Geometry WarmUp - A Simpler Integer Triangle Problem

While we're waiting for the 60° integer triangle problem, here's an easier one for both middle schoolers and secondary students. The only fact from geometry that is needed is the all-important triangle inequality:
Any side (in particular, the largest side) of a triangle is less than the sum of the other two sides.
Of course this refers to the lengths of the sides and one can express this in other forms, but I'll leave it at that.
This type of question has become a favorite on the SATs and other standardized tests but, more importantly, it develops clear systematic thinking - the organized list....

How many different triangles have integer side lengths and a perimeter of 5? 10? 15? 20? 25?

COMMENTS/INSTRUCTIONAL HINTS:

• There are really five separate questions here. The instructor can give some or all of these depending on the time allotted. To help the group get started and for clarification, it may be helpful to demonstrate the first question for the group: For a perimeter of 5, there is only one possible triangle, which we can symbolize as {2,2,1}. If these are older students who are comfortable with the triangle inequality, you do not necessarily have to model this one, but that's your call. By modeling the first one, you eliminate some of the ambiguity of ordering the sides.
  
• Since a primary objective here is to make an organized list, you may want to stop after the perimeter of 10 and discuss it at the board. Depending on the ability level of the group, I usually have students work independently, then check each other's work in pairs after they do a couple of these questions. Sort of a think-pair-share approach. Also, don't be afraid to provoke their thinking with questions as they begin to develop their systematic lists (which can get boring for some): "So, do you expect more triangles for a perimeter of 10? Twice as many?"
  
• As each question is reviewed, encourage students to record their results in a table:
  Perimeter..................Number of Triangles
  ......5........................................... 1 ................
  ....10.......................................... 2 ................
  This is critical for middle schoolers in particular, since tables are a basic model for functions! At some point, you can use n or p for the perimeter and symbolize the number of triangles having perimeter n or p as T(n) or T(p).
• Naturally, some students will assume there is a pattern and guess there are 3 possible triangles with a perimeter of 15 - NOT! However, it is natural for all of us to ask: "WHAT'S THE FORMULA?" Well, there is one. It's fairly sophisticated and related to partitions of numbers, but I'll let our readers do their own research for this...