Math Worth Thinking About: Using Polygons to Learn About Circles
Watch this video to see how surprise and student active-mindedness leads to deeper mathematical learning.
Watch this video to see how surprise and student active-mindedness leads to deeper mathematical learning.
The CT/Coding Advantage: Student Agency
Watch this video to see how coding gives students greater agency and how this leads to deeper learning.
Minds On: A Circular Track
1. Show students the image of the circular track shown to the right. Ask them what they notice and wonder about it. There is a good chance that someone will point out that the pictured race is not fair as the distances for the two runners are not equal. If nobody raises this issue, make sure it comes up in the subsequent discussion. Ensure that all students understand that the two lanes are different lengths. Ask how one might plan a fair race on this track. Through discussion, it will come up that staggered starts are required, with the runner on the outside track starting ahead of the runner on the inside track. Ask how long the stagger should be to create a fair race. |
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2. Provide pairs of students with paper copies of the circular track. Ask them to determine, by whatever means they have at their disposal, the lengths of the two tracks. Then have them use this information to determine the where the outside runner should start in relation to the inside runner.
Give students an appropriate amount of time to work on the problem. Facilitate a short discussion to allow them to share their strategies and conjectures. Then ask them to put this problem aside. Tell them that they will return to it after an investigation into the nature of circles. |
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Action: A Circular Investigation
3. Provide student pairs with measuring tapes and copies of A Circular Investigation. Read the instructions together, then let them work on it. 4. When all pairs have finished the activity, gather them together for a class discussion. Ask students to describe the patterns they see in their results. Some important ideas that need to come to light during this sharing are:
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Ask:
Wrap up the discussion and move onto the next stage of the math learning experience. |
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Action: Splitting Polygons
5. Tell students that they are going to use a Scratch code to analyze a variety of polygons in order to refine their understanding of circumference, but that the first step is to interpret the code itself. Distribute copies of Interpreting Splitting Polygons so that each pair has a copy. Review the instructions with them and give them an appropriate amount of time to complete the activity. 6. Debrief Interpreting Splitting Polygons with the students. Discuss the blocks in order, and allow students to contribute their ideas about the function of each. Explain that while looking at the code in its entirety may be confusing and overwhelming, breaking it down into smaller pieces allows for easier interpretation. Further explain that this is a form of decomposition, a key computational thinking skill used by people who write code.
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7. Project the Splitting Polygons Scratch code (linked here) so that all students can see it. Click the green flag. Input 600 for the polygon perimeter. Then, input 4 for the number of sides, and hit enter. Once the diagram is complete, the P/line list (wherein the perimeter is divided by the length of the diagonal shown on the diagram) will show the value 2.828427. Without resetting the code by tapping the space bar, click the green flag again. You will be prompted to select a different perimeter. Input a number far removed from 600 (e.g., 100). Input 4 for the number of sides again, but before hitting enter, ask students if they think that the quotient will be the same as or different than that of the previous trial. Allow students to share the reasons that support their opinions.
At an appropriate time during the discussion, click enter. Students will see that, despite the change in the perimeter, the quotient is 2.828427 once again. In the ensuing discussion, make sure students understand that the length of the diagonal/symmetry line changes at the same rate as the length of the perimeter. The relationship between the two, then, is proportional. The perimeter of a square will always be 2.828427 times longer than the length of its diagonal. Discuss this finding in relation to the circle. Return to the discussion to the three questions presented in step 4.
At an appropriate time during the discussion, click enter. Students will see that, despite the change in the perimeter, the quotient is 2.828427 once again. In the ensuing discussion, make sure students understand that the length of the diagonal/symmetry line changes at the same rate as the length of the perimeter. The relationship between the two, then, is proportional. The perimeter of a square will always be 2.828427 times longer than the length of its diagonal. Discuss this finding in relation to the circle. Return to the discussion to the three questions presented in step 4.
8. Provide students with the link to the Splitting Polygons Scratch code. Ask them to do the following:
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9. Discuss the results of the students’ work with the Scratch code. Consider using the following questions to guide the discussion:
10. Ask students to apply their new knowledge to the problem in A Circular Track. As they work, circulate among them and note the different strategies used. Think about how you might use them, and in what order, during the consolidation discussion. Consider having the students complete a Gallery Walk before the consolidation.
- Did the blocks of code behave as you/we predicted in the Splitting Polygons Investigation activity? If some did not, how did they function, and what was it about them that caused you/us to misinterpret them?
- Did the other regular polygons produce common quotients when their perimeters were divided by the lengths of their bisecting line segments? Did any regular polygons not behave this way? What does this lead you to believe about circles?
- What did you observe when you experimented with the polygons with even numbers of sides? With polygons with odd numbers of sides? With polygons starting with the triangle, then the square, the pentagon, etc.? What does all of this information tell you about circles?
10. Ask students to apply their new knowledge to the problem in A Circular Track. As they work, circulate among them and note the different strategies used. Think about how you might use them, and in what order, during the consolidation discussion. Consider having the students complete a Gallery Walk before the consolidation.
Consolidating the Learning
11. Gather the students together to discuss A Circular Track in light of their new learning. Highlight selected student work, in the order determined during observation of the students’ strategies. Focus on the application of the students’ understanding of Pi to determine the circumferences of the two lanes correctly, and the appropriate placement of the staggered start to ensure that both runners cover the same distance.
11. Gather the students together to discuss A Circular Track in light of their new learning. Highlight selected student work, in the order determined during observation of the students’ strategies. Focus on the application of the students’ understanding of Pi to determine the circumferences of the two lanes correctly, and the appropriate placement of the staggered start to ensure that both runners cover the same distance.
Extensions: Coding Tracks
12. Challenge students to represent the track in A Circular Track as a Scratch code. Their code should model both lanes (or, at the very least, the inside lines of both lanes) as well as the staggered start. An example of what they might develop is found here.
13. While the idea of a circular track helps students develop an understanding of circular measurement, it is somewhat of a silly context. Circular tracks, if they exist at all, are simply not common (and for good reason; endless turning while running puts additional strain on the runners’ joints). Now that students have a good understanding of Pi and circumference, challenge them with a composite figure that contains circular elements. A standard track (basically a rectangle with a semi-circle on each side) is a logical choice. The inside lane of a competitive running track is 400 m long, and the diameter of the circle is 73.6 m. Lanes can be as wide as 122 cm (4 ft) but many tracks feature narrower lanes. Challenge students to use this information to develop models of standard tracks in Scratch. Aspects of the code provided in step 12 (or parts of their own codes if they have created any) could be remixed for this purpose.
12. Challenge students to represent the track in A Circular Track as a Scratch code. Their code should model both lanes (or, at the very least, the inside lines of both lanes) as well as the staggered start. An example of what they might develop is found here.
13. While the idea of a circular track helps students develop an understanding of circular measurement, it is somewhat of a silly context. Circular tracks, if they exist at all, are simply not common (and for good reason; endless turning while running puts additional strain on the runners’ joints). Now that students have a good understanding of Pi and circumference, challenge them with a composite figure that contains circular elements. A standard track (basically a rectangle with a semi-circle on each side) is a logical choice. The inside lane of a competitive running track is 400 m long, and the diameter of the circle is 73.6 m. Lanes can be as wide as 122 cm (4 ft) but many tracks feature narrower lanes. Challenge students to use this information to develop models of standard tracks in Scratch. Aspects of the code provided in step 12 (or parts of their own codes if they have created any) could be remixed for this purpose.
Curriculum Expectations
As the table below shows, In Search of Pi addresses several grade eight Measurement expectations, but none of the grade six and grade seven expectations for this strand. However, in multi-grade classes that included grade eight students, this content will be taught, and it is likely that the students from the other grade(s) will come in contact with it. Their work with this content can be linked to other non-Measurement expectations in their grade levels. For instance, there is a great deal of proportionality involved in circular measurement. Younger students’ inclusion in this math learning experience could be geared toward their achievement of Number Sense and Numeration expectations pertaining to proportionality.
In Search of Pi addresses the following curriculum expectations:
As the table below shows, In Search of Pi addresses several grade eight Measurement expectations, but none of the grade six and grade seven expectations for this strand. However, in multi-grade classes that included grade eight students, this content will be taught, and it is likely that the students from the other grade(s) will come in contact with it. Their work with this content can be linked to other non-Measurement expectations in their grade levels. For instance, there is a great deal of proportionality involved in circular measurement. Younger students’ inclusion in this math learning experience could be geared toward their achievement of Number Sense and Numeration expectations pertaining to proportionality.
In Search of Pi addresses the following curriculum expectations:
Acknowledgements
In Search of Pi is partially based on "Running Tracks" by Cre8ate Maths (full citation below) and material from Unit 3: From Powers to Circles in TIPS4RM (full citation below).
In Search of Pi is partially based on "Running Tracks" by Cre8ate Maths (full citation below) and material from Unit 3: From Powers to Circles in TIPS4RM (full citation below).
References
All-weather running track. (2018, August 08). Retrieved August 12, 2018, from https://en.wikipedia.org/wiki/All-weather_running_track
Cre8ate Maths. (n.d.). Running tracks. Retrieved August 9, 2018, from https://www.stem.org.uk/rxssd
Lockhart, P. (2002). A mathematician's lament. Retrieved from http://www.gang.umass.edu/~franz/lament.pdf
Mathematics: Revised: The Ontario curriculum: Grades 1-8. (2005). Toronto: Ontario Ministry of Education.
Unit 3: From Powers to Circles - EduGAINs. (n.d.). Retrieved August 11, 2018, from http://www.edugains.ca/resourcesMath/CE/LessonsSupports/TIPS4RM/Grade8English/Unit3_FromPowersToCircles.pdf
All-weather running track. (2018, August 08). Retrieved August 12, 2018, from https://en.wikipedia.org/wiki/All-weather_running_track
Cre8ate Maths. (n.d.). Running tracks. Retrieved August 9, 2018, from https://www.stem.org.uk/rxssd
Lockhart, P. (2002). A mathematician's lament. Retrieved from http://www.gang.umass.edu/~franz/lament.pdf
Mathematics: Revised: The Ontario curriculum: Grades 1-8. (2005). Toronto: Ontario Ministry of Education.
Unit 3: From Powers to Circles - EduGAINs. (n.d.). Retrieved August 11, 2018, from http://www.edugains.ca/resourcesMath/CE/LessonsSupports/TIPS4RM/Grade8English/Unit3_FromPowersToCircles.pdf