A Story of Ratios

© 2012 Common Core, Inc. All rights reserved. commoncore.org

NYS COMMON CORE MATHE MATICS CURRI CULUM A Story of Ratios

A Story of RatiosTape Diagrams


N YS CO MM O N CO R E M AT H E MAT I C S C U R R I C ULU M A Story of Ratios

Session Objectives

2

• Experience how proficiency in the tape diagram method can be developed in students and colleagues.



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.

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.



Using Tape Diagrams

5

• 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?”



Forms of the Tape Diagram

6

8

5 ?

8

5

?



Foundations for Tape Diagrams in PK–1

7

Sara has 2 apples. Jon has 5 apples. How many apples do they have altogether? How many more apples does Jon have than Sara?



8

Example 1: Sara has 5 stamps. Mark brings her 4 more stamps. How many stamps does Sara have now?



9

Example 2: Sara has 16 stamps. Mark brings her 4 more stamps. How many stamps does Sara have now?



10

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?



11

Example 4: Matteo has 5 toy cars. Josiah has 2 more than Matteo. How many toy cars do Matteo and Josiah have altogether?



12

Example 5: Jasmine had 328 gumballs. Then, she gave 132 gumballs to her friend. How many gumballs does Jasmine have now?



13

Example 6: Jose has 4 paper clips. Harry has twice as many paper clips as Jose. How many paper clips does Harry have?



14

Example 7: Jose has 4 paper clips. Harry has twice as many paper clips as Jose. How many paper clips do they have altogether?



15

Example 8: William’s weight is 40 kg. He is 4 times as heavy as his youngest brother Sean. What is Sean’s weight?



16

Example 9: Jamal has 8 more marbles than Thomas. They have 20 marbles altogether. How many marbles does Thomas have?



17

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.



18

Example 11: Two pears and a pineapple cost $2. Two pears and three pineapples cost $4.50. Find the cost of a pineapple.



19

Example 12: David spent 2/5 of his money on a storybook. The storybook cost $20 how much did he have at first?



20

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?



21

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?



22

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?



23

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?



24

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?



25

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?



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 1: ____________________________________________________________________________________________________________________________________________________________________________________



27


Option 2: ____________________________________________________________________________________________________________________________________________________________________________________



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Option 3: ____________________________________________________________________________________________________________________________________________________________________________________



29

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?



30

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?



31

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?



32

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?



33

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?

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

34

20

Boys Girls

Wore

goggles

Children at

swim camp

Did not wear

goggles

54

34

Wore goggles

14



Key Points

37

• 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: ____________________________________



Next Steps

38

• 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?

Documents

A Story of Ratios