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# 1. Surface Area and Volume

Students will design and create a 3-dimensional object with equivalent values of surface area and volume.

Standards for Mathematical Practice
CCSS.MATH.PRACTICE.MP1: Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2: Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3: Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4: Model with mathematics.
CCSS.MATH.PRACTICE.MP5: Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6: Attend to precision.
CCSS.MATH.PRACTICE.MP7: Look for and make use of structure.

Learning Objectives

• Students will design and create a 3-dimensional object with equivalent values of surface area and volume.
• Students will use linear functions to represent the equivalence of surface area and volume.

Recommendations
Group Size: 3 to 4 students, depending on how many variations you would like students to compare within the class setting.

Class Size: up to 36 students

Materials Required

• At least one computer per group, loaded with SketchUp or Autodesk123D (For this particular lesson, AutoDesk123D is a great tool with its ability to round the edges and customize the design)
• Paper and pencil for drafting
• Airwolf 3D Printer
• Ruler to verify dimensions
• Placeholder (STL file)
• Placeholder (SketchUp file)

Introduction

• Students have a good understanding of 3D modeling. Prior to incorporating this lesson into a unit, it is recommended that students have had training on Google Sketchup.
• Students have a good understanding of finding the surface area of an object
• Students have a good understanding of finding the volume of an object

https://youtu.be/gsfH_cyXa1o

Something to note about the 3D printer itself:

Anything that students build needs to be supported. Therefore, some designs are not easily designed on a 3D printer. To ensure that students’ designs will be printed, check to verify that all unsupported structures are designed at an angle less than or equal to 45 degrees.

To introduce the class to the project, have two separate containers and have students determine the surface area of each. Based on this, have them determine the volume capacity of each. Assuming that the containers do not meet the challenge requirements, move directly into the project’s challenge.

The Challenge
Create a container that has the same values for surface area and its volume

Prior to working on the design, students will need to show proof that they understand the relationship between surface area and volume. In an effort to assist students, providing the formulas may be beneficial. Some figures, mostly regular 3D objects, lend themselves to a simple setup. Be aware that very few 3D figures will cooperate with this method.

EX: For a sphere (which is highly unlikely to be printed)

= 4π2
= 4/3π3

To find out the value in which the surface area and volume are equivalent, students will need to set them equal to each other and isolate the variable.

2 = 4/3π3

Therefore, in this instance, the radius of the sphere would be 3. Whether that is 3 millimeters, 3 centimeters, inches, it is based on how large students will need their object to be. This is one of the easier formulas to reduce, yet it will be the most difficult example for the printer to produce, so it is highly recommended that students do not use the sphere.

Due to the fact that the container will need to be filled with water to verify volume, ensure that students have left one face open in their design and printing, yet they find the surface area of the object as if it were closed in.

The Meat
On paper, students sketch their design and include the work within their notebook that confirms their accuracy. Dimensions need to be designated on the drawing and appropriate justification as to why the group feels like their design is accurate must be included in their preliminary draft.

Using a 3D modeling program such as Google Sketchup, students create an object that represents the sketch in their notebook. Prior to printing, measurements need to be verified by a partner, then by the instructor. Once verified, student will send the product for printing. Within Matter Control, students will need to generate their layers and ensure that their product is hollow to test for volume. Instructor must verify that this is the case prior to the start of printing.

To lower the cost and time of printing, making the walls of the bridge 1 layer would be effective.

After printing, have students use a ruler to measure the dimensions. Mark these dimensions into student notebooks for reference.

Discussion
Using a measuring cup, students will fill up their product with water to determine its volume. Considering that there is a thickness to the objects, there will need to be an acceptable margin of error. Within the class, have students discuss why this is the case and how it could be modified to reduce the margin of error.

Once students have tested their own container, rotate around the class and have groups verify the others’ products by finding the surface area and volume. If time permits, have students create and complete a table with the various designs that classmates have created and include the values for surface area and volume.

Desired Outcomes
Students should have created a product that has the same numerical value for the surface area and the volume. The units will be different, square versus cubic. Outside of that, the expectation is that students rotate throughout the room to see what other groups have done to meet the challenge.

Some Possible Extensions/Modifications
Place a limit on the amount of filament that shall be used for the printing of the product.

Require students to have a unique design. There are plenty of good options, and a place to start is here. However, there are many others that are irregular that will spark the creativity of some students.

For students who are struggling, or have chosen an obscure design, lead them to this website. To encourage more students to be successful, make this a challenge to see which groups can achieve the smallest ratio of surface area to volume, understanding that it is quite a demanding task.

Provide the students with the tools necessary to complete the task. For example, have students use a right triangular prism (Wolfram Alpha applet). Select an arbitrary a, b, c (triangle sides) and h (height of object). Using the appropriate formulas, determine the surface area and volume. Then, using those values, find the ratio of surface area to volume. Cube the new ratio and multiply it to the volume. Square the new ratio and multiply it to the surface area. Caution: this may get sloppy. (To avoid all of this, a right triangle with dimensions 12, 16, 20 and height of 4 will work).

If there are a small number of students in the class, printing time and material won’t be as much of an issue, so the specifications may be adjusted to account for more creativity and a higher level of support.

Questions to Ponder

• What would have happened if the height was changed in proportion to the base?
• What was the most challenging portion of the project?
• Where could you see this used in a real life setting?

Once students are finished with the project, encourage them to repurpose their products by giving them out to teachers or office staff as gifts of appreciation. After all, we all love a thoughtful and creative gift!

Content & Instruction Developed by:
John Stevens – Airwolf 3D STEM Consultant
Instructional Coach – Technology
Chaffey Joint Union High School District
CUE Rockstar Faculty & Organizer

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