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 1/4 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:123

Visualizing the Problem Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:1 Lawns

Variations Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:1 Lawns

Combining Rates Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:123 Lawns 465

Variations Joe & Matt together: How long to finish 5 lawns? Joe Matt Hour:1 23 Lawns 456

Revisiting the Algebra: Rates Joe: 3 hours for 1 lawn Matt: 2 hours for 1 lawn Joe Matt Hour:123 Joe’s rate: R J = 1 / 3 Matt’s rate: R M = 1 / 2

Revisiting: Combined Rates Joe Matt 1 Hour Joe and Matt combined: Hourly rate is R = R J + R M = 5 / 6

Revisiting: Setup and Solution At 5 / 6 lawns per hour, how many hours for 5 lawns? Hour:1 2 Lawns … (R J + R M )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:123

From Concrete to Abstract Maria Dusti Minute: Goal: 35 balloons Rate: 1 1 / 6 per minute 6 min  7 balloons 30 min  35 balloons 7 / 6 m = 35 m = 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 blocks Students draw their own pictures

Does This Technique “Work”? Research Design Control: Classes received traditional procedural instruction only (n = 26) Experimental: Classes used manipulative discovery technique (n = 49) Data Collection Pre-test Post-test (immediately after instruction) Retest (6 weeks after instruction)

Results 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 statistically significant

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

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