1. Understanding Ratio and Proportion in Middle School Mathematics
Iowa Common Core Standards for Mathematics
Developed by: Becky Thorson and Bonnie Spaight

2. “What Time Is It?”
How did you use proportional reasoning to solve this problem?

3. Line of Learning – Journal Entry #1
What do I know about teaching ratio and proportion?

4. Outcomes Participants will:
Understand the Iowa Core Ratio and Proportional Relationships Standards in 6th and 7th 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

5. 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.

6. 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.

7. 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.

8. Ratios and Proportions – It’s Critical!
Read Paragraph #1 from the Critical Areas in 6th grade (pg. 41) and 7th grade (pg. 47).
Record:
Key ideas or concepts
Questions that you may have
Share your ideas and questions
with your group.
Groups share out

9. Let’s 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.

10. Let’s 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?

11. 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?

12. Every day, Robin and Tim each save at a constant rate
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?
“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.” - 6th Grade Critical Areas #1, Iowa Core Mathematics pg. 41

13. 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…

14. Ratios Same Measures Different Measures Rate Part-to-Whole Ratio
Part-to-Part
Ratio
When using fraction notation
for part-to-part ratios, label
the parts to avoid confusing
ratios with fractions.
Fraction Notation
(preferred)
“3 out of 7 pets are cats” 3: 7
Colon Notation
(preferred)
“3 cats for every 4 dogs”
3: cats
4 dogs

15. Line of Learning – Journal Entry #2
What can I add about teaching ratio and proportion?

16. Break

17. Problem Types Examples Rationale
Part-Part-Whole
25 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 Sets
3 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).

18. Really?!?
So which problem type seems to be the struggle for this person?

19. How Can We Build Better Understanding of Proportional Relationships?
Read the Iowa Core Ratio and Proportion Standards for 6th and 7th Grade. Jot down all the different types of representations that are listed in the standards.

20. Representation #1 – Double Number Line

21. Felipe walks 2 ¼ miles in 45 minutes at a constant rate. Use the model
below to answer the questions about how far Felipe walks.
How far does Felipe walk in 15 minutes?
How far does Felipe walk in 1 hour?
How long does it take Felipe to walk 4 ½ miles?
How long does it take for Felipe to walk 3 ¼ miles?

22. Break!

23. Dueling Discounts
Make an argument about which coupon is the better deal.
Tell us under which circumstances it is better.
-adapted from Dan Meyer’s “A Problem in Three Acts” website

24. 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

25. 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%
20%
20%
20%
20%

26. Now it’s 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.

27. Representation #3 – Graphing
Exploring Similar Rectangles on a Coordinate Grid
Launch
Explore
Summarize

28. The Incredible Shrinking Dollar

29. “The Big Mix-Up” 7.RP.2

30. BRAKE!!!

31. 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.

32. Pizza Palace Peppy’s House of Pizza 15 large pizzas for $195
Problem adapted from CMP3 – “Comparing and Scaling”
Create a rate table to answer the following questions:
How much will 53 pizzas from Peppy’s House of Pizza cost?
How much will 27 pizzas from Pizza Palace cost?
How many pizzas can you buy from Peppy’s for $400? What if you only have $96?

33. 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.

34. 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).

35. 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

36. “The ability to reason proportionally develops in students throughout grade 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)

37. “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)

38. Line of Learning – Journal Entry #4
What can I add about the importance of learning ratio and proportion?
M.S.
Daily
Life
Higher Ed
H.S.

39. Outcomes Participants will:
Understand the Iowa Core Ratio and Proportional Relationships Standards in 6th and 7th 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.