Developing Proportional Reasoning Through Gears Investigation

Acknowledgments:

• Dr. Matt Roscoe

• Emily J. Wilson

Background:

• A foundational understanding of ratio and proportion is

essential for success in higher mathematics and

science.

• To understand ratio and proportion concepts, students

must be exposed to multiple contexts and

interpretations.

• Research has identified gear-pair analysis as a

promising manipulative that supports students'

conceptual understanding of ratio and proportion in

small-group (1-3 pupils) settings. (Lobato and Ellis,

2010)

Results of Investigation:

During structured stage of exploration:

• Students demonstrated a mastery of tabular representations of data

• Students demonstrated a proficiency for arithmetic presentations of

data

• Students demonstrated a low understanding of how to use

graphical representations of data as a tool for analysis

During the unstructured stage of intervention:

• Students demonstrated a strong preference for tabular and

arithmetic representations of data.

• Students used graphical interpretations ineffectively.

Discussion:

• Students were able to answer open-ended ratio problem in a

gear-pair setting using familiar strategies for representing

data

• The intervention was ineffective in guiding students toward

new strategies for representing and comparing data

• After the initial guided exploration of gears, students were not

able to make the abstract connection between a gear-pair

ratio and a line on a graph

• This activity reaffirmed student strengths in tabular and

arithmetic interpretations of data

• Specific modifications to the two-phase intervention could

better support learning targets aligned with standard 7.RP.2 in

order to better inform future classroom instruction

Developing Proportional Reasoning Through Gears Investigation

Colt Davidson

Elementary Education, University of Montana

Setting and Method:

• Research was conducted in 7

grade math classes at Washington

Elementary School. Four classes participated – three regular

classes and one honors class. A total of 93 students participated.

• Structured and unstructured activities were used in these seventh-

grade classrooms. to facilitate this investigation

• During the initial investigation of gears, students used manipulatives

to complete a guided worksheet exploring the relationships of how

different gear pairs interact.

• Next, students were given an open-ended problem designed to test

their understanding of ratio in a bicycle gear-pair context

• After the intervention, an analysis was made of the classroom

artifacts created by students during both stages of the gears

investigation.

• Data was analyzed using iterative qualitative analysis of patterns of

student activity, leading to categories and counts of recurring

patterns of response

Ability of Students to Represent

And Interpret Data in Proportional Setting

Using Gear-Pairs

Informing Classroom Instruction:

• To more effectively implement this intervention in a whole-

class setting, teachers should use structured collaborative

group work during initial investigation to guide use of

graphical interpretation of proportional relationships between

gear ratios.

• Teachers should scale axes to prevent different ratio settings

from appearing as y=x.

• Teachers should design guided questions to encourage

visual analysis of data.

• Teachers should include point (0, 0) in ratio tables to use

students’ strength with tabular analysis to promote graphical

connection.

Montana Common Core Standard:

7.RP.2 – Recognize and represent proportional relationships

between quantities including those represented in Montana

American Indian cultural contexts.

1. Decide whether two quantities are in a proportional

relationship, e.g., by testing for equivalent ratios in a table

or graphing on a coordinate plane and observing whether

the graph is a straight line through the origin.

2. Identify the constant of proportionality (unit rate) in tables,

graphs, equations, diagrams, and verbal descriptions of

proportional relationships.

3. Represent proportional relationships by equations.

4. Explain what a point (x, y) on the graph of a proportional

relationship means in terms of the situation, with special

attention to the points (0, 0) and (1, r), where r is the unit

rate.

100

Examples of Student

Work:

= proficient

= emergent

Level of skill

demonstrated:

Example of guided exploration:

Student response to open-ended gear-pairing

problem:

Student response to open-ended gear-pairing problem:

Tabular

Arithmetic

Graphical

Student scales two separate ratios to look identical to y=x. These

gear-pair ratios cannot be meaningfully compared with

this type of representation

Student correctly fills in

missing values in the

table

Student correctly

identifies equation of

graph

Student scales graph to look like y=x,

rendering meaningful graphical analysis

impossible

Student demonstrates preference for

arithmetic analysis of data, despite

difficult to interpret presentation

Research Questions:

1. Does structured investigation of gear-pairs lead to

students’ ability to abstract ratio settings?

2. Does unstructured investigation of gear-pairing

possibilities show evidence supporting students’ ability to

analyze ratio settings in pursuit of a “best solution?”

3. Does gear-pair analysis scale to whole classroom

instruction?

References:

Lobato, J., & Ellis, A. B. (2010). Developing Essential

Understanding of Ratios, Proportions, and Proportional

Reasoning. National Council of Teachers of Mathematics.