I was just helping a student with his AoPS homework, when I came across the following related problems:
Eight people, including Fred, are in a club. They decide to form a 3 person committee. How many possible committees can be formed?
So we are choosing 3 people out of 8 or .
How many possible committees include Fred?
Since Fred is taking one committee seat, that means we need to choose 2 more people from the remaining seven, or .
How many possible committees do not include Fred?
Since we can’t choose Fred, we need to choose 3 members from the remaining 7 or .
Since the total number of committees is equal to the number of of committees with Fred plus the number of committees without Fred, then we can say
Generally we call this Pascal’s Rule: .
I recently twice on the AMCs came across this problem solving technique for finding side lengths in a triangle with a cevian. (A cevian is a segment from one vertex of a triangle to the opposite side.) It uses the law of cosines, but the good news is it may not require an angle measure because it takes advantage of the fact that . Note in the diagram that the cevian creates these 2 angles. By the law cosines and . Both equations have the length of the cevian, , and the angle , so you can eliminate a variable and solve.
I recently reviewed a problem involving a bug on a cylinder. We need to find the shortest distance from the bug’s original position, A, to B which is on the opposite side of the cylinder. The bug can only crawl on the surface, and we’re given the height (8) and circumference (12). One student wanted the bug to crawl along the lower circumference and then crawl straight up the side.
You can more easily visualize the shortest path by using your imagination to “cut” the cylinder along one side starting at point A, then “flatten” the cylinder into a 12×8 rectangle. Now the shortest path is easily seen as the hypotenuse of the triangle with legs of 6 and 8. (A 3-4-5 triangle hiding in plain sight!)
Most students are familiar with triangles whose sides are Pythagorean triples like 3-4-5 and 5-12-13. They may also want to add the 13-14-15 triangle to their list. It contains both a 5-12-13 triangle and a 3-4-5 triangle (disguised as a 9-12-15, and sharing the altitude of length 12). This triangle appears often on the AMCs and you’ll save yourself some time if you can recognize it early.
I worked this problem with some 5th graders last year. Since I was unsure about their exposure to angles, I began by asking if they knew the degree measure of a right angle, from 12 to 3 on the clock. From there, I told them I would divide up that 90 degrees into 3 sections, drawing in the 1 and 2 on the clock. How many degrees between the 12 and 1?
Then I followed the solution provided, and continued by asking them for the degree measure of the angle between 12 and 8, which is the degrees between the 2 hands at 8:00. Then we stepped through how many degrees we lose by moving the minute hand from the 12 to between the 5 and 6 after 24 minutes. Then we need to add the few degrees the hour hand moved in 24 minutes.
This approach reminded me that there are two approaches to teaching math. One way is to repeat the same problem over and over, which little change until the algorithm in memorized. Another more fun way is to begin with an elementary problem, and do only a small number of those before immediately continuing to progressively more difficult problems.
After I asked them how many degrees were in a right angle, the students then unprompted told me the degree measure from 12 to 6 and then 12 to 9, pretty much by rote. But I didn’t want to hear their answers to elementary problems. I wanted to quickly propel them to more difficult problems: the degree measure from 12 to 8 and then at 8:24. I think they had fun solving this problem and internalized the concept better by solving 1 more difficult problem than a page full of identical easier problems.
(Source: Math Olympiad for Elementary and Middle School)
If/then statements can be replaced with (read: “p implies q”). Such a statement is equivalent to its contrapositive (, read: “not q implies not p”). Abstracting the statements by replacing them with variables makes it easier to quickly identify the contrapositive among the statements.