Here’s an interesting variation of a typical (MS) problem I found by following the Five Triangles ‘blog: http://fivetriangles.blogspot.com/2013/09/97-no-triangle.html .
(Note: If you sign up on this or other ‘blogs, you can get lots of problems emailed to you every time they are added.)
I know this question can absolutely be solved without using technology, but when a colleague asked if it was appropriate to use technology here (my school is one-to-one with tablet laptops), I thought it would be cool to share with her the ease and power of Desmos. You can enter the equations from the problem exactly as given (no need to solve for y), or you can set up a graph in advance for your students and email them a direct link to an already-started problem.
If you follow this link, you can see how I used a slider (a crazy-simple addition on Desmos) to help students discover the missing value of a.
I suggest in this case that playing with this problem graphically would grant insight for many students into the critical role (for this problem) of the intersection point of the two explicitly defined lines. With or without technology support, you could then lead your students to determine the coordinates of that intersection point and thereby the value of a.
Keeping with my CAS theme, you could determine those coordinates using GeoGebra’s brand new CAS View:
Substituting the now known values of x and y into the last equation in the problem gives the desired value of a.
NOTE: I could have done the sliders in GeoGebra, too, but I wanted to show off the ease of my two favorite (and free!) online math tools.
Thoughts? What other ideas or problems could be enhanced by a properly balanced use of technology?
As an extension to this particular problem, I’m now wondering about the area of triangle formed for any value of a. I haven’t played with it yet, but it looks potentially interesting. I see both tech and non-tech ways to approach it.