## Planning a Playground

- Lesson

In this lesson, students will design a playground using manipulatives and multiple representations. Maximum area with a given perimeter will be explored using tickets. The playground will include equipment with given dimensions, which decreases the maximum area that can be created. This is an interesting demonstration of how a real-world context can change a purely mathematical result. Finally, scale models will be created on graph paper and a presentation will be made to a playground planning committee for approval.

Before beginning this activity, collect all necessary materials and have the tickets counted out in strings of 200.

To begin, place students in groups of 2 or 3. Each group represents a company that will make a bid to design the playground for Animal Crackers Preschool, so group members should work collaboratively. Assign jobs to each group member to allow for individual responsibility and greater efficiency in completing the activity. Possible jobs can include the following:

Member 1 is responsible for obtaining and returning necessary materials.Member 2 is the recorder and responsible for ensuring the activity sheets are completed correctly.

Member 3 is in charge of keeping the group on task, watching the time, and ensuring all tasks are completed.

Animal Crackers Playground Activity Sheet

Pass out the Animal Crackers Playground Activity Sheet to each student, not just each group – each student should be responsible for their own activity sheet. Allow time for students to read over the problem on the first page and discuss with the class ways to plan a playground. After ideas have been brainstormed, read through the second page of the activity sheet with students and address any questions about what needs to be accomplished. Pay special attention to the last 3 paragraphs, in which the details of their presentation are set out. Set clear deadlines for completing the activity sheet and planning and making the presentations. This activity can be completed in one period with presentations at the end of the period. But if time permits, 2 periods is optimal – use the first for working through the activity sheet and the second for making the presentations to the class.

Distribute or make available all materials necessary to complete
the activity. Students should work through and complete the entire
activity sheet, including the presentation. As students are working,
circulate and assist as needed. If students struggle to find the
maximum area in Question 1, give them some guidance, such as suggesting
they make a table of the possible lengths, widths, and area. If they
change one dimension at a time incrementally, the maximum value should
become apparent. The maximum area with no equipment is 2500 ft^{2}, which occurs when the length = width = 50 ft.

As students move on to the Question 2, emphasize they should be using the use-area dimensions. Discuss with students why there is a difference in the dimensions of a piece of equipment and its use area dimensions. Help students understand that the use area includes a buffer to allow the equipment to be used. The swings are the best example of this because students can readily understand why extra space is needed when they are being used.

Encourage students to communicate with one another and use the graph paper and tickets together to more easily reach a solution. They should explore a variety of placements of the equipment in the 50’×50’ square playground before changing the dimensions. Once they understand why all the equipment cannot fit in the square, allow them to explore the other dimensions. If students created a table earlier, the table may facilitate their exploration of different playground dimensions. Some students may elect to work directly with the graph paper manipulatives to try to determine the largest possible area in which all the playground equipment will fit. If they opt for this method, make sure they use the tickets to go back and check that they have enough fence to enclose the equipment.

Below is one solution that allows for the maximum possible area, which has dimensions 52’×48’, but this is not the only solution. The equipment will not fit in any of the larger possible spaces but there is more than one arrangement that will fit in this area. If students find an arrangement in the 52’×48’ area quickly, challenge them to find other arrangements that will work in the same space. They may choose to present a different arrangement based on their own preferences for where playground equipment works best.

As students move into planning their presentations, they will need to sketch their design on graph paper using an appropriate scale. Discuss with students what makes a scale "appropriate." Since these models will be presented to the class, they should be large enough for people to see all the necessary components. They should also be careful to use the same scale for the entire model. After completing a model of the playground, students can use their own creativity to decide how to present their findings to the class. Remind students to adhere to the outline provided on the activity sheet, and that the class will act as the playground committee.

For the presentations, provide students with a rubric. You can create the rubric ahead of time, or create it with the class. Make sure students have access to it so they know the required elements, which you may also use as an assessment of the activity. Have the committee (class) complete the rubric as peer review during each presentation. After all groups have presented their playground plans, go over the committees reviews and comments and have students vote on a winner. You may wish to distribute certificates to the members of the winning group.

- Roll of tickets (enough for each group to have a string of 200)
- Calculator
- Tape
- Graph paper
- Scissors
- Certificate for winning playground plan (optional)
- Animal Crackers Playground Activity Sheet

**Assessment Options**

- As students are working, circulate among them to check individual progress and ensure every member is contributing to the group. Take notes on student progress to gauge understanding.
- Students could write a convincing argument for why the playground equipment will or will not fit into the rectangle with the greatest area.
- Have students present their results, assessing them using criteria such as group participation, organization, justification of proposal and choices, accurate calculations, accurate scales, and accurate dimensions.
- Create a written assessment asking students to complete a similar problem such as redesigning their cafeteria/classroom or their own living space.
- Have groups create a written report addressed to the playground committee instead of or to go along with their oral presentation.

**Extensions**

- Have students incorporate other equipment they think should be in a playground, or choose some for them. If technology is available, students can research different kinds of equipment, their size, and their use area. If students create their playgrounds on their own, include parameters such as necessary accommodations or categories to choose from. Avoid a situation where a student could have a playground of 10 picnic tables.
- Have students create an indoor playground, where height is restricted. Students could look up the height of the pieces of equipment and create 3-dimensional models. Keep in mind that this could take a lot of time and will require additional materials.
- Students could research the safety requirements for playgrounds in their area, such as the required thickness of loose material (e.g., wood chips) that must cover the playground. Have students calculate the amount of material needed and the cost.

**Questions for Students**

1. What is the shape of the maximum area with no equipment?

[A square.]

2. Is your arrangement the only one that works? Can you find another arrangement of the equipment that fits within your chosen rectangle?

[There are several arrangements that work.]

3. Could you list all the possible dimensions of the fenced in area using the 200 feet of fencing if it was not limited to a rectangular shape or the fencing did not come in 1 ft sections?

[No. By eliminating these requirements, the playground could be any shape with any number of sides, even a circle.]

[Yes. Without equipment, the playground with the maximum area would be a circle with diameter = 200/π≈ 63.7 ft and area ≈ 3,183.1 ft^{2}.]

[You need to add extra space around the dimensions of each piece of equipment to account for the actual area that's needed when the equipment is in use (for example, the area needed when swinging on a swing). Also, the use areas are based on regulated safety standards, which must be followed.]

[Have the class generate a list. The list may include a visual representation of the playground, a justification for why their setup is the best, and a summary of the elements or features that set their proposal apart from the others.]

**Teacher Reflection**

- How did your lesson address auditory, tactile, and visual learning styles?
- How did students demonstrate understanding of the materials presented? Was this clear in their presentations?
- What were some of the nonverbal ways in which students illustrated that they were actively engaged in the learning process?
- How did students perform in relation to the stated behavioral objectives? Were all students behaving appropriately with their groups?
- Did you set clear expectations so students knew what was expected of them? If not, how can you make them clearer?
- What worked with classroom behavior management? What didn’t work? How could you modify classroom management practices to make them more effective?

### Learning Objectives

Students will:

- Maximize a playground area with given perimeter.
- Maximize a playground area with specific equipment included.
- Create a scale model of the playground on graph paper.
- Make a convincing argument about the results of their explorations in a presentation to the class.

### NCTM Standards and Expectations

- Use visual tools such as networks to represent and solve problems.

- Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

- Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

### Common Core State Standards – Mathematics

Grade 7, Geometry

- CCSS.Math.Content.7.G.A.1

Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP3

Construct viable arguments and critique the reasoning of others.