Exploring Circumference

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Objective

SWBAT: • Measure an object to the nearest tenth of a centimeter. • Identify and find the radius, diameter, and circumference of a circle. • Explain pi in your own words. • Explain the formula for circumference and why it works.

Big Idea

What is pi, anyway? Students explore the relationship between circumference and diameter by measuring a variety of circles and use it to derive the formula for circumference.

Do Now

5 minutes

Notes:

  • I created this lesson so that students can discover the relationship between a circle’s diameter and circumference.  In my state, sixth grade students need to be able to work with circles and calculate their circumference and area. 
  • Some students may already be familiar with the formula for circumference, but they probably don’t understand where it comes from.  I want students to understand these relationships so that they can come up with the formula for circumference on their own.  I hope that this will help students to remember the formula and apply it correctly. 

See my Do Now in my Strategy folder that explains my beginning of class routines.

Often, I create do nows that have problems that connect to the task that students will be working on that day.  Here, I want students to spend a couple minutes jotting down what they know and want to know about circles.  I ask students to share their knowledge and ideas with the class.

Brain Pop Video: Circles

5 minutes

Circles Video: http://www.brainpop.com/math/geometryandmeasurement/circles/

I pause the video so that students can fill in their notes.  I do not show the entire video!  Students will be deriving the circumference formula later in the lesson, so I stop the video at 1:23 after Tim shares that the circumference is the perimeter of a circle, or the distance around it. 

Calculating Diameter and Radius

5 minutes

Now that students have been exposed to the definitions of radius and diameter, I want them to practice knowing one and finding the other.  A common mistake is that students confuse diameter with radius and vice versa.  It helps students to draw what is given first, and then figure out the missing information.

Exploring Circumference

18 minutes

I have a volunteer read the directions.  I go over group expectations and what to do when they finish measuring an item.   I stress the importance of accurately measuring the items.  I suggest that group members check each other’s measurements to ensure accuracy.  Students will be engaging in MP6: Attend to precision and MP8: Look for and express regularity in repeated reasoning.

For this activity, I let the students choose their groups of 3-4.  I give them 30 seconds to pick their groups and sit down.  If they cannot handle that, then I will have students count off and sit down.  I have volunteers pass out materials and Group Work Rubric to each group.

While students are working, I walk around and monitor student progress and behavior.  When a group has measured five objects, I check in with them briefly and then have them move on to the graph.

Some students may struggle to make connections on question 3.  This is okay.  I want students to work together to see what they can figure out.  If a group asks for a calculator, I give them one to work with. 

If a group completes the graph and questions, I ask them to explain what they noticed and their predictions for 3c and 3d.  If there is still time, the group can measure extra items and then plot the new measurements on the graph.

With about seven minutes left, I stop groups and have them come together for a class discussion.  I ask 2-3 volunteers from different groups to show their graphs under the document camera.  I ask, “What did your group notice?”  I want students to notice that the circumference of a circle is about 3 times bigger than a circle’s diameter.  I ask students to share their predictions to 3c and 3d and explain them.

Brain Pop Video: Pi

5 minutes

Pi Video: http://www.brainpop.com/math/numbersandoperations/pi/

I pause the video so that students can fill in their notes.  I do not show the entire video!  Students will be deriving the circumference formula next, so I stop the video at 1:15 after Tim shares that pi is an irrational number.

If I have time, I go to http://avoision.com/experiments/pi10k to show students what pi “sounds like” as a musical sequence.

How do we find the circumference of any circle?

7 minutes

This will serve as the Closure for the lesson.  We go through these questions together as a class.   I want students to be able to explain that pi is the ratio of a circle’s circumference and its diameter.  I have students Think Pair Share about what they think the formula for circumference of a circle is.  Students are engaging in MP 8: Look for and express regularity in repeated reasoning.

Then I have students use their formula to answer 1 and 2.  For 1a, students can just put 8 times pi.  Rather than using 3.14 to calculate the circumference, I just want them to estimate.  If I have extra time, I give students problems with bigger (circumference of 150 cm) and smaller (diameter of 2.5 cm) measurements and have students make estimates of the diameter and circumference.

Ticket to Go and Homework

5 minutes

With the last few minutes of class I give a Ticket to Go for students to independently complete.  I pass out the HW Exploring Circumference at the end of class.