Proportional Reasoning Grade Six

Transcription

1 Ohio Standards Connection Number and Number Systems Benchmark D Use models and pictures to relate concepts of ratio, proportion and percent. Benchmark I Use a variety of strategies, including proportional reasoning, to estimate, compute, solve and explain solutions to problems involving integers, fractions, decimals and percents. Indicator 14 Use proportional reasoning, ratios and percents to represent problem situations and determine the reasonableness of solutions. Mathematical Processes Benchmarks A. Clarify problemsolving situation and identify potential solution processes; e.g., consider different strategies and approaches to a problem, restate problem from various perspectives. H. Use representations to organize and communicate mathematical thinking and problem solutions. Lesson Summary: In this lesson, students focus on proportional reasoning. This is an introductory lesson to proportional thinking and reasoning. The pre-assessment determines students understanding of ratios. Students discuss methods used to solve problems with other students. Students solve problems using proportional reasoning in the post-assessments using various methods to find the solution to problem situations. Estimated Duration: Two hours Commentary: Proportional reasoning is based on an understanding of ratios which expresses a mathematical relationship involving multiplication. Students develop proportional reasoning informally. Since proportional reasoning is complex, it needs to be developed over several years. Students need to develop proportional reasoning before the introduction of the crossmultiplication algorithm. (National Research Council, 2001) Pre-Assessment: The pre-assessment is a paper-and-pencil assessment to determine the students prior knowledge of ratios from pictures and problem situations. Distribute Pre-Assessment on Ratios, Attachment A to individual students. Circulate around the room to observe student responses. Collect and evaluate the pre-assessment. Pay attention to how students write the ratios, such as what is being compared. Discuss the problems before beginning the lesson. Provide intervention to students as needed. Scoring Guidelines: Use the guidelines to determine the students understanding of ratios. Adequate Understanding Partial Understanding Needs Intervention Provides correct ratio for each situation showing understanding of part-to-part and part-to whole relationships Provides correct ratios for most situations, but may have errors in showing part-to-part or part-to-whole relationships Provides correct ratios for one or two situations. Shows inadequate understanding of part-to-part and part-to-whole relationships 1

2 Post-Assessment: Distribute Proportional Reasoning Post-Assessment, Attachment C, to each student. Collect for evaluation using Proportional Reasoning Post-Assessment Solutions, Attachment D. Scoring Guidelines: Look for common errors and misconceptions for re-teaching purposes. Three out of four correct responses show understanding of the concepts. Instructional Procedures: Part One Instructional Tip: Before moving to an algorithm for proportions, students need to compare ratios and predict or produce equivalent ratios. 1. Provide cubes of two different colors to pairs of students. Present a situation such as, A package of pens contains one red pen and five black pens. a. Ask students to generate a model with the cubes then create possible ratios to describe the package of pens. (one red pen to five black pens, five black pens to six pens, etc.) b. Select students to share the ratios and record on the board. 2. Ask students to represent two packages of the pens using the cubes to make a model and ratios to describe the pens. Select students to share the ratios. Pose scenarios to extend the situation such as, If 20 black pens are purchased, how many red pens are purchased? The secretary told the manager that she ordered 42 pens. Determine the number of red pens and black pens purchased. 3. Write the following problem on the board. Mrs. Howard planted flowers in her garden. For every marigold she planted, she planted 3 petunias. Her garden has 36 petunias. What is the ratio of marigolds to petunias? How many marigolds did Mrs. Howard plant? (12 marigold plants) a. Have students solve the problems individually. Circulate as they work, noting the strategies students use to solve the problem. Encourage students to draw pictures or use manipulatives to support the answer. b. Select students to share their answers and strategies with the class. Have students write strategy used on the board or overhead projector and explain to the class. Expect students to use tables, diagrams, addition or multiplication to solve the problem. 4. Present another situation to the class. The grocery store is selling 3 oranges for $1. Cheryl has $5. What is the ratio of oranges to dollars? How many oranges can Cheryl buy? (15 oranges) a. Observe students as they solve the problem. b. Select students to explain their strategy and solution to the class. 5. Explain that when two ratios are equal they can be written as an equation. a. Write equations representing the previous scenarios. For example: 1redpen 2 redpens 1marigold 12 marigolds 3 oranges 15 oranges = = = 5 blackpens 10 blackpens 3 petunias 36 petunias $1 $5 2

3 b. Introduce the term proportion and explain that equations showing two equal ratios are called proportions. Proportions can be used to solve problems involving ratios like the scenarios already presented in the lesson. 6. Present the following situation to students to help students understand how to set up and read proportions. A one-ounce bag of candy contains 25 pieces. How many one-ounce bags of candy are required for 250 pieces of candy? Which proportion would help find the number of oneounce bags? a. = 25 b 25 b b. = c. = b 25 1 b d. = a. Have students select the proportion that represents the situation and explain their choice. Assist students by identifying the ratio and how the ratio is expressed as a fraction. Explain that the equivalent ratio has to be expressed the same. For example: the numerator expresses the bags and the denominator expresses the number of pieces in a bag. The fraction representing the equivalent ratio should be the same. Since the unknown is the number of bags, choice d, is correct. b. Select students to explain their choices. Ask questions to get at students understanding. Note and discuss any misconceptions. 7. Have students write a proportion for the following situation. It costs $35 for 2 hours of horseback riding. How much will it cost for 5 hours? 35 c 2 35 ( = OR = ) c a. Have students write the proportion before discussing with a partner. Circulate as students discuss the proportion, listening for misconceptions. b. Select students to write their proportion on the board or overhead projector. Have students explain the rationale for the relationships in the proportion. Use a method to allow the class to agree or disagree with the proportion. Select students to explain why they agree or disagree with the proportion. How is the proportion set up differently than the bags of candy scenario? c. Continue presenting situations with students writing the proportion as needed. 8. Use Writing Proportions, Attachment E and have students write the proportion and an explanation for writing the proportion. a. Collect the paper and evaluate for understanding, using Writing Proportions, Attachment F. b. Return the papers and have a class discussion on the problems. 9. Have students explain the relationships in a proportion in their journals. Check journal entries for understanding. Part Two 10. Present different situations to students such as: a. Malik has 18 ink pens. For every red pen, he has 5 black pens. What is the ratio of red pens to black pens? How many pens of each color could Malik have? 3

4 b. The ratio of adults to students is 1 to 10 for the party. There were 45 adults and 50 students. Explain whether there were enough students for the number of adults at the party. Determine the number of adults needed for a party of 50 students. Have students solve problems individually before sharing with a partner. Select students to explain strategies and answers to the class. 11. Use situations in which students collect data, create tables and determine the relationship between the pairs in the table. Provide students with a sheet of different figures (rectangles or triangles) with the dimension so that each is similar to two or more rectangles or triangles. Have students groups the figures that are similar and create tables showing the ratios of width to length, width to width, length to length, etc. Have students describe the relationships of the ratios in each group. 12. Provide situations and contexts for students to express using proportions. Differentiated Instructional Support: Instruction is differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). Assign peer tutors. Use a variety of visual aids and manipulatives. Use a calculator. Give more examples as deemed necessary and/or instruction based on individual needs. Home Connections: Have students bring their favorite recipes from home and have them determine the proportions of the ingredients to feed a large group or a group smaller than the recipe lists. Have students look for examples of scale drawings of maps, blue prints, etc. and create a class list of examples. Materials and Resources: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not endorse any particular resource. The Web addresses listed are for a given site s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. For the teacher: Chart paper, board or overhead projector, cubes or other manipulatives of two different colors For the student: Pencil, journal, manipulatives 4

5 Vocabulary: equivalent proportion proportional reasoning ratio unit Proportional Reasoning Grade Six Technology Connection: Use the Internet to conduct a virtual field trip to compare prices to find the best buy. Research Connections: National Research Council. Adding It Up: Helping Children Learn Mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, D.C.: National Academy Press, Sousa, David A. How the Brain Learns: A Classroom Teacher s Guide. Reston, Va: NASSP, General Tips: Research has shown that if middle school students are taught to only use the algorithm (crossmultiplication) approach, students might not develop proportional reasoning. Attachments: Attachment A, Proportional Reasoning Grade Six Pre-Assessment Attachment B, Proportional Reasoning Grade Six Pre-Assessment Solutions Attachment C, Proportional Reasoning Grade Six Post-Assessment Attachment D, Proportional Reasoning Grade Six Post-Assessment Sample Solutions Attachment E, Writing Proportions Attachment F, Writing Proportions Answer Key 5

6 Attachment A Pre-Assessment Name Date Directions: Write a ratio for each problem. 1. A fruit stand at the store had apples, oranges and bananas at the end of the day. a. write a ratio for apples to oranges b. write a ratio for oranges to all fruit 2. The students in Ms. Davis s class at King Elementary School walk or ride the bus to school. The number of students who ride the bus is 18 and the number of students who walk is 12. What is the ratio of the number of students who walk to school to the number of students who ride the bus. 3. Mary had 24 pencils. She had 8 green pencils, 10 red pencils and 6 yellow pencils. a. What is the ratio of red pencils to the total number of pencils? b. What is the ratio of green pencils to red pencils? 4. Carmen made a trail mix of nuts, raisins and chocolate chips. She used 2 cups of nuts, 3 cups of raisins and 5 cups of chocolate chips. a. What is the ratio of nuts to chocolate chips? b. What is the ratio of nuts to raisins? c. What is the ratio of chocolate chips to trail mix? d. What is the ratio of nuts and raisins to chocolate chips? 6

7 Directions: Write a ratio for each problem. Attachment B Pre-Assessment Answer Key 1. A fruit stand at the store had apples, oranges and bananas at the end of the day. a. The ratio of apples to oranges is 4 to 3 or 4:3 or 3 4 b. The ratio of bananas to all fruit is 6 to 13 or 6:13 or The students in Ms. Davis s class at King Elementary School walk or ride the bus to school. The number of students who ride the bus is 18 and the number of students who walk is 12. What is the ratio of the number of students who walk to school to the number of students who ride the bus. The ratio of students who walk to students who ride the bus is 12:18 or 2:3. 3. Mary had 24 pencils. She had 8 green pencils, 10 red pencils and 6 yellow pencils. a. The ratio of red pencils to the total number of pencils is 10 to 24. b. The ratio of green pencils to red pencils is 8:10 or 4:5. 4. Carmen made a trail mix of nuts, raisins and chocolate chips. She used 2 cups of nuts, 3 cup of raisins and 5 cups of chocolate chips. a. The ratio of nuts to chocolate chips is 2 to 5. b. The ratio of nuts to trail mix is c. The ratio of chocolate chips to trail mix is 5 to 10 or 1 to 2. d. The ratio of nuts and raisins to chocolate chips is 5:5 or 1 to 1. 7

8 Attachment C Post-Assessment Name Date Directions: Solve each problem. Use pictures, words or a table to support your answer. 1. Chad has 1 blue pencil for every 4 green pencils. How many blue pencils does Chad have if he has 24 green pencils? 2. Madison bought 24 apples for $6. How many apples could Madison buy for $2? 3. The 5 th grade classes of 125 students are going on a picnic. They need 3 adults for every 25 students. How many adults are needed at the picnic? 4. The cafeteria made 250 cookies. Each student can buy 4 cookies for $1. How many cookies did the cafeteria sell, if the cafeteria made $40? 5. Sharita has 20 pencils. For every yellow pencil, she has 5 purple pencils. How many pencil of each color could Sharita have? 8

9 Attachment D Post-Assessment Sample Solutions Directions: Solve each problem. Use pictures, words or a table to support your answer. 1. Chad has 1 blue pencil for every 4 green pencils. How many blue pencils does Chad have if he has 24 green pencils? Blue Pencils Green Pencils is a multiple of 4 since 4 6 = 24. Chad has 6 green pencils. = Madison bought 24 apples for $6. How many apples could Madison buy for $2? 24 8 Madison can buy 4 apples for $1, therefore she can buy 8 apples for $2. = $6 $2 3. The 5 th grade classes of 125 students are going on a picnic. They need 3 adults for every 25 students. How many adults are needed at the picnic? Number of Adults Number of Students Fifteen (15) adults are needed at the picnic. 4. The cafeteria made 250 cookies. Each student can buy 4 cookies for $1. How many cookies did the cafeteria sell, if the cafeteria made $40? = 40 and 40 4 =160 = The cafeteria sold 160 cookies. $1 $40 5. Sharita has 20 pencils. For every yellow pencil, she has 5 purple pencils. How many pencils of each color could Sharita have? 1 yellow pencil 4 yellow pencils 1 yellow 4 yellow = = 4 purple pencils 16 purple pencils 5 pencils 20 pencils 9

10 Attachment E Writing Proportions Name Date Directions: Write a proportion for each situation. Provide an explanation for the proportion. 1. The candy maker discovered that for every 5 pieces of wrapped candy, 2 were not wrapped. How many pieces of candy were not wrapped if 2,750 were wrapped? 2. A bag contains 10 oranges. How many bags are needed for 350 oranges? 3. The bookstore has books on sale at 4 for $10. What is the cost of 14 books at this rate? 4. The volleyball team won 15 out of 18 games. At this rate, how many games will they win if they play 72 games? 10

11 Attachment F Writing Proportions Answer Key Directions: Write a proportion for each situation. Provide an explanation for the proportion. 1. The candy maker discovered that for every 5 pieces of wrapped candy, 2 were not wrapped. How many pieces of candy were not wrapped if 2,750 were wrapped? = 2 n Each ratio shows the relationship of candies that are wrapped to candies that are not wrapped. 2. A bag contains 10 oranges. How many bags are needed for 350 oranges? 1 b = Each ratio shows the relationship of the number of bags to the number of oranges in a bag. 3. The bookstore has books on sale at 4 for $10. What is the cost of 14 books at this rate? 4 14 = 10 b Each ratio shows the relationship of the number of books to the cost of the books. 4. The volleyball team won 15 out of 18 games. At this rate, how many games will they win if they play 72 games? 10 w = Each ratio shows the relationship of the number of games won to the number of games played. 11