Saturday, June 6, 2009
Two Geometry Problems To Sharpen The Mind - Never Too Late In the Year For That!
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):
1) 13
2) 11
Feel free to challenge these answers or express agreement!
Comments
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?
Posted by Dave Marain at 7:48 AM 9 comments
Labels: critical thinking, geometry, math contest problems, more, SAT strategies, SAT-type problems
Friday, July 6, 2007
If the difference of 2 numbers is less than the sum of the 2 numbers... Developing logical reasoning in our students
If the difference of 2 numbers is less than the sum of the 2 numbers, which of the following must be true?
(A) Exactly one of the numbers is positive
(B) At least one of the numbers is positive
(C) Both numbers are positive
(D) At least one of the numbers is negative
The answer given in the original source was (B). Do you agree? See notes below for further discussion of the wording of the question (before you react!).
This SAT-type question was posted about a year ago in my discussion group, MathShare (which is still extant but possibly being phased out). On that forum, I discussed how students struggled with the subtleties of logic in their analyses of the problem. That online discussion led to a meaningful debate (involving some exceptionally thoughtful educators) about how and when logical thinking needs to be developed in our students. All agreed that critical thinking and logical reasoning must begin when children enter school, long before the formalism of an axiomatic approach. Do you believe this is currently happening in most elementary schools? What materials are being used by those districts or teachers who are infusing critical thinking and logic? If we move toward a more standardized curriculum nationally, how important is this? I'm sure you know how I feel!
Other Notes about the problem above:
(1) Is the question ambiguous or flawed because the term difference fails to specify in what order the numbers are subtracted? Should the domain of numbers be specified (would integers be better?). On the SAT, it is understood that the domain is always real numbers.
(2) Why do you think so many students (these were strong SAT prep students) struggled with this and had great difficulty accepting that (B) was the correct choice? Do you think phrases like at least one and exactly one are problematic for many students?
(3) What methods do you think were used by students? There were several approaches as I recall.
(4) Do you think students should be encouraged to use an algebraic approach here rather than plugging in numbers and testing various cases?
(5) Would restating the question in its contrapositive form make it easier to grasp? (how many students remember this from geometry?!?)
(6) Would this question lead to a richer discussion if it were open-ended, i.e., no choices given?
(7) Could this question be given to middle-schoolers after they have learned the rules of integers or do you believe they do not have sufficient maturity to handle the logic?
Your thoughts....
Posted by Dave Marain at 6:41 AM 10 comments
Labels: critical thinking, logic, SAT-type problems