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Contents Domain 1 Ratios and Proportional Relationships. . . . . 4 Common Core State Standards Lesson 1 Computing Unit Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7.RP.1 Lesson 2 Identifying Proportional Relationships . . . . . . . . . . . . . . . . 10 7.RP.2.a, 7.RP.2.b Lesson 3 Representing Proportional Relationships. . . . . . . . . . . . . . 14 7.RP.2.c, 7.RP.2.d Lesson 4 Word Problems with Ratio and Percent. . . . . . . . . . . . . . . . 18 7.RP.3 Domain 1 Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Domain 2 The Number System . . . . . . . . . . . . . . . . . . . . . . . 28 Lesson 5 Adding and Subtracting Rational Numbers. . . . . . . . . 30 Lesson 6 Applying Properties of Operations to Add and Subtract Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 36 7.NS.1.a, 7.NS.1.b, 7.NS.1.c 7.NS.1.d Lesson 7 Multiplying Rational Numbers . . . . . . . . . . . . . . . . . . . 40 7.NS.2.a, 7.NS.2.c Lesson 8 Dividing Rational Numbers. . . . . . . . . . . . . . . . . . . . . . 46 7.NS.2.b, 7.NS.2.c Lesson 9 Converting Rational Numbers to Decimals. . . . . . . . . . . . . 52 7.NS.2.d Lesson 10 Problem Solving: Complex Fractions. . . . . . . . . . . . . . 56 7.NS.3 Lesson 11 Problem Solving: Rational Numbers. . . . . . . . . . . . . . . 60 7.NS.3 Domain 2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Lesson 12 Writing Equivalent Expressions. . . . . . . . . . . . . . . . . . . . . . 70 7.EE.1, 7.EE.2 Lesson 13 Factoring and Expanding Linear Expressions. . . . . . . . . . . 74 7.EE.1 Lesson 14 Adding and Subtracting Algebraic Expressions. . . . . . . . . 78 7.EE.1 Problem Solving: Algebraic Expressions and Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.EE.3 Lesson 15 Lesson 16 Word Problems with Equations. . . . . . . . . . . . . . . 86 7.EE.4.a Lesson 17 Word Problems with Inequalities . . . . . . . . . . . . . . . . . 92 7.EE.4.b Domain 3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2 CC12_MTH_G7_SE_FM_Final.indd 2 Problem Solving Fluency Lesson Duplicating any part of this book is prohibited by law. Domain 3 Expressions and Equations . . . . . . . . . . . . . . . . 68 Performance Task 15/06/12 9:31 AM Common Core State Standards Domain 4 Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Lesson 18 Scale Drawings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.G.1 Lesson 19 Drawing Geometric Shapes. . . . . . . . . . . . . . . . . . . . . . . . 110 7.G.2 Lesson 20 Examining Cross Sections of Three-Dimensional Figures . . . . . . . . . . . . . . . . . . . . . . . . 114 7.G.3 Lesson 21 Area and Circumference of Circles. . . . . . . . . . . . . . . 118 7.G.4 Lesson 22 Angle Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.G.5 Lesson 23 Problem Solving: Area and Surface Area of Composite Figures. . . . . . . . . . . . . . . . . . . . . . . . . 130 7.G.6 Problem Solving: Volume of Three-Dimensional Figures. . . . . . . . . . . . . . . . . . . . . 134 7.G.6 Lesson 24 Domain 4 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Domain 5 Statistics and Probability . . . . . . . . . . . . . . . . . 142 Lesson 25 Understanding Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . 144 7.SP.1, 7.SP.2 Lesson 26 Using Mean and Mean Absolute Deviation . . . . . . . . . . . 150 7.SP.3, 7.SP.4 Making Comparative Inferences about Two Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.SP.3, 7.SP.4 Lesson 28 Understanding Probability. . . . . . . . . . . . . . . . . . . . . . . . . 160 7.SP.5, 7.SP.6, Lesson 27 7.SP.7.b Lesson 29 Probabilities of Simple Events. . . . . . . . . . . . . . . . . . . . . . 164 7.SP.7.a, 7.SP.7.b Lesson 30 Probabilities of Compound Events . . . . . . . . . . . . . . . . . . 170 7.SP.8.a, 7.SP.8.b Lesson 31 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.SP.8.c Duplicating any part of this book is prohibited by law. Domain 5 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Math Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 CC12_MTH_G7_SE_FM_Final.indd 3 3 15/06/12 9:31 AM LE SS O N 1 Computing Unit Rate A ratio is a comparison of two quantities. A rate is a type of ratio that compares two quantities that have different units of measure. For example, $30 for 5 pounds is a rate that compares dollars to pounds. If a rate is a unit rate, the second quantity in the comparison is 1 unit. For example, $6 per pound is a unit rate because it compares $6 to 1 pound. If a rate involves comparing two fractions, you can use a complex fraction to represent the rate. A complex fraction is a fraction in which the numerator and/or the denominator is a fraction or a mixed number. 5 1 __ EXAMPLE A Evan walks __ 8 mile every 4 hour. Express his speed as a unit rate in miles per hour. 1 Write the rate as a complex fraction. 5 5 __ 8 mi __ 8 ____ __1 5 __ 1 4 h __ 4 1 The denominator is __ 4 , but in a unit rate, the denominator will be 1. 2 Rewrite the complex fraction so that it has a denominator of 1. 1 Since the denominator is __ 4 , you can multiply it times 4 to get a denominator of 1. 1 1 __ 4 __ __ 4 3 4 5 4 3 1 5 1 Multiply both the numerator and the denominator times 4. 5 4 20 The result is a denominator of 1. So, the ratio now shows a unit rate. 20 TRY 1 5 __ Ariella jogs 3 __ 8 miles in 8 hour. Express her speed as a unit rate in miles per hour. 6 5 ___ 8 mi __ 2 mi 1 _____ __ 1 h 5 ____ 1 h 5 2 2 miles per hour ▸ Evan’s rate of speed is 2 __21 miles per hour. Duplicating any part of this book is prohibited by law. 5 __ 8 4 __ 8 3 __1 ___ 8 __ __ ____ __1 3 4 5 __1 __4 5 __ 1 4 4 3 1 Domain 1: Ratios and Proportional Relationships CC12_MTH_G7_SE_D1_Final.indd 6 15/06/12 9:32 AM 3 1 __ EXAMPLE B A recipe calls for 1 __ 2 cups of flour for every 4 cup of sugar used. How many cups of flour are needed for each cup of sugar used? 1 Write the ratio as a complex fraction. 3 2 3 __ __ 2 c flour 2 __ The ratio is: _______ 5 3 3 . __ 4 c sugar __ 4 3 1 1 _______ 1 3 2 1 1 __ __ Convert 1 __ 5 2 2 to an improper fraction: 1 2 5 2 Use a fraction model. 3 __ 3 __ 3 2 __ Since __ 3 is the same as the quotient 2 4 4 , you can use a __ 4 3 fraction model to find the answer. Shade __ 2 of two squares. 3 2 3 4 Then divide the squares into fourths. Move the top 3 shaded rectangles to the right of the other rectangles. 3 3 1 __ __ The diagram shows that __ 2 (or 1 2 ) is the same as multiplying 4 by 2. ▸ 3 4 3 __ 2 __ The quotient __3 5 2, so 2 cups of flour are needed for each cup 4 2 of sugar used. 3 1 __ EXAMPLE C A motorized scooter can travel 5 __ 4 miles on 5 gallon of gasoline. How many miles per gallon does the scooter get? Duplicating any part of this book is prohibited by law. 1 Write the rate as a complex fraction. 3 Convert 5 __ 4 to an improper fraction: 5 3 4 1 3 ___ 3 ________ 23 5 __ 5 4 4 5 4 23 23 ___ 4 mi ___ 4 __ The rate is: _____ 5 1 1 . __ __ 5 gal 5 2 Write the complex fraction as a unit rate. 1 5 __ Since the denominator is __ 5 , multiply by 5 to get a denominator of 1. 23 23 5 115 ___ 4 5 ___ 4 3 __1 ___ 4 __ _____ __1 3 __ 5 5 ___ 5 1 5 1 __ 5 5 3 __1 115 ___ 4 mi So, the unit rate is: _____ . 1 gal 115 ___ 4 mi 3 _____ 1 gal 5 28 __ 4 miles per gallon C HE C K Explain how you could use multiplication to check the answer OVERSET for Example C. ▸ The scooter gets 28 __34 miles per gallon. Lesson 1: Computing Unit Rate CC12_MTH_G7_SE_D1_Final.indd 7 7 15/06/12 9:32 AM Practice Classify each rate. Write unit rate or not a unit rate. 1. 1 3 __ 2 feet per minute $1.60 per pound 2. 3. 1 1 __ 4 mile every __ 6 hour REMEMBER A unit rate, written as a fraction, has a denominator of 1. Find each unit rate by simplifying the given complex fraction. Show your work. 4. 3 1 __ A novelist can write 2 __ 4 pages in 4 hour. 5. Aaron can run 2 kilometers in 1 14 __ 2 minutes. Find his rate of speed in kilometers per minute. 2 km _____ ___ 5 29 2 min Express her writing speed as a unit rate. 9 __ 4 pages ______ __3 5 4 hour pages per hour Fill in the blanks with an appropriate word or phrase. 6. A(n) is a ratio of two quantities that have different units of measure. 7. A(n) is a ratio in which the second quantity in the comparison is 1 unit. 8. A(n) fraction has a fraction in the numerator, the denominator, or both. 9. If you multiply the numerator and the denominator of a fraction by the same number, the result will be a(n) fraction. Choose the best answer. 1 A. __ 10 mile 1 C. __ 3 mile 8 5 B. __ 18 mile 3 D. 3 __ 5 miles 1 1 __ 11. A satellite travels 29 __ 2 miles every 4 3 seconds. What is its unit rate of speed? 21 A. 6 ___ 26 miles per second 1 B. 29 __ 2 miles per second 5 C. 33 __ 6 miles per second 5 D. 127 __ 6 miles per second Duplicating any part of this book is prohibited by law. 1 10. For every __ 6 mile that a ship travels 3 north, it travels __ 5 mile west. How many miles does the ship travel north for every mile it travels west? Domain 1: Ratios and Proportional Relationships CC12_MTH_G7_SE_D1_Final.indd 8 15/06/12 9:32 AM Find each unit rate. Show your work. 1 12. Renting an office costs $486 per month. The office has an area of 202 __ 2 square feet. What is the monthly cost per square foot to rent the office? 3 2 __ 13. A recipe calls for using __ 4 cup of brown sugar for each 3 cup of white sugar. How many cups of brown sugar are used per cup of white sugar? 1 1 __ 14. Lauren bikes 1 __ 3 miles in 10 hour. What is her rate of speed in miles per hour? 1 1 __ 15. Oliver reads 28 __ 2 pages of a book in 1 6 hours. Express his reading speed in pages per hour. Solve. 16. SHOW On mountainous terrain, 2 1 17. 2 EXPLAIN A car travels __ 5 mile in 1 __ a semi-truck travels 2 __ 3 miles on 2 gallon __ 2 minute. What is the car’s speed of fuel. How many miles can the truck in miles per hour? Explain how you travel per gallon of fuel? Use drawings determined your answer. Duplicating any part of this book is prohibited by law. or equations to show your work. Lesson 1: Computing Unit Rate CC12_MTH_G7_SE_D1_Final.indd 9 9 15/06/12 9:32 AM Contents Domain Assessment—Ratios and Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . . 4 Domain Assessment—The Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Domain Assessment—Expressions and Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Domain Assessment—Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Domain Assessment—Statistics and Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Duplicating any part of this book is prohibited by law. Summative Assessment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 CC12_MTH_G7_NAA_FM_Final.indd 3 22/06/12 9:35 AM Domain Assessment • Ratios and Proportional Relationships 1. Which of the following tables represents a proportional relationship between x and y? A. B. C. D. x y 1 2 2 1 3 4 4 3 x y 1 24 2 23 3 22 4 21 2. Which of the following graphs represents a proportional relationship between x and y? A. y 6 5 4 3 2 x y 1 3 2 6 3 9 4 12 x y 1 4 2 8 3 10 4 12 1 –6 –5 –4 –3 –2 –1 0 –1 –2 –3 1 2 3 4 5 6 x 1 2 3 4 5 6 x 1 2 3 4 5 6 x 1 2 3 4 5 6 x –4 –5 –6 B. y 6 5 4 3 2 1 –6 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5 –6 C. y 6 5 4 3 2 1 –6 –5 –4 –3 –2 –1 0 –1 –2 –3 D. y 6 5 4 3 2 1 –6 –5 –4 –3 –2 –1 0 –1 –2 –3 –4 –5 –6 Go On Duplicating any part of this book is prohibited by law. –4 –5 –6 4 CC12_MTH_G7_NAA_Final.indd 4 22/06/12 9:32 AM A scientist is mixing a chemical solution for an experiment. The solution contains 3 1 __ 8 ounce of a chemical and __ 6 ounce saline solution. What is the unit rate of chemical to saline solution? 6. A factory produces c cans of paint in h hours according to the equation c 5 50h. Which of the following represents the unit rate of cans produced per hour? A. 50 Ratios and Proportional Relationships 3. B. h 1 A. __ 16 C. c 3 B. __ 4 h D. _c 9 C. __ 4 7. D. 4 4. Candice rode her bike at a constant speed. 1 She traveled 36 miles in 1 __ 2 hours. How can she find her unit rate, in miles per hour? 1 B. add 1 __ 2 to 36 1 D. divide 36 by 1 __ 2 Duplicating any part of this book is prohibited by law. 1 Peyton reads 25 pages in __ 2 hour. Which equation below represents the relationship between the number of pages Peyton reads and how much time he spends reading? Let p 5 number of pages and t 5 number of hours. 1 2 8 3 12 5 20 6 24 9 36 A. They are proportional, because the perimeter always increases by the same amount from column to column. 1 C. multiply 36 by 1 __ 2 A. p 5 50t Side Length Perimeter Which of the following best explains why these values are or are not in a proportional relationship? 1 A. subtract 1 __ 2 from 36 5. Ethan measured the side length and perimeter of five squares, as shown in the table below. B. They are proportional, because the ratio of perimeter to side length is always 4:1. C. They are not proportional, because the side length increases by varying amounts from column to column. D. They are not proportional, because the difference between side length and perimeter is different for each column. B. p 5 ___ 50 t 1 C. p 5 12 __ 2 t 2 D. p 5 ___ 25 t Go On 5 CC12_MTH_G7_NAA_Final.indd 5 22/06/12 9:32 AM Leah solved 40 arithmetic problems in 1 2 __ 2 minutes and 80 arithmetic problems in 5 minutes. Which of the following best represents her unit rate? 10. The line below represents the relationship between time since tickets to a concert at Madison Square Garden went on sale and the number of tickets sold. y B. 8 problems per minute C. 16 problems per minute D. 40 problems per minute 9. The line graphed below represents the total distance traveled by a train over time. Distance (in miles) 500 450 400 350 300 250 200 150 100 50 0 1 2 3 4 5 6 x Time (in hours) Which of the following represents the constant of proportionality (unit rate) for this line? y 400 350 300 250 200 150 100 50 0 Number of Tickets Sold 1 A. 2 __ 2 problems per minute A. 40 B. 50 1 2 3 4 5 6 x Time (in hours) Based on this graph, which of the following statements is true? A. The train traveled 300 miles in 1 hour. B. The train traveled 300 miles in 4 hours. C. The train traveled 150 miles in 3 hours. D. The train traveled 120 miles in 2 hours. C. 80 D. 400 11. Toby bought a pair of jeans and a sweater. The pair of jeans costs $30 and the sweater costs $35. If sales tax is 6%, how much did Toby spend in total for the jeans and sweater? A. $65.12 B. $66.80 C. $68.90 D. $71.00 Go On Duplicating any part of this book is prohibited by law. 8. 6 CC12_MTH_G7_NAA_Final.indd 6 22/06/12 9:32 AM 1 15. The graph below shows the amount of water in a swimming pool over time. y Water in Pool (in gallons) 3 A. __ 8 2 B. __ 3 3 C. __ 2 8 D. __ 3 Ratios and Proportional Relationships 3 __ 12. A park is __ 4 mile long and 2 mile wide. Which fraction represents the ratio of the park’s length to its width? 500 450 400 350 300 250 200 150 100 50 0 1 2 3 4 5 6 7 x Time (in minutes) 2 1 __ 13. Jimena runs __ 3 of a mile in 6 of an hour. What is her unit rate, in miles per hour? A. 2 miles per hour B. 3 miles per hour C. 4 miles per hour D. 9 miles per hour Duplicating any part of this book is prohibited by law. 14. A scale measures with a 1.5% margin of error, which means that the measurement given by the scale may be up to 1.5% lower or higher than the actual weight of the object. If Colin uses this scale to weigh a suitcase and the scale reads 40 pounds, what is the range of possible actual weights for the suitcase? What does the unit rate represent in this relationship? A. The pool is being filled at a rate of 25 gallons per minute. B. The pool is being emptied at a rate of 25 gallons per minute. C. The pool is being filled at a rate of 50 gallons per minute. D. The pool is being emptied at a rate of 50 gallons per minute. A. 34 pounds to 46 pounds B. 35.5 pounds to 44.5 pounds C. 38.5 pounds to 41.5 pounds D. 39.4 pounds to 40.6 pounds Go On 7 CC12_MTH_G7_NAA_Final.indd 7 22/06/12 9:32 AM 1 1 __ 16. A recipe for seafood gumbo calls for 2 __ 2 quarts of fish stock and 1 2 pounds of shrimp. What is the unit rate of fish stock to shrimp in the gumbo? Go On Duplicating any part of this book is prohibited by law. 17. A corporation has agreed to donate money to a charity organization each time one of its employees donates to that charity organization. The corporation donates $3 for every $2 donated by an employee. The total amount the corporation donates to the charity organization is proportional to the total amount that its employees donate. Write an equation representing the relationship between c, the amount the corporation donates, and e, the amount its employees donate. 8 CC12_MTH_G7_NAA_Final.indd 8 22/06/12 9:32 AM 1 60 Time (t) (in hours) Distance (d ) (in miles) 2 120 4 240 6 360 Ratios and Proportional Relationships 18. Naresh drove at a steady speed during a road trip. He recorded the amount of time he drove and the distance he covered, as shown in the table below. A. Plot these points on the graph below and connect them with a straight line. Distance (in miles) d 400 350 300 250 200 150 100 50 0 1 2 3 4 5 6 7 t Time (in hours) B. What are the coordinates of the point on the line where t 5 3? Duplicating any part of this book is prohibited by law. C. What do the coordinates of the point you found in part B specifically tell you about Naresh’s progress at that point in the trip? Go On 9 CC12_MTH_G7_NAA_Final.indd 9 22/06/12 9:32 AM 19. Rachel earns 2% simple annual interest on the money she has in her savings account. Last year she did not deposit to or withdraw from her savings account and earned $35 in interest. A. How much money did Rachel have in her savings account at the beginning of last year? Explain how you set up an equation to solve this. Go On Duplicating any part of this book is prohibited by law. B. How much money did Rachel have in her savings account at the end of last year? 10 CC12_MTH_G7_NAA_Final.indd 10 22/06/12 9:32 AM Ratios and Proportional Relationships 20. Last year, the attendance at the homecoming football game was 300. This year, 360 people attended. A. What was the percent increase from last year to this year? Show your work. Duplicating any part of this book is prohibited by law. B. If the increase in the number of people from this year’s to next year’s homecoming football game is the same as from last year to this year, is the percent of increase the same? Explain your answer. STOP 11 CC12_MTH_G7_NAA_Final.indd 11 22/06/12 9:32 AM Contents Instructional Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Common Core State Standards Correlation Chart . . . . . . . . . . . . . . . . . 12 Domain 1 Ratios and Proportional Relationships. . . . . . . 16 Common Core State Standards Lesson 1 Computing Unit Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7.RP.1 Lesson 2 Identifying Proportional Relationships . . . . . . . . . . . . . . . . . . 20 7.RP.2.a, 7.RP.2.b Lesson 3 Representing Proportional Relationships. . . . . . . . . . . . . . . . 22 7.RP.2.c, 7.RP.2.d Lesson 4 Word Problems with Ratio and Percent. . . . . . . . . . . . . . . . . . 24 7.RP.3 Domain 2 The Number System . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Lesson 5 Adding and Subtracting Rational Numbers . . . . . . . . . . . . . . 28 7.NS.1.a, 7.NS.1.b, 7.NS.1.c Applying Properties of Operations to Add and Subtract Rational Numbers. . . . . . . . . . . . . . . . . . . . 30 7.NS.1.d Lesson 7 Multiplying Rational Numbers. . . . . . . . . . . . . . . . . . . . . . . . . 32 7.NS.2.a, 7.NS.2.c Lesson 8 Dividing Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 7.NS.2.b, 7.NS.2.c Lesson 9 Converting Rational Numbers to Decimals. . . . . . . . . . . . . . . 36 7.NS.2.d Lesson 6 Lesson 10 Problem Solving: Complex Fractions. . . . . . . . . . . . . . . . 38 7.NS.3 Lesson 11 Problem Solving: Rational Numbers. . . . . . . . . . . . . . . . . 40 7.NS.3 Lesson 12 Writing Equivalent Expressions. . . . . . . . . . . . . . . . . . . . . . . . 44 7.EE.1, 7.EE.2 Lesson 13 Factoring and Expanding Linear Expressions. . . . . . . . . . . . . 46 7.EE.1 Lesson 14 Adding and Subtracting Algebraic Expressions. . . . . . . . . . . 48 7.EE.1 Lesson 15 Problem Solving: Algebraic Expressions and Equations. . . . . . . . . . . . . . . . . . . . . . . . 50 7.EE.3 Lesson 16 Word Problems with Equations. . . . . . . . . . . . . . . . . . . . . 52 7.EE.4.a Lesson 17 Word Problems with Inequalities. . . . . . . . . . . . . . . . . . . . . . . 54 7.EE.4.b Domain 4 Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Scale Drawings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.G.1 Lesson 19 Drawing Geometric Shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.G.2 Lesson 18 Problem Solving Fluency Lesson Duplicating any part of this book is prohibited by law. Domain 3 Expressions and Equations . . . . . . . . . . . . . . . . . . . 42 Performance Task 2 CC12_MTH_G7_TM_Final.indd 2 23/07/12 3:34 PM Common Core State Standards Lesson 20 Examining Cross Sections of Three-Dimensional Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7.G.3 Area and Circumference of Circles . . . . . . . . . . . . . . . . . . . . . 64 7.G.4 Lesson 22 Angle Pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.G.5 Problem Solving: Area and Surface Area of Composite Figures . . . . . . . . . . . . . . . . . . . . . . . . 68 7.G.6 Problem Solving: Volume of Three-Dimensional Figures. . . . . . . . . . . . . . . . . . . . . . . . 70 7.G.6 Lesson 21 Lesson 23 Lesson 24 Domain 5 Statistics and Probability . . . . . . . . . . . . . . . . . . . . . 72 Lesson 25 Understanding Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.SP.1, 7.SP.2 Lesson 26 Using Mean and Mean Absolute Deviation . . . . . . . . . . . . . . 76 7.SP.3, 7.SP.4 Making Comparative Inferences about Two Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.SP.3, 7.SP.4 Lesson 28 Understanding Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7.SP.5, 7.SP.6, Lesson 27 7.SP.7.b Lesson 29 Probabilities of Simple Events. . . . . . . . . . . . . . . . . . . . . . . . . 82 7.SP.7.a, 7.SP.7.b Lesson 30 Probabilities of Compound Events . . . . . . . . . . . . . . . . . . . . . 84 7.SP.8.a, 7.SP.8.b Lesson 31 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.SP.8.c Answer Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Math Tools. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Appendix A: Fluency Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Duplicating any part of this book is prohibited by law. Appendix B: Standards for Mathematical Practice. . . . . . . . . . . . . . . . . . B 3 CC12_MTH_G7_TM_Final.indd 3 23/07/12 3:34 PM LE SS O N 1 Computing Unit Rate Learning Objective • Students will set up and simplify complex fractions to find unit rates. Vocabulary complex fraction a fraction in which the numerator and/or denominator is a fraction rate a ratio that compares two quantities that have different units of measure a ratio a comparison of two numbers, written as __ b , a:b, or a to b unit rate a rate in which the second quantity in the comparison is 1 unit Common Core State Standard 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For 1 example, if a person walks __ 2 mile 1 in each __ 4 hour, compute the unit rate as the complex fraction Before the Lesson Review the fact that a ratio compares two quantities and that a rate compares two quantities that compare different units of measure. Pose this problem to the class: Suppose you will bring your favorite beverage to a family picnic. You can ask students to identify their favorite beverages, if you wish. Then ask: Suppose a 2-liter bottle of that beverage costs $4 and a 3-liter bottle of that beverage costs $4.50. Which is the better buy? Give students a chance to determine their answers and then discuss. Students may have used proportional reasoning. For example, they may reason that buying three 2-liter bottles means buying 6 liters for $12 and buying two 3-liter bottles means buying 6 liters for $9, so the 3-liter bottle is the better buy. Other students may have found the unit price per liter for each bottle: 1 __ 2 __ 1 miles per hour, equivalently __ 4 2 miles per hour. So, the 3-liter bottle is the better buy. However students determine the answer, be sure to mention that one way is to determine the unit price for each bottle and compare them. Explain that a unit price is an example of a unit rate and that one way to determine a unit rate is to write a fraction to represent the rate and then simplify it. Segue into the lesson, which shows how to write complex fractions to determine unit rates. Duplicating any part of this book is prohibited by law. $4 _____ 5 $4 4 2L 5 $2.00 per liter 2 liters $4.50 _____ 5 $4.50 4 3 L 5 $1.50 per liter 3 liters 18 CC12_MTH_G7_TM_Final.indd 18 23/07/12 3:35 PM EXAMPLE A This example introduces how to write and simplify a complex fraction to find a unit rate. Use Step 1 to show how to set up the complex fraction. Then use Step 2 to show that in order to get a denominator of 1, students must multiply both the numerator and the denominator by 4. Ask some questions, such as: If the denominator were 1 __ 3 instead, by what number would you multiply? Be 3 1 sure that students see that since __ 3 3 __ 1 5 1, in that case, they would need to multiply by 3 in order 5 __ 2 mi to get a denominator of 1. Finally, review how ____ 1 h 5 1 shows a unit rate of __ 2 miles per hour or 2 __ 2 miles 5 per hour. If necessary, review how to convert __ 2 to 5 1 __ a mixed number: __ 2 5 5 4 2 5 2R1 5 2 2 . TRY MP1 Explain that this problem is a little more 1 complicated. Students must convert 3 __ 8 to an 25 improper fraction, ___ 8 . Since the denominator is not a unit fraction, encourage students to use the EXAMPLE C This example is similar to Example B, since it involves writing a complex fraction with a mixed number as the numerator and a proper fraction as the denominator. The difference is that this problem is solved using a computational method, not a fraction model. CHECK MP3 To reinforce the relationship between multiplication and division, have students use multiplication to check their work. Explanations may vary. Possible explanation: The 23 ___ 4 3 23 __ 3 1 ___ __ __1 5 28 __ rate is __ 4 . This is the same as 4 4 5 5 28 4 . 5 So, to check the answer, I can multiply: 3 1 115 1 115 23 __ ___ __ ___ ___ 28 __ 4 3 5 5 4 3 5 5 20 5 4 fact that a fraction represents division to find the Practice as ___ 8 4 __ 8 . 5 miles per hour 25 1 ___ Work may vary. Possible work: 3 __ 8 5 8 ; As students are working, pay special attention to problems 12–15, which require students not only to find unit rates but also to show their work. When reviewing answers, be sure to review the work that students did to find those answers. For answers, see page 88. 25 ___ 8 quotient. They can rewrite the complex fraction __ __5 8 25 5 25 Duplicating any part of this book is prohibited by law. First, draw 2 squares and divide them in halves 3 by drawing vertical line segments. Shade __ 2 . Then show how those halves can be divided into fourths by drawing horizontal line segments. Note that dividing into halves and then into fourths actually results in squares divided into eighths. Explain 3 that together, all the shaded squares show __ 2 , so students can move the top 3 shaded squares to the right of the other rectangles and still have the model 3 show __ 2 . Demonstrate that this now shows that 3 __ 3 3 3 __ __ 2 3 __ 4 5 2 , so 2 4 4 5 2. ___ 8 x __ __5 5 __ 1 ; 8 25 5 ___ 25 __ 5 8 __ __ ___ 8 4 8 5 8 3 5 5 1 EXAMPLE B This example shows how to write a complex fraction with a mixed number as the numerator and a proper fraction as the denominator. Then it illustrates how to use a fraction model to divide the numerator by the denominator and find the unit rate. Use the fraction model to 3 3 4 . 2 by __ review the conceptual meaning of dividing __ Domain 1 Examples Common Errors When the complex fraction that represents a unit rate includes a mixed number, some students may not rewrite the complex fraction as an improper fraction before computing an answer. Be sure that students understand the importance of this step. 19 CC12_MTH_G7_TM_Final.indd 19 23/07/12 3:35 PM Identifying Proportional Relationships Learning Objectives • Students will show that ratios are equivalent and are therefore proportional and will identify the constant of proportionality for quantities in a proportional relationship. • Students will represent ratios as points on a coordinate grid to show that the ratios are equivalent and will demonstrate that an equation represents a directly proportional relationship. Vocabulary constant of proportionality the constant ratio by which two quantities co-vary in a proportion; also called the unit rate or constant ratio origin the point named by (0, 0) on a coordinate grid, where the axes intersect proportion an equation that shows that two ratios are equivalent Before the Lesson Review the fact that a ratio compares two quantities. Pose this problem: You are in a bike shop looking at bicycles. The number of tires you see in the showroom depends on the number of bicycles in the showroom. How many tires does 1 bicycle have? Draw a table on the board, labeling one row Bicycles and the second row Tires. In the bicycles row, write 2, 3, 4, and 5. Then ask: What is the total number of tires if there are 2 bicycles? 3 bicycles? 4? 5? Fill in the table as shown. Bicycles 1 2 3 4 5 Tires 2 4 6 8 10 Understand Common Core State Standards 7.RP.2 Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b.Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Explain that each column of the table shows a ratio that compares the total number of bicycles to the total number of tires. Ask: If more bicycles are added to the showroom, will the number of tires in the showroom also increase? Use this to help students understand that when ratios are in a proportional relationship, as one quantity increases, the other quantity will also increase. In the case of the above example, as the number of bicycles increases by 1, the number of tires increases by 2. Explain that students can show that quantities in a table show a proportional relationship. Segue into the lesson. Connect Tables can be used to help students recognize proportional relationships. To help develop conceptual understanding, show that the pairs of values in each column of the table showing Tina’s Earnings can be written as a ratio. Use step 1 to 1 show that all of these ratios simplify to __ 12 , so they are all equivalent ratios. Explain that since the ratios are equivalent, the table shows a directly proportional relationship. Explain that Tina’s hourly wage shows the number of dollars she earns if she works 1 hour and that this is also the constant of proportionality. Explain that the constant of proportionality is the Duplicating any part of this book is prohibited by law. LE SS O N 2 20 CC12_MTH_G7_TM_Final.indd 20 23/07/12 3:35 PM Duplicating any part of this book is prohibited by law. Connect students’ understanding of the constant of proportionality to this graph. On page 10, students recognize that the constant of proportionality showed Tina’s hourly wage, $12 per hour. Explain that her hourly wage is a unit rate, because it is the $12 ratio ___ 1 h . Connect this to the graph by showing that the point (1, 12) shows that the constant of proportionality is 12. Remind students that if a graph shows a directly proportional relationship, the point (1, k) shows k, the constant of proportionality. TRY MP2 Use this to illustrate that directly proportional relationships can be represented as equations, as well as in tables and graphs. Answers may vary. Possible answer: Each pair of values makes the equation true. (1, 12): 12 5 12(1) is true. (2, 24): 24 5 12(2) is true. (3, 36): 36 5 12(3) is true. (4, 48): 48 5 12(4) is true. (5, 60): 60 5 12(5) is true. (6, 72): 72 5 12(6) is true. The equation y 5 12x is in the form y 5 kx, so it represents a directly proportional relationship. Since k 5 12, the constant of proportionality is 12. Practice As students are working, pay special attention to problems 2, 5, and 6, which provide an opportunity for students to recognize that not all tables, graphs, and equations show proportional relationships. It is critical that students understand why these representations do not show proportional relationships. Also pay careful attention to questions 9–12, since these are the first times students are asked to determine if verbal statements or mapping diagrams show proportional relationships. Again, be sure that students understand why problems 10 and 11 do not show proportional relationships. For answers, see pages 88 and 89. Domain 1 amount by which each x-value must be multiplied to get each y-value. So, if Tina works for 1 hour, she earns 1 3 12, or 12, dollars. If she works for 2 hours, she earns 2 3 12, or 24, dollars. The amount by which each x-value is multiplied is always the same, 12. Ask: Does it make sense that the constant is 12 given that this is an hourly wage? Have students discuss the fact that the amount Tina earned can be found by multiplying $12 by the number of hours she works. So, it makes sense that the constant of proportionality is also her hourly wage. To extend students’ understanding of the concept that values in a table can represent a directly proportional relationship, show that the pairs of values in the table on page 10 can be plotted as points. Use steps 1 and 2 to illustrate that the points can be connected to form a straight line that passes through the origin, and that this means they show a directly proportional relationship. Common Errors Some students may mistakenly think that the constant of proportionality, k, must be a whole number. If so, they may think that the equation in 1 problem 8, y 5 __ 5 k, does not show a proportional relationship. Be sure to point out that k does not need to be a whole number. For example, if a child earned $0.20 for each cookie she sold, then one could multiply the number of cookies sold by $0.20 1 5 , this could to find the total earned. Since 0.20 5 __ 1 be represented as y 5 0.2x or y 5 __ 5 x. You could also generate points for that situation and graph them to show that they form a straight line. 21 CC12_MTH_G7_TM_Final.indd 21 23/07/12 3:35 PM LE SS O N 3 Representing Proportional Relationships Learning Objectives • Students will write equations to represent proportional relationships. • Students will examine graphs of proportional relationships, interpreting the meanings of points and identifying unit rates. Before the Lesson Explain that one way to represent a word problem is using an expression or equation. Then remind students that a directly proportional relationship can be represented in the form y 5 kx, where k is the constant of proportionality. Because this lesson focuses on directly proportional relationships, the equations should all involve multiplication. Ask: What key words, when they appear in a problem, signal multiplication? Which key words signal an equal sign? Make a chart showing the words students suggest on the board. Include examples of how these words may be used. The chart may look similar to this. Key Word(s) In Words In Symbols times 5 times x 5 3 x or 5x twice twice as much as a number 2 3 n or 2n is equal to Cara's age, c, is equal to twice Justin's age, j c 5 2j is, are, was, were Sam's height, s, is 3 times Betty's height, b s 5 3b Common Core State Standards 7.RP.2 Recognize and represent proportional relationships between quantities. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t 5 pn. d.Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Understand Connect If a relationship is directly proportional, all the (x, y) pairs are related in the same way: y 5 kx, and the constant of proportionality, k, holds for each ordered pair in the relationship. To help develop conceptual understanding, emphasize that although a directly proportional relationship could be displayed in different ways, the equation that represents this relationship is the same. For example, use the given verbal description about charges for a baby-sitting job. Ask: How would the same relationship be displayed in a table of x and y values? Ask a volunteer to express that relationship by making a table of values in which the Hours Worked column increases from 1 to 4 in increments of 1 and the Total Charges column increases from 10 to 40 in increments of 10. Similarly, ask a volunteer to create a graph by plotting the ordered pairs (1, 10), (2, 20), (3, 30), and (4, 40) to express this relationship. Duplicating any part of this book is prohibited by law. Continue to list as many key words and examples as students can think of. Explain that because proportional relationships are always in the form y 5 kx, the key words that indicate multiplication (3) and equality (5) will be useful in this lesson. 22 CC12_MTH_G7_TM_Final.indd 22 23/07/12 3:35 PM As students are working, pay special attention to problems 6 and 7, which provide an opportunity for students to interpret a point on the graph in the context of the given situation and to use the graph to identify the unit rate for this situation. For answers, see pages 89 and 90. Common Errors Some students may make errors when writing equations to represent quantities in a table if they do not pay careful attention to how the x-values and y-values are increasing. For example, in problem 11, 5 if students only recognize that each y is __ 2 more than the previous one, they may mistakenly choose D, 5 y 5 __ 2 x, as the answer. Be sure to point out that 1 since the x-values are increasing by __ 2 s, students cannot simply look at how the y-values are increasing to determine the answer. Ask: How could you determine the relationship in this case? Students may suggest rewriting the table so it only lists values when x is 1, 2, or 3. x y 1 5 2 10 3 15 This shows that as the x-values increase by 1s, the y-values increase by 5s. So, C, y 5 5x, is the correct answer. Duplicating any part of this book is prohibited by law. Practice Domain 1 Although the relationship is displayed in three different ways, the same equation is used for each representation: y 5 kx, where k 5 10, the constant of proportionality. To connect the concept to procedural understanding, stress the importance of determining the meaning of the (x, y) relationship in terms of the context of the situation. Use Step 1 on the Connect page to show that each point on the graph can be interpreted in the context of the given problem. For example, (4, 8) indicates that after 4 minutes, there will be 8 gallons of water in the tank. To test for understanding, ask additional questions: What does the point (2, 4) on the graph represent? Then use Step 2 to show that whatever point on the graph has a first coordinate of 1 will have a second coordinate that shows the unit rate. Connect the knowledge that (1, r) shows the unit rate, r, to the fact that students know that a point (1, k) on the graph of a proportional relationship shows the constant of proportionality, k. Remind students that the unit rate is also the constant of proportionality. TRY MP4 MP6 Use this problem to highlight the fact that the point (1, r) shows a unit rate, but a point (x, 1) does not. Make sure that students understand 1 2 , 1 is that although one of the coordinates of __ a 1, since the first coordinate is not 1, the second coordinate does not show a unit rate. Also use this problem to illustrate that every point on the line graph has a meaning in the context of the problem situation. Even though the graph 1 2 , 1 , it is on the line, so it has shows no dot __ meaning. 1 1 2 , 1 shows that after __ 2 minute, there is The point __ 1 gallon of water in the tub. 23 CC12_MTH_G7_TM_Final.indd 23 23/07/12 3:35 PM