Geometry Challenge for SAT Prep or Review for Final Exam

In the coordinate plane, what is the area of ΔPQR given the coordinates P(4.5,4.5), Q(8.5,8.5), and R(6,0)?

Comments

• Example of "Grid-In" or student-constructed response question on the SAT
• This question seems more difficult than it really is. Students often give up on questions near the end of a section. DON'T!!
• Hopefully you will view the utility of questions like this as I do:
  Students become more competent problem solvers only when challenged with nonroutine problems which are not always to be found in the textbooks. Questions like these should become more routine in our texts and in our classes (not only honors!).

For further strategies, answer/solution, discussion, click Read more...


Answer: 12
Solution (no explanation, details omitted):
(1/2)(6)(8.5) - (1/2)(6)(4.5) = 12

Discussion Points

• What are some problem-solving strategies we need to review with our students here? Draw a diagram for sure but what are some other general attack strategies students should employ in triangle area problems?
• Although advanced theorems could be used here, the actual solution given above is efficient and fairly basic. But what insights are needed to use that approach? What geometry or algebra standards are being tested here?
  
• I chose this problem because coordinate geometry problems connect many important ideas in geometry and algebra. Not to mention that they are becoming more common on standardized tests like SATs, ACTs and state assessments. Besides, I enjoyed writing the question! Sometimes I'll get the germ of an idea, re-work it many times and then the question takes on a life of its own.
• If you find an error in my work or want to share your thoughts, please add a comment!
  

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Geometry Problem Requiring Critical Reading & Thinking

Exactly two of the sides of a triangle are congruent and one of the equal angles is known to have degree measure greater than 50. How many integer values are possible for the measure of the remaining angle (the one that is different from the base angles)?

Comments:
(1) This is not intended to be a significant challenge. Rather it is meant as a warmup or for a slightly more extended discussion.
(2) Since middle school students know the sum of the angles of a triangle and can be told the basic fact about the base angles of an isosceles triangle, this problem is appropriate for them too.
(3) How would you expect most students to approach this? Do you think the majority would start by plugging in 51, 52, 53, etc.?
(4) Does this type of question promote important problem-solving skills and strategies? Do students recognize the significance of the 'boundary values' 50 and 90, values that are not in the domain of the base angles yet can be critical for the analysis?
(5) There are at least two 'traps' set in this problem that are intended to help students become more critical thinkers and not jump too quickly to conclusions. After all, what is a trap? If one is circumspect, details are not so easily overlooked.

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...