Modeling Prompt

Having physical materials to make prototypes with can help students make connections between the formulas for the volumes of geometric shapes and the dimensions of physical objects. We recommend making materials available, for example cardboard, aluminum foil, posterboard, and plastic containers. If students make their prototype out of water-resistant material, they can check their volume calculations by filling their prototype with water and measuring its volume. Dynamic digital geometry tools can also help students visualize their designs.

Once students have completed the task, invite them to look at each other’s designs, for example by having a gallery walk. Here are some possible questions for discussion:

• "Which basic geometric shapes were the containers made of?" (Prisms, cones, pyramids, hemispheres)
• "What strategy did you use to make sure your container would have the right volume?" (I wasn’t sure how tall to make my container, so I wrote a formula for the volume where \(x\) represented the height, and then I graphed it to see which \(x\) value gave the right volume.)
• "Did you see other strategies for designing a container that you like more than your strategy? What was good about the other strategies?" (Someone else used geometry software to represent their design and the shape changed when they changed the side lengths. I liked how you could see what the shape would be.)

A juice company wants a new container design, and you have been commissioned to create it. It should look different from most containers so it will stand out to customers—it could be a prism, cone, pyramid, a combination of more than one type of shape, or any three-dimensional geometric figure you can describe. It needs to have a volume of 16 fluid ounces.

Once you have your design, you’ll need to present it to the company. For your presentation you will need:

• A prototype of the container, or an image showing what it will look like and what its dimensions will be
• Calculations that prove the container will hold 16 fluid ounces

Your mathematical work should be as clear as possible. Remember that you’re explaining your design to people who may not be comfortable with math, so you’ll need to help them understand where your results came from.

Having physical materials to make prototypes with can help students make connections between the formulas for the volumes of geometric shapes and the dimensions of physical objects. We recommend making materials available, for example cardboard, aluminum foil, posterboard, and plastic containers. If students make their prototype out of water-resistant material, they can check their volume calculations by filling their prototype with water and measuring its volume. Dynamic digital geometry tools can also help students visualize their designs.

Once students have completed the task, invite them to look at each other’s designs, for example by having a gallery walk. Here are some possible questions for discussion:

• "Which basic geometric shapes were the containers made of?" (Prisms, cones, pyramids, hemispheres)
• "What strategy did you use to make sure your container would have the right volume?" (I wasn’t sure how tall to make my container, so I wrote a formula for the volume where \(x\) represented the height, and then I graphed it to see which \(x\) value gave the right volume.)
• "Did you see other strategies for designing a container that you like more than your strategy? What was good about the other strategies?" (Someone else used geometry software to represent their design and the shape changed when they changed the side lengths. I liked how you could see what the shape would be.)

A juice company wants a new container design, and you have been commissioned to create it. It should look different from most containers so it will stand out to customers. It should be some combination of 2 different geometric shapes. It needs to have a volume of 16 fluid ounces. 1 fluid ounce is approximately 29.57 cubic centimeters and approximately 1.80 cubic inches.

The containers the company currently uses are cans that are 6.19 inches tall (15.72 cm) and have a diameter of 2.43 inches (6.17 cm).

Once you have your design, you’ll need to present it to the company. For your presentation you will need:

• A prototype of the container, or an image showing what it will look like and what its dimensions will be
• Calculations that prove the container will hold 16 fluid ounces

Your mathematical work should be as clear as possible. Remember that you’re explaining your design to people who may not be comfortable with math, so you’ll need to help them understand where your results came from.