Transcript Document

NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
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
• 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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
Forms of the Tape Diagram
8
5
?
5
8
?
6
© 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
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Example 8: William’s weight is 40 kg. He is 4 times as
heavy as his youngest brother Sean. What is Sean’s
weight?
15
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
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?
21
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
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?
22
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
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?
29
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
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?
32
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
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
goggles
Did not wear
goggles
Wore
goggles
20
14
34
NYS COMMON CORE MATHEMATICS CURRICULUM
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
NYS COMMON CORE MATHEMATICS CURRICULUM
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