1-Centimeter Grid Paper

Transcription

1 1-Centimeter Grid Paper TEKSING TOWARD STAAR 2014 Page 1

2 1 2 Centimeter Grid Paper TEKSING TOWARD STAAR 2014 Page 2

3 Student Activity 1 Materials: Centimeter grid paper, 1 2 centimeter grid paper, four circles that are the same and labeled A, B, C, D, E, F, G or H (whichever ones were given to your group), a safety compass, a centimeter ruler, scissors, one circular object per group of 4 (jar lid or coffee can) plus 3 extra circular objects, one measuring tape per group of 4 students. Problems: What is the approximate area of a circle using a model? What is the approximate circumference of a circle using a model? Part 1: Procedure: Work in groups of 4. Identify Student 1, Student 2, Student 3 and Student 4. Each member of your group needs one of the circles from the baggie given to your group, a safety compass, centimeter ruler, and scissors. Student 1 and Student 4 need centimeter grid paper and Student 2 and Student 3 need 1 2 centimeter grid paper. Working independently to find the length of the diameter and radius to the nearest tenth of a centimeter for your circle and record your measurements on the Circle Measurements Record. Be sure to record which circle you measure in the title of the table. Use a safety compass to inscribe a circle with the same measurements as your circle in a square on centimeter grid paper or the 1 2 centimeter grid paper. Use the grid paper to find an estimate for the area of the circle and record your estimate on the Circle Measurements Record. (Do not use any formulas to find the area.) After you have recorded your estimate, work with the other 3 people in your group to share the measurements everyone made independently and record on the Circle Measurements Record. Circle Measurements Record for Circle Student #1 #2 #3 #4 Group Data Diameter (in cm) Radius ( in cm) Area (in cm 2 ) Discuss the questions below as a group and record the answers. Then decide upon the group data. Record your decisions on the group data in the last row of the table. How did you find the diameter of your circle? How did you find the radius of your circle? Explain the method you used to determine the area of your circle. How could you get a better estimate of this area? How will you determine which data to report as the group data? TEKSING TOWARD STAAR 2014 Page 3

4 Your teacher will put a Circle Pi Projection Master on the overhead projector. Student #4 transfers the data from his/her table to the appropriate table on the transparency. Record the group data from the transparency in the table below, then answer the questions following the table. Group Data Circle Measurements Record Circle Diameter (in cm) Radius ( in cm) Area (in cm 2 ) A B C D E F G H What is the ratio of the diameter of circle A to the diameter of circle E? How does the ratio of the diameters compare to the ratio of the areas of circle A and E? What is the ratio of the area of circle G to the area of circle B? How does this compare to the ratio of the radius of circle G to the radius of circle B? Do you find a similar pattern between the radii (or diameters) of other circles and their areas? Explain. TEKSING TOWARD STAAR 2014 Page 4

5 Transfer the data from the Group Data Circle Measurements Record to graph the ordered pairs (radius, area) on the grid below. Title and label the graph. Answer the following questions: What do you observe about the shape of this graph? Use a pencil to extend the graph so that predictions can be made. Is it reasonable for this graph to contain the origin? Why or why not? Use the graph to: predict the area of a circle with a radius of 8 cm. find the radius of a circle with an area of 20 cm 2. 2 The Staar Formula Chart has the area of a circle formula to be A = r. How does your approximated area compare to the area you would get using the formula for your circle? Our approximated area was and the formula area would be using 3.14 for. Part 2: Round 1: Student 1 will select a circular object and a tape measure from the table where the teacher has placed the objects. The group needs to decide how you will determine the center of your circular object. Discuss how to do this effectively. Student 2 will measure the diameter using the tape measure or ruler. When each member agrees with the measurement, record it in the table below. Student 3 will measure the circumference of the circle using the tape measure. When each member agrees with the measurement, record it in the table below. Student 4 will calculate an approximate ratio of C : d. TEKSING TOWARD STAAR 2014 Page 5

6 Round 2: Student 4 will return the first circular object to the table and will select a second circular object from the table where the teacher has placed the objects. The group needs to decide how you will determine the center of your circular object. Discuss how to do this effectively. Student 3 will measure the diameter using the tape measure or ruler. When each member agrees with the measurement, record it in the table below. Student 2 will measure the circumference of the circle using the tape measure. When each member agrees with the measurement, record it in the table below. Student 1 will calculate an approximate ratio of C : d. Round 3: Student 3 will return the first circular object to the table and will select a second circular object from the table where the teacher has placed the objects. The group needs to decide how you will determine the center of your circular object. Discuss how to do this effectively. Student 4 will measure the diameter using the tape measure or ruler. When each member agrees with the measurement, record it in the table below. Student 1 will measure the circumference of the circle using the tape measure. When each member agrees with the measurement, record it in the table below. Student 2 will calculate an approximate ratio of C : d. Round 1 Round 2 Round 3 Diameter Circumference Ratio of C : d Do you see any relationship between the diameter and the circumference of a circle from comparing the ratios in the last column of the table? This ratio should be close to 3. Is your group s ratio close to 3 for each circular object? This ratio is a constant and is called (pi). Pi is an irrational number and is often approximated with a rational number of 3.14 or Even 3 can be used for a less accurate approximation. The STAAR formula chart has the formula for the circumference of a circle to be C = d. Do your measurements support this formula? Explain. Using the formula, what would be the circumference of a circle with a diameter of 22 inches? TEKSING TOWARD STAAR 2014 Page 6

7 Student Activity 2 Work with your partner to answer the following questions. Problem 1: Write three formulas that can be used working problems with circles. Problem 2: Look at the circle M below. M N M is the of the circle. Segment MN is a of the circle. If MN = 8 units, the diameter of the circle will be units. If MN = 10 units, the circumference of the circle will be units or about units. If MN = 5 units, the area of the circle will be square units or about square units. Problem 3: Find the circumference and area of a circle with the given diameter. Find them in terms of and approximated to the nearest hundredth if needed. Show the formula you used and the work you did. 8 units 12 units TEKSING TOWARD STAAR 2014 Page 7

8 Problem 4: Decide if the following statements are true or false. T or F 1. The ratio of the circumference to the diameter of a circle is approximately The diameter of a circle is twice the radius of a circle. 3. The circumference of a circle is about 3 times a radius of the circle. 4. To find the area of a circle, square the value of the radius and add to the value. Problem 5: Find the value of the following numbers when they are squared Problem 6: Determine what number was squared to give the value below. 16 is the square of. 49 is the square of. 64 is the square of. Problem 7: The area of a circle is 100 square inches. What is the radius of the circle? 2 The area of the circle 100 which is the value of r. If 100 = r 2, the divides off each side. This gives us the equation 100 =. What number is 100 the square of? The radius of the circle is inches. Problem 8: Find the circumference of a circle with a radius of 12 inches. Give a answer and an approximated answer. Show your work. Problem 9: A circle dining table has a diameter of 5 feet. What is the area of the circle in square feet? Show your work. TEKSING TOWARD STAAR 2014 Page 8

9 Problem 10: Benji drew two circles. The smaller circle had a radius of 5 inches. The larger circle had a radius of 9 inches. Show your work. What is the difference in the circles circumferences? What is the difference in the circles areas? Problem 11: A semicircle has half of a circumference and a diameter for its boundary. Draw a sketch of a semicircle. How would you find the area of a semicircle? If a semicircle has a diameter of 12 units, what would be the area of the semicircle? Problem 12: Look at the figure below. The larger circle has a diameter of 8 feet. The two smaller circles have radii of 5 feet. What is the total area of the figure? Show your work. TEKSING TOWARD STAAR 2014 Page 9

10 NAME DATE SCORE /5 7.8C/7.5B/7.9B Skills and Concepts Homework 1 1. What is the ratio of the circumference of a circle to the diameter of the circle? What is the ratio of the circumference of a circle to the radius of the circle? 2. Draw a circle and label its radius, diameter, and circumference. 3. Give the value of the diameter of a circle if the radius is: 15 inches. 20 inches. 4. Give the value or approximated value of the diameter of a circle if its circumference is: 38 centimeters. 38 centimeters. 5. Explain how to find the area of a circle. Using your ruler measure the radius of the circle to the nearest centimeter. The radius is centimeters. The formula for the area is. The area is square centimeters. TEKSING TOWARD STAAR 2014 Page 10

11 NAME DATE SCORE /5 7.8C/7.5B/7.9B Skills and Concepts Homework 2 1. Find the area of a circle with a diameter of 14 inches. Show your work. Use 3.14 for. 2. Find the circumference of a circle with a radius of 12 feet. Show your work. Use 3.14 for. 3. Which of the following statements are true about a circle with a radius of 9 inches? Write T or NT. 1. The diameter of the circle will be 18 inches. 2. The circumference of the circle will be 18 inches. 3. The area of the circle will be more than 1,000 square inches. 4. Find the circumference of a cylinder who base is a circle with a radius of 20 inches. Show your work. Use 3.14 for. 5. Mary cut congruent circles with a 5 centimeter radius to decorate a bulletin board in her classroom. She uses 20 circles for the board. How many square centimeters of paper will she need for the decoration? TEKSING TOWARD STAAR 2014 Page 11