Sand Under the Swing Set

Task

The 7th graders at Sunview Middle School were helping to renovate a playground for the kindergartners at a nearby elementary school. City regulations require that the sand underneath the swings be at least 15 inches deep. The sand under both swing sets was only 12 inches deep when they started.

The rectangular area under the small swing set measures 9 feet by 12 feet and required 40 bags of sand to increase the depth by 3 inches. How many bags of sand will the students need to cover the rectangular area under the large swing set if it is 1.5 times as long and 1.5 times as wide as the area under the small swing set?

IM Commentary

The purpose of this task is for students to solve a contextual problem where there are multiple entry points to this geometry based concept. The student can choose to solve the problem using a scale factor or a unit rate, but must first must analyze the context of the problem to understand the situation and choose their approach. This task provides opportunities for students to reason about their computations to see if they make sense.  This task could be used as an assessment question or for guided instruction on scale factoring and/or unit rate. 

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, the commentary will spotlight one practice connection in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This task helps illustrate Mathematical Practice 1, “Make sense of problems and persevere in solving them.” Students must read the problem carefully and analyze the givens, constraints, relationships, and goals while continuously verifying if the answers make sense based on their initial analysis. In the task at hand, students explain to themselves the meaning of the problem, look for entry points to begin their work on the problem, and select a solution pathway. If they can’t make sense of a problem, they ask themselves questions that will help them get started such as, “What do we know?” “How might I organize the information?” “How will I use the information about the area under the small swing set and the number of sand bags required for the small swing set to help me find how many bags are needed for the large swing set?” “What operation(s) will I use to solve this problem?”  Once students have a solution, they look back at the problem to determine if their answer is reasonable and accurate. They might check their answer using a different approach.

This task could be used as a formative assessment by giving it in a pre-post situation prior to beginning a geometry or ratio/proportion unit to see which solution methods the students use most often to solve the problem. After formal instruction, the task could be posed again to see if the student’s solution method has changed or improved based on new understanding.

Solutions

Solution: Finding the scale factor the hard way

3 inches is $\frac14=0.25$ foot, so the volume of sand that was used is $$0.25 \times 9 \times 12 = 27$$ cubic feet. The amount of sand needed for an area that is 1.5 times as long and 1.5 times as wide would be $$0.25 \times (1.5 \cdot 9) \times (1.5 \cdot \times 12) = 60.75$$ cubic feet.

We know that 40 bags covers 27 cubic feet. Since the amount of sand for the large swing set is $$60.75 \div 27 = 2.25$$ times as large, they will need 2.25 times as many bags. Since $2.25 \times 40 =90$, they will need 90 bags of sand for the large swing set.

Solution: Finding the scale factor the easy way

Since we have to multiply both the length and the width by 1.5, the area that needs to be covered is $$1.5^2 = 2.25$$ times as large. Since the depth of sand is the same, the amount of sand needed for the large swing set is 2.25 times as much as is needed for the small swing set, and they will need 2.25 times as many bags. Since $2.25 \times 40 =90$, they will need 90 bags of sand for the large swing set.

Solution: Using a unit rate

The area they cover under the small swing set is $9 \times 12 = 108$ square feet. Since the depth is the same everywhere, and we know that 40 bags covers 108 square feet, they can cover $108 \div 40 = 2.7$ square feet per bag.

The area they need to cover under the large swing set is $$1.5^2 = 2.25$$ times as big as the area under the small swing set, which is $$2.25 \times 108 = 243$$ square feet. If we divide the number of square feet we need to cover by the area covered per bag, we will get the total number of bags we need: $$243 \div 2.7 = 90$$ So they will need 90 bags of sand for the large swing set.

Solution: The other unit rate

The area they cover under the small swing set is $9 \times 12 = 108$ square feet. Since the depth is the same everywhere, and we know that 40 bags covers 108 square feet, they can cover $40 \div 108 = \frac{10}{27}$ bags per square foot.

The area they need to cover under the large swing set is $$(\frac32)^2 = \frac94$$ times as big as the area under the small swing set, which is $$\frac94 \times 108 = 243$$ square feet. If we multiply the number of square feet they need to cover by the number of bags needed per square foot, we will get the total number of bags we need: $$243 \times \frac{10}{27}= 90$$ So they will need 90 bags of sand for the large swing set.