Title: Counting Triangles
Pete Morey
Math, grade 6/7
Prerequisites: Familiarity with problem solving
process, counting problems.
Materials: Polya’s
problem solving process, 2 worksheets
(one to create various sized triangle cutouts), scissors.
Time required: Each lesson needs 60-90 minutes.
Lesson procedure:
Day one: On overhead, readdress four-step problem solving process. Pass out sheet one.Work with a partner and determine how many triangles there are (5 minutes). Call on students for answers and discuss; use cutouts from sheet two to make sure that students are counting larger and upside-down triangles. When confident we have counted them all, announce that our problem is to try to achieve a formula that will tell us how many triangles there are based on the length of the side. Discuss step one (understand the problem). For example, note that the side increases by one as the figure moves to the next size. With partner, complete steps two and three. Now class compares strategies and success in predicting. Have groups use Excel to show class comparisons of formula consistency as n increases.
Day two: Students put their formulas in symbolic form.Compare findings. Recall closed form solution to triangular numbers. Does it look related? Brainstorm for a way to us this to predict number of triangles. Note notational difficulties (can’t use Tn for both triangular numbers and number of triangles).
Explorations and extensions: A similar lesson
involving a pattern of squares is easier both to see and to calculate, and
would work with younger students or as an introductory experience. Alternately, striking the top triangle
of this design poses a significantly more complex trapezoid-counting problem,
even more so if the length of the top and the bottom are both variables. This is because the number of possible
shapes varies in height and width independently.
Assessment/evaluation opportunities: The level of
independent work enables the teacher to informally evaluate student grasp on
the problem-solving process, symbols, and fluency in Excel.
Appropriate content standards connections (Maryland
Learning Outcomes):
3.1.Use the results of a statistical investigation to answer a question.
5.1.Use information to identify and define the question(s)
within a problem.
Mar2003