© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Tape Diagrams

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Session Objectives 2 Experience how proficiency in the tape diagram method can be developed in students and colleagues.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Opening Exercise 3 If you have any tape diagramming experience, try to solve this problem using tape diagrams. If not, try to solve it algebraically. 94 children are in a reading club. One-third of the boys and three-sevenths of the girls prefer fiction. If 36 students prefer fiction, how many girls prefer fiction? If you finish before time is up, re-write the problem as a ratio problem.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Using Tape Diagrams 4 Promote perseverance in reasoning through problems. Develop students’ independence in asking themselves: “Can I draw something?” “What can I label?” “What do I see?” “What can I learn from my drawing?”

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Forms of the Tape Diagram ? 8 5 ?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Foundations for Tape Diagrams in PK–1 6 Sara has 2 apples. Jon has 5 apples. How many apples do they have altogether? How many more apples does Jon have than Sara?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 7 Example 1: Sara has 5 stamps. Mark brings her 4 more stamps. How many stamps does Sara have now?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 8 Example 2: Sara has 16 stamps. Mark brings her 4 more stamps. How many stamps does Sara have now?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 9 Example 3: Sara brought 4 apples to school. After Mark brings her some more apples, she has 9 apples altogether. How many apples did Mark bring her?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 10 Example 4: Matteo has 5 toy cars. Josiah has 2 more than Matteo. How many toy cars do Matteo and Josiah have altogether?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 11 Example 5: Jasmine had 328 gumballs. Then, she gave 132 gumballs to her friend. How many gumballs does Jasmine have now?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 12 Example 6: Jose has 4 paper clips. Harry has twice as many paper clips as Jose. How many paper clips does Harry have?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 13 Example 7: Jose has 4 paper clips. Harry has twice as many paper clips as Jose. How many paper clips do they have altogether?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 14 Example 8: William’s weight is 40 kg. He is 4 times as heavy as his youngest brother Sean. What is Sean’s weight?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 15 Example 9: Jamal has 8 more marbles than Thomas. They have 20 marbles altogether. How many marbles does Thomas have?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 16 Example 10: The total weight of a football and 10 tennis balls is 1 kg. If the weight of each tennis ball is 60 g, find the weight of the football.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 17 Example 11: Two pears and a pineapple cost $2. Two pears and three pineapples cost $4.50. Find the cost of a pineapple.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 18 Example 13: Alex bought some chairs. One third of them were red and one fourth of them were blue. The remaining chairs were yellow. What fraction of the chairs were yellow?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 19 Example 12: David spent 2/5 of his money on a storybook. The storybook cost $20 how much did he have at first?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 20 Example 14: Jim had 360 stamps. He sold 1/3 of them on Monday and ¼ of the remainder on Tuesday. How many stamps did he sell on Tuesday?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 21 Example 15: Max spent 3/5 of his money in a shop and ¼ of the remainder in another shop. What fraction of his money was left? If he had $90 left, how much did he have at first?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 22 Example 16: Henry bought 280 blue and red paper cups. He used 1/3 of the blue ones and 1/2 of the red ones at a party. If he had an equal number of blue cups and red cups left, how many cups did he use altogether?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 23 Example 17: A club had 600 members. 60% of them were males. When 200 new members joined the club, the percentage of male members was reduced to 50%. How many of the new members were males?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 24 Example 18: Meagan had $1780 and Lisa had $1910. Lisa gave some money to Meagan. In the end Meagan had twice as much money as Lisa. How much money did Lisa give to Meagan? Meagan Lisa $1780 $1910 $3690 Meagan Lisa

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 25 Example 19: Ingrid is mixing yellow and green paint together for a large art project. She uses a ratio of 2 pints of yellow paint for every 3 pints of green paint. Option 1: _____________________________________________ _____________________________________________ _____________________________________________ _____________________________________________

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 26 Example 19: Ingrid is mixing yellow and green paint together for a large art project. She uses a ratio of 2 pints of yellow paint for every 3 pints of green paint. Option 2: _____________________________________________ _____________________________________________ _____________________________________________ _____________________________________________

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 27 Example 19: Ingrid is mixing yellow and green paint together for a large art project. She uses a ratio of 2 pints of yellow paint for every 3 pints of green paint. Option 3: _____________________________________________ _____________________________________________ _____________________________________________ _____________________________________________

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 28 Example 20: The ratio of the length of Tom’s rope to the length of Jan’s rope was 3:1. The ratio of the length of Maxwell’s rope to the length of Jan’s rope was 4:1. If Tom, Maxwell and Jan have 80 feet of rope altogether, how many feet of rope does Tom have?

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 29 Example 21: Lena finds two boxes of printer paper in the teacher supply room. The ratio of the packs of paper in Box A to the packs of paper in Box B is 4:3. If half of the paper in Box A is moved to Box B, what is the new ratio of packs of paper in Box A to Box B? X A B The new ratio of A to B is 2 to 5.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 30 Example 22: Sana and Amy collect bottle caps. The ratio of the number of bottle caps Sana has to the number Amy has is 2:3. The ratio became 5:6 when Sana added 8 more bottle caps to her collection. How many bottle caps does Amy have? Sana Amy 8 Amy has 56 bottle caps.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 31 Example 23: The ratio of songs on Jessa’s phone to songs on Tessie’s phone is 2 to 3. Tessie deletes half of her songs and now has 60 fewer songs than Jessa. How many songs does Jessa have? Jessa Tessie Jessa Tessie 60 Jessa has 240 songs.

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios 32 Opening Exercise: 94 children are in a reading club. One-third of the boys and three-sevenths of the girls prefer fiction. If 36 students prefer fiction, how many girls prefer fiction?

Boys Girls Prefer fiction Children in reading club Prefer non-fiction Prefer fiction 14

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Key Points 34 When building proficiency in tape diagraming skills start with simple accessible situations and add complexities one at a time. Develop habits of mind in students to reflect on the size of bars relative to one another. Part-whole models are more helpful when modeling situations where:_____________________________ Compare to models are best when: ____________________________________

© 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Next Steps 35 How will you share your understanding of modeling with your colleagues? How will you share your understanding of coherence through application with your colleagues? How will your school address standards involving the use of tape diagrams with students new to the process?