Understanding Ratio and Proportion in Middle School Mathematics
Iowa Common Core Standards for Mathematics
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What Time Is It? How did you use proportional reasoning to
solve this problem?
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Line of Learning Journal Entry #1 What do I know about teaching
ratio and proportion?
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Outcomes Participants will: Understand the Iowa Core Ratio and
Proportional Relationships Standards in 6 th and 7 th Grade
Experience a variety of meaningful strategies and models to develop
ratios and proportional relationships Become familiar with the 6-7,
Ratio and Proportional Relationships Progressions document Build an
awareness of the instructional shifts necessary to implement the
Iowa Common Core Ratio and Proportional Relationship standards
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Take a moment to reflect on one grade level that you teach, and
jot down the 4 most important math ideas that you currently focus
on in that grade level.
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Mathematics | Grade 6 In Grade 6, instructional time should
focus on four critical areas: (1) connecting ratio and rate to
whole number multiplication and division and using concepts of
ratio and rate to solve problems; (2) completing understanding of
division of fractions and extending the notion of number to the
system of rational numbers, which includes negative numbers; (3)
writing, interpreting, and using expressions and equations; and (4)
developing understanding of statistical thinking.
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Mathematics | Grade 7 In Grade 7, instructional time should
focus on four critical areas: (1) developing understanding of and
applying proportional relationships; (2) developing understanding
of operations with rational numbers and working with expressions
and linear equations; (3) solving problems involving scale drawings
and informal geometric constructions, and working with two- and
three-dimensional shapes to solve problems involving area, surface
area, and volume; and (4) drawing inferences about populations
based on samples.
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Ratios and Proportions Its Critical! Read Paragraph #1 from the
Critical Areas in 6 th grade (pg. 41) and 7 th grade (pg. 47).
Record: Key ideas or concepts Questions that you may have Share
your ideas and questions with your group. Groups share out
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Lets Hear from the Experts! At your table is an envelope of
quotes and cited research about ratio and proportional reasoning.
Share the slips of paper around the table and discuss the
implications of each in relation to instruction of ratio and
proportion. When groups are finished, we will share out some of the
big ideas.
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Lets begin with a problem Tawnya and Julie are running equally
fast around a track. Tawnya started first. When she had run 9 laps,
Julie had run 3 laps. When Julie had completed 15 laps, how many
laps had Tawnya run?
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Listen to a group of teachers discussing this problem How did
the cross-products method keep some of these teachers from solving
the problem correctly? How do the methods we use to teach
proportional reasoning either hinder or develop the way students
understand these concepts?
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By viewing equivalent ratios and rates as derived from, and
extending, pairs of rows (or columns) in the multiplication table
students connect their understanding of multiplication and division
with ratios and rates. - 6 th Grade Critical Areas #1, Iowa Core
Mathematics pg. 41 Every day, Robin and Tim each save at a constant
rate. If, on a certain day, Robin has $6 and Tim has $10, then how
much will Tim have when Robin has $21?
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What are Ratios? 6.RP.1 Understand the concept of a ratio and
use ratio language to describe a ratio relationship between two
quantities. A diagram might help us to make sense of the different
types of ratios
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Ratios Different Measures Same Measures Rate Part-to-Part Ratio
Part-to-Whole Ratio Colon Notation (preferred) 3 cats for every 4
dogs 3:4 3 cats 4 dogs Fraction Notation (preferred) 3 out of 7
pets are cats 3:7 3 7 When using fraction notation for part-to-part
ratios, label the parts to avoid confusing ratios with
fractions.
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Line of Learning Journal Entry #2 What can I add about teaching
ratio and proportion?
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Problem TypesExamplesRationale Part-Part-Whole25 students in
groups of 5; 3 girls in each group. How many boys in the class?
Students are inclined to use informal methods of reasoning since
these problems lend themselves to counting, matching, and building
up strategies. Associated Sets3 balloons for $2. How much would 24
balloons cost? Students use a high level of proportional reasoning
since they are forced to think of two sets that are not typically
associated as a composite Well-Known Measures 6 gallons to drive
156 miles. At this rate, can he drive 561 miles on 21 gallons? The
familiar language may allow them to mask their understanding. They
may have learned formulas to solve these problems, but they may not
see the proportional relationship. For example, in miles per hour
problems, they may not understand that it tells you how far you go
in each hour. Growth (Stretching and Shrinking) A 6x8 photo is
enlarged. The width changes from 8 inches to 12 inches. What is the
height on the new photo? Research says these tend to be the most
difficult because the quantities are continuous rather than
discrete (more difficult to represent with objects or
pictures).
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Really?!? So which problem type seems to be the struggle for
this person? http://www.youtube.com/watch?v=Qhm7-LEBznk
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How Can We Build Better Understanding of Proportional
Relationships? Read the Iowa Core Ratio and Proportion Standards
for 6 th and 7 th Grade. Jot down all the different types of
representations that are listed in the standards.
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Representation #1 Double Number Line
http://threeacts.mrmeyer.com/nana/
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A.How far does Felipe walk in 15 minutes? B.How far does Felipe
walk in 1 hour? C.How long does it take Felipe to walk 4 miles?
D.How long does it take for Felipe to walk 3 miles? Felipe walks 2
miles in 45 minutes at a constant rate. Use the model below to
answer the questions about how far Felipe walks.
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Dueling Discounts Make an argument about which coupon is the
better deal. Tell us under which circumstances it is better. -
adapted from Dan Meyers A Problem in Three Acts website
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Representation #2 Tape Diagrams Oobleck is made by mixing glue
and liquid laundry starch in a ratio of 3 to 2. How much glue and
how much starch is needed to make 85 cups of Oobleck? Glue: Starch:
How does this tape diagram help students make sense of the problem?
85 cups
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Representation #2 Tape Diagrams After a 20% discount, the price
of a SuperSick Skateboard is $140. What was the price before the
discount? How does this tape diagram help students make sense of
the problem? 20%
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Now its your turn After you solve the problems, compare your
answers to pages 10 and 11 in the 6-7 Ratio and Proportional
Relationships Progressions document.
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Representation #3 Graphing Exploring Similar Rectangles on a
Coordinate Grid Launch Explore Summarize
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The Incredible Shrinking Dollar
http://mrmeyer.com/threeacts/shrinkingdollar/
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The Big Mix-Up 7.RP.2
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Representation #4 Ratio Tables There are 64 pretzels in a
16-ounce bag of chocolate-covered pretzels. Sketch and use a table
to find the number of pretzels in a 5 ounce bag. Ounces Pretzels
What is the number of chocolate-covered pretzels per ounce? How
many ounces does each pretzel weigh? Write an equation that relates
the number of pretzels to the number of ounces.
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Pizza Palace 15 large pizzas for $195 Peppys House of Pizza 10
large pizzas for $120 Create a rate table to answer the following
questions: How much will 53 pizzas from Peppys House of Pizza cost?
How much will 27 pizzas from Pizza Palace cost? How many pizzas can
you buy from Peppys for $400? What if you only have $96? Problem
adapted from CMP3 Comparing and Scaling
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Line of Learning Journal Entry #3 We have worked through 4
types of representations double number line, tape diagram,
graphing, and ratio tables. Reflect and write about the value of
using these representations with students rather than introducing
the cross-multiplication algorithm.
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Proportional reasoning has been described as the capstone of
elementary and the gateway to higher mathematics, including
algebra, geometry, probability, statistics, and certain areas of
discrete mathematics. -NRC. Adding it Up, (2001).
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Proportional Reasoning - Where Will I Ever Use This? Take a few
minutes to list as many applications for proportional reasoning as
you can on your Give One, Get One handout. Move around the room and
exchange ideas with others. Read and discuss handouts
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The ability to reason proportionally develops in students
throughout grade 5- 8. It is of such great importance that it
merits whatever time and effort must be expended to assure its
careful development. Students need to see many problem situations
that can be modeled and then solved through proportional reasoning.
(NCTM 1989, pg. 82) Proportional reasoning pervades the secondary
school curriculum. The authors of Principles and Standards for
School Mathematics identify the following topics in the math
curriculum as ones involving proportional reasoning: ratio,
percent, similar figures and their area and volume relationships,
scaling, linear equations, rational numbers and expressions, slope,
relative-frequency histograms, and probability. (NCTM 2000, pg.
212)
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Furthermore, proportional reasoning is necessary for solving
problems in all branches of science. (Post, Behr, and Lesh 1988) It
is important to realize that much time and many experiences are
needed for students to build up a web of knowledge, since these
ideas are mathematically complex. While as teachers we want to
strive for precision in language and classification, our most
important job is to help students make sense of this complex topic.
(Math Matters, pg. 189)
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Line of Learning Journal Entry #4 What can I add about the
importance of learning ratio and proportion? M.S. H.S. Higher Ed
Daily Life
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Outcomes Participants will: Understand the Iowa Core Ratio and
Proportional Relationships Standards in 6 th and 7 th Grade.
Experience a variety of meaningful strategies and models to develop
ratios and proportional relationships. Become familiar with the
6-7, Ratio and Proportional Relationships Progressions document.
Build an awareness of the instructional shifts necessary to
implement the ICC Ratio and Proportional Relationship
Standards.
