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## The Dreaded “Work” Problems Revisited: Connections through Problem Solving from Basic Fractions to Calculus

Using a Journal Article as a Professional Development Experience
FoY 2009-2010 Connections

Title:        The Dreaded “Work” Problems Revisited: Connections through Problem Solving from Basic Fractions to Calculus
Authors:   Felice S. Shore and Matthew Pascal
Journal:    Mathematics Teacher
Issue:       March 2008, Volume 101, Issue 7, 504 – 511

Rationale/Suggestions for Use
The focus of this article is to uncover multiple approaches to algebra rate problems while making connections between ratio, proportion, fractional and inverse relationships, and algebra using multiple representations.  Evidence in student work is highlighted to show connections between fractional relationships needed to solve the problems and the infinite series concepts in calculus.

Materials

• Copy of each of the two problems for the participants (or use  PDF ):

(a)  Deneen and Briana have to mow their parents’ lawn. If Deneen usually takes 4 hours to mow it alone, and Briana can do it in 2 hours alone, how long will it take them to mow the whole lawn working together?

(b)  If Mary can mow the lawn in 3 hours working alone, and Joe can do the same in 5 hours, how long would it take them to mow the whole lawn working together?
• Copy of the article for each participant
• PDF of the problems and solutions (Figures 2-4,6)
• Chart paper, markers or a document camera to display participants’ work
• Manipulatives: Fraction Bars and square tiles (a set for each work group)

Procedures/ Questions

1. Introduce Problem (a) to the participants.  They are to think and solve the problem individually using at least two approaches (e.g., algebraic and pictorial).
• Initially, participants share and compare their approaches and thinking in small groups.  Have volunteers explain their method to the whole group.
• Discuss with the group the differences in each approach and how each method addresses the mathematical content.
• Discuss the importance of understanding multiple representations.
• Discussion: How would you address this statement from a student:  “All you have to do is average the number of times it would take to complete the job alone?”
2. Analyze the student work from pp. 506-507; Figures 2 & 3, Part (a) (see  PDF ).
Discuss the following questions:
• How is the student work either similar or different to the work you just shared?
• What do you have to understand about the problem in order to interpret the student work?
• What specifically can you point to from the students’ work that provides evidence of their understanding of particular mathematics concepts?
• What are the mathematical connections between the two examples?
• Is one representation more enlightening in understanding their depth of knowledge?
• What are the potential misconceptions that students might have with respect to this problem?
3. Introduce Problem (b) to the participants.  They are to think and solve the problem using similar approaches as above.
• Participants share and compare their solutions in their small groups.
• Discussion:  What happened to the approaches students took to solving the problem by changing the numbers?
• Analyze the student work from pp. 506-507; Figures 2 & 3, Part (b) (see  PDF ).
• Ask the same questions as listed above for Problem (a).
4. Hand out copies of the article.
5. Read the article, starting on page 506, “One Problem, Multiple Solutions” to page 508 stopping before, “Enter Calculus.”
• Reflect on the analysis that the authors made on the students’ work.
• What did you learn about the mathematics in this problem?
• How can you integrate the use of multiple representations in your instruction?

Note to Facilitator: To continue into the following session, the group will need to be familiar with concepts of pre-calculus and calculus.
6. Analyze the student work in Figure 4, page 508 (see  PDF ).
• What mathematical approach did the student use?
• Read the rest of the article explaining the approach and its application to calculus.
• Compare the student work in Figure 4 (p. 508) and the pictorial representation of Figure 5 (p. 509).
• Compare your solution to Problem (b) and the solution in Figure 6, p. 510.