July 17th, 2015

Students will engage directly with solids in a guided activity meant to deepen their understanding of the formulas for surface area and volume. From creating a set of cylinders with the same volume but different surface areas, to exploring the variables in each formula, students will gain further insight into what goes in and comes out of these formulas.

### Volumize Activity 4: Formula Exploration

A deep dive into surface area and volume formulas for basic geometric solids. ### Expected Activity Time

Formula Exploration (20-40 minutes)

### Materials and Prep

• Formula Exploration student sheets.

### Activity Prompt Intro: You may have learned about surface area and volume before, but what do the formulas for basic solids really tell us? Can five different cylinders share the same volume but have different surface areas? What would six distinctive shapes with the same volume look like? Could those same six shapes keep the same volume but look different than the first set? When companies need to package their products, they want to use materials in the most efficient way possible. Why do you think it’s important for companies to think about shape, volume and surface area when considering which packaging to go with?

This activity will dive into these questions and you will come out of it with a better understanding of geometric solids, their formulas for surface area and volume, and how this can help you investigate and design three-dimensional objects in the real world.

Formula Exploration: There are a few steps to understanding surface area and volume formulas in this activity: 1) exploring each solid individually, 2) exploring across different geometric solids, and 3) reflecting on how the formulas incorporate linear (1D) measurements, area (2D) measurements and volume (3D) information about the solids.

### To Do

Formula Exploration (20-40 minutes)

• Draw the six two-dimensional solids on the board or interactive white board:
• 1) rectangular prism, 2) cylinder, 3) cone, 4) sphere, 5) triangular prism and 6) square-based pyramid.
• Have students draw the 2D shapes on their paper and take pictures of their drawings.

Once each group has drawn and photographed the six shapes, students will build on top of those flat images with 3D representations in Volumize by following along with the student sheet.

• The first step is to set a scale so that the height of their shape is 1m. This will be helpful to manage data as the lesson unfolds.
• Students will be prompted to create a rectangular prism with a volume of 1m-cubed. Then, they will be prompted to create two more that look different but also have a volume of 1m-cubed.
• Have students note in the table the varying surface areas for each prism that has the same volume. They will be prompted to reflect on these differences as well.
• For each new solid, have students start a new project in Volumize and repeat the process. If working in groups, have each student take responsibility for one of the shapes.
• Once groups have worked through each geometric solid, have them create another new Volumize project in which they create six different solids, all with the same surface area, but taking note of their different volumes. Tip: Start by creating a base (a rectangular prism works well) that each of the six solids is built off of. You will not need this for calculation or data purposes but it allows for all the shapes to be seen in the same frame, yet not connected to each other (i.e., each individual shape will be connected to the base.)
• Leave time for students to explore and reflect on the data they have gathered. Use student sheet questions to guide the conversation and ask them to deconstruct the formulas, shedding light on how each one works.

Extensions and Inquiring Further

This activity would extend well to explorations of geometrically similar figures and comparing their surface areas and volumes.  This activity would likely incorporate ratios and proportions as well.

or:

Conversion between metric units and feet and inches could work well with the data created by this activity.  Choose one or more data sets and explore the conversions.  Have students convert from meters-cubed to liters, for example, and feet-cubed to gallons.  Spend some time with the histories of these different units. Discuss various situations that might be meaningful to students, like mph vs. km/h, or gallons of gas vs. liters.

### Volumize Activity 4: Formula Exploration  In 2D, begin by drawing the six geometric solids in the provided table.  You will build off of these drawings using the Volumize app.

### To Do:

1. Open Volumize and tap Get Building. You will be prompted to take a picture of something that you would like to model.  Take a picture of the sketch you made of the first shape, a rectangular prism.
2. Set the scale so that the height of your prism is 1 meter (1m).
3. Add a 3D rectangular prism and align it closely with the sketch you made. Now edit one of the handles — length, height or width, so that the volume is 1m-cubed.  (Use the info panel on the right to see the volume of the prism as you make adjustments)
4. Add a second rectangular prism to one of the sides, and manipulate it to look different from your first shape, but maintain a volume of 1 meter cubed. Experiment with making it narrow or wide to get to 1 meter-cubed.
5. Add one more rectangular prism, and change it so that it looks different from the first two, but also has a volume of 1 meter-cubed.
6. Now tap on Surface Area, notice the varying values, and fill in the data table in your handout.
7. Go back to the Volumize start page, and start a new project, this time use the cylinder, and repeat steps 2-6. Then do this for the other solids that you made sketches of.
9. Create one more Volumize project. Start with a picture of the blank wall or tabletop.   Set the scale again to be around 1m for roughly half the screen.
10. Add a rectangular prism, and modify the size of the sides until the surface area is 3 meters squared. Then add each of the other solids, making them all arrive at the same surface area.  Open the info panels for each solid and note their volumes in the data table.
11. Go back and edit each solid so that they look different but maintain a 3 meter-squared surface area. Note the new volumes in the Volume 2 column in the data table.
12. Be prepared for a class discussion about the things you noticed were at play.

### Volumize Activity 4: Formula Exploration

1. Why are linear measurements defined as meters, surface area as meters squared and volume as meters cubed?
1. In the space below, draw the following shapes:
 Rectangular prism Cylinder Cone Sphere Triangular Prism Square-based Pyramid
1. As you work through the activity, save each project and fill in the table with the data from the side-panel of Volumize.
 Shape Volume Surface area Rectangular prism 1 1 m cubed Rectangular prism 2 1 m cubed Rectangular prism 3 1 m cubed

 Shape Volume Surface area Cylinder 1 1 m cubed Cylinder 2 1 m cubed Cylinder 3 1 m cubed

 Shape Volume Surface area Cone 1 1 m cubed Cone 2 1 m cubed Cone 3 1 m cubed

 Shape Volume Surface area Sphere 1 1 m cubed Sphere 2 1 m cubed Sphere 3 1 m cubed

 Shape Volume Surface area Triangular prism 1 1 m cubed Triangular prism 2 1 m cubed Triangular prism 3 1 m cubed

 Shape Volume Surface area Pyramid 1 1 m cubed Pyramid 2 1 m cubed Pyramid 3 1 m cubed

1. Why is it that shapes with the same volume can have different surface areas?
1. Create six different shapes in Volumize that have the same surface area of 3 meters-squared, and fill in the table below. Note their volumes in Volume 1.  Then, edit each shape by changing one dimension (height, radius, etc.) but keeping the surface area at 3 m-squared.  Then take note of the new volumes as Volume 2.
 Shape Volume 1 Volume 2 Surface area Rectangular prism 3 m-squared Cylinder 3 m-squared Cone 3 m-squared Sphere 3 m-squared Triangular prism 3 m-squared Pyramid 3 m-squared Duration: 0-20 mins
Prep: Easy

#### Big Idea

Formulas are often given to students to plug in values and find correct answers. Rarely do they get the time to explore where formulas come from and what variables are at work while actively changing dimensions of objects. This activity explores what each variable means when calculating surface area and volume of basic solids. It will also highlight the difference between the two-dimensional surface area measurements and three-dimensional volumes. Students will design and investigate a myriad of solids that all have a constant volume or all share the same surface area. This will give them the opportunity to explore what dimensions are at play and how everything connects back to the formulas. For example, students may investigate the ways in which five different cylinders, all with the same volume, have different surface areas, or how six different shapes with the same total surface area may look similar or different.

The six geometric solids in Volumize are:

• Rectangular Prism
• Cylinder
• Sphere
• Cone
• Triangular Prism
• Square-Based Pyramid

Students are able to manipulate each shape in the app with simple handles and watch the data of the shape change dynamically in real time. This ability makes for an efficient deep dive into the surface area and volume formulas for each solid.

#### Learning Objectives

Students will gain an understanding of:

• Calculating surface area and volume of different solids.
• The function of variables in each formula.
• How mathematical nets are related to the 3D objects that they fold into and how the nets help one understand total surface area.
• How changing scale affects surface area and volume.

Common Core State Standards-Math

Geometry

6.G.A.1                 HSG.GMD.A.3

6.G.A.2                 HSG.MG.A.1

6.G.A.4

7.G.A.3

7.G.B.4

7.G.B.6

8.G.C.9

Common Core State Standards-Math

Mathematical practices.

MP2: Reason abstractly and quantitatively.

Students create a dance visually and then have to determine the quantitative moves before making it virtual.

MP4: Model with mathematics.

Students outline their dance using the angles of rotation and coordinate notation for the translation.

#### Vocabulary

• Scale
• Mathematical Nets
• Length
• Surface Area
• Volume
• Dimension, 2D and 3D
• Rectangular Prism
• Triangular Prism
• Cone
• Sphere
• Cylinder
• Pyramid

#### Device Strategies

Single-device implementation

Exploring formulas with one iPad will work well because of the dynamic nature of the shapes and data in Volumize. Project the app in front of the class and work through the activity as defined below, allowing student volunteers to take turns creating and manipulating the shapes. Even if the app is only used to project the lesson in front of the class, teachers will find that they can lead a discussion on how to go about calculating the surface area and volume of the basic geometric solids and the variables at play that go beyond most lessons using static images. That said, giving student volunteers as much engagement as possible with the app will provide a sense of agency over the solids, the variables and the formulas.

Multiple-device implementation

If one iPad per student is not possible, groups of two, three or four students per iPad will work just as well. In 1:1 situations, students can simply follow the activity as defined below. For groups, be sure to facilitate students taking turns and sharing tasks in the activity, perhaps taking on one shape each and following the guidelines provided for the activity.

#### Tips & Ideas

Getting a handle on orienting the 3D models in the app can be tricky. Here are some helpful techniques:- Tap the lock icon on the bottom left of the modeling screen to enable rotating the base shape (that's the first shape you add to the scene).- Moving your model on one axis often helps maneuver it into its desired orientation.- Try swiping up or down, or right or left, and watch how your swiping affects the orientation.- Zoom in and out on models can help to orient the scene to your liking.- Double tap at any point will return your model to the starting position. This can be very helpful for making modifications to your construction.