• Checkpoint 1: Evaluate understanding of parts of a circle before moving to circle calculations.
• Checkpoint 2: Evaluate students’ ability to understand and use circle formulas to calculate the circumference and area of a circle prior to real-world applications.
• Evaluate student understanding during work time and presentations of the Partner Circles Practice and Circle Application Cards activities.
• Exit Ticket Lesson 1 is a direct and individual assessment of student understanding to be used at the end of the lesson.

Note: Your discretion should be used in deciding which vocabulary words are necessary to teach and which may not be necessary.

Before students arrive, draw a large circle on the board or chart paper, or use the circle diagram transparency (M-7-6-1_Circle Diagram.docx). Cut apart and place the following circle terms in a container: arc (major), arc (minor), center, central angle, chord, circle, diameter, radius, secant, sector, semi-circle, tangent (M-7-6-1_Circle Terms.docx). Also place the vocabulary terms area and perimeter on the front board. Begin class by asking students to share what they know already about circles. Give each student two or three vocabulary journal pages to use throughout the unit (M-7-6_Vocabulary Journal.docx). [IS.4 - Struggling Learners]

“Raise your hand if you can describe the word circle in your own words to the class.” Call on one or more students to share their ideas. Emphasize correct concepts and clarify misconceptions.

“If we combine the ideas we just shared we can conclude that the definition of a circle is the set of all points the same distance from a specific point, which is the center of the circle. Today we will look at many aspects of circles, beginning with the definition of its parts and moving toward calculations and problem solving.”

Parts of a Circle Activity

Hand out one circle diagram to each student (M-7-6-1_Circle Diagram.docx). Place students in small groups of three or four. Provide each group with a bag or envelope of circle terms (M-7-6-1_Circle Terms.docx).

“I am going to give you a few minutes to locate and place the term labels for several parts of a circle onto your circle diagram with your group. Place the terms you find in your envelope on just one of the circles in your group. If you are able to finish placing all of the terms before the time is up, raise your hand to have me check your work. If you have trouble with some terms, you can get help in 3 or 4 minutes when we discuss all of the terms.”

Walk around the room observing which terms students are having difficulty with to help guide the class discussion. Separate the set of terms in your container into two piles, those most students know, and those most did not know. Place those that most students know back into the container.

After 3 to 4 minutes, call the student’s attention back to the front of the room.

“It seems that most of the groups were familiar with the terms _____, ______, ______,… already (name the terms you noticed while walking around the room).

“Let’s quickly review those. Each of you should write these labels on your own circle as we go, including the symbols used to represent it, and any notes that might help you remember it.” [IS.5 - Struggling Learners]

Draw a term out of your container. “I would like someone to explain this term to the class.” Either randomly select a person or group to describe the term, or allow students to volunteer.

Continue with this process until you have reviewed the terms students already know. Add details to descriptions that are not clear or complete. Also indicate the symbols used to represent each part of a circle. Once students are done with the terms in the container, discuss the remaining terms using the Circle Terms Reference sheet (M-7-6-1_Circle Terms Reference.docx).

“Are there any questions on the names and symbols we use to describe the parts of a circle?” Address any questions that arise.

Checkpoint 1: Cover any display you have which shows the circle term names and symbols. Draw a circle on the board or chart paper. One at a time, add the parts of a circle. Randomly select students to state the name and symbol used to represent that portion of the circle before adding the next part. Clarify any errors as you proceed. You could also draw a term out of your container and have a randomly selected student come to the board and draw that part of the circle and label it.

“Before we move on to the next part of our lesson, I want you to notice the last two circle terms that I have listed on the board right here (point to circumference and area). The circumference of the circle is the distance around the circle. The area is the space inside of the circle measured in square units. We are going to learn how to use these measurements to answer many types of problems later in this lesson. We need to start by learning about a very special value known as pi.”

Calculating Circumference and Area [IS.6 - Struggling Learners]

“Can someone please review for the class how we can find the circumference of a circle?” (Use the formula C = π • d.)

“So, to get the circumference, we take p and multiply by the diameter. Remember, p is just a number; it is about 3.14.”

Note: An optional activity may be used here to show students where pi comes from (M-7-6-1_Optional Activity—Discovering Pi and Circumference.docx).

Optional example: Ask, “When the diameter of a circle is 7 inches, what is the circumference of the circle? To get circumference, we multiply diameter by 3.14, so take 7 and multiply by 3.14. We get 21.98 inches.”

“How could we use this formula to find the length of the diameter if we know the circumference?

C = π • d

Let us get d alone, to solve for diameter. We use the inverse operation to eliminate the p.

So, we will divide each side of the equation by p, and we get:

C/π = d

Dividing the circumference by 3.14 will give you the diameter.”

Optional example: “When the circumference of a circle is 65 cm, what is the diameter of the circle? To find the diameter, we take the circumference and divide by 3.14. So let us find 65 divided by 3.14. We end up with a diameter of 20.70 cm.”

“The circumference means the distance around the circle. Sometimes we need to measure the space inside of a circle, the area. We use pi to calculate the area in square units. The formula is similar to the circumference formula, A = π • r². Notice that we need to use the radius for this formula instead of the diameter.”

“How can we find the radius if we only know the diameter?” (diameter divided by 2)

Optional example: “When the diameter of a circle is 12 feet, what is the radius of the circle? We take the diameter and divide by 2 which gives us a radius of 6 feet.”

“What if we only know the circumference, can we still find the radius to use it for the area?” (circumference divided by 3.14 will give us the diameter, which we can divide by 2 to get the radius)

“You will do some problem solving like this with a partner in a few minutes. Let’s do a few simple examples together first.” Practice finding the circumference and area of two or three example circles.

Example answers:

1. r = 5 cm                              2.   d = 3.2 in.                         3.   C = 40.82 cm

C = 31.4 cm                            C = 10.048 in.                         d = 13 cm, r = 6.5 cm

A = 78.5 cm²                           A = 8.0384 in.²                        A = 132.665 cm²

In pairs, students will complete the Partner Circles Practice (M-7-6-1_Partner Circles Practice and KEY.docx). Walk around the room assisting students. Let them know which place value you would like them to use for rounding their responses. Rounding to the nearest hundredth is suggested. Go over results (M-7-6-1_Partner Circles Practice and KEY.docx).

Checkpoint 2: Select students randomly to answer area and circumference questions, or have students answer them in the Think-Pair-Share format to check for understanding. You could use questions similar to those found in the Partner Circles Practice activity.

Checkpoint 2 Answers:

1. r = 8 in.                             2.   d = 9 cm                            3.   C = 109.9 ft

C = 50.24 in.                           C = 28.26 cm                          d = 35 ft, r = 17.5 ft

A =200.96 in.²                         A = 63.585 cm²                       A = 961.625 ft²

Answer student questions and correct any errors in students’ thinking.

Circle Applications Activity and Presentations

Ask students to share ideas on when area and circumference of a circle are needed in real life. Record several ideas on chart paper. “The last part of our lesson is to apply all that you have learned about circles to real-world problems. Hand out the problem cards (M-7-6-1_Circle Application Cards and KEY.doc), chart paper, and markers. Each group should get one to three of the problems depending on your time frame.

“You will work with ______ other people. You will have _______ minutes to work. Be prepared to explain your strategies and solutions with the class at that time. You may use a calculator, but show your work. Please round your answers to the nearest _______.”

As groups present, encourage them to correct any errors in logic or calculations. Also encourage the student audience to use the presented strategies and solutions to reflect on their own work. Allow students to revise or add to their work.

Additional, optional application problem:

Use the figure below to answer the questions.

1. What is the radius of the circle? (9 in.)
2. What is the area of the circle? (81π in. ≈ 254.34 in.²)
3. What is the area of the square? (324 in.²)
4. What is the area of the shaded parts? (69.66 in.²)

At the end of the lesson, have each student complete an Exit Ticket to evaluate his/her level of understanding (M-7-6-1_Exit Ticket Lesson 1 and KEY.docx).

Extension:

• Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. During this lesson, the following terms should be entered into students’ Vocabulary Journals: area, central angle, chord, circle, circumference, diameter, major arc, minor arc, pi, radius, sector. Keep a supply of Vocabulary Journal pages on hand so students can add pages as needed. Bring up instances of area and circumference as seen throughout the school year. Ask students to bring up circle calculation examples that they see outside of class and discuss the use and meaning in each particular context. Distinguish the difference in labeling lengths such as circumference with standard units and areas with square units as they are used throughout the year. Always require students to use appropriate labeling in both verbal and written responses.
• Students who might need an opportunity for additional learning can participate in small groups during an Area Review Activity. Use this activity for the entire class or for small groups of students who might need additional practice with finding the circumference and area of a circle.
• If a computer is available, visit Web site https://www.youtube.com/watch?v=lWDha0wqbcI&feature=related

This is a 1½ minute song and video clip that reviews the basic parts of a circle and the formulas for circumference and area. If time permits, play it a second time. For visual and musical learners, this clip may help them remember these facts more easily.

• Review with students:
  • radius is half the diameter.
  • diameter is 2 times the radius.
  • practice calculating the circumference.
  • practice calculating the area.

Next, if computers are available, have students go to any or all of these sites to get additional practice with circle terms and calculations with immediate feedback:

• Interactive practice for parts of a circle

http://www.ixl.com/math/practice/grade-7-parts-of-a-circle

• Interactive circle word problems

http://www.ixl.com/math/practice/grade-7-circle-word-problems

• Interactive circumference and area questions

http://www.ixl.com/math/practice/grade-7-circles-calculate-area-circumference-radius-and-diameter

If computers are not available, use index cards to write down 5–10 circle vocabulary, circumference, and area questions (involving whole number values for radius and diameter). Give one card to each student in the small group. Have students record their work and answer on a piece of paper, then rotate cards. Repeat these steps. The number of practice cards students complete will depend on the amount of time available and the students’ proficiency. Students who finish before the group is ready to rotate the cards can get an extra card from you. Work one-on-one with students who are having difficulty using the formulas to calculate.

• Station: Exploring Patterns of Change: Allow students to explore how the circumference and area of a variety of circles are related as the radius changes. Have students select a radius for a circle and calculate the diameter, area and circumference for that circle. Next, they should double the radius and again calculate the diameter, circumference, and area. Repeat these steps again for tripling and cutting the radius in half. Ask students to make observations of the relations between the scale used to increase or decrease the radius and the corresponding increase or decrease in the diameter, circumference, and radius. The pattern of change they should discover is that diameter and circumference increase or decrease by the same scale factor as the radius, while the area will increase by the square of that scale factor.
• Technology: Create Your Own Applications Activity: Students may work individually or in groups. Provide paper, pencils, markers, and computer access. Students will write a minimum of five real-world problems involving the area and circumference of circles. Allow use of the computer or media center to research circle topics and circle measures in real-life situations. Students will turn in a product format of their choice using their questions, such as a worksheet, quiz, or trivia game. Students should provide an answer key as well. To assist with making initial calculations or checking calculations allow students to use a site like http://www.calculatorsoup.com/calculators/geometry-plane/circle.php. This Web site allows students to:
  • enter radius to get both area and circumference.
  • enter area to get both circumference and radius.
  • enter circumference to get both area and radius.

If time permits, have students or groups exchange end products (game, quiz, or worksheet). Have the student or group who created the product answer questions the user may have and score the results using the answer key they have created. Provide students time to engage in a discussion about the reasoning, errors, and other possible ways of calculating the solutions after they have exchanged and graded each other’s work.

• Expansion: Dividing It Up!: Tell students, “You will need to use some additional problem-solving strategies for this activity. One new strategy you will need to consider is finding the area of a sector. Recall that a sector is a part of the inner area of a circle. It is bounded by two radii and an arc. Use the central angle to determine what fractional part of the area is in the sector. Also consider the length of the arc on the rim of the circle. Similar to the area of the sector, the arc will be a fractional part of the circumference proportional to the central angle size. Can anyone describe how we might get started on a problem involving arc length or the area of a sector?” (To find the area of a sector, start by calculating the area of the entire circle, or find the circumference for an arc problem. Then, if you know the central angle for the sector you can calculate the fraction or percent of the full circle you are working wit based on the full 360˚. Multiply the entire area by this fraction or percent to get the area of the required sector. Do the same for the fractional part of the circumference for the arc.)

Provide students with a copy of the Dividing It Up! worksheet (M-7-6-1_Dividing It Up.docx and M-7-6-1_Dividing It Up KEY.docx). Allow students to work individually or in small groups.

If time permits, have students write one or more real-world problems of their own that require arc and/or sector calculations. Ask them to include labeled illustrations and instructions.