Unit 9 Cycle 1 Circle and Angle Geometry

Transcription

1 Vocabulary Unit 9 Cycle 1 Circle and Angle Geometry Lesson Pi and the Parts of Circle Circle Circumference Radius Chord Diameter Pi Active Instruction ta(1) Trace circular objects or use a compass to make circles of different sizes. Then use a string and a ruler to measure the diameter, d, and the circumference, C, for each circle. th(1) What is the diameter of Circle A? Team Mastery (4) Kenyatta draws a circle for an art project. She uses string to measure the diameter and circumference of her circle: diameter = 4.7 in circumference = 15 in Do you think Kenyatta measured the circle correctly? Explain your thinking. Lesson Quick Look Team Name Team Complete? Team Did Not Agree On Questions Write the vocabulary introduced in this lesson: Today we learned about circles. Knowing the vocabulary that goes with circles makes it a lot easier to talk about circles. We also learned about the relationships between the parts of the circle! The diameter and radius are related. We can see on the circle above that the radius goes from the center to the edge of the circle. We can also see that the diameter is a straight line that goes from a point on the edge of the circle to another point on the edge of the circle AND goes through the center of the circle. When a line goes from point to point without passing through the center, it is a chord. This means that the diameter is twice the length of the radius. So if our radius measured 4 inches, then the diameter would measure 8 inches. We also started to learn about pi. We use the symbol,, to represent pi. We find pi by the ratio of the circumference of a circle to the diameter of the circle: Pi = This ratio always has the same value and today we learned that that value is a little more than 3. So if a circle has a diameter of 8 inches, then the circumference will be a little more than 8 inches 3 = 24 inches! 1 P a g e

2 Lesson Homework 1) Marco uses string to measure the diameter and circumference of a circle: diameter = 230 mm and the circumference = 1,150 mm. Do you think he measured the circle correctly? Why or why not? 2) Terry measured the diameter, d, and the circumference, C, of a circular plate using a string. Which of the following measures most likely represents the measures he took? Circle 1 Circle 2 Circle 3 C = in. C = 36.4 in. C = 4.1 in. d = 8.25 in. d = 9.1 in. d = 12.8 in. 3) Dante measured the diameter of a traffic circle to be 16.8 yards. What is the measure of the radius of Dante s circle? 4) What does the 1 represent on Circle S? Mixed Review 7) Which of the following are proportional to the ratio 40 : 160? a. 1 to 4 c b. 22:88 d. 16:64 8) Simplify the expression. y y y 9) Find the difference. 10) Evaluate. (5.6 2) + [( ) (4 2) 4 ] 5) Kylie draws a circle for part of his art project. He used string to measure the diameter and circumference of the circle: diameter = 1.7 inches and the circumference = 5.3 inches. Do you think Kyle s measured the circle correctly? Why or why not? 6) The directions say to draw a circle with a radius of 8.6 inches. What will be the diameter? 2 P a g e

3 Unit 9 Cycle 1 Circle and Angle Geometry Lesson Circumference Active Instruction th(1) When you use accurate tools to measure the circumference and diameter, you can get a more accurate value for pi. A mathematician used a more accurate tool, computer software, to measure a circle. She got the following measures: Circumference, C = cm Diameter, d = 6 cm Use the mathematician's measures and your toolkit to get a more accurate value for pi. ta(1) Find the circumference of each circle using the value Round your answer to the nearest hundredth. (2) Ann's Bakery made a cake. The diameter of the circular bottom layer of the cake is 18.9 inches. What is the circumference of the bottom layer of the cake? th(1) The wheel of a school bus has a radius of inches. How far does the bus travel each time the wheel rotates completely? Show your work. Use the value 3.14 for and round to the nearest hundredth. Team Mastery (3) What is the radius of Circle P? Show your work. Use the value 3.14 for and round to the nearest hundredth. (1) Lesson Quick Look Team Name Team Complete? Team Did Not Agree On Questions Write the vocabulary introduced in this lesson: Today you learned more about the circumference of a circle and the constant pi! Yesterday you learned that the ratio of the circumference to the diameter of a circle is about 3. Today you learned that that ratio, pi, is always the same value and that we approximate that value with the decimal To find the circumference of a circle given the diameter, d, use: C = d For example, a circle has a diameter of 15 centimeters. To find the circumference: C = cm C = 47.1 cm To find the circumference of a circle given the radius, r, use: C = 2 r For example, a circle has a radius of 4 feet. To figure out the circumference: C = ft C = ft You can also use the formulas for circumference to find the missing diameter or radius when we already know the circumference. You just plug in the circumference to the formula and solve for the missing radius or diameter! 3 P a g e

4 Lesson Homework Directions for questions 1 6: Show your work. Use the value 3.14 for and round to the nearest hundredth. 1) Marilyn is decorating a coffee can, the diameter of the can is 16.5 centimeters. She wants to trim the top edge of the coffee can with lace. How much lace does she need? 2) California s giant sequoias (Sequoiadendron giganteum) are among the world s largest trees. One giant sequoia has a circumference of 113 feet. What is its diameter? 3) Mrs. Trainer measured the circular top of her dining table and found it was 42 inches in diameter. What is the circumference of the table? 8) Of the 36 students who tried out for the team, 15 were accepted. What percent of those who tried out made the team? 9) A factory worker can assemble boards in 3 4 dozen circuit hour. Write a unit rate that describes how many circuit boards the factory worker can make in an hour. 10) A small aircraft propeller has a length of 86 inches. What distance does the tip of a propeller blade travel in one rotation of the propeller? Show your work and explain your thinking. 4) A circle has a circumference of cm. What is its radius? 5) What is the diameter of Circle H? 6) If the radius of a circle is 75.8 miles, what is the circumference? Mixed Review 7) Evaluate the expression. Show your work. ( ) ( 5 + 8) 3 4 P a g e

5 Unit 9 Cycle 1 Circle and Angle Geometry Lesson Area of Circles Active Instruction ta(1) Area is a measurement of how many square units fit in a given space. About what is the area of this circle? 3.14 for and round your answer to the nearest hundredth. Team Mastery (4) What is the area of Circle D? Show your work. Use 3.14 for and round your answer to the nearest hundredth. th(1) A giant telescope is being constructed. The telescope will use seven large circular mirrors to collect a lot of light to make better images. Each mirror is 8.4 meters in diameter. What is the area of each mirror? What is the total area of all seven mirrors? Show your work. Use Lesson Quick Look Team Name Team Complete? Team Did Not Agree On Questions Write the vocabulary introduced in this lesson: Today we learned how to find the area of a circle using the formula A = _r2. We found the area of various circles when given the radius or the diameter. We also recalled that the units for the area of any shape are always in units squared. To find the area of the circle: A = r 2 A = 3.14 (6.3 in.) 2 A = in. 2 Now that you know the formulas for circumference and area, you might get a little mixed up about which one is which. Here s a simple mnemonic, or memory device, to help you remember: Cherry pies delicious! (C = d) Apple pies are too! (A = r 2 ) 5 P a g e

6 Lesson Homework Directions for questions 1 6: Show your work. Use 3.14 for and round your answer to the nearest hundredth. 5) What is the area of Circle S? 1) What is the area of Circle B? 2) Cole is making a chicken and spinach pizza for his friends. The pizza has a diameter of centimeters. What is the area of the pizza? 3) A circle has a radius of 24 feet. What is the area of the circle? 4) Sun drew a design for a campaign button. She needs to order the correct size button to make. What is the area of the button Sun is ordering? 6) A round pencil holder has a diameter of 1.25 inches. A typical pencil has a diameter of 0.25 inch. About how many pencils will fit in the round pencil holder? Mixed Review 7) A florist shop sold 162 flower arrangements in the month of June. In July the shop sold 180 arrangements. What is the percent of change in sales from June to July? 8) What is the perimeter of a rectangular room that measures 12.7 feet long and 10.6 feet wide? 9) Find the difference. 10) Evaluate the expression. Show your work. (15.2 5) + [( ) (6) 3 ] 6 P a g e

7 Active Instruction ta(1) The community center is putting a new circular whirlpool in for its members. The allotted space for the whirlpool is 15 feet long and 12 feet wide. The community center wants at least 1 foot of space around the whirlpool. (1) If 1 person takes up about 2.5 feet of space to sit comfortable, about how many people can sit around the circumference of the whirlpool? (2) During the winter months, the community center wants to cover the whirlpool with a tarp. The local hardware store will make tarps that are 75 square feet, 100 square feet, or 150 square feet. Which should the community center get? th(1) Delia and her sister get on an observation wheel ride at the state fair. The seats are 34 feet from the center of the wheel. Lesson Quick Look Team Name Unit 9 Cycle 1 Circle and Angle Geometry Lesson Problem Solving with Circles a) How many feet will Delia and her sister travel in 10 revolutions of the wheel ride? b) How many revolutions would Delia and her sister make if they traveled 1 mile on the wheel ride? c) If the wheel ride goes around at a constant speed and it takes 3.5 minutes for 1 revolution, at what rate of speed are Delia and her sister traveling in feet per second? Team Mastery (3) A pet owner is building a circular puppy play yard around a tree for shade. The circumference of the base of the tree is 2.42 feet. The pet owner has purchased 100 feet of wire fencing. What is the area of the largest circular play yard the pet owner can make? Show your work. Use 3.14 for and round your answer to the nearest hundredth. Team Complete? Team Did Not Agree On Questions Write the vocabulary introduced in this lesson: Today we used area and circumference of circles to answer multiple step and real-world problems. If 1 person takes up about 2.5 feet of space to sit comfortably in a whirlpool hot tub, about how many people can sit around the circumference of the whirlpool? C = d C = C = 31.4 feet 31.4 feet 2.5 feet per person = 12.56, or 12 people can sit around the circumference of the whirlpool. 7 P a g e

8 Directions for questions 1 and 2: Use 3.14 for round your answer to the nearest hundredth. Lesson Homework and 1) A skateboard store is selling skateboard wheels with different diameters: small wheels for flip tricks, medium wheels for ramps, and larger wheels for cruising. The largest wheels have a diameter of 60 millimeters. If a skateboard with the largest wheels travels 4 meters, about how many rotations have the wheels made? Show your work. Explain your thinking. 2) Marta set up her checker board to play a game with her cousin Finn. Mixed Review 3) Find the sum ) Find the quotient ) The starting principal for a savings account is $300. If there are no additional deposits or withdrawals, at a simple interest rate of 3.5%, what will be the account balance after 5 years? 6) Lily, Louis, and Ben went to an arcade and won tokens for prizes. Louis won 4.5 times as many as Ben. Lily won 8 more tokens than Louis. Write an expression that describes how many tokens Lily has won. Be sure to define any variables you use. The checkerboard is a square with a width of 40 centimeters. Each checker playing piece has a circumference that is centimeters. a. When the checkerboard is first set up and no moves have been made yet, what is the area of the board that is NOT covered by a checkers? b. When the checkerboard is not covered with any checkers, what area of the checkerboard is black? What area is gray? c. When the checkers are in place on the board (before the game has started), what area of black is showing on the checkerboard? What area of gray is showing? 8 P a g e

9 Vocabulary Unit 9 Cycle 2 Circle and Angle Geometry Lesson Identify Types of Angles angle acute angle obtuse angle right angle straight angle reflex angle Active Instruction ta(1) How can we describe these angles? What is wrong with her thinking? ta(2) How can we describe these angles? ta(3) Draw an angle that measures 270. th(1) Kendal said that is a reflex angle. Team Mastery (5) Which of these angles is an obtuse angle? a) ABC measuring 83 b) XYZ measuring c) 4 measuring 180 d) Q measuring 90 e) JKL measuring f) 2 measuring 270 Lesson Quick Look Team Name Team Complete? Team Did Not Agree On Questions Write the vocabulary introduced in this lesson: Today we drew and identified angles by their measures and by how they relate to 90, 180, and 360. Here is a chart that summarizes the angles. Angle type Characteristics Right angle is exactly 90 Acute angle is greater than 0 and less than 90 Obtuse angle is greater than 90 and less than 180 Straight angle is exactly 180 Reflex angle is greater than 180 and less than 360 We measure and draw angles with a protractor. The two lines that make up an angle have a common point, or vertex. Angles are often named by the three points that make up the angle. 9 P a g e

10 1) 1) What type of angle is JKL? What is its measure? Lesson Homework 5) What type of angle is XYZ? What is the measure of XYZ? 2) t measures What is the measure of t s reflex angle? 3) Which of the following is a straight angle? a. 3 measures b. 6) 5 s reflex angle measures 250. What is the measure of 5? What type of angle is 5? Explain your thinking. Mixed Review 7) If the diameter of a circle is 7.4 inches, then what is the circumference? 8) Divide. c. d measures 90 d. G measures 180 4) Draw BCE which measures 245. What type of angle have you drawn? 9) Evaluate the expression when a = a 4.7a + (a + 6.9) 10 P a g e

11 Unit 9 Cycle 2 Circle and Angle Geometry Lesson Complementary and Supplementary Angles Vocabulary Complementary angles Supplementary angles Active Instruction ta(1) What is the measure of DEF? th(1) What is the measure of DBE? Explain your thinking. ta(2) What is the measure of ABD? Team Mastery (3) What is the measure of x? Explain your thinking. Lesson Quick Look Team Name Team Complete? Team Did Not Agree On Questions Write the vocabulary introduced in this lesson: Today you learned to solve problems involving complementary and supplementary angles. Complementary angles are two angles whose measures equal 90. Supplementary angles are two angles whose measures equal 180. If you know the value of one of the angles, you can find the unknown measure by subtraction. 1 and 2 are complementary angles. If 1 measures 29, then 2 measures 61! 3 and 4 are supplementary angles. If 3 measures 76, then 4 measures 104. What are some pairs of angles are complementary? How about supplementary? 11 P a g e

12 1) CDE measures 169. What is the measure of an angle that is supplementary to CDE? 2) What is the measure of 1? Explain your thinking. Lesson Homework Mixed Review 5) Which of the following is an acute angle? a. An angle that measures 90 b. An angle that measures 243 c. d. 6) A circle has a diameter that measures 8.2 inches. What is the area of the circle? 3) What is the measure of x? 7) Find the difference. 8) 34 is what percent of 57? 4) If two angles are complementary and the first angle equals 46.5, then what is the measure of the second angle? 9) An architect designs a porch support that forms a 42.5 angle with the root. What is the measure of the angle s complement? What is the measure of its supplement? 12 P a g e

13 Vocabulary Adjacent angles Active Instruction Unit 9 Cycle 2 Circle and Angle Geometry Lesson Adjacent and Vertical Angles Vertical angles ta(1) What names can we use to describe ABC, CBD, and ABD? a) Name two angles that are adjacent. b) Name two angles that are vertical. ta(2) What is the value of y in the image? Team Mastery (4) Describe ABD and DBC as vertical or adjacent. What is the measure of DBC? th (1) Use the diagram to answer each question. Lesson Quick Look Team Name Team Complete? Team Did Not Agree On Questions Write the vocabulary introduced in this lesson: Today you answered questions and solved problems involving adjacent and vertical angles. Angle type Adjacent angles Vertical angles Characteristics Share a vertex and one side Share a vertex; are formed from the same two lines; always have the same measure In this image we see examples of both adjacent and vertical angles. One pair of adjacent angles is COB and BOD. Can you find any other pairs of adjacent angles? One pair of vertical angles is AOC and BOD. Vertical angles are equal, so if AOC measures 55.35, then BOD has the same measure! 13 P a g e

14 1) ABC measures DEF is vertical to ABC. What is the measure of DEF? Lesson Homework 7) An angle measures 177 ; what is the measure of its reflex angle? 2) Describe 1 and 2 as adjacent angles are vertical angles. What is the measure of 2? 8) Kyle is covering the square surface of his driveway with concrete. The length of his driveway is yards. What is the area of the driveway that needs to be covered with concrete? 9) Use the map of Gianni s neighborhood to help you answer the question. 3) Draw a pair of angles that are both adjacent and supplementary. 4) What is the measure of BOD? Mixed Review 5) Find the product. Gianni told his friend he was going to take a walk on the streets that formed a vertical angle to EBA. Name the angle on the streets he will be walking. 6) Simplify the expression. 2x (x 2) + x P a g e

15 Unit 9 Cycle 2 Circle and Angle Geometry Lesson Problem Solve with Angles Active Instruction ta(1) What is the value of x? a) What is the measure of x? How could you describe the relationship between x and the 45 angle? b) What is the measure of y? c) What is the measure of z? Team Mastery (3) Use the diagram to answer parts a, b, and c. th (1) Keisha drew a pattern for a mosaic she wants to make. Now Keisha needs to figure out the angle measure for the pieces so she knows how to cut the pieces of tile for the mosaic. b) Natasha says that x and y are supplementary angles; Vladimir says they are adjacent angles. Who is correct? Explain our thinking. Lesson Quick Look Team Name Team Complete? Team Did Not Agree On Questions Write the vocabulary introduced in this lesson: Today you used your knowledge of angles to find the missing measure(s) of angle(s). You used information about the types of angles based on their measures: right, acute, obtuse, straight, and reflex. You also used your knowledge of how angles compare to each other. Remember that the measures of complementary angles add to 90, and the measures of supplementary angles add to 180. We also used vertical angles and adjacent angles to help us find missing measures. Here is an example! Mrs. Johnson cut an apple pie. Mr. Johnson, not being very hungry, took slice b from the pie. The angle of slice b measures 15. His son was much more hungry and took slice a. What is the measure of slice a? 15 P a g e

16 We know that the angles on slice a and slice b are complementary angles because the form a right angle. So to find the angle on slice a we subtract the measure of the angle on slice b from = 75, so the angle of slice a measures 75 1) Use the clues to draw the shape described. Angles a and b are supplementary and adjacent Angles a and c are vertical angles Angles a and z are adjacent angles 2) Wanda is building a picnic table for her yard. Here is her design: a. If ABC is 72, then what is the measure of CBG? b. If JCH measures 108, then what is the measure of BCH? c. Corey says that ABC and JCH are supplementary angles. Laura says that they are adjacent angles. Who is correct? Explain your thinking. Lesson Homework Mixed Review 3) The diameter of an apple pie is 9.4 inches. What is its circumference? 4) What is 25% of the area of the apple pie in question 3? 5) Find the quotient ) A 32-ounce bottle of sports drink costs $7.99. What is the unit price of the sports drink? 16 P a g e