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chapter 5

chapter 5. Proportions and Similarity. During this chapter, Students will…. distinguish between situations that are proportional or not proportional use proportions to solve problems apply proportionality to measurement in multiple contexts, including scale drawings and constant speed

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chapter 5

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  1. chapter 5 Proportions and Similarity

  2. During this chapter, Students will… • distinguish between situations that are proportional or not proportional • use proportions to solve problems • apply proportionality to measurement in multiple contexts, including scale drawings and constant speed • solve problems involving similar figures • determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures

  3. Standards & Vocabfor 5-1-B: Rates • GLE 0706.2.3- Develop an understanding and apply proportionality • GLE0706.2.4- Use ratios, rates, and percents to solve single-and multi-step problems in various contexts • SPI 0706.2.7- Use ratios and proportions to solve problems. Main Idea: Determine unit rates. • rate • unit rate

  4. Your goals!!! • I will be able to show what I know about finding unit rates by correctly solving at least 5 out of the 7 real world problems shown in the Power Point. • With a partner, I will be able to create a visual to explain how to find the unit rate from a newspaper ad. I will explain my answer in sentence form and with a mathematical expression. • I will be able to explain the difference between rate and unit rate after this lesson. • Tonight, I will get at least 75% of the homework problems correct. • Next week, I will get at least 80% of the rate problems correct on our Friday Quiz!

  5. Explore! • Do you know where to find your pulse? • Your neck or your wrist • For two minutes, count the number of beats. • Then, write the ratio of beats to minutes as a fraction. A ratio that compares two quantities with different kinds of units is called a RATE! When a rate is simplified so that it has a denominator of 1 unit, it is called a UNIT RATE

  6. 5-1-b rates

  7. find a unit rate Adrienne biked 24 miles in 4 hours. If she biked at a constant speed, how many miles did she ride in one hour? Adrienne biked 6 miles in 1 hour. Find each unit rate. Round to the nearest hundredth if necessary. $300 for 6 hours 220 miles on 8 gallons Answers: $50 per hour 27.5 miles per gallon

  8. find a unit rate Find the unit price if it cost $2 for eight juice boxes. Round to the nearest cent if necessary. The unit price is $0.25 per juice box! Practice!!! Find the unit price. Round to the nearest hundredth if necessary. Find the unit price if a 4-pack of mixed fruit sells for $2.12. Julia read 52 pages in 2 hours. What is the average number of pages she reads per hour? Find the unit price per can if it costs $3 for 6 cans of soda. Answers: $0.53 26 pages/hr $0.50 per can

  9. compare with unit rates Mrs. Bybee is shopping for Layla’s dog food. The prices of 3 different bags of dog food are given in the table. Mrs. Bybee wants to save some money so she need to know which size has the lowest price per pound? 40 lb bag - $1.225 per pound 20 lb bag - $1.172 per pound 8 lb bag- $1.235 per pound So…the 20 pound bag is the best buy! HINT: Find out the unit price for each….the price per pound!

  10. practice comparing Ms. Holloway wants to buy some peanut butter to donate to the Second Harvest Food Bank so that her homeroom will win the food drive. If Ms. Holloway wants to save as much money as possible, which brand should she buy? Peter Pan will be the best buy! The results of a swim meet are shown. Who swam the fastest? Be sure to show all of your work! Self Assessment: Try p. 268 #1-6 on your own. Then you may check with a partner.

  11. Standards & vocab for 5-1-C: Relationships • GLE 0706.2.3- Develop an understanding and apply proportionality • GLE0706.2.4- Use ratios, rates, and percents to solve single-and multi-step problems in various contexts • SPI 0706.1.3 Recognize whether information given in a table, graph, or formula suggests a directly proportional, liner, inversely proportional, or other nonlinear relationship. • SPI 0706.2.7- Use ratios and proportions to solve problems. Main Idea: Identify proportional and non proportional relationships • proportional • non proportional

  12. 5-1-c Proportional and Non proportional Relationships Mrs. Bybee and Ms. Holloway are planning a year-end pizza party . Little Italy Pizza offers delivery and charges $8 for each medium pizza. For each number of pizzas, we are going to write the relationship of the cost and number of pizzas as a ratio in simplest form. What do you notice? The cost of an order is proportional to the number of pizzas ordered. Two quantities are proportional if they have a constant ratio. If the relationship in which the ratio is not constant, the quantities are nonproportional.

  13. proportional or nonproportional? Papa John’s sells medium pizzas for $7 each but charges a $3 delivery fee per order. Is the cost of an order proportional to the number of pizzas ordered? Explain. For each number of pizzas, write the relationship of the cost and number of pizzas as a ratio in simplest form. Since the ratios of the two quantities are NOT the same, the cost of an order is NOT PROPORTIONAL to the number of pizzas ordered.

  14. proportional or nonproportional? You can use the recipe shown to make a healthier version of a popular beverage. Is the amount of mix used proportional to the amount of sugar used? Explain. Remember that when you are dividing a mixed number, you can change it to an improper fraction. Don’t forget to multiply by the reciprocal! On your own, simplify each of the ratios written above. Are they equal? Is this a proportional or non proportional relationship?

  15. proportional or nonproportional? Look at the chart to the right. Is the amount of sugar used proportional to the amount of water used? Show all of your work on your paper! At the beginning of the year, Isabel had $120 in the bank. Each week, she deposits another $20. Is her account balance proportional to the number of weeks of deposits? This time, create your own chart and then find the ratios! A cleaning service charges $45 for the first hour and $30 for each additional hour. Is this fee proportional to the number of hours worked? Make a table of values to explain your reasoning. Self Assessment: Try p. 273 #1-4 on your own. Then you may check with a partner.

  16. Standards & Vocab for 5-1-D: Solving proportions • SPI 0706.1.1 Use proportional reasoning to solve mixture/concentration problems • SPI 0706.2.7- Use ratios and proportions to solve problems. • GLE 0706.2.3- Develop an understanding and apply proportionality • GLE0706.2.4- Use ratios, rates, and percents to solve single-and multi-step problems in various contexts Main Idea: Use proportions to solve problems. • equivalent ratios • proportion • cross products

  17. 5-1-d Solve proportions Kohl’s advertised a sale as shown at the left. Write a ratio in simplest form that compares the cost to the number of bottles of nail polish. Suppose Kate and some friends wanted to buy 6 bottles of polish . Write a ratio comparing the cost to the number of bottles of polish. Is the cost proportional to the number of bottles of polish purchased? Explain. The ratios of the cost to the number of bottles of polish for two and six bottles are both equal to 5/2. They are equivalent ratios because they have the same value!

  18. proportions There are two ways to tell if two ratios form a proportion. Either you must: Show that cross products are equal Show that they simplify into equivalent fractions.

  19. write and solve a proportion After 2 hours, the air temperature had risen 7°F. Write and solve a proportion to find the amount of time it will take at this rate for the temperature to rise an additional 13°F. Solve each proportion below. a.) 3.6 b.) 85 c.) 4.9

  20. solve using proportions FYI: There are four different blood types: A, B, AB, and O. People with Type O are considered universal donors. Their blood can be transfused into people with any blood type. During a blood drive, the ratio of Type O donors to non-Type O donors was 37:43. About how many Type O donors would you expect in a group of 300 donors?

  21. Solve using proportions Janie can decorate 8 T-shirts in 3 hours. Write and solve a proportion to find the time it will take her to decorate 20 T-shirts at this rate. A recipe serves 10 people and calls for 3 cups of flour. If you want to make the recipe for 15 people, how many cups of flour should you use? Recycling 2,000 pounds of paper saves about 17 trees. Write and solve a proportion to determine how many trees you would save by recycling 5,000 pounds of paper.

  22. Write and use an equation Beth bought 8 gallons of gasoline for $31.12. Write an equation relating the cost to the number of gallons of gasoline. How much would Beth pay for 11 gallons at this same rate? So…How much would Beth pay for 11 gallons at this same rate? Olivia typed 2 pages in 15 minutes. Write an equation relating the number of minutes m to the number of pages p typed. If she continues typing at this rate, how many minutes will it take her to type 1- pages? to type 25 pages? Self Assessment: Try p. 278 #1-5 on your own. Then you may check with a partner.

  23. Standards & Vocabfor Wildlife Sampling • SPI 0706.2.7- Use ratios and proportions to solve problems. • GLE 0706.2.3- Develop an understanding and apply proportionality Main Idea: Use proportions to estimate populations.

  24. 5-1-d Extend:Wildlife Sampling Naturalists can estimate the population in a wildlife preserve by using the capture-recapture technique. You will model this technique using dried beans in a bowl to represent bears in a forest. Fill a small bowl with dried beans Scoop out some of the beans. These represent the original captured bears. Count and record the number of beans. Mark each bean with an X on both sides. Then return these beans to the bowl and mix well. Scoop another cup of beans from the bowl and count them. This is the sample for Trial A. Count the beans with the X’s. These are the recaptured bears. Record both numbers in a table. Use the proportion below to estimate the total number of beans in the bowl. This represents the total population P. Record P in the table. Return all of the beans to the bowl. Repeat steps 2-4 nine times!

  25. Standards & Vocab for 5-2-b: scale drawings • SPI 0706.1.4- Use scales to read maps • .GLE 0706.2.3- Develop an understanding and apply proportionality • SPI 0706.2.7- Use ratios and proportions to solve problems Main Idea: Solve problems involving scale drawings. • scale drawing • scale model • scale • scale factor

  26. 5-2-b scale drawings Scale drawings and scale models are used to represent objects that are too large or too small to be drawn or built at actual size. The scale gives the ratio that compares the measurement of the drawing or model to the measurements of the real object. The measurements on a drawing or model are proportional to the measurements on the actual object. What is the actual distance between Hagerstown and Annapolis? You would first need to use a centimeter ruler to find the distance on the map between the two cities. Then, you would write and solve a proportion using the scale.

  27. use a map scale On the map of Arkansas shown, find the actual distance between Clarksville and Little Rock. Use a proportion to solve. (The ruler measures 4cm.) Refer the the map of South Carolina. What is the actual distance between Columbia and Charleston. Use a proportion to solve. (The ruler measures 3.8 cm.)

  28. use a scale model A graphic artist is creating an advertisement for a new cell phone. If she uses a scale of 5 inches = 1 inch, what is the length of the cell phone on the advertisement? TRY THIS ONE! A scooter is 3 ½ feet long. Find the length of a scale model of the scooter if the scale is 1 inch = ¾ feet. 4 2/3 inches.

  29. scale factor SCALE FACTOR- A scale written as a ratio without units in simplest form Find the scale factor of a model sailboat if the scale is 1 inch = 6 feet. Tip: Scale factors can be used to calculate actual distances from the distances shown in a scale drawing or map. If, for example, a drawing has a scale factor of 1/96, then something that measures 1 inch in the drawing will actually measure 96 inches, or 8 feet! Find the scale factor of a model car if the scale is 1 inch = 2 feet. Find the scale factor of a blueprint if the scale is ½ inch = 3 feet. Answers: 1/72; 1/24; 1/72

  30. construct a scale model Zara is making a model of a Ferris wheel that is 60 feet tall. The model is 15 inches tall. Zara is also making a model of the sky needle ride that is 100 feet tall using the same scale. How tall is the model? Try This One! Julianne is constructing a scale model of her family room to decide how to redecorate it. The room is 14 feet long by 18 feet wide. If she wants the model to be 8 inches long, about how wide will it be? Self Assessment: Try p. 287 #1-8 on your own. Then you may check with a partner.

  31. Standards & Vocab for 5-3-a: similar figures • GLE 0706.4.1 Understand the application of proportionality with similar triangles. • SPI 0706.4.1 Solve contextual problems involving similar triangles Main Idea: Solve problems involving similar figures • similar figures • corresponding sides • corresponding angles • indirect measurement • Side-Side-Side Similarity (SSS) • Angle-Angle Similarity (AA) • Side-Angle-Side Similarity (SAS)

  32. 5-3-A Similar figures SIMILAR FIGURES- Figures that have the same shape but not necessarily the same size

  33. similar figures • Congruent or Similar? • Congruent figures are the same SIZE AND SHAPE • Similar figures are the SAME SIZE but not necessarily the same shape. So… since corresponding sides are proportional, if you have to find a missing side length, write and solve a proportion.

  34. find missing measures In the second example, the triangles are positioned differently. You might want to re-draw the figures to help you set up the proportions correctly!

  35. indirect measurement Old Faithful in Yellowstone National Park shoots water 60 feet into the air and casts a shadow of 42 feet. What is the height of a nearby tree that casts a shadow of 63 feet long? Assume the triangles are similar. Daley wants to resize a 4-inch-wide by 5-inch-long photograph so that it will fit in a space that is 2 inches wide. What is the new length? 2.5 in

  36. indirect measurement At a certain time of day, a cabbage palm tree that is 71 feet high casts a shadow that is 42.6 feet long. At the same time, a nearby flagpole casts a shadow that is 15 feet long. How tall is the flagpole? Self Assessment: Try p. 296 #1-4 on your own. Then you may check with a partner.

  37. Standards & Vocabfor 5-3-B: Perimeter & Area of Similar Figures • GLE 0706.4.3- Understand and use scale factor to describe the relationships between length, area, and volume. Main Idea: Find the relationship between perimeters and areas of similar figures • perimeter • area

  38. 5-2-b perimeter and area of similar figures Suppose you double each dimension of the rectangle at the right. The new rectangle is similar to the original rectangle with a scale factor of 2. What is the perimeter of the original rectangle? What is the perimeter of the new rectangle? How is the perimeter of the new rectangle related to the perimeter of the original rectangle and the scale factor? In SIMILAR FIGURES, the perimeters are related by the scale factor! What about the area? Use the example rectangle above to think about what happens to the area? So…the area of the new rectangle is equal to the area of the original rectangle times the square of the scale factor!

  39. Perimeter and area of similar figures

  40. determine perimeter • Two rectangles are similar. One has a length of 6 inches and a perimeter of • 24 inches. The other has a length of 7 inches. What is the perimeter of this rectangle? • First, think: What is the scale factor. • Next, multiply the perimeter by the scale factor. Triangle LMN is similar to triangle PQR. If the perimeter of ΔLMN is 64 meters, what is the perimeter of ΔPQR? 48m

  41. determine area The Eddingtons have a 5-food by 8-food porch on the front of their house. They are building a similar porch on the back with double the dimensions. Find the area of the back porch. Think: What is the scale factor? What is the original area? How is the area affected by the scale factor?

  42. practice! Reminders: What is the scale factor? Are you finding what happens to the PERIMETER or AREA? Look @ your notes and think about if you multiply by the scale factor or (scale factor)2. Two rectangles are similar. One has a length of 10 inches and a perimeter of 36 inches. The other rectangle has a length of 7.5 inches. What is the perimeter of this rectangle? Maria is painting a mural on her bedroom wall. The image she is reproducing is 1/20 of her wall and has an area of 36 square inches. Find the area of the mural. The Coopers bought a 6-food by 9-foot rectangular rug. They would like to buy a similar rug with double the dimensions. What will be the area of a new rug? Self Assessment: Try p. 301 #1-5 on your own. Then you may check with a partner.

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