N YS CO MMO N COR E MAT H E MAT I C S C U RR I C U LU M A Story of Ratios A Story of Ratios Tape Diagrams 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Session Objectives Experience how proficiency in the tape diagram

method can be developed in students and colleagues. 2 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Opening Exercise If you have any tape diagramming experience, try to solve this

problem using tape diagrams. If not, try to solve it algebraically. 88 children were in swimming camp. One-third of the boys and three-sevenths of the girls wore goggles. If 34 students wore goggles, how many girls wore goggles? If you finish before time is up, re-write the problem as a ratio problem. 3 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Using Tape Diagrams 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?

5 2012 Common Core, Inc. All rights reserved. commoncore.org A Story of Ratios N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Forms of the Tape Diagram 8 5 ?

5 8 ? 6 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Foundations for Tape Diagrams in PK1 Sara has 2 apples. Jon has 5 apples.

How many apples do they have altogether? How many more apples does Jon have than Sara? 7 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Example 1: Sara has 5 stamps. Mark brings her 4 more

stamps. How many stamps does Sara have now? 8 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Example 2: Sara has 16 stamps. Mark brings her 4 more stamps. How many stamps does Sara have now? 9

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios 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? 10 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Ratios Example 4: Matteo has 5 toy cars. Josiah has 2 more than Matteo. How many toy cars do Matteo and Josiah have altogether? 11 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios

Example 5: Jasmine had 328 gumballs. Then, she gave 132 gumballs to her friend. How many gumballs does Jasmine have now? 12 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Example 6: Jose has 4 paper clips. Harry has twice as many paper clips as Jose. How many paper clips does Harry have?

13 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Example 7: Jose has 4 paper clips. Harry has twice as many paper clips as Jose. How many paper clips do they have altogether? 14

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Example 8: Williams weight is 40 kg. He is 4 times as heavy as his youngest brother Sean. What is Seans weight? 15 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Ratios Example 9: Jamal has 8 more marbles than Thomas. They have 20 marbles altogether. How many marbles does Thomas have? 16 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios

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. 17 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Example 11: Two pears and a pineapple cost $2. Two pears and three pineapples cost $4.50. Find the cost of a pineapple.

18 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Example 12: David spent 2/5 of his money on a storybook. The storybook cost $20 how much did he have at first? 19

2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios 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? 20 2012 Common Core, Inc. All rights reserved. commoncore.org

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Example 14: Jim had 360 stamps. He sold 1/3 of them on Monday and of the remainder on Tuesday. How many stamps did he sell on Tuesday? 21 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios

Example 15: Max spent 3/5 of his money in a shop and 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? 22 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios 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? 23 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios 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? 24 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios 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?

25 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios 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: ______________________________________________ ______________________________________________ ______________________________________________

__________________________________________ 26 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios 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: ______________________________________________

______________________________________________ ______________________________________________ __________________________________________ 27 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios 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: ______________________________________________ ______________________________________________ ______________________________________________ __________________________________________ 28 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Example 20: The ratio of the length of Toms rope to the

length of Jans rope was 3:1. The ratio of the length of Maxwells rope to the length of Jans rope was 4:1. If Tom, Maxwell and Jan have 80 feet of rope altogether, how many feet of rope does Tom have? 29 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios 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? 30 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios 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? 31 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Example 23: The ratio of songs on Jessas phone to songs on Tessies 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?

32 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Opening Exercise: 88 children were in swimming camp. One-third of the boys and three-sevenths of the girls wore goggles. If 34 students wore goggles, how many girls wore goggles? 33

2012 Common Core, Inc. All rights reserved. commoncore.org 88 children were in swimming camp. One-third of the boys and three-sevenths of the girls wore goggles. If 34 students wore goggles, how many girls wore goggles? 88 Children at swim camp Boys Girls 34

54 Wore Did not wear goggles goggles Wore goggles 20 14 34

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Key Points 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: ____________________________________ 37 2012 Common Core, Inc. All rights reserved. commoncore.org N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Ratios Next Steps

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? 38 2012 Common Core, Inc. All rights reserved. commoncore.org