Plan for a Making Connections Workshop

Plan for a Making Connections Workshop

Plan for a Making Connections Workshop

California Team (Beck Creekmur, Evan Fuller, Ifah Grady, Lawrence Yee)

June 2008, Institute for Mathematics and Education

Activity

Full day seminar with approximately 10 to 30 teachers. (Mathematicians/“math experts” will be available to provide mathematical guidance).

Goal

Show teachers new ways to present problems. Help teachers create new problems that incorporate the mathematical and pedagogical issues.

Program

Morning Session (3 hours)

Present Texting Quandary problem to a group of teachers. The problem will be initially presented with a discussion regarding problem (#1). Teachers will work in small groups (3-5 teachers) to answer question #2. After working together, the entire group will be brought together to discuss/compare their results. Questions #3 and #4 would then be presented together to the teachers and discussed. Then, Questions #5-#8 will be presented together to the teachers. Question #9 will be a discussion of the methods, techniques and approaches to solving the problems that helps identify the mathematical and pedagogical issues.

Lunch Break

Afternoon Session (4 hours)

Group Work (2 hours).

Teachers work in groups.After reviewing the mathematical and pedagogical issues brought up through the presentations of solutions, teachers will break into groups and will design their own problems that incorporate any of the issues discussed.

Presentations (1 hour).

Teachers will share their work with the entire group. Receive critiques and comments from the teachers and mathematicians.

Final revisions (1 hour)

Teachers break into their groups again to modify their problems based upon the comments made during the presentation hour.

Texting Quandary

Jorge wants to buy a new texting plan from Verizon Wireless. When he goes to the store they give him four options:

  1. $19.98 Unlimited texting
  2. $10 for 200 texts, 10¢ per additional text
  3. $8 for 100 texts, 5¢ per additional text
  4. 7¢ per text rate

Jorge texts 300 times a month.

  1. Make a prediction about which would be the cheapest plan and explain your reasoning.
  2. Check your prediction. Which plan will be the cheapest?
  3. If Jorge’s plan (in question 2) was raised by 1¢ per text would it still be the cheapest plan to use?
  4. How many texts would Jorge have to make so that the unlimited plan is the cheapest option?
  5. For what range of texts used would plan a. be the cheapest?
  6. For what range of texts used would plan b. be the cheapest?
  7. For what range of texts used would plan c. be the cheapest?
  8. For what range of texts used would plan d. be the cheapest?
  9. Discuss the implications that this task has in the realm of Algebra. More specifically, how will the methods you used in this problem help you use prediction in other contexts?

Mathematical Focus

Our goal is to enable students to use numerical experimentation that necessitates algebraic manipulations. We want to encourage students to use algebra as a powerful tool for answering questions while abstracting the whole process of solving. At the same time, we want students to practice algebraic thinking about the processes and theories beneath the surface. Some concepts that show up in this particular problem include: finding intersection points of piecewise linear functions. Implicitly, the problem requires partitioning the interval into regions where inequalities are known. The concept of domain also comes into play because we only use non-negative integers (in the sense that there is no such thing as negative texts and fractional texts). It would also be helpful if students recognize the latent concept of slope through their exploration.

Pedagogical Focus

We want students to be able to feel the need to use algebra in their daily lives. We want them to be able to correctly talk about the math and convert the language of English into math. We want to make sure they can express themselves and their thinking in a rigorous and precise way. We also want students to reflect on their knowledge and learning. In order to discover the common mistakes and misconceptions, we want to produce a problem that correctly identifies them and brings them to the surface. We also want a problem that can be a springboard for discussion and provide a rich assortment of other topics to which we can jump. We also want students to practice their prediction skills so they can extrapolate and predict what changing the original problem would do and have the ability to project what the function does in the future. We want students to be able to use multiple representations including graphing, using tables, and ultimately using algebraic equations to calculate points of intersection. We want students to develop a heuristic for solving these types of problems so that they can have a basis for which to predict.

Example Problem:

Hamster House

Jennifer wants to build a3-Dimensional rectangular enclosure for her pet hamster. She wants to have the length of the enclosure 3 units longer than the height. She also wants the area of the base to be 9 square units smaller than the square of the height.

  1. What information, if any, is needed to find the dimensions of the enclosure?
  2. If the length is twice the width, write an expression for the volume of the enclosure.
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