Houston County School System Mathematics

Transcription

1 Student Name: Teacher Name: Grade: 6 th Unit #: 4a Unit Title: Rates, Ratios, and Proportional Reasoning Approximate Start Date of Unit: Approximate End Date (and Test Date) of Unit: The following Statements and examples show the skills, concepts, and understandings that I will gain before the end of this unit. I CAN identify a ratio as a number that relates two quantities or measures within a given situation in a multiplicative relationship. I CAN understand that the order of the numbers is important to the meaning of the ratio. Switching the numbers changes the relationship. The description of the ratio relationship tells us the correct order for the numbers in the ratio. I CAN recognize that ratios appear in a variety of different contexts: part-to-whole, part-to-part, and rates. 1. Using the floor tiles design shown below, create 4 different ratios related to the image. Describe the ratio relationship and write the ratio in the form A:B or the form A to B. 2. Billy wanted to write a ratio of the number of cars to the number of trucks in the police parking lot. He wrote 1: 3. Did Billy write the ratio correctly? Explain your answer. 3. At the 6 th grade school dance, there are 132 boys, 89 girls, and 14 adults. a. Write the ratio of number of boys to number of girls. b. Write the same ratio using another form (A: B vs. A to B). 4. In the cafeteria, 100 milk cartons were put out for breakfast. At the end of breakfast, 27 remained. a. What is the ratio of milk cartons taken to total milk cartons? c. Write the ratio of number of boys to number of adults. d. Write the same ratio using another form. b. What is the ratio of milk cartons remaining to milk cartons taken? 1

2 5. A food company that produces peanut butter decides to try out a new version of its peanut butter that is extra crunchy, using twice the number of peanut chunks as normal. The company hosts a sampling of its new product at grocery stores and finds that 5 out of every 9 customers prefer the new extra crunchy version. a. Make a list of ratios that might be relevant for this situation. i. The ratio of number preferring new extra crunchy to total number surveyed is. ii. The ratio of number preferring regular crunchy to the total number surveyed is. iii. The ratio of number preferring regular crunchy to number preferring new extra crunchy is. iv. The ratio of number preferring new extra crunchy to number preferring regular crunchy is. b. Use the value of each ratio to make multiplicative comparisons for each of the ratios we described here. i. The number preferring new extra crunchy is of the total number surveyed. ii. The number preferring regular crunchy is of the total number surveyed. iii. The number preferring regular crunchy is of those preferring new extra crunchy. iv. The number preferring new extra crunchy is of those preferring regular crunchy. c. If the company normally sells a total of 90,000 containers of regular crunchy peanut butter, how many containers of new extra crunchy peanut butter should it produce, and how many containers of regular crunchy peanut butter should it produce? What would be helpful in solving this problem? Does one of our comparison statements above help us? Student Notes/Comments/Questions: 2

3 I can identify and calculate a unit rate. I can use appropriate math terminology as related to rate. I can analyze the relationship between a ratio a:b and a unit rate a/b where b A publishing company is looking for new employees to type novels that will soon be published. The publishing company wants to find someone who can type at least 45 words per minute. Dominique discovered she can type at a constant rate of 704 words in 16 minutes. Does Dominique type at a fast enough rate to qualify for the job? Explain why or why not. 7. Potatoes are on sale at both Grocery Mart and Baldwin Hills Market. At Grocery Mart, a 5 lb. bag of potatoes cost $2.85, and at Baldwin Hills Market a 7 lb. bag of potatoes costs $4.20. Which store offers the best deal on potatoes? How do you know? How much better is the deal? 8. Write each ratio as a rate. a. The ratio of miles to hours is 434 to 7. b. The ratio of laps to minutes is 5 to Dave can clean pools at a constant rate of 3 5 pools/hour. a. What is the ratio of pools to hours? b. How many pools can Dave clean in 10 hours? c. How long does it take Dave to clean 15 pools? 10. Chandra is an editor at the New York Gazette. Her job is to read each article before it is printed in the newspaper. If Chandra can read 10 words/second, how many words can she read in 60 seconds? 11. Who walks at a faster rate: someone who walks 60 feet in 10 seconds or someone who walks 42 feet in 6 seconds? 12. Mallory is on a budget and wants to determine which cereal is a better buy. A The table below shows the amount of calories Mike burns as he runs. 3

4 ounce box of cereal costs $2.79, and a 13- ounce box of the same cereal costs $3.99. a. Which box of cereal should Mallory buy? Number of Miles Ran Number of Calories Burned ,440 b. What is the difference between the two unit prices? Fill in the missing part of the table 14. The grocery store is selling lettuce at a cost of 5 pounds for $4.00. Part A Carol needs 2 pounds of lettuce and can buy it at the advertised price. If there is no tax on food, is $2.00 enough to buy the lettuce she needs? Show your work. Part B What is the unit rate for lettuce at this grocery store? Show your work and state your answer in a complete sentence. Part C How many pounds of lettuce can you buy for $12.00? Show your work. 15. Lloyd knows that the unit price for dry pinto beans is $0.13 per cup. Explain whether or not Lloyd can buy 14.5 cups of pinto beans for $2.00 or less. 16. Ellen paid $20 to download 25 songs. She stated that this is a rate of $1.25 per download. Explain whether or not Ellen's statement is true. Student Notes/Comments/Questions: I can make a table of equivalent ratios using whole numbers. I can find the missing values in a table of equivalent ratios. 4

5 17. Javier has a new job designing websites. He is paid at a rate of $700 for every 3 pages of web content that he builds. Create a ratio table to show the total amount of money Javier has earned in ratio to the number of pages he has built. Total Pages Built Total Money Earned Javier is saving up to purchase a used car that costs $4,300. How many web pages will Javier need to build before he can pay for the car? 18. Spraying plants with cornmeal juice is a natural way to prevent fungal growth on the plants. It is made by soaking cornmeal in water, using two cups of cornmeal for every nine gallons of water. Complete the ratio table to answer the questions that follow. How many cups of cornmeal should be added to 45 gallons of water? Paul has only 8 cups of cornmeal. How many gallons of water should he add if he wants to make as much cornmeal juice as he can? Cups of Cornmeal Gallons of Water What can you say about the values of the ratios in the table? 19. Assume each of the following represents a table of equivalent ratios. Fill in the missing values Student Notes/Comments/Questions:

6 I can solve real-world and mathematical problems involving ratio and rate, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 20. Josie took a long multiple-choice, end-of-year vocabulary test. The ratio of the number of problems Josie got incorrect to the number of problems she got correct is 2:9. If Josie missed 8 questions, how many did she get right? Draw a tape diagram or table to demonstrate how you found the answer. 21. In a bag of mixed walnuts and cashews, the ratio of number of walnuts to number of cashews is 5:6. Determine the amount of walnuts that are in the bag if there are 54 cashews. Use a tape diagram to support your work. Justify your answer by showing that the new ratio you created of number of walnuts to number of cashews is equivalent to 5: The ratio of Isabella s money to Shane s money is 3:11. If Isabella has $33, how much money do Shane and Isabella have together? Use diagrams to illustrate your answer. 23. At a country concert, the ratio of the number of boys to the number of girls is 2:7. If there are 250 more girls than boys, how many boys are at the concert? 24. A sporting goods store ordered new bikes and scooters. For every 3 bikes ordered, 4 scooters were ordered. However, bikes were way more popular than scooters, so the store changed its next order. The new ratio of the number of bikes ordered to the number of scooters ordered was 5:2. If the same amount of sporting equipment was ordered in both orders and 64 scooters were ordered originally, how many bikes were ordered as part of the new order? 25. Frank has been driving at a constant speed for 3 hours, during which time he traveled 195 miles. Frank would like to know how long it will take him to complete the remaining 455 miles, assuming he maintains the same constant speed. Help Frank determine how long the remainder of the trip will take. Include a table or diagram to support your answer. 26. Most television shows use 13 minutes of every hour for commercials, leaving the remaining 47 minutes for the actual show. One popular television show wants to change the ratio of commercial time to show time to be 3:7. Create two ratio tables, one for the normal ratio of commercials to programming and another for the proposed ratio of commercials to programming. Use the ratio tables to make a statement about which ratio would mean fewer commercials for viewers watching 2 hours of television. Student Notes/Comments/Questions 6

7 I CAN understand that percents are related to part-to-whole ratios and rates where the whole is 100. I CAN model percents. I CAN write ratios as fraction, decimals, and percents interchangeably. 27. One hundred offices need to be painted. The workers choose between yellow, blue, or beige paint. They decide that 45% of the offices will be painted yellow; 20% will be painted blue, and the remaining offices will be painted beige. Create a model that shows the percent of offices that will be painted by each color. Write the amounts as decimals and fractions. Color % Fraction Decimal Yellow Blue Beige 28. Complete the conversions from fraction to decimal to percent. Fraction Decimal Percent % Use the 10x10 grid to express the fraction 11 as a percent.. Use a tape diagram to relate the fraction to a percent. How are the diagrams related? What decimal is also related to the fraction? Student Notes/Comments/Questions I can find a percent of a number as a rate per

8 I can solve real-world problems involving finding the whole, given a part and a percent. 30. What is 15% of 60? Create a model to prove your answer. 31. The Sparkling House Cleaning Company has cleaned houses this week. If this number represents of the total number of houses the company is contracted to clean, how many total houses will the company clean by the end of the week? 32. Of the 25 girls on the Alden Middle School soccer team, 40% also play on a travel team. How many of the girls on the middle school team also play on a travel team? 33. The Alden Middle School girls soccer team won 80% of their games this season. If the team won 12 games, how many games did they play? Solve the question using at least two different methods. 34. Duplicate 35. Micah read 16 pages of his book. If this is 10% of the book, how many pages are in the book? 36. Marissa got an 85% on her math quiz. She had 34 questions right. How many questions were on the quiz? 37. Lucas read 30% of his book containing 480 pages. What page is he going to read next? 38. A pair of jeans was 30% off the original price. The sale resulted in a $24 discount. a. Is the original price of the jeans considered the whole, part, or percent? b. What was the original cost of the jeans before the sale? 39. Mrs. Bennett has graded 20% of her 150 students papers. How many papers does she still need to finish grading? 40.. Jennifer had a coupon for 20% off and a coupon for $15 off at Old Navy. She found a dress to buy for $80. Which coupon should she use? Explain why. 8

9 41. An airplane left Atlanta with 52 empty seats. If the airplane was 75% full, what was the total number of seats on the airplane? Student Notes/Comments/Questions: I can apply ratio reasoning to convert measurement units in real-world and mathematical problems. 42. Beau buys a 3-pound bag of trail mix for a hike. He wants to make one-ounce bags for his friends with whom he is hiking. How many one-ounce bags can he make? 43. Banana slugs can move at 17 cm per minute. If a banana slug travels for 5 hours, how far will it travel? 44. A striped marlin can swim at a rate of 70 miles per hour. Is this a faster or slower rate than a sailfish, which takes 30 minutes to swim 40 miles? 45.. A recipe calls for 2 cups of butter to make 1 layer of a cake. Denise is making a 5-layer cake. How many pints of butter will she need for her cake? 46. A grasshopper can jump 2 feet. A football field is 100 yards long. How many jumps will it take a grasshopper to travel the length of the football field? 47. Rebekah measured two line segments using metric units. The first line segment is 30 centimeters long. The second line segment is 500 millimeters long. What is the total length, in centimeters, of the two line segments. 9