Area vs. Circumference

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Objective

SWBAT: • Identify and find the radius and diameter of a circle. • Estimate the area of a circle. • Find the area and circumference of a circle using a formula. • Determine whether to find circumference or area of a circle given a situation.

Big Idea

Make an estimate for the area of each circle. How can we calculate a more exact area? Students develop an estimate for a circle’s area and learn the formula. Then students must apply their knowledge to determine whether they calculate area or circumferen

Do Now

5 minutes

Note:

  • Common Core has students work on circles during 7th grade.  I teach in Massachusetts where 6th grade students are expected to identify parts of a circle as well as calculate circumference and area.

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.  Today I want students to start thinking about finding the area of a circle.  Some students may initially complain that they don’t know how to find the area of a circle.  This is okay!  I tell them that I want an estimate of how many square units the circle covers.  Some students my count the whole and partial squares inside of the circle.  Other students may draw a square around the circle and find its area.  Some students may estimate ¼ of ½ of the circle and then multiply their estimate.   

Rather than share out strategies, I call on students to share out their estimate for the area of the circle.  I record the estimates on the board.  Students will have a chance to further develop and share their strategies later in the lesson.

 

Review: Calculating Diameter and Radius

3 minutes

Before we continue working on area I want students to have a solid grasp of radius and diameter.  I encourage students to draw a circle for each problem.  A common mistake is that students confuse diameter with radius and vice versa.  We quickly review the answers before moving on.

Area of a Circle

15 minutes

Notes:

  • Before this lesson make copies of the circles with radius squares.
  • I give each student a copy of these circles and a pair of scissors.
  • I have glue sticks available if students like to glue their radius squares down.

I read the description of radius squares.  I describe the task and pass out materials.  I walk around and monitor student strategies and behavior.

 Strategy 1: Unit 7.5 Four Radius Squares.jpg

Some students may cover the circle with four squares and then cut off the corners of the squares.  This means that the circle takes up less than 4 squares, or a bit more than 3 squares. 

Strategy 2: Unit 7.5 Three Radius Squares A.jpg

Some students may use three squares to cover ¾ of the circle.  The next step is to cut off the corners of the squares and see how much space they can fill. 

Strategy 3: Unit 7.5 Three Radius Squares B.jpg

Other students may place three squares differently.  The next step is to cut off the corners fill in the empty space in the circle.

I want students to notice that each circle, despite its difference in size, requires a little bit more than three radius squares to cover it.  Students are engaging in MP5: Use appropriate tools strategically, MP7: Look for and make use of structure, and MP8: Look for and express regularity in repeated reasoning.  I strategically choose students to share these strategies under the document camera.  If one of these strategies does not show up in the class, I present it and ask students what they think.  Students are engaging in MP3: Construct viable arguments and critique the reasoning of others.

I label the radius in the blank circle at the bottom of the page.  I ask students, “How could we represent the area of this radius square?”  I want students to see that without any measurements the most specific we can get is r times r, or r squared.   I reveal the formula for area of a circle.  I ask students, “How does the formula connect to your work with radius squares?”  I want students to see the connection between their work with pi in the previous lesson with their work with radius squares and area today.  My hope is that by spending time having students estimate and find a pattern on their own they will be more likely to understand the formula and remember it.

Area or Circumference?

6 minutes

Note:

  • For this part of the lesson, I have a calculator available for each student.  I want students to focus reasoning about what to calculate, rather than spending a lot of time multiplying decimals.

I read over the directions.  Students work independently to read over the questions and identify if they are going to have to find the circumference or the area.  We come together and share out ideas.  I push students to comment on whether they agree or disagree with their classmates and why. 

I explain that to be practical we are going to use 3.14 to represent pi in our calculations.  I explain my reasoning for giving them calculators and I review my expectations for using them.  Students work in partners to solve problem a and b.  A common mistake is for students to use the incorrect measurement when calculating area or circumference.  Another common mistake for calculating area is to forget to square the radius.  If I see these mistakes I address them as a class.  These mistakes will also come up next in the Error Analysis problems.

I call on students to share out their answers.  I push students to include the appropriate units and explain why their answer makes sense.  Students are engaging in MP5: Use appropriate tools strategically and MP6: Attend to precision.

Error Analysis & Practice

13 minutes

Note:

Students start working independently.  I allow students to continue to use their calculators.  Students are engaging in MP5: Use appropriate tools strategically and MP6: Attend to precision.

As students work I walk around to monitor student progress and behavior.  If students are struggling, I may ask them one or more of the following questions:

  • What is going on in the problem?
  • Are you finding area or circumference?  How do you know?
  • How do you calculate the circumference/area?
  • What is the diameter/radius of the circle?  How do you know?

When students complete their work, they raise their hands.  I quickly scan their work.  If they are on track, I send them to check with the key.  If there are problems, I tell students what they need to revise.  If students successfully complete the chart they can work on the challenge questions. 

Closure and Ticket to Go

8 minutes

For the Closure, I tell students to flip to problem 4.  I ask students to share what we need to calculate for part a and part b.  I ask students, “How can we calculate the area and the circumference of this shape?”  Students participate in a Think Pair Share.  I want students to recognize that since the shape is ½ of a circle, they can find the circumference/area and then divide it by two.  Students share their answers.

If I have time, I have students revisit the do now question.   I tell students to use what they know now to calculate the area of the circle and compare it with their estimate at the beginning of the lesson. 

I pass out the Ticket to Go and the HW Area vs. Circumference.