Hexagons, Hexagons, & More Hexagons.
Problematic Mathematics with a focus on Algebra:
Hexagons, Hexagons, and More Hexagons is a task that requires students to think about mathematics in a general sense, ultimately leading to the use of variables. What is a pattern, how can I use my knowledge of mathematics to solve this problem, and what do I need to know in order to solve this problem; are all questions that will be asked by the students as they develop different solution methods. The instructor should be careful when choosing which questions to answer and which to leave for the students to think about. Since according to John Dewey "If teachers tell too much, students will not need to develop their own problem-solving abilities; if teachers tell too little; students will not make much progress" (Hiebert et al., 1997, p. 36). Thus, let students brainstorm and think about the task before intervening.
Reason for Implementing Task:
- "By learning problem solving mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom" (PSSM, 2000).
- "Research indicates a variety of student difficulties with the concept of [the] variable, so developing an understanding of [the] variable over the grades is important." Therefore, since "a thorough understanding of [the] variable develops over a long time, it needs to be grounded in extensive experiences" (PSSM, 2000).
The Lesson Procedure:
- Split students up into groups of four so that the students can have a dialogue about the task. Be sure that the groups are divided in a way the encourages effective communication.
- Hand out the task
- The Task/Problem:
- Below are five trains with a hexagon pattern. The first train is simply a regular hexagon and each train after that one adds one additional hexagon to it's pattern.
- What you will need to do...
- Compute the perimeter of each of the five trains. Note: each side of each hexagon represents 1 unit.
- Determine the perimeter of the twentieth train without constructing it.
- Explain how someone could find and compute the perimeter of any train that uses this pattern.
- Can your solution be adapted for any polygon train pattern? What if the train was in the shape of octagon? A heptagon? Explain your answer in complete sentences, using diagrams and explanations to thoroughly answer the question.
- What you will need to do...
- What you will need to hand in...
- At the end of this period you should have all of your ideas and explanations written on a piece of paper. Please include all of your solution methods even if they did not work.
- What you will need to hand in...
- 3. After the task has been handed out and explained, circulate around the room.
- 4. Then hold a class discussion (Be sure to ask for multiple solution methods and have the students thoroughly explain their solution methods to the class).
- 5. Collect their work for review.
Back to: Unit 2: Designing Problematic Tasks
References and Resources:
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D. , Murray, H.,…Human, P.(1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
Pattern block algebra: hexagons.(2008, January). Retrieved from http://curriculum.dpsk12.org/math_science/math/ms/add_tools/6_algebra_patterns.pdf
Principles and Standards for School Mathematics. (PSSM). National Council of Teachers of Mathematics Standards. (2000). Retrieved April 22, 2011, fromhttp://standardstrial.nctm.org/document/chapter2/index.htm.