WORD LADDERS - a cool way to shift thinking for verifying identities in precalc

A word ladder is a sequence of words formed by changing just one letter each time, it’s a fun word puzzle invented by Lewis Carroll, the author of Alice in Wonderland. Doing these puzzles in PreCalculus gives students an example of how something must be changed "one step at a time".  In our next unit of “AnalyticTrigonometry” we verify identities, where we change a trig identity one step at a time to create a new identity. (word ladders we change words, one letter at a time.)

For example: changing BACK to FIRE could look like this. One letter was changed at a time, each time a new word was formed.

I personally find these quite challenging! (surprisingly maybe, I am a big reader).  I like the ones that have clues to help me out.

But even with clues this one is difficult!

How do I incorporate these into my busy precalculus curriculum? Students get this packet after out unit test (today & tomorrow in my classes) and are asked to do at least two pages for homework. Some students get really hooked and complete the whole thing.

Solving Trigonometric Equations In PreCalculus part one

Well this year we have rearranged our Pre-Calculus curriculum (honors level) a bit since Algebra 2 is now covering a fair amount of our old topics. Our "new" unit 4 is limited to 4 topics: sinusoidal graphs, inverse trig functions, applications of sinusoidal graphs and the basics of solving trigonometric equations. Seems pretty short but we do have an every-other-day block schedule and graphing took a few classes. But I am confident my students are good at graphing all kinds of transformations on sine or cosine graph and comparing their graphs to the original parent functions, woo-hoo!

One thing they are not so good at is remembering their trig ratios of basic special and quadrantal angles. Well, they're pretty good at it. They've been under a little pressure this year because every day we've had a 5 question mini-quiz asking to evaluate some trig ratios. Questions read by me aloud to the class with 5-10 seconds for students to write down the response. Do they all get 5s every time (especially after doing this for over a month)? Sadly, no. But I keep telling them - you have to know these, they come up again & again in this course and future math courses. And lo & behold when we were evaluating inverse trig functions - there they were. Now when we solve trig equations, yup here they are again.

Okay, enough rambling - on to solving trig equations.

I like for students to have an understanding of what the heck it is they are learning and how it relates to everything else we've been doing. We just recently finished evaluating inverse trig functions but now we are going to solve trig equations that work with the full trig function. I do a lot of repeating myself in making it clear that we are visualizing all the solutions on the trig function graph, NOT only considering the little inverse function graph and its limited range of angles. So we start with some questions - what's the graph of the sine function? How would we find all instances where the sine ratio equals 0.2? 

Our conversation was supported by this ISN foldable.

We looked at solutions as defined in the standard period interval of [0. 2pi) and the general form that indicates ALL the solutions that infinitely appear on the sine curve as it oscillates off to both negative & positive infinity. That's a little tricky for students, in past years I get a lot of complaining about not being able to remember which way to write the solutions (a clear indication they have no idea what the +2kpi means and how the solution relates to the sine curve, sigh). If I had more time I think I would have them map out a longer version of a coordinate plane that has maybe 6 periods and find all the solutions for say sin(x) = 1/2 and then discuss a shorter way to illustrate those solutions in a general way. But we ran out of time this year (no excuses but I had to be out for my daughter's wisdom teeth removal for a few days). 

Then we moved on to special angle solving (which means knowing those trig ratios). I used the inside of the ISN to go over some examples.

Then we finally looked at using some algebra skills to solve some trig equations that had a little more to them (no identities yet, that's our next unit).  But before doing that I wrote some review equation solving examples on the board. Not to teach how to do these but just to refresh everyone's memory. 

And I always love to tell students how much I did not appreciate the zero product property when I was a student. 

Always seemed like an obvious little statement but is really so cool and so powerful when it comes to equation solving. Students tend to forget about it when they do their little short cut of solving something like (x + 3)(x - 2) = 0 when they say that x = -3 or 2. They forget (or never really learned?) how they can get those solutions from those two factors. 

I used this ISN to illustrate examples that use these algebra skills.

inside left:

inside right:

Our textbook does have some good homework problems that I assign, but they don't have any of the basic equations to solve. I created a worksheet for that with equation solving using the calculator (graph) instructions on the back.

All my files can be found HERE. 

see above for a cool basic example, but I also draw a unit circle to show that perspective too.  http://www.slideshare.net/timschmitz/higher-maths-123-trigonometric-functions-358346