# Pasta Circle

Sharpen your students’ measurement skills and area-finding abilities with this pasta-themed challenge:

### Challenges

1. How many pieces of pasta are in the cooking pot?

### Setting

I started using Andrew Stadel’s Estimation 180 challenges about two years ago. A few months in, and my “estimation radar” was about a hundred times more sensitive. If you’re a heavy user of the site, just try walking through Costco without your estimation sense going nuts.

One such trip down the pasta aisle inspired this challenge. Contrived? Absolutely. Useful (in a context-specific sense)? Absolutely not. Engaging? I hope so. And that was the goal: to grab students’ attention with an unusual image that would provide us with an opportunity to explore estimation, measurement, and circle area.

### Lesson Notes

Display the following image:

Ask students to write down 2-3 they notice. After a moment or two, collect responses from students and record them on the board. Next, ask students to write down at least one thing they wonder. If they need help, consider framing it this way: “What is a math question we could ask about this picture?” After another moment or two, collect (and record) student responses.

Depending on what you would like to see happen with the lesson, either select a question to explore as a class, or steer them toward the following question:

Have students write down a too low, too high, and “just right” estimate. Then ask them what information would be helpful in answering that question. If you are able to provide the information they request, do so. There is likely more than one approach that will yield a reasonable answer. For the sake of these lesson notes, I’ll suggest the following two approaches.

(Note: Before launching into either of the approaches below, invite students to suggest and/or develop their own strategies. From there, you can decide whether to pursue their ideas, redirect to an approach described below, or to use a blended approach.)

#### Approach 1

Provide each student with a ruler and a copy of the student handout (available for download in the Resources section below).

Next, have students complete the following steps:

1. Draw a 1-inch by 1-inch square on the image (on the pasta portion)
2. Count the number of pieces of pasta visible in your square
3. Draw a second 1-inch by 1-inch square, count pieces in square

Our goal here is find a reasonable estimate for the average number of pieces of pasta per square inch. Because of human error, camera angle distortion, shadows, and other reasons, one measurement is likely insufficient. Each student will have two measurements. As a class, there will be dozens. You can decide the best way to gather these together. A few ideas include:

• Have students create a table of values with their two measurements, as well as the two measurements from at least three classmates
• Have students write their measurements on the board, to be used or left alone as individuals and groups see fit
• Have students find the average value of the measurements taken at their table (or in the whole class)

Next, students will need to find:

• The radius of the circle, and from that…
• The area of the circle, and from that…
• A calculation of the total number of pieces of pasta in the container

For the “reveal,” show the following:

With 433 pieces per package, and 26 packages used, I estimate that there were 11,258 pieces all together.

#### Approach 2

For a slight variation on the approach described above, instead of providing students with the handout and asking them to make the measurements, display some or all of the following images:

While this will save time, streamline the “measurement/data collection” part of the lesson, and lead to more uniform answers, I would recommend using Approach 1 if time allows.

Note that the measurements in Approach 1 (using the handout) will differ from those in Approach 2, due to the less-than-100% scale factor involved in printing the handout. While the individual measurements will differ from the images, the proportional relationships remain in tact, and students should end up with an answer close to the one I’ve calculated above.

### Extensions

1. How much does all that pasta weigh?
2. How much did the pasta cost?
3. How many pieces of pasta would fit in a pot with half that diameter?
4. How large a container would you need to fit \$100 worth of pasta?
5. How many pieces of pasta are touching the inner wall of the pot?

Some of the extensions will require additional information, including:

### Key Standards

• CCSS 7.G.B
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

### Supporting Standards

• CCSS 7.RP.A
Analyze proportional relationships and use them to solve real-world and mathematical problems.