Students will apply their knowledge of the circumference of a circle to determine how many times a person has to fully turn the handle of a can opener to open a can.

As is true for any task, the intentionality or learning objective can vary depending on what mathematical thinking you are hoping to elicit.

The purpose of this task is to strengthen students’ application skills involving the formula for the circumference of a circle and support the use of spatial reasoning through visualization.

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.

Show students this video of a can being opened.

Ask students to engage in a notice and wonder protocol. ANYTHING and EVERYTHING that comes to mind is fair game.

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.

The question we will land on for this task is:

How many cranks of the can opener will it take to open the can?

Have students estimate the the number of cranks it will take to open the can.

Students should estimate a time that is too high and a time that is too low before making their “best estimate”.

Have students share their high, low, and best estimates with a class elbow partner and then to the whole class.

Ask students what information they might need in order to figure out the answer.

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.

Share the following image:

Give students an opportunity to turn and talk with a partner and discuss what strategies they may need to determine how many cranks it would take to open the can.

Ask students to use this information to update their estimates.

Monitor student thinking by circulating around the room and listening to the mathematical discourse. Select and sequence some of the student strategies and ask a student from the selected groups to share with the class from:

• most accessible to least accessible solution strategies and representations;
• most common/frequent to least common/frequent strategies and representations; or,
• choose another approach to selecting and sequencing student work.

The tools and representations you might see students using to convince their peers and/or the teacher include:

• Concrete representations such as paper cut outs of a circle and folding or cutting the "piece" of the circle that was "covered" by the 2 cranks spatially.
• Drawings.
• Double number line.
• Ratio table.
• Symbolic.

Have students share their strategies and reasoning for determining the number of cranks required to open the can. Ask them to convince you and their peers that their answers are correct by sharing mathematical models. Discuss their strategies and elicit student thinking during your consolidation to build off of their current prior knowledge and understanding rather than “fixing” or “funnelling” student thinking to a strategy and/or model that does not connect to their strategy and/or approach.

I estimate that the can is opened about 1 fifth of the way. So we have 4 fifths left to open. I can skip count 5 times [2, 4, 6, 8, 10] to get a total of 10 cranks to open the can.

OR

I can scale up 2 cranks by 5 to get a total of 10 cranks.

I know that the circumference of a circle is just over 3 times longer than the diameter. So, I know that the circumference should be a little more than 24 cm long. I estimate the circumference to be about 25 cm. Then I can use a ratio table to organize my thinking:

Number of cranks   Length of can opened (cm)

1                                    2.3

2                                    4.6

4                                    9.2

8                                    18.4

10                                  23

11                                  25.3

It will take about 11 cranks to open the can.

I used the formula for circumference of a circle to determine the length that needs to be opened.

C = πd

= 3.14(8 cm)

= 25.12 cm

Since we’ve already cranked 4.6 cm, I know it’ll take at least 5 sets of 2 cranks because 5 cm × 5 = 25 cm. We have a little less than 5 cm opened, so we’ll need more cranks.

I then divided 25.12 cm by 4.6 cm to determine how many units of 2 cranks we’ll need to open the can.

25.12 cm / 4.6 cm = 5.46 units of 2 cranks

5.46 units of 2 cranks = 5.46 × 2 cranks = 10.92 cranks

So it should take approximately 11 cranks to open the can.

At first glance this task might seem relatively simple, however it does require students to work with two separate proportional relationships:

1. The relationship between the circumference length and diameter length; and,
2. The relationship between the length of can opened and number of cranks.

This means that you should anticipate the possibility of some students struggling to get started and or to be able to come up with a strategy initially. This productive struggle is what we want in our math classrooms to Make Math Moments, however we must be prepared with purposeful questions to ensure students do not lose confidence and give up too quickly.

As you walk around the room to monitor student thinking through observation and conversation, be sure to ask students about what information they might need in order to help them come up with the number of cranks. If students say "the distance around the can", then be sure to ask questions about what they know about the distance around the can. Questions like:

• How might we estimate what the distance around the can is?
• I wonder if you were to try to make a model of the lid using paper and scissors if that would help you get closer?
• What might be helpful to measure the relationship between the diameter and the circumference?

Ensuring students have a clear visual of the relationship between the length of the circumference and the diameter (i.e.: π) is very important and may call for you to help students recognize this using string around different circles and/or using a visual like the following:

Video: Consolidating Circumference

As you select and sequence student work for the consolidation, you may want to also include the following visual after sharing student samples in order to ensure all students make the connection that we can scale the number of cranks and the amount of can opened as a composed unit in tandem using ratio reasoning:

Video: Consolidating Number of Cranks

If students haven't used a double number line and/or a ratio table for their thinking, you may choose to use this visual to support during consolidation as well:

Video: Double Number Line

Want more support regarding proportional relationships including ratio and rate reasoning? Be sure to join our course The Concept Holding Your Students Back: Unlocking Key Understandings in Proportional Relationships.

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.

Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

Consolidation Prompt #1:

In your own words, summarize the relationship between the diameter of the can and the Circumference of the can.

How might this relationship be helpful when working with measuring circles?

Consolidation Prompt #2:

A large can of tomato sauce has a radius of 12 cm.

How many cranks will it take you to open this can using the same can opener?

Consolidation Prompt #3:

Your younger sister has opened a can of soup 65% of the way until her hands get tired and she asks you for help.

What is the diameter of the can if she’s cranked the can opener approximately 6 times using the same opener?

We suggest collecting these reflections 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.

...determining the number of cranks needed to open the can:

You could … provide students with a scale printout of the top view of the can. Then, provide students with string so they can cut pieces into 4.6 cm lengths in order to determine approximately how many iterations of 4.6 cm they will need to complete the circle.

You could … ask students to revisit an exploration of the circumference of a circle and the diameter in order to reveal the approximate triple/third relationship between the circumference and the diameter.

Watch for students who are able to clearly articulate and can represent their calculations algebraically using the circumference formula and are able to organize their thinking to either scale in tandem (how many iterations of the composed unit of 2 cranks and 4.6 cm it will take to cover the distance around the can) or use the constant of proportionality (dividing the circumference by the rate of 2.3 cm per crank of the can opener) to determine the total number of cranks it will take to open the can.