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 practice of 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,...)???
Posts
9 comments:
When I first did these problems in highschool I disliked them because
there seemed to be no straightforward or efficient way to solve them.
Over time I developed a systematic approach for myself that goes like this:
In these types of problems there are always two quantities q1, q2
and a multiplier for each of them m1, m2. Additionaly the two q have
a unit f=U/u. Also m1 and m2 are either of unit U or unit u.
Now, the problem is always given in two points:
a) a worded problem that needs to be translated into an equation like this:
q1*f*t1 + q2*f*t2
Examples:
#2: q1=70, m1=4, q2=90, m2=6, f=[averagepts]/[test]
#3: q1=40, m1=2, q2=60, m2=2, f=[mi]/[hr]
#4: q1=40, m1=4, q2=60, m2=2, f=[mi]/[hr]
#5: q1=40, m1=120, q2=60, m2=120, f=[mi]/[hr]
b) a value that needs to be determined.
Case 1: Find the average q for m1+m2 where m1,m2 are of unit u.
Solution: (q1*m1 + q2*m2) / (m1+m2)
Examples:
#2: (70*4+90*6)/(4+6) = 820/10 = 82
#3: (40*2+60*2)/(2+2) = 200/4 = 50
#4: (40*4+60*2)/(4+2) = 280/6 = 46
Case 2: Find the average q for m1+m2 where m1,m2 are of unit U.
Solution: (m1 + m2) / (m1/q1 + m2/q2)
Example:
#5: (120+120) / (120/40+120/60) = 240/5 = 48.
Clearly, the answer is not 5 because m1 != m2, which means that (q1+q1)/2 is no solution.
Please note that I ommited the units in the calculations for simplicity.
What I like about this approach is that it requires only some thinking
to form the problem into an equation. The rest of the calculation process
is then pretty intuitive because all values will fall into place. The units
will be correct and the solution will have a senseful value and unit. And
it is easier to find your own error sbecause the translation of the problem
as an equation and the calculation are almost strictly seperated. Just my 2 cents of course ;)
Correction: 50 is no solution because m1/q1 != m2/q2
Apologies for breaking the flow, but I'm checking in here as a new subscriber. I'm a math teacher in Texas, teaching precal and stat at the moment, so I'm interested in instructional strategies for math.
Janet
Florian--
Excellent ideas! I gave serious consideration to approaching all of these using weighted averages. In the case of different rates and equal distances, we can use the fact that if the rates are in the ratio 40:60 or 2:3, then their respective times are in the ratio 3:2, since the times and rates vary inversely in this case. This also leads to the harmonic mean I mentioned. Developing these ideas into a coherent lesson with guided exercises and strategically posed questions is the real challenge however...
Hi Janet! Thanks for joining us. You might look through the index in the right sidebar for any posts relating to trig, advanced algebra, precalculus or particular topics. Not too much on STAT however (other than combinatorial math if that helps). I assume we are talking about high school students here. Are you looking for general strategies for the course or ways to develop specific topics? Using multiple representations (verbal, symbolic, numerical, graphical)? If you'd like you can always email me directly at dmarain at geeemail dot com. There are many experienced educators who visit here, so you may want to be more specific about topics you'd like to see. Do you need this for courses you're teaching in Sep?
Dave, what are your thoughts on the
idea of seperating the understanding
of the problem and the act of applying
a technique to solve it?
It appears to me that many students
will start to form all kinds of ratios
until they find a the ones that
solve the problem. Showing a great deal of
confusion and lack of understanding
of the actual solution of the problem. Or not?
I suggest that you graph speed against time, calling the function s. You get a step function. Then, define the average of s as the integral of s divided by the integral of 1.
You can even discuss this sort of integral without any calculus; you just use the area instead. After all, integrals of step functions are what you need in discussing Lesbegue integrals, but high school students couldn't be expected to know that.
You can draw the same sort of pictures in probability theory too.
Florian--
Your question is central to the whole issue of teaching and learning.
For each lesson I would make a decision, based on the group I had, whether to
(A) guide them through the derivation of the formula/algorithm OR
(B) to state the method and show them how to apply it.
I preferred the former but that was not always possible. In all cases, I attempted to provide some motivation for the formula. Needless to say, I spent considerable time planning each lesson to reduce the possibility of confusion. However, students absolutely need some time to 'mess around' with ideas. On a limited controlled basis this can be done IN the classroom and then continued outside when they grapple with other questions for homework.
What I have observed in many classrooms and from what we see in most texts, is almost exclusively (B). I could suggest a few reasons for this but I'll leave it to my readers to speculate...
Eric--
For the upper-level students, particularly those with some physics background, your approach would make sense and, in fact, I stressed this in calculus:
The average velocity of a particle (defined as "delta s" over "delta t") is the same as the average value of the velocity function (defined as an integral).
The special case of the step function is very nice and I think I would have used it for this type of problem if I had thought of it!
Introducing average 'speed' to middle schoolers is a bit different however, IMO.
Eric, Florian:
Did you think my development would make sense to them or would be too confusing? of course, one could only be sure if we see it 'live'!
I think your approch makes sense. The nature
of the material, availabe teaching time and
the level of the students seem to be the factors
that dictate which method should be used.
I don't think there is anything wrong with textbooks
that use method (B) to present material. They are
a great addition to the in class lecture because
they allow fast reviewing and training of the
new material - if the concepts have been understood
in class.
Students should be instructed using both methods
It shows them that problems can be approached from
a different angle (=part of studyskills).
Dave,
for the last six years I have been teaching traditional word problems in algebra 1 for 9th graders. I have felt a bit embarrassed as I justified this...
But lately I have changed my tune. Why drop these problems?
1. They are contrived. But so are all of their substitutes, except for things too complicated to correctly model. The artificial context on NY State's exams drives me nuts, and I assume it is bad all over.
2. They are hard. Right. Even though the kids know that these problems are variations on R*T=D, they remain challenging. For fractional equations t/r1 + t/r2 = 1 (work problems) are even harder. If we already have tough stuff where kids are taught directly what to do, why yank away that last bit of support and ask them to decide which kind of math to use in unfamiliar situations? Doesn't make sense.
Average rate problems are great, and your suggestions for building up to them should be used, as nec'y.
By the way, since I still talk about properties and that sort of thing, I find operations on real numbers that may or may not be associative, commutative, or both. I like A max B is the greater, A left B = A, A avg B = (A+B)/2...
There's that famous study where many teachers had trouble finding operations that were commutative but not associative...
Jonathan
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