**Contents**

Theme

Common Core State Standards

Introduction

Symbolic Representation

Interviews with students

Mathematical Issues

Teaching Strategies

References

Your Experience

# Theme

Describing Analysis of Change

# Common Core State Standards

7RP2

# Introduction

Proportionality is a mathematical relationship between two quantities. Oftentimes the cross-multiplication algorithm is used to calculate the proportionality between two quantities; however, this algorithm does not help students to make connections between the relationships of the quantities. Instead, students should be given the opportunity to use informal reasoning to explore relationships between the objects. Proportional reasoning forms the basis for understanding rates of change and slope. (See also Analysis of Change: Understanding Rate of Change, Analysis of Change: Understanding Speed as Rate of Change, and Analysis of Change: Understanding Slope.)

# Symbolic Representation

A 7th Grade student is given a 2x3 inch picture and is asked to enlarge it. He doubles the dimensions and creates a 4x6 inch picture. He is then asked to enlarge the picture again so that the long side is 9 inches long. His first attempt is to double the sides of his enlargement and he gets an 8x12 picture. Since this is too long he goes back and adds 3 to each side of his enlargement and gets a 7x9.

# Interviews with students

# Mathematical Issues

Calculating rates of change and proportional reasoning are difficult for both students and adults. Understanding proportional reasoning is important for understanding rates of change.

Proportional reasoning is a pivotal concept between elementary school arithmetic and middle and high school algebra. Use of the rote algorithm of cross multiplication A/B = x/D. AD = Bx, x = AD/B does not promote proportional reasoning on the part of the student. Research shows that:

1. The algorithm is poorly understood by students

2. Using the algorithm is seldom a naturally generated solution method

3. The algorithm is used by students to avoid proportional reasoning.

Instead, people typically reason proportionally in one of two ways: scalar reasoning and functional reasoning. For example, suppose I can bake 16 cookies in two hours. There are two ways I can find how many cookies I can bake in 4 hours. Using functional reasoning, I see that an input of two hours gives me 16 cookies, so doubling the input means doubling the output, and I can make 2 times 16, or 32 cookies. The scalar-oriented way to solve the problem is to notice that 2 hours by 8 cookies per hour yields 16 cookies, so multiplying 4 hours by 8 cookies per hour yields 32 cookies.

# Teaching Strategies

Students should be given the opportunity to use informal reasoning when exploring problems of proportionality . It is important for students to discuss the reasonableness of their solutions and use mathematical arguments to justify their findings. The context of the proportional reasoning problem can affect the difficulty. If students are familiar with the setting for the problem, they are more likely to be successful.

Models for proportional reasoning are the double number line and ratio tables. They both offer a systematic way of writing down the relationship between two variables. The position of the points on a double number line is meaningful, and the columns in a ratio table can be in any order. Both models function as a tool to break down complicated calculations into smaller manageable steps.

# References

Lesh, R., Post, T. R., & Behr, M. J. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number Concepts and Operations in the Middle Grades (pp. 93-118). Reston, VA: NCTM.

Gravemeijer, K., van Galen, F., & Keijzer, R. (2005). Designing Instruction on Proportional Reasoning with Average Speed. Paper presented at The 39th International Conference of the International Group for the Psychology of Mathematics Education.

Lesh, R., Post, T. R., & Behr, M. J. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number Concepts and Operations in the Middle Grades (pp. 93-118). Reston, VA: NCTM.

# Your Experience

- What have you learned about your students’ thinking regarding this topic?
- What have you learned that is an effective teaching strategy to help students understand this topic?