Transcript File
RATIOS AND RATES Objective: RP.01 I can describe two quantities using a ratio. RP.02: I can use a ratio relationship to understand unit rate. Key Vocabulary: (Skip a line between words. • Ratio: an ordered pair of non-negative numbers, which are not both zero. • • • • • Relationship: For every ___, there are ____ Rate: a ratio comparing two different units Units: a fixed quantity used to measure Measurement: the quantity, length, or capacity of something Quantities: amounts Key vocabulary con’t. Key Vocabulary: (Skip a line between words. • • • • Unit: a fixed quantity Numerator: tells how many equal parts are described – (top number in a fraction) Denominator: tells the whole amount being described – (bottom number in a fraction) Reciprocals: two numbers that have a product of 1: ¾ and 4/3 are reciprocals because they equal 12/12 or 1. Essential Questions Skip 3 lines between questions. 1. What is a ratio? How is a ratio different from a fraction? 2. What is a unit rate? How does it compare two quantities? 3. How can a ratio be used to solve for a missing value? Notes: A ratio is an ordered pair of non-negative numbers, which are not both zero. Ratios are written as 3:2, 3 to 2, 3/2. The order of the pair of numbers matters. The description of the ratio relationship determines the correct order of the numbers. Check It Out! Example 1 The Knox soccer team has four times as many boys on it as it has girls. We say the ratio of the number of boys to the number of girls on the team is 4:1. We read this as “four to one.” Let’s make a table to show the possibilities of the number of boys and girls on the soccer team. Discuss in your groups some possibilities. Check It Out! Example 1: Table # of boys 4 # of girls Total # of players 1 5 What are some other options that show four times as many boys as girls or a ratio of boys to girls of 4 to 1? Add your options to your table. Suppose the ratio of number of boys to girls on the team is 3 to 2. Create a new table to show these ratio options. Notes: Another Way to Show Ratios: Tape Diagram or Bar Model: One bar for each number. Boys There are 4 boys to every 2 girls: Girls Class Ratios Find the ratio of boys to girls in our class. Write your ratios in 3 ways: Is the ratio of the number of girls to boys the same as the ratio of boys to girls? When writing Ratios: ORDER MATTERS!! Class Ratios: Group Practice Record a ratio for each of the examples Mrs. Tanaka provides. 1.Find the ratio of boys to girls in our class. 2.You traveled out of state this summer. 3.You are an only child. 4.Your favorite class is math. 5.You have at least one sibling. 6.Your favorite food is spaghetti. Group Work: Using words, describe a ratio that represents each ratio below. Example:1 to 12: for every one year, there are twelve months A.12 to 1 B.2 to 5 C.5 to 2 D.10 to 2 E.2 to 10 Group Discussion: Summarize Your Learning: Answer Essential Questions • What is a ratio? • How is a ratio written? • Does the order of the ratios matter? New Learning: Equivalent Ratios Notes Ratios that make the same comparison are equivalent ratios. Equivalent ratios represent the same point on the number line. To check whether two ratios are equivalent, you can write both in simplest form. Example : Determining Whether Two Ratios Are Equivalent Simplify to tell whether the ratios are equivalent. A. 3 and 2 27 18 3 3 ÷ 3 =1 = 27 27 ÷ 3 9 2 2 ÷ 2 =1 = 18 18 ÷ 2 9 B. 12 and 27 12 = 12 ÷ 3 = 4 15 36 15 15 ÷ 3 5 27 = 27 ÷ 9 3 36 36 ÷ 9 = 4 1= 1 Since , 9 9 the ratios are equivalent. Since 4 3 , 5 4 the ratios are not equivalent. Practice: Are they Equivalent? 13 16 and 39 48 21 28 and 49 56 Shanni and Mel are using ribbon to decorate a project in their art class. The ratio of the length of Shanni’s ribbon to the length of Mel’s ribbon is 7:3. Draw a tape diagram to represent this ratio. Shanni Mel What does each unit on the tape diagram represent? What if each unit on the tape represents 1 inch? What are the 7:3, 7 to 3, 7/3 lengths of the ribbons now? Write the ratio 3 ways. What if each unit represents 3 inches? Write the ratio 3 ways. 21:6, 21 to 6, 21/6 Group Practice: Mason and Laney ran laps for the long-distance running team. The ratio of the number of laps Mason ran to the number of laps Laney ran was 2 to 3. Draw a tape diagram. If Mason ran 4 miles, how far did Laney run? Draw a tape diagram to demonstrate how you found your answer. 6 miles If Laney ran 930 meters, how far did Mason run? Draw a tape diagram to determine how you found your answer. 620 m Are these ratios equivalent? Discuss in your group. Notes: Ratio Relationships Part to Part, Part to Whole, Whole to Part Part to Part: Comparing two parts Part to Whole: Comparing one part to the total amount Whole to Part: Comparing the whole amount to one part Example: Ratio Relationships Part to Part, Part to Whole, Whole to Part Gretchen checked out 3 mystery novels and 2 adventure novels from the library. Part to Part: 3:2 and 2:3 Part to Whole: 3 to 5 and 2 to 5 Whole to Part: 5/3 and 5/2 Group Practice Mrs. Tanaka has 25 students in her math class. 16 of those students are boys and 9 students are girls. Write ratios for the following: Part to Part: Part to Whole: 16 to 9, 9 to 16 Whole to Part: 16 to 25, 9 to 25 25: 16, 25:9 RP.02: I can use a ratio relationship to understand unit rate. A rate is a comparison of two quantities that have different units that do not cancel out. A unit rate is one in which the denominator is 1. Rates are often written using a slash (/) which is read “per”. Examples: 50 miles per hour = 50mi/h(mph) 32 miles per gallon = 32mi/gal(mpg) 20 dollars per hour = $20/h Notes: Unit Rates •A unit price is the ratio of price to the number of units. •Example: •John went to McDonald’s and paid $40 for 5 hamburgers. What was the cost of each hamburger? What do we know? $40 = ????? 5 hamburgers 1 hamburger $40 ÷ 5 = $8 Each hamburger cost $8.00. Another Example A baker buys 25 lb of flour for $74.75. What is the rate or unit price in dollars per pound? Since we are asked for the rate in dollars per pound, the monetary amount must be in the numerator. Price $74.75 74.75 dollars Unit Price = 25 lb Number of units 25 lb Unit Rate: 2.99 dollars per pound or $2.99/lb One pound of flour will cost $2.99 per pound. Notes: Unit Rates Finding unit rates does not always involve money. Example: It took a pet store 10 weeks to sell 80 cats. What is the rate sold per week? 80 cats = ???cats 10 weeks 1 week Group Work: Find the unit rate of each problem. (Use your whiteboards) A jogger travelled 50 kilometers in 5 days. What is the rate he travelled per day? 10 1 For every _______ kilometers travelled, it took ____ day/s. A fair owner made 18 dollars when a group of 3 people entered, 6 which is a rate of _______ per person. A candy company used 8 gallons of syrup to make 4 batches of candy. What is the rate of syrup per batch? 2 Unit prices often vary with the size of the item being sold. Many factors can contribute to determining unit pricing in food, such as variations in store pricing and special discounts. Compare unit prices to determine the best buy for a certain item that is sold in various size containers. Example Find the unit price of a 32 oz bottle of household cleaner and then decide which is the best purchase based on the unit price per ounce. How much will one ounce cost? Size 8 oz 12 oz 16 oz 32 oz Price $1.99 $2.99 $3.49 $6.29 Unit Price 24.875 ¢/oz 24.917 ¢/oz 21.813 ¢/oz 19.656 ¢/oz Based on unit price alone the 32-oz size is the best buy. The unit price for the 32-oz size is given by $6.29 629 cents 629 cents 19.656 cents per ounce 32 oz 32 oz 32 oz Laundry Detergent Comparison A box of Brand A laundry detergent washes 20 loads of laundry and costs $6. A box of Brand B laundry detergent washes 15 loads of laundry and costs $5. What are some equivalent loads? Brand A Loads washed 20 Cost $6 Brand B Loads washed 15 Cost $5 Study sheet for Unit Rates Test: Wednesday A unit price is the ratio of price to the number of units. Practice: If a baker makes 2 dozen donuts in half an hour, how many dozens of donuts can he make in three hours? Make a tape diagram to solve. Wal-Mart sells a 14 ounce of Cheerios for $3.98. What would be the unit price of each ounce. Round if needed. More Practice: Kneaders sells cinnamon muffins for $0.75 each. Mostly Muffins sells a dozen muffins for $7.20. Compare the unit prices and determine the difference in cost. Which place offers the better buy? Daja’s hamster gained 5 ounces in 4 weeks. If the hamster gained the same amount of weight each week, write a ratio to calculate how many ounces the hamster would have gained in 7 weeks. Explain your reasoning in a complete sentence. Draw a bar diagram to solve. More Practice Saul went shopping for marbles. He finds a container of 3 bags for $3.45 and a container of 7 bags for $9.73. Find the unit rate of each container and then state which container is the better buy. Explain your reasoning in complete sentences.