Back on the MTBoS Bandwagon!

The first quarter of the school year is over and grades are done, which means (among other things) that I finally have time to blog again! Time to catch up with some of the great MTBoS missions of the past few weeks. I'm responding in this post to MTBoS Mission #3. It's been a while.

I've played with the Desmos online graphing calculator before and liked it, so I used this mission to explore the Daily Desmos blog. Part of the reason I didn't explore Desmos much in the past was that it didn't seem to do anything GeoGebra couldn't. (Full disclosure: I am a huge GeoGebra fan.) But Daily Desmos gave me some pretty good evidence that the simple interface and streamlined visuals of the Desmos graphing calculator make it a potentially better independent exploration tool than GeoGebra.

Every few days the folks at Daily Desmos post a pair of related graphs, one Basic and one Advanced, with the deceptively simple instructions to use Desmos to recreate the graphs. The easy problems can often be solved with Algebra 1 level knowledge of linear and quadratic functions, while the difficult problems can be time-consuming even for undergraduate math majors. This, of course, means serious bragging rights if a student in your high school classroom manages to solve one of the Advanced problems.

The Daily Desmos problem that hooked me and subsequently ate up an hour of my Sunday afternoon is here, and it's based on a (much, much) simpler problem here. The Basic version of the problem requires graphing ten different linear equations which can be found by simply finding the slopes and y-intercepts of the lines in the graph.

Way more fun than a set of ten practice problems, but essentially the same thing.

The Advanced version requires finding the envelope curve for the family of lines in the Basic version. The given graph is below.

Looks like a fun Sunday afternoon, right?

When I saw this graph, I thought to myself, "Piece of cake. Just generalize the equations in the Basic version!" I did that, and then I had to parametrize the equation, then combine the resulting parametric equations back into a function, then square the function and make it implicit to allow for both positive and negative square roots. [SPOILER ALERT] After some serious algebraic gymnastics, I ended up with the equation and graph shown here. It turns out it's just a parabola with an oblique axis of symmetry!

So what did I take away from this that could help my teaching? Well, for students naturally inclined toward math, the Advanced problems on the Daily Desmos site are engrossing. And for those not naturally inclined, the simplicity of the Daily Desmos approach is brilliant. It doesn't give these students anything they don't need. It just gives them a picture and a simple question: "Can you make yours look like this?" For some of these problems there may be more than one solution, leading to interesting and heated debates among students. Really, I've seen this happen. Even if there's only one final solution, there may be many routes to get there. Daily Desmos problems require ingenuity and develop problem-solving abilities. But unlike some problems I toss at students, they know for certain that they've reached the end. If their graph looks exactly like the one they were aiming at, they're done! That's very appealing for them.

For now, I'd like to use Daily Desmos problems for extra credit or for independent work on those weird half-day schedules, but the approach could lead to some seriously cool unit design.

The math clock

The clock below was given to me by my wife's parents for my birthday this year. It makes a great addition to my classroom, and my students get a kick out of it. There's just one small mathematical inaccuracy...

I pointed out the expression for 9:00 to my department chair, and she posed a question: what if I could fix it? Or even better, what if I could get my students to fix it? We could add some more complex expressions for the 12:00, 2:00, and 8:00 slots, and maybe get rid of that pesky long division symbol at 11:00, 3:00 and 5:00. Since one of my classes is currently working on solving multi-step linear equations, an opportunity presents itself...

Fix the Math Clock!

First, I'll provide each of my students with a list of the expressions and equations already on the clock, and tell them that there's one that's not quite accurate. I'll ask them to simplify or solve each one (whichever word applies--a great time to review the difference between the two terms) to figure out which expression/equation doesn't correspond exactly to the number it's supposed to represent. Once they figure it out, I'll also lament loudly that the equations and expressions are way too easy for my students, and they deserve more of a challenge when they want to know what time it is.

Then I'll give each student circular piece of paper cut to fit my lovely clock, and a fistful of markers. Their job will be to create twelve equations, one for each clock number. These equations must take more than one step to solve, and they must require the use of the distributive property, combining like terms, or adding variable terms to both sides of the equation. Then they'll swap equations with a neighbor and check each other to make sure the solutions of these equations come out correctly. Once they're good to go, they'll create a new clock face featuring their equations along with any other creative touches they care to add.

The best one will replace the clock face that's already there!

Any thoughts to make this task richer?