Every now and then I get an e-mail from a friend’s son asking for help in algebra problems. When it’s about solving word problems, the email will start with “How about just telling me which one is the x and I’ll figure out the rest”. The follow-up email will open with “Done it. Thanks. All I need is the equation and I can solve the problem”. The third and final e-mail will be “Cool”. Of course I let this happen only when I’m very busy. Most times I try to explain to him how to represent the problem and set-up an equation. Here’s our latest exchange.
Josh: What is the measure of an angle if twice its supplement is 30 degrees wider than five times its complement? All I need is to know which one’s the x.
Me: How about sending me a drawing of the angle with its complement and supplement?
Josh: Is this ok?
Me: Great. Let me use your drawing to make a dynamic version using GeoGebra. Explore the applet below by dragging the point in the slider. What do you notice about the values of the angles? Which angle depend on which angle for its measure? If one of the measure of one of the angles is represented by x, how will you represent the other angles? (Click here for the procedure of embedding applet]
[iframe https://math4teaching.com/wp-content/uploads/2011/07/angle_pairs1.html 650 435]
Josh: They are all changing. The blue angle depends on the green angle. Their sum is 90 degrees. The red angle also depends on the green angle. Their sum is 180 degrees. The measure of the red angle also depends on blue angle.
Me: Excellent. Which of the three angles should be your x so that you can represent the others in terms of x also? Show it in the drawing.
Josh: I guess the green one should be x. The blue should be 90-x and the red angle should be 180-x.
Me: Good. The problem says that twice the measure of the supplement is 30 degrees wider than five times the complement. Which symbol >, <, or = goes to the blank and why, to describe the relationship between the representations of twice the supplement and five times the complement:
2(180-x) _____ 5(90-x)
Josh: > because it is 30 degrees more.
Me: Good. Now, what will you do so that they balance, that is make them equal? Remember that 2(180-x) is “bigger” by 30 degrees? What would the equation look like?
Josh: I can take away 30 degrees from 180-x. My equation would be (180-x) -30 = 5(90-x)?
Me: Is that the only way of making them equal?
Josh: Of course I can add 30 to 5(90-x). I will have 180-x = 5(90-x)+30.
Me: You said you can do the rest. Try it using both equations and tell me the value of your x and the measures of the three angles.
Josh: x = 40. That’s the angle. It’s complement measures 50 degrees and its supplement is 140 degrees. They’re the same for both equations.
Me: Does it makes sense? Do you think it satisfies the condition set in the problem?
Josh: 2(140) = 280. 5(50) = 250. 280 is 30 degrees wider than an angle of 250 degrees. Cool.
Me: What if you make A’DC your x? Do you think you will get the same answer?
No reply. I guess I’ll have to wait till the teacher give another homework to get another e-mail from him.
I don’t know if the questions I asked Josh will work with other students. Try it yourself. Please share or send this post to your co-teachers. Thanks. I will appreciate feedback.
Problem solving is the heart of mathematics yet it is one of the least emphasized activity. Solving problems are usually relegated at the end of the textbooks and chapters.