# Systems of lines

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.)

INITIAL SOLUTION

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.

FOLLOW-UP

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.

CONCLUSION

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.

### 2 responses to “Systems of lines”

1. Your question about the triangle’s area got me curious. As a function of a, the area is 7/6 * (a+2)^2. I decided to go the tech route — powered up GeoGebra, set up the triangle and plotted a point (a, poly1) with the trace on. That was enough for me to see the shape was a parabola (mental check: yeah, we’re working with area so a quadratic makes sense) and identify a good value to check if it was the minimum; from that and one other point I came up with the function above, plotted it and it matched the traces!
At this point, I don’t even want to think about how I would find that area by hand, looking just at the algebraic equations…
Here’s a related essay by Bret Victor that addresses this type of question (how might a balanced use of technology be used to efficiently accomplish a goal) http://worrydream.com/ScrubbingCalculator/.

• Thanks so much for your response. This is exactly what I always hope will happen when I pose “what ifs” online and in my class. And I couldn’t agree more about the the power of technology to help us explore really interesting questions.

What’s so stunning to me is how little effort it takes to extend problems into this “interesting” realm. Perhaps it takes a little experience and confidence in looking at what else you can do in a situation, but once you start, you start seeing alternatives everywhere!

Now for a hard-core math-type, seeing that your so-very-clean answer is $Area(a)=\frac{7}{6}(a+2)^2$ convinces me that there must be a super-nice, purely algebraic way to accomplish this that somehow captures the essence of the problem situation. Clean answers almost always hint to me of hidden handwriting–something more than I see on the surface. It makes me want to know more. There has to be a reason why this works out so nicely! That said, without the technology driving exploration and discovery, we might not have ever seen the pretty solution driving it. It is absolutely the balanced use of technology that can take us to much deeper understanding. I’ll leave alone for a few days to see if anyone wants it. Otherwise, that could be another cool ‘blog post.

Thanks for joining the discussion!