MathNotations

CONTEST IS OFFICIALLY OVER AND THE WINNER IS ----- NO ONE! Guess I should have offered a 64GB 3G IPad! to be awarded on Black Friday...


The floor is now open for David, Curmudgeon, and my other faithful readers to offer their own solutions.


And the next contest is...




This is a contest so students must work alone and this needs to be verified by a teacher or parent. No answer will be posted at this time. Deadline is Wed 11-17-10 at 4 PM EST.






Here's a variation on the classic motion-type problems we don't see as often in Algebra I/II but still appear on the SATs. I found this in some long-forgotten source of excellent word problems to challenge NINTH graders!

Barry walks barefoot in the snow to school in the AM and back over the same route in the PM. The trip to school first goes uphill for a distance, then on level ground for a distance and finally a distance downhill. Barry's rate on any uphill slope is 2 mi/hr, any downhill slope is 6 mi/hr and 3 mi/hr on level ground. If the round trip took 6 hours (hey, these are the old days in the 'outback'), what was the total number of miles walked?


First five correct answers with complete detailed solutions emailed to me at dmarain@gmail.com will receive a downloaded copy of my new book of Challenge Problems for the SATs and Beyond when it becomes available. Both the student and teacher(s) will receive this. (Illegal to reproduce or send electronically!). Read further...

Submission by email must include (Number these in your email and copy the validation as well).


1. Answer and complete detailed solution. If answer is correct but method is sketchy or flawed, the submission will be rejected.
2. Full name of student
3. Grade of student
4. Math course(s) currently taking
5. Math teacher's name(s) and parent's name(s)
6. Name, Complete Address of School; Principal's Name & Email address (if known)
7. Email addresses of teacher(s), parents, student
8. Phone number (in case I need to call you) - Optional
9. How your or your teacher or parent became aware of MathNotations.




VALIDATION


I certify that my student (child) did the work independently.




--------------------------------------------------------------------------------


Name of Teacher or Parent (if work done at home)



"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

"You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear. You've got to be carefully taught." --from South Pacific

Posted by Dave Marain at 12:41 PM 5 comments

Labels: algebra, contest, math contest, online math contest, rate-time-distance problems

Have you submitted your vote yet in the MathNotations poll in the sidebar?

Target audience for this investigation: Our readers and algebra students (advanced prealgebra students can sometimes find a clever way to solve these).

Let's resurrect for the moment those ever popular rate/time/distance classics. Hang in there -- there's a more interesting purpose here!

We'll start by using fictitious presidential candidates running in a 'race.' Any resemblance to actual candidates is purely coincidental.

R and J are running on a huge circular track. J can run a lap in one month whereas it takes R twelve months to run the same lap. To be nice, J gives R a 3-month head start. After how many months will J 'catch up' to (overtake) R?

Are those of us who were trained to solve these feeling a bit nostalgic? Do you believe that our current generation of students has had the same exposure to these kinds of 'motion' problems or have most of these been relegated to the scrap heap of non-real world problems that serve no useful purpose. Well, they still appear on the SATs, a weak excuse for teaching them, perhaps, but I can certainly see other benefits from solving these. Can you?

Ok, there are many approaches to the problem above. Scroll down a ways to see a couple of methods (don't look at these yet if you want to try it on your own):







Method I: Standard Approach (using chart)

..............RATE ...x.......TIME .....=.....DISTANCE
............(laps/mo).....(months)............(laps)

R.........1/12....................t........................t/12

J................1....................t...........................t

Equation Model (verbal): At the instant when J 'catches up' to R:
Distance (laps) covered by J = Head Start + Distance covered by R

Equation: t = 1/4 + t/12 [Note: The 1/4 comes from the fact that R covers 1/4 of a lap in 3 months]

Solving: 12t = 3 +t --> 11t = 3 --> t = 3/11 months.

Check:
In 3/11 months, J covers 3/11 of a lap.
In the same time, R covers (1/12) (3/11) = 1/44 lap. Adding the extra 1/4 lap, we have 1/4 + 1/44 = 12/44 = 3/11. Check!

[Of course, we all know these fractions would present as much difficulty for students as the setup of the problem, but we won't go there, will we!]

Method II: Relativity Approach
Ever notice when you're zipping along at 65 mph and the car in the next lane is going the same speed, it appears from your vehicle that the other car is not moving, that is, its speed relative to yours is zero! However, if you're traveling at 65 mph and the vehicle in front is going 75 mph, the distance between the 2 cars is ever increasing. In fact, the speedier vehicle will gain 10 miles each hour! This 75-65 calculation is really a vector calculation of course, but, in relativity terms, one can think of it this way:
From the point of view of a passenger in the the slower vehicle, that person is not moving (speed is zero) and the faster vehicle is going 10 miles per hour. We can say the relative speeds are 0 and 10 mph.

Ok, let's apply that to the 'race':

If R's relative speed is regarded as zero, then J's relative speed will be 1 - 1/12 = 11/12 laps/month.
Since R is not 'moving', J only needs to cover the head-start distance to catch up:
(11/12)t = 1/4 --> t = (1/4)(12/11) = 3/11 months. Check!
[Note: Like any higher-order abstract approach, some students will latch on to this immediately and others will have that glazed look in their eyes. It may take some time for the ideas to set in. This method is just an option...]

There are other methods one could devise, particularly if we change the units (e.g., working in degrees rather than laps). Have you figured out how all of this will be related to those famous clock problems? Helping students make connections is not an easy task. One has to plan for this as opposed to hoping it will happen fortuitously.

Here is the analogous problem for clocks:

At exactly what time between 3:00 and 4:00, will the hour and minute hands of a clock be together?

Notes:
(1) I will not post an answer or solution at this time. I'm sure the correct answers and alternate methods will soon appear in the comments.
(2) A single problem like this does not an investigation make. How might one extend or generalize this question? Again, these are well-known problems and I'm sure many of you have seen numerous variations on clock problems. Share your favorites!
(3) Isn't it nice that analog watches have come back into fashion so we can recycle these wonderful word problems!
(4) For many problem-solvers, part of the difficulty with clock problems is deciding what units to use for distance (rotations, minute-spaces, some measure of arc length, degrees, etc.). This is a critical issue and some time is needed to explore different choices here.

Posted by Dave Marain at 6:20 AM 6 comments

Labels: algebra, clock problems, percent word problem, rate-time-distance problems