Visualizing Algebraic Relationships: Solving Rate Problems with Pattern Blocks

Dianna Spence

Robb Sinn

North Georgia College & State University

Joint Mathematics Meetings 2010

Introduction

Joe and Matt start a landscaping business together. Homes in their neighborhood have similarly-sized lawns. Typically, Joe can mow a lawn and trim all the shrubs in 3 hours. Matt usually needs 2 hours to do the same job. They decide to work together on 5 lawns. How long should it take them to finish?

Course: Modeling in Algebra

Students: K-8 pre-service teachers

Sample Problem: Combined work rate problem

Instructional Strategy

Ensure students are familiar with pattern blocks

Pose a combined rate problem and suggest modeling the problem with pattern blocks

Guided discovery

1 = = = =

1/2 1/3

1/6

Pattern Block Conventions

1/4 1/12

Recall Sample Problem

Joe and Matt start a landscaping business together. Homes in their neighborhood have similarly-sized lawns. Typically, Joe can mow a lawn and trim all the shrubs in 3 hours. Matt usually needs 2 hours to do the same job. They decide to work together on 5 lawns. How long should it take them to finish?

Rate Representation

Joe: 3 hours for 1 lawn Matt: 2 hours for 1 lawn

Joe

Matt

Hour: 1 2 3

Visualizing the Problem

Joe & Matt together: How long to finish 5 lawns?

Joe

Matt

Hour: 1

Lawns

2 3

4 5 6

Variations

Joe & Matt together: How long to finish 5 lawns?

Joe

Matt

Hour: 1

Lawns

2 3

4 5 6

Combining Rates

Joe & Matt together: How long to finish 5 lawns?

Joe

Matt

Hour: 1 2 3

Lawns

4 65

Variations

Joe & Matt together: How long to finish 5 lawns?

Joe

Matt

Hour: 1 2 3

Lawns

4 5 6

Revisiting the Algebra: Rates

Joe: 3 hours for 1 lawn

Matt: 2 hours for 1 lawn

Joe

Matt

Hour: 1 2 3

Joe’s rate: RJ= 1/3

Matt’s rate: RM = 1/2

Revisiting: Combined Rates

Joe

Matt

1 Hour

Joe and Matt combined:

Hourly rate is

R = RJ + RM = 5/6

Revisiting: Setup and Solution

At 5/6 lawns per hour, how many hours for 5 lawns?

Hour: 1 2

Lawns

…

(RJ + RM)h = 5

5/6 h = 5

h = 6

Extending the Reasoning

Maria and Dusti are decorating the gym with helium balloons. Maria can inflate and tie off 2 balloons every 3 minutes. Dusti requires 2 minutes to finish 1 balloon. Working together, how long will it take them have a batch of 35 balloons ready?

Rate Setup

Maria: 2 balloons every 3 minutes Dusti: 2 minutes for 1 balloon.

Maria

Dusti

Minute: 1 2 3

From Concrete to Abstract

Maria

Dusti

Minute: 1 32

4 5 6

Goal: 35 balloonsRate: 11/6 per minute6 min 7 balloons30 min 35 balloons

7/6 m = 35m = 30 minutes

Extending & Generalizing

Progression:Situations with fractional answer

(e.g., 7½ minutes)Change of question: “How many lawns

could they mow in 9 hours?”Situations with fractions that don’t lend

themselves to pattern blocksStudents draw their own pictures

Does This Technique “Work”?

Research DesignControl: Classes received traditional

procedural instruction only (n = 26)Experimental: Classes used manipulative

discovery technique (n = 49)Data CollectionPre-testPost-test (immediately after instruction)Retest (6 weeks after instruction)

Results

Comparison of Gains

12

14

16

18

20

Post-test Re-test

Control

Treatment

Gains are defined as improvement from pre-test

Scores are out of 30 points total• 3 items each scored with 10-point scoring rubric

Results were encouraging,but not statisticallysignificant

Final Notes

Mitigating Factors Relatively small samples Very limited instruction time (1 class period)

Not enough time for full discovery Insufficient followup: generalizing, formalizing

Our Interpretation Method shows potential, especially

to improve long-term outcomes A better trial is warranted

Questions