Making Connections: An Algebra Problem -- Or Is it Geometry? Or Both?

Lines L and M are perpendicular. Line L contains (0,0). Lines L and M intersect at A(5,2). Find x.

Comments:
(1) Would students in geometry be more likely to consider some or all of the following: similar triangles, altitudes on hypotenuse theorems, areas, Pythagorean approaches? If this were presented as a coordinate problem in Algebra 2, what would be the most likely approach? Would some students use the distance formula?
(2) As always, it is our role as educators to present these kinds of challenges and to encourage students to think more deeply. Making connections between algebra and geometry happens naturally for some, but certainly not for all! We must enable this dialogue via the classroomm environment we establish and the kinds of questions we ask. It is important to first decide the goal. What concepts are we trying to develop? Slope? Ratios of corresponding parts? How many students never make the connection between the two!
(3) The answer is 29/5 or 5.8. Think of at least two methods!

Helping Students Think 'Outside the Box'...

[Update: I'm adding an additional problem at the end. This question seems to be of interest to some since I've seen it in a Google search for awhile now. Answers and solutions to some of the problems will shortly appear in the comments.]


The issue for me as a math educator has always been:
How do we enable children to think conceptually?

Here are some standardized types of questions each of which can be solved by a variety of methods.

In each of the problems below, there is a conceptual approach that requires skill, knowledge and some insight. We all know as educators we can do something about the first two (provided students are given enough practice and they do it!), but how do we develop insight? I can only tell you how my insight improves: When I tackle harder problems or those requiring me to 'think differently'. I am sure there are those out there who are able to invent these methods on their own, but, as for me, I have to work at it and think about it!

The first 3 questions involve ratios. Since we know how the current generation feels about fractions, these may cause students to feel some frustration!

See if you can find an 'insightful' or conceptual approach. Also, ask yourself how most middle or secondary students would approach these:

1. Background Terminology: For those of the younger generation who may not have heard of the term proper fraction, it means a ratio of positive integers in which the denominator is larger than the numerator. Thus, 4/3 and 3/3 are improper; 2/3 and 1/3 are proper. Also, the phrase 'in lowest terms' means that the greatest common factor of the numerator and denominator is 1. (But everyone knew that, of course!)

If 10/n, 14/n and 15/n are proper fractions in lowest terms, what is the least possible positive integer value of n which is not prime?

2. For how many integer values of n is 11/n between 1/9 and 1/10?
Note: Is this an algebra problem? A guess-test 'plug-in' problem? A calculator problem? Or just a fraction 'exercise'?

3. Fahrenheit temperature is related to Celsius by the equation: F = (9/5)C + 32.
An increase of 36 degree F. is equivalent to an increase of how many degrees Celsius?
Note: Many students struggle with an approach here. Some try it algebraically, most plug in some initial temperature, virtually none I have observed think conceptually about the meaning of ratios.

4. A cube 6 inches on each edge is sliced 'horizontally' to form 2 congruent rectangular solids. If these 2 solids are joined to form a rectangular solid which is not a cube, the surface area of this resulting solid is how many more square inches than the surface area of the original cube?
Note: Some youngsters simply 'see' this with little or no calculation! I guess you could say, they really 'think outside the box!' (sorry 'bout that...)

5. What is the smallest positive integer having exactly seven factors?