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## Proportionality: A Unifying Theme for the Middle Grades

Using a Journal Article as a Professional Development Experience

FoY 2009-2010 Connections

Author:          Cynthia S. Lanius and Susan E. Williams

Journal:         Mathematics Teaching in the Middle School

Issue:             April 2003, Volume 8, Issue 8, pp. 392-396

Rationale for Use

The article addresses the NCTM Connections Process Standard by making connections among mathematical ideas and applying mathematics in contexts.

Proportional reasoning is one of the big mathematical ideas of the middle school curriculum.  “Almost every page of the Principles and Standards stresses the importance of proportionality in grades 6-8” (Lanius & Williams, 2003, p. 392). Also, the report from the National Math Panel identifies proportional reasoning as a benchmark for the critical foundation of Algebra. Teachers will benefit from experiences that help them deepen their own understanding about it and learn how to help their students to do so.

The author cites Hoffer and Hoffer (1988), who state, “Not only do these skills emerge more slowly than originally suggested, but there is evidence that a large segment of our society never acquires them at all” (p. 301).

Procedures/Questions

1. Ask teachers in your learning community what they perceive as big ideas in middle grades mathematics. Make a list of their ideas on chart paper. How quickly does proportional reasoning emerge?
2. Of the list that is generated, which topics do students struggle with the most?
3. Put the topics in order of which are most easily related to outside applications using contexts that interest your students.  Does proportional reasoning lead the list?
4. Before reading the article – solve the following problem without using paper and pencil: If Mona saves \$5 per week:
• How much would Mona save in 3 years? How long will it take her to save \$1000?
• How did you solve the problem? (e.g., some participants may make mental picture, cross-products)
• Share with a partner your approach. Did you both use a similar or different approach?
• Compare and contrast participants’ approaches for solving the problem.
• Discuss the approaches to ascertain which ones are better for developing proportional reasoning.
1. Take a set of orange and white connecting cubes and use them to develop the following proportions described in the Comparing and Scaling book of Connected Mathematics Program, 2009.  If there are four mixtures of orange juice:
• Mix A: 2 cups of orange concentrate, 3 cups of water
• Mix B: 5 cups of orange concentrate, 3 cups of water
• Mix C: 1 cups of orange concentrate, 2 cups of water and
• Mix D: 3 cups of orange concentrate, 5 cups water.

Which mixture is most orangey?  Least orangey?*

*PBS has a website where you can test different mixtures. The website link is http://www.pbs.org/teacherline/resources/activities/mixing_orange_juice/comparemix/comparemix.htm

When you get to the website you have to click on the “Mixing Orange Juice” applet.

1. In a 1996 research article by Kamii she tried the following problem with students:  Fish B eats two times as much as Fish A and Fish C eats three times as much as Fish A.
• Suppose Fish A is given 2 pinches of food, how many pinches of food do Fish B and Fish C receive?
• Suppose Fish A is given 3 pinches of food, how many pinches of food do Fish B and Fish C receive?
• Suppose Fish B is given 8 pinches of food, how many pinches of food do Fish A and Fish C receive?
• Suppose Fish C is given 18 pinches of food, how many pinches of food do Fish A and Fish B receive?
• Which problem is the easiest to solve? Most difficult to solve? Why do you think that is the easiest and most difficult? How can you make sure you are posing problems that will allow all children to be able to access the content – yet provide challenges for all students?
1. What would you describe as an example you can use in instruction to compare additive and multiplicative thinking?
2. How would you get students to describe the different meanings of ratio?
3. How would you help students to understand the difference between proportional vs. non-proportional relationships?
4. We suggest that this is a good stopping point for reading the article.
5. Take a look at the common student misconception illustrated on p. 394, . Is this a procedural or conceptual misconception? What would you consider positive evidence that middle grades students understand the cross-multiplication algorithm?
6. The NCTM Connections Process Standard states: Instructional programs from prekindergarten through grade 12 should enable all students to--
•  recognize and use connections among mathematical ideas;
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.

Give specific examples which explain why NCTM presented each of these as a process standard.

Next Steps

1. What type of professional development is required to better teach proportional reasoning?
2. Think about ways to incorporate connections to proportional reasoning in all content areas of mathematics.
3. Proportionality applications can be found across all mathematics content standards.  Generate tasks that will engage students in thinking proportionally in the following content areas (explore http://balancedassessment.concord.org):
• Geometry (e.g., similarity and scale factor)
• Algebra (e.g., slope, direct variation)
• Measurement (e.g., scale, conversions)
• Probability (e.g., likelihood of an event)
• Statistics (e.g., sample size and validity, population density, comparison of groups—males:females)

Could you argue that using tasks that emphasize connections with proportionality provides a framework that brings forth understanding all of the content standards at the middle grade level?

Connections to Other NCTM Publications

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