Objectives To introduce how to calculate degree measures of sectors; and to use a protractor to draw circle graphs. 1 materials Teaching the Lesson Key Activities Students use fractions, decimals, and percents to calculate the degree measures of sectors in a circle graph. They use a protractor to draw each sector. Math Journal 1, pp. 169–171 Student Reference Book, pp. 59, 60, and 147 Study Link 5 2 Key Concepts and Skills • Apply place-value concepts to round decimals to the nearest whole number. [Number and Numeration Goal 1] • Recognize and use equivalent names for fractions, decimals, and percents. [Number and Numeration Goal 5] • Use appropriate strategies to construct circle graphs. [Data and Chance Goal 1] • Draw and measure angles to the nearest degree. [Measurement and Reference Frames Goal 1] Transparencies (Math Masters, pp. 152 and 153; optional) compass Geometry Template/protractor calculator scissors tape board compass and protractor (for demonstration purposes) Key Vocabulary sector Ongoing Assessment: Recognizing Student Achievement Use journal page 169. [Geometry Goal 1] 2 Ongoing Learning & Practice Students calculate percents of numbers to solve sale-price problems. Students practice and maintain skills through Math Boxes and Study Link activities. materials Math Journal 1, pp. 172 and 173 Study Link Master (Math Masters, p. 154) Geometry Template/protractor calculator 3 materials Differentiation Options READINESS Students use a diagram of the full-circle protractor to find fractional parts of a circle and their degree equivalencies. ENRICHMENT Students use diagonals and apply properties of triangles to find the sums of angle measures in polygons. Student Reference Book, p. 233 Teaching Masters (Math Masters, pp. 155 and 156) color pencils or markers Technology Assessment Management System Math Message See the iTLG. 346 Unit 5 Geometry: Congruence, Constructions, and Parallel Lines Getting Started Mental Math and Reflexes Math Message Complete the Math Message on journal page 169. Students use a calculator to solve percent problems. Suggestions: 28% of 18 5.04 12.5% of 62 7.75 3% of 42 1.26 7.25% of 52 3.77 Study Link 5 2 Follow-Up Ask students to explain how they determined the angle measures. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION (Math Journal 1, p. 169) Use Problems 1 and 2 of the Math Message to check students’ understanding of the number of degrees in a circle. Students will need to know that there are 360 degrees in a circle when they calculate the degree measure of each sector in a circle graph. Math Message Ongoing Assessment: Recognizing Student Achievement Use the Math Message to assess students’ ability to apply the definitions of supplementary and vertical angles. Students are making adequate progress if they use the given angle measure (150°) to find the measures of vertical angles and their supplements. [Geometry Goal 1] Student Page Date LESSON Calculating the Degree 5 3 WHOLE-CLASS ACTIVITY Measure of a Sector (Math Journal 1, pp. 169 and 170; Student Reference Book, pp. 59, 60, and 147; Math Masters, pp. 152 and 153) Time Degree Measures of Sectors Math Message 163 1. Find the measures of the following angles in circle O without using a protractor. P a. mQOR 150 b. mPOS c. mPOQ d. mSOR 150 30 30 Q O ° 150° S ° R ° 2. Find the sum of the angle measures in Problem 1. Use Problem 3 on journal page 169 to review strategies for renaming fractions as percents, an essential skill for this lesson. Circulate and assist as needed. 150 mPOS mPOQ mSOR 360 ° 3. Connie, Josh, and Manuel were running for student council representative. The table below shows the number of votes that each candidate received. Complete the table. Candidate Number of Votes Received Connie 7 Josh 6 59 60 147 Fraction of Percent of Votes Received Votes Received 7 25 28% 24% 48% 100% Manuel 12 6 25 12 25 Total 25 25 25 4. Use the percents from the table above to calculate the degree measure of the sector representing each candidate. Candidate Percent of Votes Received Degree Measure of Sector (to nearest degree) 28% 0.28 360 100.8 艐 101 Manuel 24% 48% 86 173 Total 100% 360 Connie Josh 169 Math Journal 1, p. 169 Lesson 5 3 347 Student Page Date Time LESSON Draw and shade in a sector of a circle on the board. Explain that in previous lessons, students used the Percent Circle on their Geometry Template to draw each wedge-shaped piece, or sector, of a circle graph. In this lesson, they will learn how to use a protractor to draw each sector. Drawing Circle Graphs with a Protractor 5 3 Mr. Li surveyed the students in his class to find out what kinds of pets they owned and how many of each kind they had. The results are shown in the first two columns of the table below. Fraction Decimal of Total Equivalent Number (to nearest of Pets thousandth) Percent of Total Number of Pets 59 60 147 Degree Measure of Sector Kind of Pet Number of Pets Dog 8 8 24 0.333 333% Cat 6 6 24 0.25 25% Guinea pig or hamster 3 3 24 0.125 12.5% 0.125 360° 45 Bird 3 3 24 0.125 12.5% 0.125 360° 45 4 4 24 Other 0.167 1. Complete the table above. 1 3 1 360 1 4 1 6 16.7% Copy the table in Problem 4 on the board or use a transparency of Math Masters, page 152. (See margin.) Discuss solution strategies for finding the degree measure of each sector. 90 360° One approach is to express the data as fractions and find those fractions of 360°. For example, Connie received 7 out of 25 7 votes, or 25 of the votes. 60 360° Pets Owned by Mr. Li’s Students Study the first row. 7 25 2. At the right, or on a separate Bird sheet of paper, use a compass and a protractor to make a circle graph of the data in the table. If you need to, tape your completed circle graph on this page. Write a title for the graph. Dog Cat 4 7 of 360° 2 5 360° 100 5 ° 6 Other Similarly, Josh received 25 of the votes. Guinea pig or hamster 6 25 2 6 of 360° 2 5 360° 86 5 ° 12 Manuel received twice as many votes as Josh (2 5 ), so the 2 measure of the sector representing Manuel is 865° 2 4 1725°. Since the protractors that students use are marked in whole-degree increments, round each degree measure to the 4 2 4 nearest whole degree: 1005° → 101°; 865°→ 86°; 1725° → 173°. 170 Math Journal 1, p. 170 Another approach is to express the data as fractions, rename the fractions as decimals, and then multiply the decimal by 360°. For example: 7 7 25 0.28 and 0.28 360° 100.8° 25 NOTE The table on Math Masters, page 152 is identical to the table in Problem 4 on journal page 169. 6 25 6 25 0.24 and 0.24 360° 86.4° 12 25 12 25 0.48 and 0.48 360° 172.8° Similarly, each degree measure should be rounded to the nearest whole degree. Student Page Date Time LESSON Drawing Circle Graphs with a Protractor 5 3 cont. 3. Sixth-grade students at Hawthorn School took a survey about after-school activities. Students answering the survey named the activity on which they spent the most time after school. The results are shown in the table below. Complete the table. Activity Number of Students Fraction of Students Music 12 12 60 Math Club 28 28 60 Art 5 Decimal Equivalent Percent of Students (to nearest percent) 59 60 147 Size of Sector 0.2 20% 72 47% 168 5 60 0.4– 6 0.08– 3 8% 30 13% 48 5% 18 7% 24 Sports 8 8 60 0.1– 3 Computers 3 3 60 None 4 4 60 0.05 – 0.06 When students have completed Problem 4 on journal page 169, demonstrate how to make the circle graph of the data. Draw a circle on the board or use the circle on the transparency (Math Masters, p. 152) and mark off sectors with a protractor. In order to make the graph as accurate as possible, graph the smallest category (Josh’s votes) first; graph the largest category (Manuel’s votes) last. Allow students to make their circle graphs on a separate sheet of paper as you demonstrate the appropriate procedures. When the graphs have been completed and titled, have students check them with the Percent Circle. 4. In the space below, or on a separate sheet of paper, use a compass and Votes Received in the Student Council Election a protractor to make a circle graph of the data in the table. If you need to, tape your completed circle graph on this page. Write a title for the graph. After-School Activities of Sixth Graders Art Sports Computers e Non Music Manuel Math Club Connie 171 Math Journal 1, p. 171 348 Unit 5 Geometry: Congruence, Constructions, and Parallel Lines Josh Student Page Next, work through the table on journal page 170 with the class. Have students complete the table as you do the same on the board or transparency (Math Masters, p. 153). The first row of the table has already been filled in. Because some data are easier to work with as fractions and other data as decimals, ask students to share and explain the approach that they find most efficient for finding the degree measures of each sector. When students have completed the table, ask them to use a compass and a protractor to make a circle graph of the survey data. Students may construct the graph on the bottom of journal page 170 or on a separate sheet of paper. Date Time LESSON Calculating Sale Price 5 3 Study each method for calculating sale price. Two-Step Method Regular Price: $35.50 Discount: 20% Find the sale price. One-Step Method Regular Price: $35.50 Discount: 20% Find the sale price. Step 1: Find the discount in dollars: 20% of $35.50. 0.2 $35.50 $7.10 The discount is 20%, so the amount of the regular price that remains is 100% – 20%, or 80% of the regular price. Step 2: Subtract the discount amount from the regular price. Find 80% of $35.50. 80% of $35.50 0.8 $35.50 $28.40 Regular Price Discount Sale Price The sale price is $28.40. $35.50 $7.10 $28.40 The sale price is $28.40. Use either method to find the sale price. 1. Regular Price: $99.00 Discount: 30% Sale Price: 2. Regular Price: $45.00 $69.30 3. Regular Price: $435.00 Drawing a Circle Graph PARTNER ACTIVITY Students find fraction, decimal, and percent equivalencies, calculate degree measures of sectors, and use a compass and a protractor to create a circle graph of the after-school activities of sixth graders. Remind students that to make the graph as accurate as possible, they should graph the smallest category (Computers) first; they should graph the largest category (Math Club) last. 2 Ongoing Learning & Practice $38.25 4. Regular Price: $348.50 $413.25 5. Regular Price: $4,380 Discount: 18% Sale Price: (Math Journal 1, p. 171) Calculating Sale Price Discount: 5% Sale Price: Discount: 15% Sale Price: Discount: 20% Sale Price: $278.80 6. Regular Price: $25,125 $3,591.60 Discount: 12% Sale Price: $22,110 172 Math Journal 1, p. 172 NOTE A small error in a big percentage is relatively less important than a small error in a small percentage. For example, if a category is only 2% of the total and the sector is off by 1%, that is a fairly large error (50%). However, if a category is 34% of the total and the sector is off by 1%, the error is much less significant (only about 3%). INDEPENDENT ACTIVITY (Math Journal 1, p. 172) The problems on journal page 172 provide practice using a one- or two-step method for finding a sale price. Student Page Date Time LESSON Math Boxes 5 3 1. Estimate the degree measure of QRS. 2. Use your full-circle protractor to draw an Sample estimate: Math Boxes 5 3 INDEPENDENT ACTIVITY angle measuring 330. Label it NOP. ° Estimate 200 Then use your full-circle protractor to measure QRS to the nearest degree. N O Q R (Math Journal 1, p. 173) P S Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 5-1. The skills in Problems 5 and 6 preview Unit 6 content. ° QRS measures about 240 . 230–232 232 3. Solve mentally. 2 a. of 45 9 4. Rewrite each fraction pair using a common denominator. 10 6 of 100 80 75% of 32 24 1 b. 33 % of 18 3 Study Link 5 3 4 c. 5 INDEPENDENT ACTIVITY (Math Masters, p. 154) Home Connection Students make a circle graph. They then interpret the graph and draw conclusions. Remind students to take home a protractor or their Geometry Template. d. 49 50 87 3 2 4 5. Marta has rolls of ribbon. If there are 1 3 yards of ribbon on each roll, about how 3 Sample answers: 3 8 a. and 10 25 15 50 and 16 50 7 19 b. and 15 45 21 45 and 19 45 4 3 c. and 5 9 36 45 and 15 45 6 d. 27 12 54 and 7 54 and 7 54 6. Insert parentheses to make each sentence true. many yards of ribbon does Marta have? Circle the best estimate. a. 20 16 / 2 15 30 1 A. yard 2 b. 12 6 22 / 5 10 B. 6 yards c. 72 / 8 4 / 6 1 (( ) ) (( ) ( ( 1 C. 6 yards 2 D. 9 yards 79 ) )) ( ( )) d. 95 10 / 3 2 93 90 247 173 Math Journal 1, p. 173 Lesson 5 3 349 Study Link Master Name Date STUDY LINK Time The table below shows a breakdown, by age group, of adults who listen to classical music. 1. 3 Differentiation Options Circle Graphs 53 a. Calculate the degree measure of each sector to the nearest degree. b. Use a protractor to make a circle graph. Do not use the Percent Circle. Write a title for the graph. 49 147 Age Percent of Listeners Degree Measure Age of Adult Classical Music Listeners 18–24 11% 25–34 18% 35–44 24% 45–54 20% 40 65 86 72 40 58 18– 65+ 24 55 yrs yrs 25–34 yrs–64 yrs 45–54 yrs 35–44 yrs 55–64 11% 65 16% Finding Fractions of 360° 5–15 Min (Math Masters, p. 155) Source: USA Today, Snapshot On average, about 8 million adults listen to classical music on the radio each day. 2. INDEPENDENT ACTIVITY READINESS a. Estimate how many adults between the ages of 35 and 44 listen to classical music on the radio each day. About 1,920,000 adults b. Estimate how many adults at least 45 years old listen to classical music on the radio each day. About 3,760,000 adults (unit) To provide experience finding fractions of 360°, have students use Math Masters, page 155 and coloring pencils or markers. Ask students to shade fractional parts of a full-circle protractor diagram. Then have them use the shaded regions to determine the number of degrees in each fractional part. (unit) INDEPENDENT ACTIVITY ENRICHMENT Practice Order each set of numbers from least to greatest. Finding Sums of Angle 7, 0, 0.07, 0.7, 7 3. 7, 0.07, 7, 0.7, 0 4. 0.25, 0.75, 0.2, , , 0.06, 0.18, 5 4 10 1 10 4 4 1 0.06, , 0.18, 0.2, 0.25, 0.75, 45, 44 5–15 Min Measures in Polygons Math Masters, p. 154 (Math Masters, p. 156; Student Reference Book, p. 233) To further extend their knowledge of angle relationships, students find the sums of angle measures in various polygons. They use a pattern to calculate the sums of angle measures in a heptagon, nonagon, and dodecagon. Teaching Master Teaching Master Name Date Time Fractions of 360ⴗ 53 53 Shade each fractional part. Then record the number of degrees in each shaded region. 1 2 Shade of the circle. 280 29 260 0 250 24 0 280 29 270 30 0 260 250 0 23 0 23 270 30 31 00 0 280 29 250 280 29 0 260 250 24 0 180 20 1 23 0 13 0 170 31 0 6 º1801,080 10 8 8 º1801,440 Study your completed table. Use any patterns you notice to write a formula to find the sum of the angle measures in any polygon (n-gon). 0 13 0 14 0 0 22 15 0 15 0 160 190 20 of 360 0 0 0 160 12 100 11 0 2. a. Formula 160 280 29 6 40 90 0 180 20 1 170 0 22 190 20 0 15 270 30 0 260 30 0 260 14 0 22 190 20 270 30 31 00 0 10 20 14 0 0 180 20 1 170 0 22 190 20 250 24 0 0 180 20 1 170 24 8 160 0 0 160 0 23 15 15 0 23 0 0 0 0 4 º180 720 70 80 280 29 14 24 14 0 22 190 20 31 0 180 20 1 170 350 0 0 360° 33 60 270 30 0 4 50 12 0 13 0 100 11 0 24 6 0 90 0 0 70 80 60 260 250 0 0 12 50 0 180 20 1 170 23 3 º180 540 100 11 13 0 340 0 270 ° 0 22 190 20 Math Masters, p. 155 350 3 Decagon 3 4 3 4 5 Octagon ° Shade of the circle. 32 ° 2 º 180 360 Hexagon 120 of 360 40 30 40 90 0 30 30 70 80 0 of 360 0 0 360° 60 12 100 11 6. 2 Pentagon 50 90 1 3 4 10 20 33 10 20 34 0 0 33 360° 13 0 70 80 60 1 12 0 50 ° 1 12 0 350 0 13 0 0 32 Shade of the circle. 0 diagonal ° 1 3 40 Sum of Angle Measures Example: Quadrangle Shade of the circle. 350 340 60 of 360 32 12 0 30 90 of 360 10 20 0 0 33 360° Number of Number of Sides (n) Triangles Polygon 100 11 0 1 4 4. Draw diagonals from the given vertex to separate each polygon into triangles. Then complete the table. 40 90 12 100 11 0 30 70 80 90 ° 1 6 32 1. 10 20 350 0 0 33 360° 60 70 80 60 180 of 360 350 340 5. 270 30 31 00 0 0 32 160 31 0 340 40 15 30 50 0 0 33 360° Shade of the circle. 1 6 Shade of the circle. 50 1 2 1 4 Sums of Angle Measures in Polygons 0 0 32 2. 10 20 Time A diagonal is a line segment that connects two vertices of a polygon and is not a side. You can draw diagonals from one vertex to separate polygons into triangles. 14 350 340 3. Date LESSON LESSON 1. Name Unit 5 Geometry: Congruence, Constructions, and Parallel Lines b. (n 2) º 180 Use the formula to find the sums of the angle measures in a ° ° heptagon. nonagon. dodecagon. 900 Math Masters, p. 156 1,260 1,800 °
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