Well, the June SATs have arrived today so these problems come too late for that, but these kinds of questions can be used to review basic ideas while strengthening thinking skills. Both questions below are appropriate for both middle and secondary students, although the second requires knowledge of a fundamental geometry principle regarding the sides of triangles.
There are other important principles embedded in these problems as well. In the end, I believe that students need to be exposed to many of these "contest-type" challenges to improve reading skill, learn how to pay attention to detail and think clearly. As a separate issue, performing well under testing conditions requires extensive training. You may not feel this is an important objective for math teaching in the classroom, but testing is a reality for the student...
These questions may appear fairly straightforward at first but be careful! I believe the second is more challenging than the first. These are not so different from the "gotcha" problem on our latest online contest.
1) The dimensions of a rectangle are odd integers and its perimeter is 100. How many different values are possible for its area?
2) The perimeter of an isosceles triangle is 96 and the lengths of its sides are even integers. How many noncongruent triangles satisfy these conditions?
For my "unofficial" answers, click on Read more...
Unofficial Answers (no solutions):
Feel free to challenge these answers or express agreement!
Which of the following do you believe would cause the most difficulty for students?
- The wording/terminology (e.g., noncongruent); general reading comprehension issues
- The sheer number of details (e.g., odd vs. even, perimeter vs. area, integer values)
- A precise counting/listing strategy vs. an abstract or commonsense approach
- The "square is also a rectangle", "equilateral is also isosceles" traps
- The issue of different areas for #1
- The triangle inequality for #2
- Other concerns?