amount of material you used also an optimized value?

To answer the first question, students need to compute the surface area of the prism, which is the area of the two trapezoidal bases plus the area of the four lateral rectangles. From any of the nets in (Figure 4) or (Figure 6):

Using the following values (see 10.a above), the surface area for the vertical configuration using volume V = 474.97 in³ is:

As an ordered pair, write down the values for the volume and surface area for each of the configurations and ask the students this question: Do you see something peculiar or special in these values?

The expected answer: The prism with the more volume has less surface area, and vice versa; the prism with the less volume has more surface area. Ask the class: Is this reasonable? Is this what you were expecting? It is not reasonable to think that with more volume you will have more surface area? Should surface area be an optimized value?

To answer this question, ask this follow-up question: How did you find the optimal value for the volume? Their answer: By finding the maximum of the graph of the volume function. Now, ask the students: What you can do in this case? Their answer: Find the same value for the surface area. Have students organize themselves in groups of three, and instruct them to find an x-function for the surface area, graph it, and verify if the area of the prism with maximum volume is optimal or not. Have students summarize their analysis in a poster that includes a good sketch of the graph of their surface area function and that displays the point representing their prism’s area. The prism’s dimensions and its volume should also be in the poster. Students will use this poster to explain their analysis to the class:

For the vertical net: Combine equations (6) through (9) with (20) above to find the surface area function:

Graphing (21) and its value for x = 4.153, the point for which (10) has its maximum, shows the surface area for the trapezoid with maximum volume is not an optimized value (Figure 8):

Graphing (22) and its value for x = 4.361, the point for which (19) has its maximum, shows that the surface area for the trapezoid with maximum volume is not an optimized value (Figure 9):

Students may use their posters to explain why the area of the prism is not an optimal value and why the maximum of the graph cannot be used.

Note: The restrictions on the x-values that could be taken as valid entries for the volume functions are also applicable to the surface area functions. Values x > 7 may produce a negative b, B or l, for the vertical net. In the same way, values x > 9 may produce negative dimensions for the horizontal net. Students may also include this in their analysis.

Activity Scaling

• For higher grades.

Students with a background in calculus can perform this activity by optimizing the cubic polynomial using its first and second derivatives. The function’s graph is not used here:

Vertical net. Using equation (10), and multiplying the terms:

Next, find the derivative of this function and then the values that make this derivative equal to zero:

Vertical net. Using equation (19), multiply the terms:

Next, find the derivative of this function, and then the values that make this derivative equal to zero:

References

Copyright

Contributors

Supporting Program

Acknowledgements

The author expresses his thanks to Kimberly Martin, Galena Park High School Principal; Robert Derr, Galena Park High School Math Department Chair; Shamel Ali, Galena Park High School Math Specialist