Modeling Prompt

In this prompt, students encounter the concept of density in geometric modeling (MP4), while also grappling with what happens to perimeter and area when a figure is dilated using a certain scale factor. Students are asked to design a playground that can hold 3 times as many students as a given playground. They explore the density calculation of kids per square yard, and because of the openness of the modeling prompt, they can choose to find a scale factor that will increase the area by a factor of 3, or increase only one dimension by a factor of 3, or design a differently-shaped playground with a fixed area and determine a shape that will ensure the perimeter remains under budget.

This activity is designed to be done digitally because students will benefit from seeing the relationships between perimeter, area, and costs in a dynamic way and from using dynamic geometry software to design their playgrounds and using spreadsheet software to track costs for different configurations. If students don’t have access, providing calculators will help with some of the calculations.

Display this image of the original playground for all to see:

Ask, “What do you notice? What do you wonder?” After some quiet think time to notice and wonder, ask students to share with a partner. Then invite students to share what they noticed and wondered with the class, and record the responses for all to see.

Students may notice:

• This looks like a playground.
• There is a fence around the whole playground.
• There are bushes and playground equipment.
• The playground forms a pentagon.

Students may wonder:

• How big is the real playground?
• How long is the fence?
• How many kids can fit on the playground at the same time?
• How many bushes would it take to go around the whole fence?

Then display this diagram:

1 unit = 10 yards

Tell students that this is a diagram of a playground, and they will use it to design a new playground to fit different constraints. If needed, clarify that the fence of the playground only goes around the perimeter, not along the segments \(DB\) or \(EF\).