GOING IN CIRCLES [DAY 7]
EXPLORING PI: CIRCUMFERENCE & AREA OF CIRCLES
Leverage contextual problems to investigate and emerge Pi to help conceptualize formulas for both the circumference and area of a circle.
Intentionality
Spark Curiosity
Fuel Sensemaking
During Moves
Student Approaches
Next Moves
Consolidation
Reflect and Consolidation Prompts
Resources & Downloads
Educator Discussion Area
Intentionality & Unit Overview
Length of Unit: 8 Days
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Students will continue their exploration of the relationship between circumference, diameter and area of a circle to determine how many “cranks” on a can opener are needed to open the lid.
In this task, students will explore the concept of area of a circle through estimation followed by measurement. This task will allow students to develop a deeper understanding of big ideas, including the following:
- When exploring measurement relationships of any circle, there are three (3) standard measures that can be used to reveal more information about the circle:
- Circumference: the perimeter of a circle.
- Diameter: any straight line segment beginning and ending on the outer edge of a circle and passes through its centre.
- Radius: any straight line segment beginning at the centre of a circle and ending on the outer edge of the circle.
- Knowing one of the standard measures of a circle provides information about all other measures;
- There is a multiplicative relationship between the length of the diameter and the circumference of a circle. In other words, as the diameter of a circle increases, the circumference increases multiplicatively (not additively);
- For all circles, the ratio of the circumference to the diameter is the same (approximately 22:7); and,
- For all circles, the rate of each unit of circumference length per unit of diameter length is the same (approximately \(\frac{22}{7}\) or 3.14) and is known as Pi, π.
- The area of a circle can be determined by decomposing and recomposing the figure into familiar polygons (i.e.: rectangles or parallelograms); and,
- For all circles, changing the diameter or radius will affect the area and circumference in predictable ways.
Lesson Walkthrough Video
Watch a short video below that will take you through the entire lesson from start to finish.
More specific details about each part of the lesson can be found below as well.
Spark Curiosity
What Do You Notice? What Do You Wonder?
Show students this video of a can being opened.
Then, ask students:
What do you notice?
What do you wonder?
Give students 60 seconds (or more) to do a rapid write on a piece of paper.
Replaying the video and/or leaving a screenshot from the video up can be helpful here.
![Going In Circles [Day 7] - Open Sesame - 02 -Spark Image - Notice Wonder](https://learn.makemathmoments.com/wp-content/uploads/2021/04/Going-In-Circles-Day-7-Open-Sesame-02-Spark-Image-Notice-Wonder.jpeg)
For example, students might notice and wonder the following:
- I notice a can of food.
- I notice a can of cat food or tuna.
- I wonder how many “cranks” it will take to open the can?
- I noticed that he cranked the can opener 2 times.
- I wonder what kind of food it is.
- Does the person in the video have a cat?
- I wonder how much food is in the can?
- And many others…
Document what students notice and wonder with you on chart paper, whiteboards, or other visual displays in your classroom.
At this point, you can answer any wonders that you can cross off the list right away and of course, you can create your own answers for some of these wonders as you leverage your storytelling skills. For example:
- You could explain who the person is (is it you? a friend? a family member?)
- You could explain what was in the can (tuna) and what you were planning to do with it.
- You could consider adding on more context by creating a storyline of how much you enjoy (or dislike) tuna, etc.
- And so on…
Spending times to acknowledge and address specific thoughts that students shared whether a notice or a wonder is crucial to building a culture in your classroom where students know that their voice is being valued and thus encourages them to continue sharing their thoughts and opinions later in this lesson and in future lessons.
Estimation: Prompt
After we have heard students and demonstrated that we value their voice, we can land on the first question we will challenge them with:
How many cranks of the can opener will it take to open the can?
![Going In Circles [Day 7] - Open Sesame - 03 -Spark Estimate Prompt Image](https://learn.makemathmoments.com/wp-content/uploads/2021/04/Going-In-Circles-Day-7-Open-Sesame-03-Spark-Estimate-Prompt-Image.jpeg)
Have students estimate the number of cranks it will take to open the can.
You can also prompt them with the following:
Think of 3 numbers:
- a number you think is too low;
- a number you think is too high; and,
- a number you think is just right.
You can confirm that the person in the video had opened “2 cranks-worth” of the can.
![Going In Circles [Day 7] - Open Sesame - 04 -Spark Update Estimate Prompt Image](https://learn.makemathmoments.com/wp-content/uploads/2021/04/Going-In-Circles-Day-7-Open-Sesame-04-Spark-Update-Estimate-Prompt-Image.jpeg)
Fuel Sense-making
Crafting A Productive Struggle: Prompt
After students have shared their low, high and just right estimates and you have recorded them on the board, chart paper or a digital collaboration space, you can then ask students what information could be useful in order to help them become more precise.
It is likely that students will ask about the dimensions of the can and they may even ask for the length of the portion that was opened with the two cranks they saw in the video.
Facilitator Move:
Be sure to ask students to explain why they feel a particular piece of information would be useful. What are they intending to use that information for? This is helpful data to assess student thinking and where they might be headed once we reveal more information about this problem.
Now, you can share the following information revealing the diameter and the length of the can lid that can be opened with two cranks of the can opener:
How many cranks of the can opener will it take to open the can?
Update your estimate.
Convince us by modelling your thinking.
![Going In Circles [Day 7] - Open Sesame - 05 -Sense Making - Productive Struggle Prompt Image](https://learn.makemathmoments.com/wp-content/uploads/2021/04/Going-In-Circles-Day-7-Open-Sesame-05-Sense-Making-Productive-Struggle-Prompt-Image.jpeg)
We are still going to encourage students to avoid the use of a calculator and if a student is relying on a formula alone, we will prompt them to help convince us by modelling in a way that would convince even the most skeptical critic.
During Moves
While Students Are Productively Struggling…
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Student Approaches
Student Approach #1: Reasoning With Skip Counting and/or Scaling
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Student Approach #2: Ratio Table With Doubling and then Adding On
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Student Approach #3: Symbolic Using a Formula
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Next Moves
Reveal
After consolidating learning using student generated solution strategies and by extending their thinking intentionally, we can share what really happened with this video.
Revisit the student responses. Ask students why they think their answers may be different. [E.g., I assumed that 1 fifth of the can was open, but it was actually slightly less than 1 fifth, so my result was slightly lower than what really happened, etc.]
Have students discuss their thinking and what they would change if they did the task again.
Return to the questions you recorded from the Spark section, and answer any questions that have not been answered.
![Going In Circles [Day 7] - Open Sesame - 12 - After Moves - Reveal Image](https://learn.makemathmoments.com/wp-content/uploads/2021/04/Going-In-Circles-Day-7-Open-Sesame-12-After-Moves-Reveal-Image.jpeg)
Answer: Approximately 11 cranks.
Extend: Area of a Circle
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Consolidation
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Reflect and Consolidation Prompts
Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.
Consolidation Prompt #1:
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Consolidation Prompt #2:
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We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.
Resources & Downloads
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Educator Discussion Area
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