I deeply appreciate the positive comments and support I'm receiving for these problems. At some point my time, energy, creativity and resources will start to decay exponentially! I've been focusing more on high school topics but today's are more mixed. Read the comment for the 3rd question for 1-18-07. One of our astute readers clarified my sloppy wording -- thanks!!
1. The lengths of the sides of two triangles are 7,16,x and 9,20,x respectively. If x denotes the same integer value in both triangles, how many values are possible for x?
2. Borat averaged 150 mi/hr for the first 3 hours of his sojourn through Kazakhstan. What must be his average rate, in mi/hr, for the next 5 hours to average 200 mi/hr overall? (Will he get a speeding ticket in view of the fact that he is so famous?!?)
3. The square root of an integer added to the square root of one less than that integer is greater than 100. What is the least possible value of the integer?
[Comments: What % of your students would reach for the calculator for this one rather than estimate using number sense?? How many would try an algebraic solution?]
Thursday, January 18, 2007
Problems for 1-19-07
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5 comments:
Problem (1) is a good application of the triangle inequality and set theory.
I guess for problem (3), you want the least integer n such that sqrt(n)+sqrt(n-1) > 100. Things get more interesting if you make it sqrt(n)+sqrt(n-2) > 100.
In my opinion, it would be good practice to use 'n' for an integer variable rather than 'x'. Your mileage may vary.
Keep the problems coming. In general, it is more difficult to formulate good problems than to solve them.
Problem 1 also will catch a few students who will think that since 22-12 = 10 there are 10 numbers that work, when in reality, since the endpoints are included, there are 11.
Problem 2 is very similar to problems involving average that my seventh graders are working, but with the added bonus of using d = rt! It's going up on the Brain Teasers bulletin board!
Problem 3 would be a beast if solved algebraically, but winds up being pretty straightfoward to solve by "number sense" approach.
Keep the great ideas coming! I'm excited about the possibility of a "problem day" when my students get back after mid-term exams.
[In pseudo-LaTeXish.]
For number 3, the best approach is to estimate. Unfortunately, students don't learn that these days.
So, take the inequality
$$
\sqrt{n} + \sqrt{n-1} > 100
$$
and replace it with the similar
$$
\sqrt{n} + \sqrt{n} > 100
$$
In fact, replace it with the equality
$$
\sqrt{n} + \sqrt{n} = 100
$$
since your students should be able to solve it. It simplifies to $n=2500$.
Will that work in the original problem? No.
$$
\sqrt{2500} + \sqrt{2499} < 100
$$
But, bump n up to 2501, and it works.
$$
\sqrt{2501} + \sqrt{2500} > 100
$$
by simple comparison with the equality above.
In fact, your students will gain insight from seeing how close $\sqrt{2501}$ and $\sqrt{2500}$ are without the use of calculators:
$$
\sqrt{2501} - \sqrt{2500} = \frac{1}{\sqrt{2501}+\sqrt{2500}}\approx 0.01
$$
by a standard algebraic technique the students may not be familiar with.
After that, you can demonstrate why the standard quadratic formula is not always appropriate. Use it naively for $x^2 - 10000x + 1$ and one of the two roots will end up with a catastrophic subtraction. $\sqrt{2501}-\sqrt{2500}$ loses precision, but $\sqrt{2501}+\sqrt{2500}$ does not. Naive use of calculators will hide these problems from their users.
great analysis, eric...
this is why i really like this question as it encourages nonroutine thinking or should i say it encourages thinking and not merely algorithm-pushing or mindless button-pushing! imagine if textbooks contained a variety of problems in which students would need to experiment with different approaches and actually learn from the effort! your comment about the advantages of the sum of radicals compared to the difference of radicals was very powerful. In calculus we often re-write differences of radicals this way when taking limits.
Dave,
If you get a chance, find the book "Numerical Methods That [Usually] Work", by Forman Acton. There are many similar analyses, written far more entertainingly than I could hope to do. Most of them are suitable for calculus students, but you could find a few ideas for young students.
I also recommend "The Pleasures of Counting", by T.W. Korner. It is a wonderful guide to how mathematics is used.
Do you think your students would understand how computers store real numbers as floating-point, and how subtracting two nearly-equal numbers would lose precision?
I used to have some pet questions for my students:
1. Draw the graph of y = sin x near x = 0, freehand. [Worry most about the slope they draw at the origin.] If a student asks for a ruler or protractor, give it to her and praise her for her care.
2. Ask your students why the 10-foot long cockroaches one sees in monster movies or the giant humans in "The Amazing Colossal Man" and "Attack of the 50-Foot Woman" can't exist. Point out that they have the same proportions the ordinary members of the species, and ask the students what is wrong with that. Then, ask why grain silos are dangerous places. [Dusts are often explosive--surface area to volume ratio.]
3. For a calculus class, ask a student how he'd find the limit of (sin x)^8 / x^8 as x goes to 0. Then ask how a) a lazy student would do it, b) a mediocre student, c) a good student, d) an engineer. The answers should be something like:
a) Cancel the 8s.
b) Use l'Hopital's Rule, 8 times.
c) Note that polynomials are continuous everywhere, so the answer is the limit of (sin x)/x raised to the 8th power.
d) Say sin x is just x, so the answer is 1. For this sort of problem, it is.
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