Easton Elementary-Moton Building
Math, grade 6/7
Topics: Triangular numbers, recursion, mathematical symbols.
Prerequisites:
Materials: Chinese checkers, checkers, Excel,
Time required:
Lesson procedure:
Day One:Using Chinese checkers board, show how numbers can be put into a triangular array.Illustrate 1,3,6; students name and check other possibilities.Let’s describe these numbers in terms of the length of their side.T1=1, T2=3, T3=6, …Probe for a way to describe T_, as: a sum of whole numbers; a relationship involving the number after the T; an extension of the T before it.Find T8.Is there a shortcoming to our system? For example, how easy would it be to find T189?This is a job for someone who can add very quickly; use Excel to find T189 (note: depending on computer availability, all, some, or a rotation of students could discover, with guidance depending on their experience, how best to accomplish this.
Day Two: After recapping the previous day, set up the equation that Gauss, age ten, is said to have used to find sum of 1+2+3…+100.Imagine finding a way that would save that much of your time.This is actually T100.Try to guide students into deriving concept for formula.Compare recursive and closed form relationships; intuitiveness versus ease of plugging in.This is a jumping-in point for an introduction to mathematical symbolism, including subscripts and the meaning of n.Use Tn=Tn-1+Tn and Tn=1+2+3…+n to illustrate (along with Tn=[n(n+1)]/2 if student understanding allows).
Now, using a checkerboard, show some rectangular numbers (5*4, 1*2), and have students define/describe.Have them work in groups to find a recursive formula describing R in terms of n.Does this relate to Tn?Show rectangle as two triangles if hint is necessary.
Day Three: In-class assignment: What’s My Rule?Using Excel, students in groups create a two-column chart showing rows for 1, 2, 3.The second row shows the resultant number when the first number is plugged in.The class must guess the rule (operations) in column two.
Explorations and extensions: Modify closed form solution from Tn to Rn.
Assessment/evaluation opportunities:
Homework assignment.Complete worksheet showing the first four pentagonal numbers (1,5, 12, 22) in array.
Find a recursive (dependant on prior element) formula for these numbers and test it on P6.
Appropriate content standards connections (Maryland Learning Outcomes):
1.2.Describe the recursive relationship of simple arithmetic and geometric sequences in a table or graph.
5.16.Make and test generalizations based on an investigation or observation.
Mar2003