Grass Act

Est. Class Sessions: 2

Developing the Lesson

Develop a Plan. Read the Grass Act page in the Student Guide so that students understand the context of the problem. After reading the exchange between Manny and Felicia, ask students to work with a small group of students to answer Question 1. Students work together to develop a plan that will determine the number of grass blades in a lawn or the given area you have selected.

After small groups have had a chance to develop a plan for how to solve the problem, ask a few groups to share and discuss their plans with the class. This discussion should help students identify any possible difficulties in their own plans. See the Sample Dialog.

Below is a sample conversation from a class trying to find a reasonable strategy for counting the blades of grass in a field.

Teacher: Would counting the blades of grass be possible?

Peter: No, it would take too long. And how would you keep track? And I doubt it is okay that we crawl around on the field.

Teacher: Can we come up with an exact answer?

Alex: No, but we can try for a good estimate.

Harry: How about we weigh one blade of grass, cut the grass, collect and weigh the clippings, and divide by the weight of one blade of grass.

Teacher: What about this idea?

Peter: I think it would work, but I don't think the school would let us cut the grass for a math problem.

Lily: And, all blades of grass are not going to be the same size. I'm not sure we can weigh one blade of grass either. That seems hard.

Teacher: Maybe we need to find another strategy. Anyone else have any other ideas?

Jimmy: How about we count all the blades of grass in a smaller square first and then figure out the number of squares in the lawn.

Teacher: How large a square will you use? What measurement?

Jimmy: A yard? A square that is made by four yardsticks?

Teacher: What is the unit of measure for that area?

Jimmy: A square yard. [Students head outside and put 4 yard sticks down to show a square yard.]

Cassidy: This is still too many to count!

Teacher: How about a square foot? [Students put down rulers to show a square foot.]

Cassidy: This is still too many too count! [Students head back inside the classroom.]

Teacher: Any other ideas?

Cassidy: How about a square inch?

Teacher: How are we going to show a square inch?

Cassidy: Cut one out of a piece of paper. [She measures and cuts a square inch from a sheet of paper.]

Teacher: What do you think of this strategy?

Cassidy: I can count those. What about weeds?

Teacher: What do you think about weeds?

Cassidy: I think we should try to avoid them. Try to find a place with grass and not weeds. Pretend there are no weeds.

Teacher: So, you are going to count the blades in one square inch? Is one sample sufficient to find a reasonable estimate?

Jimmy: I think so. Just count the blades in a square inch and multiply by the number of square inches in the lawn.

Lily: I do not think one sample is enough. Some parts of the lawn will have thicker grass than others. We plan on counting a couple of times and then finding the average.

Teacher: Do you think we can share the work for that part?

Lily: That would be nice. I really don't want to count all those blades of grass.

Ask students to record their plan for finding the number of blades of grass (Question 2). Student plans should include:

finding the total area of the lawn;
counting the blades of grass in several sample areas of lawn—one method is to use a note card with a square-inch hole removed to isolate a square-inch “sample” of grass to count;
finding the average (median or mean) number of blades of grass in the samples; and
using the average to estimate the number of blades of grass in the larger area.

Solve the Problem. Once students have a plan for finding the number of grass blades in an area, they are ready to measure the total area and collect data. Students will get a variety of answers for the number of blades of grass in a square inch. Ask them to decide which number they are going to use. See the Content Note for how one teacher helped her students decide on the mean, median, or mode.

Mean, Median, or Mode. “Our next dilemma was how to know which number to use. The students decided that finding the average would give them the best number to use to estimate the number of blades of grass in a football field. Since the students had studied the different measures of central tendency, we discussed whether we should use the mean, median, or mode. Since we had more than one mode, that measure would not give us a good representation. The mean and median were close together, so some students decided to use the mean and some, the median.”

Chris Nugent, Dubuque, Iowa

To find the number of square inches in the total area, the class should negotiate the basic measurements of the area and then students should work in small groups to complete the problem.

As groups work on their plans, check to see that they include all the steps. Make a note of questions and support you provide students so that you can take it into consideration when you evaluate their problem solving skills. As appropriate, ask:

What do you need to know to solve the problem?
How are you going to sample the lawn?
What size sample are you going to use? Square foot? Inch? Meter?
How many samples are you going to count?
Let's imagine a sheet of paper is the lawn. How would you figure out how many blades of grass are in the lawn?
How many square units are in the sheet of paper? How do you know?

While the total area of the lawn will likely be measured in square feet or square yards, the sample size will probably be measured in square inches. Therefore, students will need to convert between units of measure. Some students will find these calculations easily. Some students will require more support. Below are some strategies to help students make the connections needed for the unit conversion calculations.

Use four 12-inch rulers to build 1 square foot then ask students to figure out how many square inches are in that one square foot. Have students “fill” the square foot with square inches or square-inch tiles.
Construct a model out of graph paper where 1 square is 1 square inch. Construct a square foot and then a square yard. See Figure 1.

Write Solutions. Before students write their solutions, facilitate a class discussion in which students develop problem-specific expectations for writing up their problem solving process. To begin, display the Math Practices Notes Master and refer students to the Math Practices page in the Student Guide Reference section.

Ask students to discuss the following with their group:

What will you need to include in your solution to meet each of the Math Practices Expectations?

As groups discuss the expectation, note which groups develop expectations that are specific to the problem and clearly state what needs to be included in a response to fully explain their strategy. Ask selected students to write their specific expectation for one of the entries on the display of the Math Practices Notes Master. Sample expectations are shown in Figure 2. Discuss the expectations with the whole class and revise them based on their input.

Choosing Appropriate Estimation Methods. Since the calculations in this lesson are estimates, it is appropriate to round numbers to ease computation. For example, Ana could have rounded the number of blades of grass in the sample to 30 blades of grass per square inch and rounded the number of square inches in the grassy area to 150,000. (sq. in.) To find the estimate, she could have simply multiplied 30 × 150,000 to get an estimate of 4,500,000 blades of grass.

At this point, ask students to individually write a paragraph describing their group's strategies. Encourage them to write their paragraph so they meet the expectations collected.

Review Student Work. Use samples on the Ana's Work and Tim's Work Masters or choose appropriate work from your class to discuss how to review others' work. Point out to students that they should be able to follow the steps in the written paragraphs to solve the problem the same way the writers of the paragraph did.

Distribute one sample of student work to each group of students so that each work sample is evaluated by at least two groups. Distribute one copy of the Grass Act Feedback Box Assessment Master to each group as well. Ask each group to use the Feedback Box to guide a review of the student work. See Figures 3–6 for student work samples and sample completed Feedback Boxes. In Figure 3, Ana estimated the total number of blades of grass in a portion of a playground. In Figure 5, Tim estimated the number of holes in a window screen.

Organize small groups into larger groups that reviewed the same student work. Ask students to share their review of the student work and discuss any discrepancies.

Revise Work. Ask students to use what they learned from the discussion of the student work to revisit and revise their own work on the Grass Act problem.

Use students written solutions to the Grass Act problem described in the Student Guide and the corresponding Feedback Box to assess students' abilities to find the mean and median [E13]; know the problem [MPE1]; find a strategy [MPE2]; check for reasonableness [MPE3]; check their calculations [MPE4]; show their work [MPE5]; and use labels [MPE6].

Student Guide: 357

Answers