Wednesday, May 7, 2008

Multiple Representations (Rule of 4) in Algebra 2 or Precalculus

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Did you overlook our Mystery Mathematician for May? I've received two correct responses thus far, but submissions can still be emailed until the 15th of the month. Don't forget to include the info requested in a previous post.

If (A+3) ÷ (B+5) ≥ 10 and B ≥ 7,
what is the least possible value of A?


The use of multiple representations of a concept or procedure in mathematics is highly recommended by NCTM and other math education experts. Also known as the Rule of Four, it suggests that instructors use some or all of the following, when introducing a new concept. This requires careful planning and considerable thought on the part of the teacher. Over time and with experience, it will flow. However, it does help to see many models of this heuristic for geometry, algebra, etc.

The Rule of Four suggests that a concept be presented
(a) Using natural language (words)
(b) Numerically (concrete examples, 'plugging in', use of data tables, etc.)
(c) Visually (e.g., using graphs, charts, concrete models)
(d) Symbolically (algebraical mode)

From my experience, many students will approach the problem at the top by ignoring the inequalities and simply plug in 7 for B. They've learned that this strategy usually works on standardized tests. It is our role as educators to challenge them to think more deeply. Create disequilibrium by provoking them with a question like,
"But to make a fraction small, don't you need to make the denominator as large as possible?" Of course this statement does not apply to this problem, but I'll wager that it would cause some to reconsider their initial answer!

Do you think that most students would quickly recognize that the relationship between A and B can be described by a linear inequality, which can be then be approached both algebraically and graphically? Do you think I need strong medication for asking you that question!

To deepen their understanding, one could ask:
How would you have to change the above problem so that one could ask for the greatest possible value of A?

I plan on posting further examples of the Rule of Four. I am aware that I have not fully demonstrated this technique for the problem above. I'm only hinting at it. More will likely come out in the comments...


Florian said...

I dont know but I think students would know
that in order to make a fraction large the
divider needs to be small. Therefore its only
natual to plug in the least possible value
for B, 7. The least possible value of A
follows naturaly.

I would be surprised if students didn't know
that if they can't see a solution right away
they might want to try to simplyfy the equation
... which immeadiately leads to the solution.

To ask for the hightest possible value of
A I would change the inequality sign to the
other direction (=< instead =>) as well
as for B so there is an upper bound for A.

Dave Marain said...

A few thoughts, Florian...

Whenever I make statements like "Many (or some) students would...", I recognize that it's foolish to generalize globally since students' responses are affected by their educational training in math. I base my statements on anecdotal evidence from listening to students in my own classes and classes I have observed from 37 years in education (35 in public education). Although, one would imagine that students would re-write the inequality fairly quickly, that doesn't always happen, even with above-average students.

The form of the division also has an effect. When presenting this problem, I would not recommend writing it in horizontal format as I did in the post. Vertical fraction format is better, but I didn't want to play with LaTeX.

Further, my primary goal in publishing this blog is to use somewhat challenging problems to demonstrate effective models of instruction, particularly the elusive art of questioning. The problem in this post was not intended to be challenging or even the focus, rather my intent was to suggest how the instructor can use multiple representations to promote conceptual understanding and problem-solving proficiency.

Making the connection to a linear inequality of the form "y ≤ ____" is not as obvious as one might suppose. Somehow, using variable names such as A and B lead "some students" to treat this problem purely as an arithmetic exercise!

Florian said...

I am not shure about which grade we
are talking here but if students are
unable to investigate a problem with
the tools they've learned in class
(like simplifing an equation) how can
they learn something about math?
At best they will be good math
technicians right?

mike mc said...

I like the idea of keeping the "Rule of Four" in mind as I teach.

For example, in this problem, asking the students to explain (using words) the relationships between the expressions in numerator and denominator might be interesting.

Understanding that the quotient could be of two positive numbers OR two negative numbers might be considered before actually solving anything, for example.

This would also support the students' understanding of basic number properties. How would they know if multiplying both sides by the denominator affected the sense of the inequality, unless they knew that the denominator HAD to be positive.

I guess I might withhold the information about the value of B at the start, so students could grapple with those ideas a little bit in the process.

Thanks for the stimulating example.