Thursday, May 13, 2010
If a hen and a half can lay an egg and a half in a day and a half...
The full version in one of its many many variations:
If a hen and a half can lay an egg and a half in a day and a half, how many eggs can three hens lay in three days? Assume that all hens are a-laying at the same rate.
Putting aside the silliness of the riddle, there really is some serious mathematics going in these kinds of rate/ratio/proportion problems. Rather than solve the "hen" problem for you, I'll leave it to my readers to solve it by their own favorite methods. By the way, the answer to this riddle is in the description of the video below on my YouTube channel. Sorry 'bout that!!
Instead, the video below, which appears on my YouTube channel, MathNotationsVids, presents a developmental approach to a more complicated ratio problem for middle schoolers and beyond. I'm far more interested in your thoughts about the teaching strategies than I am about the problem itself. Please understand, further, that I am not suggesting the method shown in the video is efficient nor would it make much sense for the upper level math or science student. See comments below the video for further discussion of this.
The Problem in the Video Below:
If 10 workers can build 3 houses in 60 days, how many workers are needed to build 5 houses in 40 days? Assume all workers build at the same rate.
[埋込みオブジェクト:http://www.youtube.com/v/P_VCYl0zdts&hl=en_US&fs=1&rel=0]
More Advanced and Efficient Algorithms
(1) We assume from the "constant rate" assumption in the problem that the number of houses (H) which can be built varies jointly as the number of workers (W) and the number of days (D).
Thus, H = kWD.
Substituting, H=3, W=10 and D=60, we obtain:
3 = k(10)(60) or k = 1/200. Note that the units of k are Houses/(Workers x Days).
We can interpret k to mean that 1/200 of a house can be built by 1 worker in 1 day. Thus, k is not only a constant but actually represents a rate. Another way of expressing this rate is
(1 House)/(200 Worker-Days) or the reciprocal version:
(200 Worker⋅Days)/(1 House)
Substituting the new set of values into the relationship H = (1/200)WD, we obtain:
5 = (1/200)(W)(40) or W = 25 workers.
(2) This can be made even more efficient using the "factor-label" (dimensional analysis, etc.) format:
(200 Worker⋅Days)/(1 House)) x (5 Houses)/(40 Days) = 25 Workers!
(3) I could also exploit the inverse variation between W and D, but that's for my readers to bring up or for another video!
I see these efficient methods as "black box" methods for some students. Developing a deeper understanding of direct and inverse variation is far more important for the younger student.
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"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)
Posted by Dave Marain at 2:37 PM 3 comments
Labels: average rates, math videos, mathnotationsvids, middle school, proportions, ratios, work problems
Tuesday, June 17, 2008
SOMETHING NEW! Instructional Strategy Series: Teaching Average Rates
The following is the first in a series of strategies for teaching concepts that often prove difficult for many students from middle school on. These are not based on carefully controlled research studies following clinical methodology for a dissertation. They are based on 30+ years of learning how to do it better!! I suspect that's why we refer to the practiceof teaching. Our readers are encouraged to share their own favorite methods that have been helpful to their students or to themselves. These ideas are intended only as suggestions. Each teacher will, of course, bring her/his own ideas and style to bear on the lesson.
Most of you know the classic algebra word problem type that has appeared frequently on standardized tests and math contests:
THE BIG QUESTION
Jack averaged 40 mi/hr going to school and 60 mi/hr returning from school over the same route. What was his average speed in mi/hr for the round trip?
Since there has been a decrease over the past 25 years in the number of word problems to which our students are exposed, some youngsters may not get to see one of these until reviewing for SATs or in their physics class.
From watching how students approach this type of question, I'm getting a sense that we need to introduce the basic concepts earlier on in middle school, which I am sure already occurs in some programs. In planning to teach methods of solving these kinds of problems, I usually tried to return to basic principles of math pedagogy - keep it simple and start with concrete numerical exercises that built on prior knowledge. What does all this jargon mean?
Start with a review of averages, then move on to combined averages before attempting to explain the round-trip rate problem!
[Concerned that such development will take too much time? There won't be enough time to review homework and provide enough practice for the homework assignment? My supervisors never threatened to fire me if a lesson lasted for more than one day and if, heaven forbid, I did not assign homework that first evening! Some ideas just cannot be rushed.]
Suggested Question #1:
Jack had a 70 avg on some tests and a 90 average on some other tests. Can his overall average be determined?
More specifically: When do you think 80 will be the correct answer? When will it not?
Comment:
Question 1 is intended to provoke thought and encourage an intuitive response, not a calculated answer!
Suggested Question #2:
Jack had a 70 average on his first 4 tests and a 90 average on his next 6 tests. What was his overall average for the 10 tests?
Comments
Note that I am suggesting beginning with problems to which middle school students may better be able to relate than a rate-time-distance question. The first question above is fundamental in developing the concept of the original rate problem.
These questions should help many students focus on the essential idea that we need to know how many are in each sub-group!
Since most students connect average to dividing a TOTAL by some quantity, they should feel comfortable in solving the average grade question as follows:
(TOTAL PTS)/(TOTAL NUMBER OF TESTS) to arrive at an average of 82.
BUT DON'T STOP THERE! Stress the UNITS of this result to build the rate concept:
Since students generally do not attach units to the 82, stress that the combined average is 82 PTS PER TEST or 82 PTS/TEST! BTW, not a bad time to mention that PER MEANS DIVIDE!!
Suggested Question #3:
Jack averaged 40 mi/hr for 2 hours, then 60 mi/hr for the next 2 hours. What was his average speed (rate), in mi/hr, for the 4 hours?
[Note the incremental development (commonly termed scaffolding in today's world!). Rather than jump to the abstraction of the original problem, we move on to the next logical step - giving them both the rates and the times for each part of the trip. In this case, we use equal times to provoke their thinking about why the result is also the simple arithmetic mean of the two rates. Each of us needs to make decisions about how many of these examples are needed before moving on to the main question.
Depending on the background and ability level of the group, you may be able to skip one or more of these suggested questions.Further, you may already be thinking of placing these questions on a worksheet for students to try alone or in pairs, stopping and reviewing as needed.
Suggested Question #4:
Jack averaged 40 mi/hr for 4 hours, then 60 mi/hr for 2 hours. What was his average rate, in mi/hr, for the 6 hours?
Suggested Question #5:
Jack averaged 40 mi/hr for the first 120 miles of a trip, then 60 mi/hr for the remaining 120 miles. What was his average rate, in mi/hr, for the entire trip?
Key question: Why does it turn out that the answer is NOT 50 mi/hr?
Comments
Do you think your students would now be ready for the BIG QUESTION near the top of this post? OR do you think they would need at least one more interim problem? Again, could these questions have just as effectively been placed on a worksheet and given to students, working in pairs?
I'll leave the rest to our readers. Pls feel free to share your ideas, comments, thoughts and questions. There's no question in my mind that some of you would develop these ideas differently! Remember you can always email me personally at dmarain at geemail dot com (the last 4 words misspelled intentionally of course!). Unfortunately, I typically get little response from posts about instruction since most readers prefer to solve a challenging problem!
Final Comment: Note that I didn't once suggest that students use a short-cut for the original round-trip problem. Ok, so it is the harmonic mean of the two rates, and can be calculated
from the formula: 2R1R2/(R1+R2).
But who would want to use that (uh, SATs, GREs, GMATs,...)???
Posted by Dave Marain at 7:28 AM 9 comments
Labels: average rates, averages, instructional strategies, Instructional Strategy Series, pedagogy