Sunday, May 6, 2012

Given the sum and product of 2 numbers...

A fairly common standardized test question for Algebra 1,2 or SATs is something like

The sum of 2 numbers is 20 and their product is 64. What is the larger number?

This question requires the student to actually find the numbers as opposed to a question with the same given info but asking for the positive difference of the numbers.

Do you suggest to students that many of these types of questions can be handled by inspection with mental math?  This is because the majority of standardized math questions involve simple integer values or adhere to the "Keep it Simple" philosophy!

From either of the given relationships students should be able to arrive at 16 and 4 as the values and proceed from there. For the 25% or so of questions which do not admit a simple solution there's always straight algebra or the "test each answer choice" strategy for Multiple Choice. By the way this is why item writers often shy away from direct "solve for x" types, preferring the "find the positive difference " type.

Please don't forget to make that critical connection to the graph of a linear-quadratic system. A quick sketch of the line x+y=20 and the rectangular hyperbola xy=64 suggests there are 2 pairs of solutions which involve the same numbers by symmetry, i.e., (4,16) and (16,4).

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