Thursday, December 17, 2015

Math problems of the week: more Exeter Math

Last week I posted the first 10 problems of the Math I (9th grade) curriculum at Phillips Exeter Academy.  This week we see, in the last 10 problems, where all this Exeter (reform? traditional?) math culminates:

864. Give an example of a line that is parallel to 2x + 5y = 17. Describe your line by means of an equation. Which form for your equation is most convenient? Now find an equation for a line that is equidistant from your line and the line 2x + 5y = 41.

865. A PEA crew training on the Squamscott River, which has a current of 3 kph, wondered what their speed r would be in still water. A mathematician in the boat suggested that they row two timed kilometers — one going upstream and one going downstream. Write an expression that represents their total time rowing these two kilometers, in terms of r.

866. Hill and Dale were out in their rowboat one day, and Hill spied a water lily. Knowing that Dale liked a mathematical challenge, Hill demonstrated how it was possible to use the plant (which was rooted to the bottom of the pond) to calculate the depth of the water under the boat. Without uprooting it, Hill gently pulled the plant sideways, causing it to disappear at a point that was 35 inches from its original position. The top of the plant originally stood 5 inches above the water surface. Use this information to calculate the depth of the water.

867. Most positive integers can be expressed as a sum of two or more consecutive positive integers. For example, 24 = 7+8+9, 36 = 1+2+3+4+5+6+7+8, and 51 = 25+26. A positive integer that cannot be expressed as a sum of two or more consecutive positive integers is therefore interesting. The simplest example of an interesting number is 1.
(a) Show that no other odd number is interesting.
(b) Show that 14 is not an interesting number.
(c) Show that 82 is not an interesting number.
(d) Find three ways to show that 190 is not an interesting number.
(e) Find three ways to show that 2004 is not an interesting number.
(f) How many interesting numbers precede 2004?

868. On a single set of coordinate axes, graph several parabolas of the form y = bx − x2. Mark the vertex on each curve. What do you notice about the configuration of all such vertices?

869. Sketch the graphs of y = 2√x and y = x − 3, and then find all points of intersection. Now solve the equation 2√x = x − 3 by first squaring both sides of the equation. Do your answers agree with those obtained from the graph?

870. Show that the solutions to ax2 + bx + a = 0 are reciprocals.

871. Consider the solutions to ax2 + bx + c = 0. What must be true in order for the solutions to be real numbers?

872. From its initial position at (−1,12), a bug crawls linearly with constant speed and direction. It passes (2, 8) after two seconds. How much time does the bug spend in the first quadrant?

873. Alex the geologist is in the desert, 6 miles from a long, straight road. On the road, Alex’s jeep can do 50 m/hr, but in the desert sands, it can only go 30 m/hr. Alex is very thirsty Alex is very thirsty and

wants to buy a Pepsi at a gas station which is 8  miles down the road from the nearest point, N, on
the road. Which route would take Alex less time: going directly from
A to P, or traveling fromAtoN toP?
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gasstationwithoutpumps said...

I got confused by 867 for a bit—the number of negations had me inverting the meaning of "interesting number". It would have been better if they had picked a more descriptive name.

lgm said...

I see the course isnt watered down. The quadratic is no longer in Algebra 1 here, its been moved to Alg. 2. The rest is quite typical for my day, with Dolciani.

Auntie Ann said...

867 is really poorly worded.

5 is interesting because it is the smallest positive integer which is the sum of two squares (1^2 + 2^2).

7 is interesting because it's the smallest one-digit prime number.

25 is interesting because it is both the sum of two squares (3^2 +4^2) and a perfect square itself (5^2), and is the smallest integer that can be described both those ways.

There is a whole field of mathematics which looks at what makes numbers interesting.

lgm said...

Perhaps the student is being given the opportunity to determine if 'interesting' includes more than the definition given. Some will, just as in first grade when they figure there must be something to the left of the zero on the number line, and they take the initiative to inquire.

Anonymous said...

I remember that some mathematician (GH Hardy?) had a proof that there are no uninteresting numbers.

If there are uninteresting numbers, one of them is the least uninteresting number.

But that makes it interesting.

So, by contradiction, there are no uninteresting numbers.