Lesson plan at a glance...

In this lesson plan…

• Lesson Overview
• Materials and Equipment
• Preparation Tasks
• The Lesson
• Learning Objectives and Standards
• Additional Information and Resources

Confirm that all computers are turned on and logged-in

1 to 3 minutes

Confirm that your projector is turned on and is projecting properly

1 to 4 minutes

Install Python 2.x (https://www.python.org/downloads) or navigate to Trinket (https://trinket.io/)

5 to 10 minutes

Pre-load Python Starter Files (or copy and paste in the code)

20 minutes

Warm-up Activity: Exploring the relationship between a circle and parallelogram

5 minutes

Activity 1: Deriving the area of a circle

10 minutes

Activity 2: Calculating the area of a circle

15 minutes

Activity 3: Programming Python to calculate the area of a circle

20 minutes

Wrap-up Activity: Building an algorithm for the formula of a circle

20 minutes

Activity:

Have students work through the following activity.

Below are three congruent circles that are sliced into six, eight, and twelve slices, respectively, and then unraveled and rearranged to form a new shape.

Q1: Describe any patterns you notice in the unraveled circles as the slices become smaller and smaller.

Q2: As the circle is cut into more and more slices, what shape does it begin to resemble?

Q3: Do you think it is possible to cut the circle into so many tiny pieces that when we unravel it, it becomes a perfect rectangle with four 90 degree angles and four straight edges?  Explain why or why not. 

Notes to the Teacher:

After students have answered the questions, project the below chart onto the board and discuss the similarities and differences between the outstretched circles and the polygons that they resemble. Have students examine the “curvature” of each figure in the chart and determine which of the three outstretched figures’ area is most similar to the area of a parallelogram.

Assessment:

A1: Answers may vary, however students should notice that as the slices become smaller, bases of the unraveled circle become flatter and flatter.

A2: A parallelogram or a rectangle.

A3: Answers may vary; the explanations are more important than the actual answers.

Notes to the Teacher:

After students answer the questions, project the chart onto the board and discuss the concept of infinite slices. Have students consider what the figure would look like if we cut the circle into infinitesimally small pieces. Discuss the concept of a “theoretical activity” vs. a “realistic activity”, and the idea that although many ideas in higher-level mathematics are based on theory, they actually work out in reality. 

Point out that in spite of the fact that our figures appear to be an approximation of a rectangle, the formula A = pi*r * r, or A = pi*r^2, is not an approximation; it is exact!

Activity:

Have students work through the following activity.

1. The figures below illustrate the relationship between the perpendicular height of the unraveled figures and the radii of the circles.

2. Type the following program into the Python editor and run it to calculate the percent difference between the perpendicular height and the radii of all four circles. Record the results in the table.

Teaching Tips:

Although students do not contribute to the writing of the program above, take a minute to explain the variables and their values. Make sure students understand that whatever value is entered when asked for the radius will be stored in the variable radius, and whatever number is entered when asked for the perpendicular height will be stored in height.  They will need to understand this concept in order to complete the programs in the following activities.

• Instruct students to keep their answers in terms of pi in order to identify patterns in the chart.
• The answers are provided below.

Activity:

Have students answer the following questions.

Q1: Describe any pattern you see between the length of the circle’s circumference and the base of the unraveled circle.

Q2: Compare the radius to the perpendicular height of each circle. Which one is longer and why?

Q3: Does the percent difference between the radius and the perpendicular height increase or decrease as we use more slices?

Q4: Imagine slicing the circle into 10,000 pieces and unraveling them. Based on the patterns you see in the figures above, would the percent change between the perpendicular height and the radius be really big or really small? Could we use the radius as an approximation of the perpendicular height? Why or why not?

Q5: Do you think we could have slices so small that the radius equals the perpendicular height?

Q6: The base of the outstretched circle is half of the circumference and the height is approximately equal to the radius. Recall, circumference equals 2(pi)(r). Using this information, write a formula for the area of the outstretched circle. The radius, r,  should be the only variable in your answer.

Assessment:

A1: The length of the base is half of the circumference.

A2: The radius is always longer because it stretches between the same two bases at a slant, whereas the perpendicular height does not slant.

A3: The percent difference decreases as the length of the slant height approaches the length of the perpendicular height.

A4: Based on the patterns in the chart above, the percent difference would be very small, so yes, it would be a good approximation of the perpendicular height.

A5: Answers may vary; the explanations are more important than the answers.

A6: Area = pi*r*r

Student Activity:

Give the following code to students to use to calculate the area of a circle and instruct them as follows.

1. Use the code below to calculate the area of a circle.
   

2. Run the code to calculate the areas of the following three circles, to the nearest ten thousandth.

3. Notice that each of the circles below is missing a piece. In the chart below, explain how you would calculate the area of each circle with a missing piece (sector):

Chart D

Notes to the Teacher:

Students can translate the algorithms they wrote in the previous activity to Python code.  Present them with the same three partial circles, and ask them to fill in each blank with a formula that Python can follow to calculate the partial circle’s area.

There is more than one correct way to solve these problems.  The solutions below represent one of several correct answers.

Activity:

Have students work through the following activity.

1. Use the method you described for each partial circle in the last activity to complete the Python programs below. Then run each program to calculate the area of each partial circle. 

2. Use the program above to calculate the area of the following partial circles (with answers below).

Q1: It is repetitive to change the program for each different sized slice. In this last program, Python asks for the radius, the total number of slices, and the number of slices remaining. In your own words, explain how it will calculate the remaining area of any circle with any number of congruent slices removed.

Assessment:

A1: Answers may vary, students should understand that in this algorithm, Python calculates the area of each slice and then multiplies that area by the number of slices remaining in the picture.

Activity:

The final question serves as an evaluation of students’ understanding of the derivation of the area of a circle from the area of a parallelogram.  Emphasize that students are not being asked to write down the formula for the area of a circle, rather they should describe the process for coming up with that formula.

Q: Imagine that you were living in the 1600s. Back then, people knew how to calculate the area of rectangles and parallelograms, however no one had figured out exactly how calculate the area of a circle. Write an algorithm that people from that time could follow to derive the formula for the area of a circle.

Learning Objectives

Standards

LO1: Find arc lengths and areas of sectors of circles.

Common Core

CCSS MATH.CONTENT.HSG.C.5: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Computer Science

CSTA L2.CT14: Examine connections between elements of mathematics and computer science including binary numbers, logic, sets and functions.

Term

Definition

For Additional Information

Parallelogram

A (non self-intersecting) quadrilateral with two pairs of parallel sides.

http://en.wikipedia.org/wiki/Parallelogram

Radius

The length of a line segment from a circle’s center to its perimeter.

http://en.wikipedia.org/wiki/Radius

Circumference

The linear distance around the edge of a closed curve or circular object.

http://en.wikipedia.org/wiki/Circumference

Concept

Definition

Pattern Recognition

The ability to find commonality in form, format, structure, method or plan.

Decomposition

To break down or regroup data to facilitate analysis.

Algorithm Development

Building a reusable description of generalized behavior.

Data Representation

Depicting and organizing data in appropriate graphs, charts, words or images

Contact info

For more info about Exploring Computational Thinking (ECT), visit the ECT website (g.co/exploringCT)

Credits

Developed by the Exploring Computational Thinking team at Google and reviewed by K-12 educators from around the world.

Last updated on

07/02/2015

Copyright info

Except as otherwise noted, the content of this document is licensed under the Creative Commons Attribution 4.0 International License, and code samples are licensed under the Apache 2.0 License.