Conversions among Fractions, Decimals, and Percents

Conversions among Fractions, Decimals, and Percents Objectives To reinforce the use of a data table; and to reinforce renaming fractions as percents using a calculator and renaming decimals as percents. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Read and use large numbers. [Number and Numeration Goal 1] Explore repeating and terminating decimals. [Number and Numeration Goal ] Use a calculator to rename fractions as percents; rename decimals as percents by multiplying by 100. [Number and Numeration Goal ] Compare two quantities with like units using division. [Operations and Computation Goal 4] Round to the nearest whole-number percent. [Operations and Computation Goal 6] Use a data table. [Data and Chance Goal 2] Key Activities Students look up country population and land area data and convert these to percents of the world population and land area. Students complete the percent column of the Equivalent Names for Fractions table on journal pages 342 and 343. Ongoing Assessment: Informing Instruction See page 746. Materials Math Journal 2, pp. 342 and 343 Student Reference Book, pp. 271 and 281 Study Link 9 4 calculator classroom world map slate Advance Preparation Updating the World Tour Math Journal 2, pp. 329 331, 336, and 337 Student Reference Book, pp. 276, 277, 281, 288, 289, 297, and 302 30 Math Masters, pp. 419 and 420 (optional) Students continue the World Tour. Angle Addition and Subtraction Math Journal 2, pp. 260A and 260B Students practice finding unknown angle measures. Math Boxes 9 Math Journal 2, p. 260 Students practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 4. [Operations and Computation Goal 4] Study Link 9 Math Masters, p. 288 Students practice and maintain skills through Study Link activities. READINESS Rounding Percents Math Masters, p. 289 Students use a curved number-line model to round percents. EXTRA PRACTICE -Minute Math -Minute Math, pp. 93 and 181 calculator Students practice conversions among fractions, decimals, and percents. Teacher s Reference Manual, Grades 4 6 pp. 62, 63, 13, 14 744 Unit 9 Fractions, Decimals, and Percents

Getting Started Mental Math and Reflexes Write fractions on the board. For each fraction, students write the equivalent decimal and percent on their slates. Have students explain their strategies for the problems. Suggestions: _ 100 0., % _ 2 0.10, 10% 20 71_ 100 0.71, 71% 12_ 0.48, 48% 2 67_ 100 0.67, 67% 1_, 00% 3 _ 0.0, % 100 _ 0., 0% 10 1_ 0.2, 2% 4 Math Message Use your calculator to rename these fractions as percents: 1_ 8, 3_ 8, _ 8, 7_ 8. Study Link 9 4 Follow-Up Review answers. Have students share the strategies they used to solve Problems 3 6. For example: Problem 3: 42_ 84_ 84_ 0 = (multiply numerator and denominator by 2); 100 100 = 84% 2_ Problem 6: 12 = 1_ (divide numerator and denominator by 2); 1_ = 20% Ask if any student can describe a solution strategy to solve Problems 7 and 8 without a calculator. For example: 23_ Problem 7: 92 = 1_ 4 Problem 8: 12_ 3_ 40 = 10 (divide numerator and denominator by 23); 1_ 4 = 2% (divide numerator and denominator by 4); 3_ 10 = 30% 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS Go over the answers: 1_ 8 = 12.%; 3_ 8 = 37.%; _ 8 = 62.%; 7_ = 87.%. Students may have renamed the fractions as percents 8 in one of two ways. Use the percent key. For example, to rename 3_ as a percent, 8 TI-1: 3 8 Display: 37. Casio fx-: 3 8 Display: 37. Divide numerator by denominator, and multiply by 100. For example, divide 3 by 8 (= 0.37) and multiply by 100 (= 37.). Remind students that multiplying a decimal by 100 can be done by moving the decimal point two digits to the right. Students may use either method, but they should be able to use both. Adjusting the Activity ELL Ask students to explain how 1_ 8, 3_ 8, _ 8, and 7_ 8 could be renamed as percents without using a calculator. Record the steps on the board. _ 8 is half of 1_ 4. Because 1_ 4 = 2% and half of 2% is 12.%, 1_ 8 = 12.%. 3_ 8 equals 1_ 4 + 1_ 8, which is 2% + 12.%, or 37.%. _ 8 and 7_ 8 are renamed in the same way. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Lesson 9 74

Facts About the World Continents are large land masses. There are seven continents on the Earth, although Europe and Asia are sometimes thought of as one continent. Most continents contain many countries, but there are no countries at all in Antarctica. A country is a territory and the people who live there under one government. The number of countries in the world often changes as countries split apart or join with other countries. At this time, there are about 200 countries in the world. Population is the number of people who live in a certain region. Population growth is the change in the population every year after all births and deaths are accounted for. The population growth rate is the increase (or decrease) in population per year, written as a percent. The world s population is now increasing by about 200,000 people per day, or about 7 million people per year. Over the last 40 years, the world s population has about doubled. It reached the 6 billion mark in 1999. World population is expected to reach about 9 billion people by the year 200. The Continents Percent of World Area Percent of Continent Population* Population (sq miles) Land Area North America 09,000,000 8.0% 8,300,000 14.8% South America 367,000,000.8 6,800,000 12.1 Europe 799,000,000 12. 4,100,000 7.3 Asia 3,797,000,000 9. 16,700,000 29.8 Africa 874,000,000 13.7 11,00,000 20. Australia 32,000,000 0. 3,300,000.9 Antarctica 0 0.0,400,000 9.6 World Totals 6,378,000,000 100.0% 6,100,000 100.0% (about 6.4 billion) *Data are for the year 2004. World population growth rate for the year 2004: about 1.2% per year Student Reference Book, p. 271 World Tour Renaming Fractions as Percents (Student Reference Book, pp. 271 and 281) PROBLEM SOLVING WHOLE-CLASS Social Studies Link Use the classroom world map to identify Russia and China. Russia has the largest land area of any country in the world. China has the largest population of any country in the world. Tell students that in this lesson they will investigate population and land area data and use their calculators to convert these to percents of the world population and land area. Ask students to use the Student Reference Book to find the population of China and the total world population. Write these populations on the board. Point out that the world population has been rounded to the nearest million. Ask students to round China s population to the nearest million. Record this estimate on the board. Ask students what fraction of the world s population lives in China. Write this fraction on the board. China Population 1,298,848,000 Population Rounded 1,299,000,000 Fraction _ 1,299,000,000 6,378,000,000 World 6,378,000,000 6,378,000,000 Australia Area: 2,967,900 sq mi Population: 19,913,000 Capital: Canberra (Pop. 373,000) Languages: English, aboriginal languages Monetary unit: Australian Dollar Bangladesh Area:,600 sq mi Population: 141,340,000 Capital: Dhaka (Pop. 11,60,000) Languages: Bangla, English Monetary unit: Taka China Area: 3,70,400 sq mi Population: 1,298,848,000 Capital: Beijing (Pop. 10,848,000) Languages: Mandarin, Gan, Wu, Haka, Yue, Minbei, Xiang, Minnan Monetary unit: Renminbi (Yuan) India Area: 1,269,300 sq mi Population: 1,06,071,000 Capital: New Delhi (Pop. 12,441,000) Canada Area: 3,81,800 sq mi Population: 32,08,000 Capital: Ottawa (Pop. 1,093,000) Languages: English, French Monetary unit: Dollar Costa Rica Area: 19,700 sq mi Population: 3,97,000 Capital: San José (Pop. 1,08,000) Language: Spanish Monetary unit: Colon Cuba Area: 42,800 sq mi Population: 11,309,000 Capital: Havana (Pop. 2,189,000) Language: Spanish Monetary unit: Peso El Salvador Area: 8,100 sq mi Population: 6,88,000 Capital: San Salvador (Pop. 1,424,000) REGION 4 Asia and Australia Languages: Hindi, English, 14 regional languages Monetary unit: Rupee Iran Area: 636,000 sq mi Population: 67,03,000 Capital: Tehran (Pop. 7,190,000) Languages: Farsi, Kurdish, Turkic, Luri Monetary unit: Rial Japan Area: 14,900 sq mi Population: 127,333,000 Capital: Tokyo (Pop. 34,997,000) Language: Japanese Monetary unit: Yen Russia Area: 6,92,800 sq mi Population: 143,782,000 Capital: Moscow (Pop. 6,468,000) Languages: Russian, many others Monetary unit: Ruble REGION North America Language: Spanish Monetary unit: Colon Guatemala Area: 42,000 sq mi Population: 14,281,000 Capital: Guatemala City (Pop. 91,000) Languages: Spanish, Mayan languages Monetary unit: Quetzal Haiti Area: 10,700 sq mi Population: 7,66,000 Capital: Port-au-Prince (Pop. 1,961,000) Languages: French, Haitian Creole Monetary unit: Gourde Jamaica Area: 4,200 sq mi Population: 2,713,000 Capital: Kingston (Pop. 7,000) Languages: English, Jamaican Creole Monetary unit: Jamaican Dollar Student Reference Book, p. 281 World Tour Thailand Area: 198,00 sq mi Population: 64,866,000 Capital: Bangkok (Pop. 6,486,000) Languages: Thai, English Monetary unit: Baht Turkey Area: 301,400 sq mi Population: 68,894,000 Capital: Ankara (Pop. 3,428,000) Languages: Turkish, Arabic, Kurdish Monetary unit: Lira Vietnam Area: 127,200 sq mi Population: 82,690,000 Capital: Hanoi (Pop. 3,977,000) Languages: Vietnamese, Chinese, French, English Monetary unit: Dong Mexico Area: 761,600 sq mi Population: 104,960,000 Capital: Mexico City (Pop. 18,660,000) Languages: Spanish, Mayan dialects Monetary unit: New Peso Panama Area: 30,200 sq mi Population: 3,000,000 Capital: Panama City (Pop. 930,000) Languages: Spanish, English Monetary unit: Balboa United States of America Area: 3,717,800 sq mi Population: 293,028,000 Capital: Washington, D.C. (Pop. 63,000) Languages: English, Spanish Monetary unit: Dollar 746 Unit 9 Fractions, Decimals, and Percents Have students use their calculators to rename this fraction as a percent. They should use both methods and get the same answer: Use the percent key. TI-1: 1299000000 6378000000 show 20.36688617. Casio fx-: 1299 6378 20.366886. Display will show Display will Divide numerator by denominator, and multiply by 100. 1299000000 6378000000 Display will show 0.2036688617. 0.2036688617 Æ 100 Display will show 20.36688617. Ongoing Assessment: Informing Instruction Watch for students who notice that, because both numbers are in millions, it is sufficient to divide 1,299 by 6,378. If students calculators, such as the Casio fx-, cannot display these large numbers, encourage them to use this strategy. Help students summarize. Round percent answers to the nearest whole-number percent. In 20.366886179, the digit in the tenths place is less than, so the number is rounded down to 20%. About 20 of every 100 people in the world live in China. Because 20% equals 1_, about 1 of every people in the world live in China.

Repeat this last routine to calculate the percent of the world s land area that is in Russia. Russia s area is about 6,92,800 square miles. The world s land area is about 7,900,000 square miles. The fraction of the world s area that belongs to Russia is about 6,93,000 7,900,000 = 0.1138687392. Multiply 0.11 100 = 11%. So about 11% of the world s area belongs to Russia. Equivalent Names for Fractions Fraction Equivalent Fractions Decimal Percent 2 0 0% 2 2 4, 3 6 2 2 1 100% 3 Completing the Equivalent Names for Fraction Table (Math Journal 2, pp. 342 and 343) Students should already have filled in the equivalent fractions and decimals columns of the table on journal pages 342 and 343. Now they will fill in the percents column. On the first page of the table, students find the percents by using the percent key. If the calculator display shows an answer with more than 3 digits, they record only the first 3 digits. For example, for the fraction _, the percent answer will 6 be displayed on the calculator as 83.33333333, but only 83.3 should be recorded in the table. On the second page of the table, students can find the percents without using a calculator and without making any actual computations. The decimal names are already recorded in the table. Students need only multiply the decimal by 100 (move the decimal point two digits to the right) to rename the decimal as a percent. As before, ask students to record only the first 3 digits for any percent name. 4 3 4 2 3 4 6 6 8 3 8 8 7 8 Math Journal 2, p. 342 2 3 Equivalent Names for Fractions continued Fraction Equivalent Fractions Decimal Percent 9 2 9 4 9 9 7 9 8 9 1 3 7 9 1 2 2 7 2 1 12 Math Journal 2, p. 343 Lesson 9 747

9 Finding Unknown Angle Measures Without using a protractor, find the measure of the unknown angle. Write an equation to show how you solved the problem. Use a variable to represent the unknown angle measure. Sample equations are given. 1. 2. D A 20 30 C B Measure of DAB = 0 Measure of HEF = 116 Equation: 20 + 30 = x Equation: 7 + 9 = x 3. 4. K L 70 x Q O x 4 M J N Measure of KJL = 2 Measure of PNO = 92 Equation: 70-4 = x Equation: 26 + x = 118. R 6. S U T SRU is a right angle. YZW is a straight angle. Measure of TRU = 60 Measure of XZY = 33 Measure of TRS = 30 Measure of XZW = 147 Equation: 90-60 = x Equation: 180 = 33 + x H E Y 7 X 9 G F P 118 26 Z W 2 Ongoing Learning & Practice Updating the World Tour (Math Journal 2, pp. 329 331, 336, and 337; Student Reference Book, pp. 276, 277, 281, 288, 289, 297, and 302 30; Math Masters, pp. 419 and 420) Social Studies Link Students follow the established World Tour routine. They update the Route Map by drawing a line segment to connect Brasília, Brazil, and Beijing, China. They use the World Tour section of the Student Reference Book to locate facts about China and Beijing, and they fill in the Country Notes pages for this country and capital. Students who are also keeping a Route Log update that as well. Math Journal 2, p. 260A Angle Addition and Subtraction (Math Journal 2, pp. 260A and 260B) Students practice using addition and subtraction to find unknown angle measures. Math Boxes 9 (Math Journal 2, p. 260) 9 Finding Unknown Angle Measures continued Find the value of x. Write an open sentence to show how you solved the problem. Sample equations are given. 7. 8. (x + 100) 110 E F x = 4 Equation: 110 + 3 = x + 100 Measure of EFG = 14 A 18 (x + 22) D x = 63 73 3 B Equation: (x + 22) + 73 = 18 Measure of ADB = 8 Try This H G C 9. Angela used her protractor to measure KJL and NJM. She found the two angles had the same measure. Angela said, Without measuring, I also know KJM has the same measure as NJL. Explain how Angela knows her statement is true. M L K Sample answer: There is only one angle in between NJM and KJL. So, the measure of NJM + the measure of MJL is the same as the measure of KJL + the measure of LJM. N J Mixed Practice Math Boxes in this lesson are linked with Math Boxes in Lessons 9-7 and 9-9. The skill in Problem 6 previews Unit 10 content. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 4 Use Math Boxes, Problem 4 to assess students ability to divide a multidigit whole number by a 1-digit divisor. Students are making adequate progress if they express the quotient as a whole number with a whole-number remainder. Some students may be able to express the remainder as a fraction. [Operations and Computation Goal 4] Study Link 9 (Math Masters, p. 288) Home Connection Students use a table of data to calculate the approximate percentage of marriages that occurred each month in 2001. Math Journal 2, p. 260B 748 Unit 9 Fractions, Decimals, and Percents

3 Differentiation Options READINESS Rounding Percents (Math Masters, p. 289) PARTNER 1 Min To explore rounding percents to the nearest whole number, have students plot numbers on a curved number line to see which way the percent will slide. Ask students to describe how they rounded their numbers. Encourage vocabulary such as top, bottom, endpoint, middle, closer, and farther. 9 1. Calculate. Math Boxes a. 10% of 70 = 7 b. % of 60 = 3 c. 2% of 28 = 7 d. 0 % of 48 = 24 e. 80 % of 2 = 20 38 39. What is the height of the parallelogram? Include the correct unit. 3. Complete the table with equivalent names. 4. Divide. Use a paper-and-pencil algorithm. Fraction Decimal Percent _ 10 20_ 100 7_ 10 2_ 0. 0% 20% 0.20 70% 0.7 0.4 40% 61 62 2. Insert parentheses to make each number sentence true. ( ) ( ( ) ) ( ) ( ) ( ) a. 6 + 2 4 = 32 b. + 7 3 = 36 c. 1 + 8 8 + 2 = 90 d. 1 + 7 8 + 2 = 80 897 6 = 149 R3, or 149 1_ 2 22 23 6. Draw the mirror image of the figure shown on the left of the vertical line. 10 EXTRA PRACTICE SMALL-GROUP -Minute Math 1 Min To offer students more experience with conversions among fractions, decimals, and percents, see -Minute Math, pages 93 and 181. Number model: Height:? 9 in. Area = 27 in 2 27 / 9 = 3 3 in. Math Journal 2, p. 260 13 106 109 Planning Ahead Remind students to bring to school the second page of Study Link 9-1 (Trivia Survey). Their survey results will be used in Lesson 9-6. Teaching Master Name 9 Rounding Percents Study Link Master Name STUDY LINK 9 Renaming Fractions as Percents 62 207 The number lines below are curved like hills. You can use them to help you round percents to the nearest whole-number percent. Example: Round 89.7% to the nearest whole-number percent. Think: Which whole-number percents are nearest to 89.7%? If I look at the number line, 89% is the whole-number percent to the left of 89.7%. If I look at the number line, 90% is the whole-number percent to the right of 89.7%. What number would be exactly halfway between 89% and 90%? 89.% Mark 89.7% on the curved number line. 89.7% Would 89.7% slide down to 89% or 90%? 89.7% rounded to the nearest whole-number 89% 90% percent is 90%. 1. Round 23.6% to the nearest whole-number percent. Label the curved number line. Mark 23.6%. 23.6% would slide down to 24%. 24% 23.6% rounded to the nearest whole-number percent is. 2. Round 92.1% to the nearest whole-number percent. Label the curved number line. Mark 92.1%. 92.1% would slide down to 92%. 92.1% rounded to the nearest whole-number percent is 92%. 92.1% 23.% 23.6% 23% 24% 92.% 92% 93% In 2001, there were about 2,317,000 marriages in the United States. The table below shows the approximate number of marriages each month. 1. Use a calculator to find the percent of the total number of marriages that occurred each month. Round the answers to the nearest whole-number percent. Approximate Approximate Month Number of Percent of Marriages Total Marriages January 147,000 6% February 19,000 7% March 166,000 7% April 166,000 7% May 189,000 8% June 237,000 10% July 244,000 11% August 22,000 10% September 224,000 10% October 217,000 9% November 191,000 8% December 12,000 7% Source: U.S. Department of Health and Human Services 2. According to the table, what is the most popular month for a wedding? What is the least popular month for a wedding? January July 3. Describe how you used your calculator to find the percent for each month. Sample answer: I divided the number of marriages for each month by the total then multiplied by 100 and rounded. Practice Math Masters, p. 289 Name all the factors of each number. 1, 3, 7, 9, 21, 63 1, 2, 4, 7, 14, 28 4. 63. 28 Math Masters, p. 288 Lesson 9 749