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

Applications of Sinusoidal Graphs

My pre-calculus students are now very good at graphing sine and cosine with all sorts of transformations. We've completed our work on inverse functions. So now we are ready to dive into applications of sinusoidal functions.

We start by revisiting the Ferris wheel. This is how I like to introduce sine and cosine graphs this unit (after spending time with the unit circle and rotations it is a great way to see how we get the sinusoidal graph from a circle, see my blog post here for details).

We begin with this scenario:

1     A Ferris wheel has a diameter of 30 m, with the center 18 m above the ground. It makes one complete rotation every 60 seconds.

I I have students draw a graph for this, showing one complete rotation. Then I pose the question, what would a sine equation for this scenario look like? How can we go about calculating that? What do we need for our equation?

  We use the standard form of the sine equation and talk about what we can use for values for a, b, c, & d.

    from there students identify the amplitude, the period and the midline. We use the lowest point ( ) to calculate our "c" value. And ta-da we have our sine equation.

    then we use this equation and what we know about the scenario to answer questions:

  

   Here is the work used to calculate these:

   Our next example looks at situations that take place over a year's time (thus fixing the period length to 365 days). We start with this group example:

     Events that are cyclic (or periodic), such as seasonal variations in temperature, can be modeled with trigonometric functions.  Regina is the capital of Saskatchewan, a province in Canada. The average temperature for Regina is hottest at 27^o C on July 28, and coolest at -16^oC on January 10.

We jump right in to finding the equation for this situation. Again using the basic sine equation and deciding what we already know (period and using the dates and temperatures we can calculate the amplitude and midline). Again we use the minimum temperature to solve for "c". The result is not as perfect as our Ferris Wheel sine equation as the real life data is not as perfect as a perfectly circular Ferris wheel rotating at a constant speed. But it's very close. (you can see the problem with it by calculating the hottest temperature using the equation, it doesn't give you exactly 27 degrees). 

    I like to draw the graph for these yearly events after I have the equation. It's easier to do that than just working with two pieces of data. I can get some other points on the graph with my equation.

Then I have students answer a few questions using the equation.

From here I now have students work on a packet of problems related to those two examples. They work together in small groups and do some problems for homework as well. This packet is used over a two day period. You will see those two class examples embedded in the worksheet (I put them there so there is a hard copy record of them, also my scanned answer key has all the details of "how to" for all the problems. This is helpful for absent students). 

On the second day (or third depending on scheduling etc( we do a little bit of summarizing with some ISN inserts for our notebooks. I have a two page spread where we model two examples. One in which the given information is the description of a scenario:

The second example uses a set of data. We "hand-calculate" the sine equation but then we compare it to the regression equation from the graphing calculator.

I have students attach the problem only on the left margin with tape (both front & back to keep it secure) so they can show the graph on the reverse side. (we don't tape it in until after they draw the graph). We show both our "hand-calculated" graph and the calculator regression graph. Very similar.

At some point I do discuss the difference between the form of the equation we use to create our equation

Versus the equation the TI graphing calculator uses to create the regression equation: 

y = a sin (bx+c)

Two big differences. First the "b" value is not factored out. We always factor out our "b" value to isolate the true "c" value which illustrates the right/left shift in our graph. In the graphing calculator model this shift is obscured by the horizontal stretch/compress action. Also the calculator uses +c and we use -c to indicate the counter-intuitive nature of the right/left shift. 

So what can happen (as seen in the work in the above example) the regression equation can be very similar to our equation in the amplitude and period and midline but be quite different in the "c" value. Even if you factor out the value of b. This comes from the fact that the calculator zeroed in on a different "standard period" than we did in our calculations. Basically my understanding is that a sine equation transformations indicate what transformations take place on the parent function of sine with the basic period of 0 to 2pi. It show where the "new standard period" shifts to. Our lines are very close, we just have different "new standard periods".  Hope this makes sense!

And finally we wrap things up with a performance task that has students collect data on number of daylight hours at various latitudes. The project introduction reads:

In this project you will develop equations and apply sinusoidal functions that model the number of hours of daylight for locations in the world at different latitudes. To study the relationship between latitude and the number of hours of daylight, data will be collected and analyzed. Using the example of the number of daylight hours, you will then investigate other sinusoidal phenomena.

This will be the second year I am using this project. It is a bit time consuming so I do give them some time in class to get started. Then students need to manage their time well to complete the project. I warn them it's a bit involved so not something they should leave until the night before it's due to work on (as so many seem to do!).

This link will bring you to all my graphing sinusoidal function files including applications. 

Inverse Trig ISN and Evaluate

We set the stage with our "exploring inverse trig functions" and now we move into how we use them in this course. We take a few minutes at the start of class to summarize what we learned with the explore by filling in a little ISN booklet of three pages (one each for sine, cosine & tangent):
   

Recall that you can "undo" any trig operation with its inverse to find the angle (such as in solving right triangles or other triangle application problems). What I am asking students to do now is to find the range value for a given domain value of a trig inverse function. 

Very subtle concept here. Students have to be very solid with their trig ratios to be successful here. Basically what we are doing is finding the angle when we are given the ratio. This can be very difficult for some students.

What I do is go over a few problems with them such as

and refer to the unit circle visuals for the inverse trig function

   

I remind them that the domain of the inverse (for sine & cosine) is very limited. We can only evaluate values between [-1, 1]. When the ratio is negative I ask them questions to help them remember that the original sine ratio is the y-coordinate in the function and where is the y-coordinate negative in the restricted unit circle visual?

Basically students came up with these "thinking steps" for evaluating:
1) look at the ratio - what special or quadrantal angle results in this ratio?
2) now be careful - what is the sign of the ratio? If it's negative think about which inverse trig ratio we are using and where the negative ratios reside in the "unit circle connection" we drew above. Use that to determine the angle.

I also ask,

 "which inverse trig ratios can have negative angles in their domain? which can only have positive angles in their domain?"

I like to keep them thinking with this one

(I ask them "what is the domain of arcsine?".....and what is the numerical value for pi?)

After a few examples students practice "how to evaluate" inverse trig ratios with this foldable:

And then I work through a few "double evaluations" such as:
        

You've got to be careful with the second one as the inverse doesn't always just undo the operation with the result being the given angle. 

And then to keep them on their toes I give them one of these. All kinds of confusion zings around the room as students try to remember what special angle has that trig ratio. When I confirm that there isn't one, I tell them that we really don't need to know the angle. We are ultimately just finding the trig ratio for cosine of some mystery angle. 

with some prompting the hope is that someone will remember that the tangent ratio is y/x and the cosine ratio is x/r so all they need to do is figure out r with the Pythagorean relationship for x, y & r.

Finally students use this foldable on the page opposite to practice these evaluations.

this double evaluation goes on top of the evaluation page - I glue the evaluation page onto the notebook page and this page gets taped across the top creating a little flap over the evaluation page. 

I also have a worksheet of practice for students. Preview of first few problems:

All my files on Inverse Trig Functions can be found HERE (and this is everything - introduction sheets, evaluate sheets, ISN materials, answer keys)

Introducing Inverse Trig Functions

So I've been reading a bit about inverse trig functions on line and how people approach teaching them. It got me thinking about what my focus is with inverse trig functions and inverse functions in general in PreCalculus. My thinking is that students in Algebra 2 learn about inverse functions and the mechanics of inverse functions (how to create graphs, how to create inverse function equations, what they mean in terms of real life situations). PreCalculus is a time to dissect the idea of inverse functions more deeply in a bit of a more theoretical way. Are there limitations to the inverses of functions? Is every inverse of a function a function itself? We explored these questions back in the fall and it was hard for students. I have a good set of classes, honors level but they have been used to learning processes and procedures and not really developing a deeper understanding.

I like to start out these lessons on inverse trig functions by having a discussion about the difference between inverse operations and inverse functions.  This sort of maps out our discussion (I lead the discussion with questions - what is an operation? what is the inverse of that operation? Let's consider an operation we all know pretty well - "squaring" - what does that look like as an operation? as a function? as an inverse operation? as an inverse operation?).

Then I steer the discussion to trig ratios as operations, as functions, etc. I try to do this by asking as many questions as possible.

Then after the discussion I distribute an "explore" packet to students. We do the first page together (it's just a review of inverses with quadratics and the domain & range restrictions). Then students work together in their small groups and using their graphing calculators work on visually representing  and defining domains and ranges of inverse trig functions.  I put a big focus on the visual (it's how I learn best and I find students often learn better that way too),

I thought I'd visually go through what we do & students do with the explore packet. You can open a blank electronic version of this worksheet at the end of the post.

We use TI-84 or TI-83 graphing calculators. We graph a basic quadratic and state the domain & range:

Then using the DRAW menu we can actually see the true inverse without restrictions of this function. We do that and draw it. Discussion ensues with "what's wrong with this picture?"

From there we remember that the inverse of the quadratic is the square root function. We draw that. We carefully examine the graph and the function equation to be clear about the domain and range. 

A little "mantra' we chant is the the domain of the inverse comes from the range of the original. This is verified and we see how it restricts the range. All this is review of something we did back in the fall.

Now students get to work on their own doing a similar process to sine, cosine and tangent. These bits below show ideally what they should be coming up with. If there is time I have three small groups present what they come up with for each and explain. Then I lead the discussion into how these relate to the unit circle. That will continue in my next post (we pull it all together in some ISN inserts).

SINE:

COSINE:

TANGENT:

Is this good math teaching? I hope so. I just remember when I first started teaching PreCalculus 10+ years ago. I was equipped with my previous learning as a precalculus student many many years previous and a bunch of textbooks. No real curriculum. When I got to inverse trig functions I had to ask myself - what are these really? What do I want to know about them? How can I make sense of them in relationship to everything else I know about mathematics? I have a hard time using problem sets in textbooks to drive my teaching. I like having a deeper understanding of how things work rather than just know how to solve given problems.

We do then go onto apply these ideas to problems. But I'd like to think of another way to assess understanding than just evaluate inverse trig functions. Maybe some writing prompt for the students to respond to.

All my files on Inverse Trig Functions can be found HERE (and this is everything - introduction sheets, evaluate sheets, ISN materials)