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Middle Grades Problem Solving Name
Figurate Numbers
Figurate numbers are those that come from using dots to represent regular polygons. For example, the first four triangular numbers are shown in the picture below:
Work in a group of two or three students. Can you find the number of dots in the next three triangular numbers? What strategy did you use first? Can you use a different method to find the same result? Could you convince a friend of the number of dots in the 100th triangular number without drawing it? What about the nth triangular number (can you represent the number in general)?
Extension: What is the next regular polygon that could be represented as a figurate number? Can you list the first 7 figurate numbers, the 100th figurate number, and the nth figurate number for this polygon?
Middle Grades Problem Solving Lesson Plan: Figurate Numbers
Teacher Notes
Submitted by David Erickson, The University of Montana,
With triangular numbers, the first seven numbers are 1, 3, 6, 10, 15, 21, and 28. One way to find these is to draw them, adding another row of dots for each successive number of triangles formed. Moving from the first to second triangle, we add two to the first triangular number (one), for a total of 3 dots. Next, we add three more for a total of six, and so forth. Knowing the previous total and the number of triangles allows us to recursively get the result, so the 100th would be the total for the 99th triangle plus 100. The difficulty is in knowing the total dots in the 99th figure. So, finding an explicit way to calculate these is a nice solution, not as intuitive as the recursive method most all middle school students see, but equally valid. Students can explore and find this technique using the dots. For instance, if we consider that the current figurate number (2) is slid together with a copy of itself (this means we have twice the dots we really need), it makes a rectangle with dimensions n by (n+1), but this has twice the number of dots required, so we must divide this product in half, therefore, [n*(n+1)]/2. This representation comes from the figures as shown below.
The simplified version, (n^2 + n)/2 may be seen from the figures shown below.
Notice the square on the left and the vertical number of dots (n) added on. Students will find both representations and a discussion about equivalence should follow as they begin to work in representing values with variables.
Moving into the next regular polygon, a square, leads to work with square numbers, and the explicit pattern, n^2 is much easier to find for most middle school students.