Modeling Prompt 3
Scaling a Playground
Integrated Math 2
In-Class Launch
Use after Unit 2, Lesson 1
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 accommodate three times as many children as a given playground. Students explore the density calculation of children per square yard, and because of the openness of the modeling prompt, they can choose to increase the area by a factor of 3, increase only one dimension by a factor of 3, or design a different-shaped playground, with a fixed area, and determine a shape that will ensure the perimeter remains under budget.
This activity is designed to be completed digitally because students 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 spreadsheet software to track costs for different configurations. If students don’t have access, providing calculators will help them with the calculations.
Display this image of the original playground 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 children 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 this is a diagram of a playground that they will use as a guide to design a new playground to fit different constraints. If needed, clarify that the fence of the playground goes around the perimeter only, not along the segments \(DB\) and \(EF\).
Teacher Instructions
Tell students that their new playground doesn’t have to be the same shape as the given playground.
If possible, display the finished playground designs for all to see and compare. For example, invite the class to complete a Gallery Walk.
Here’s a playground for a school of 360 children in Springfield.
1 unit = 10 yards
- The fence around the playground costs 90 dollars per 50-foot roll. Laying 1-foot squares of sod across the area costs 400 dollars per 500 square feet. How much does the fencing and the grass for this playground cost?
- How many children per square yard can Springfield’s playground hold?
- There’s a new playground going up in nearby Wintermeadow, which has a budget about 3 times greater than Springfield’s. Recommend a playground shape and size that would fit Wintermeadow’s budget and hold at least 3 times as many children as Springfield’s playground at the same density of children per square yard. How many children can your recommended playground hold?
Lift Analysis
Teacher Instructions
Tell students that their new playground doesn’t have to be the same shape as the given playground.
Consider pausing the class after the second problem to discuss why the scaled-up playground ended up being too big.
If possible, display the finished playground designs for all to see and compare. For example, invite the class to complete a Gallery Walk.
Here’s a playground for a school of 360 children in Springfield.
1 unit = 10 yards
- The fence around the playground costs 90 dollars per 50-foot roll. Laying 1-foot squares of sod across the area costs 400 dollars per 500 square feet. How much does the fencing and the grass for this playground cost?
- There’s a new playground going up in nearby Wintermeadow, which has a budget about 3 times greater than Springfield’s. Wintermeadow is building a playground by scaling up Springfield’s playground dimensions by a factor of 3. Calculate the cost of the grass and the fencing for a playground that has been dilated, using a scale factor of 3. Will Wintermeadow’s budget cover the costs?
- How many children could play on the playground that is scaled up from Springfield’s playground by a factor of 3, at the same density of children per square yard?
- Recommend a playground shape and size that would fit Wintermeadow’s budget and hold at least 3 times as many children as Springfield’s playground. How many children can your recommended playground hold?