visualizing algebraic relationships: solving rate problems with pattern blocks dianna spence robb...
Post on 17-Dec-2015
215 Views
Preview:
TRANSCRIPT
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
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?
Visualizing the Problem
Joe & Matt together: How long to finish 5 lawns?
Joe
Matt
Hour: 1
Lawns
2 3
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: 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
top related