Some more "why you Twitter" evidence here. Our gracious host for TMC14 is teaching Pre-Cal and every so often wants to know connections to Calculus. Maybe you do to?

I've taught Pre-Cal for a while now and Pre-Cal/Calculus concurrently for several years. I have learned so much about how to teach Pre-Cal by observing what really happens the next year (assuming AP Calc specifically). Sidenote: throw your on campus Stats teacher a bone and carve out some time for a Stats primer too, yeah?

In no particular order, here's what I have found to actually matter in Pre-Calculus:

Rational Functions

The concept of discontinuity comes up here. The key points are horizontal/vertical asymptotes and removable discontinuities (holes). You can talk about continuity in general here. I describe holes as glitches that can be compensated for (there's a problem at x=1, but is that true at x=0.99999?).

Piecewise Functions

Strictly for the notation and the idea that functions can have multiple behaviors. Piecewise functions aren't particularly practical things on their own other than giving you an opportunity to talk about increasing, decreasing, constant, and undefined behaviors over intervals. Piecewise functions pop up when determining if a function is continuous/differentiable at a point. It's also quite common to use piecewise functions to demonstrate the additive properties of integrals.

Radian-based Trig

Trig is not the focus of Calculus, but it is an assumption that a student has a working knowledge of trig functions. They should know that sin/cos/tan/sec are things with a unique class of behavior. They should be able to spot the graphs of those four easily. They should also be aware that sin(π/4) and other trig values of special angles can be simplified. This is about all the unit circle you need to know:


While students are in Pre-Cal with me they have free use of a full unit circle diagram. In Calc that's not the case and I have them learn this particular subset. It covers 95% of the random trig values you need to apply along the way. Knowing how to find sin(4π) or cos(-π) is a smart move too.

Exponential/Logarithmic Functions

Exponentials and natural logs are real late-game Calculus, the curriculum has an affinity for them when it comes to separable differential equations. Also a fan favorite with volume problems where it may be necessary to solve something like y = e^(2x) for x. The general use case is a need to know how to integrate/derive functions that make use of them. I use these functions as a part of some algebra review, really just as a "hey, remember these things?" Basic knowledge of their graphs is useful.

Polynomial Sketching

Function behavior and the connections between f, f', and f'' are the bulk of Calc AB. While it is not necessary to try and introduce the idea of a derivative, it is helpful if a student can roughly sketch f(x) = x (x - 3) (x + 5)^2, note the type of zeros present, and see how the function being even or odd helps them with the sketch. Solving some kooky polynomial for all its real/imaginary roots is not a thing that will happen.


If you want to save your local Calc teacher some time, covering Limits is a great way to spend a week and a half at the end of the year. It's a good time to bring back continuity from Rational Functions and discuss how Limits are concerned with the "neighborhood" around a function. Removable discontinuities as merely a "glitch" gets justified here since we can now talk about how right-side and left-side behavior agree, letting us "remove" the offending point from the discussion. There is not need for an exhaustive study. Finding limits graphically, algebraically, one-sided, and at infinity (end behavior! horizontal asymptotes! same thing!) does the job.

Stuff That Only Matters to Calc BC

If you cover all of the previous topics, you will still have a lot of time during the year. And you might be wondering "what about [blank] which so and so SWEARS we have to teach?" Well, it's probably in the list of things that only matter to students taking Calc BC. Roughly:

Vectors, polar coordinates, parametric functions, summations, converging/diverging series, partial fractions, and inverse trig functions.

Before you freak out, I am not saying you are free to ignore those topics completely. Among them are some of my favorite things to teach in Pre-Cal. Depending on the size of your potential Calc BC population, you aren't doing any harm in condensing or focusing these topics into something you like to teach. If partial fractions leave a bad taste in your mouth and someone's trying to tell you students will suffer if we don't spend three weeks on it, kindly tell them to chill out.

AuthorJonathan Claydon

A continuing series of posts on how I implement topics in my classroom. Last year we looked at Pre-Calculus. Let's explore some Calculus, shall we?


Students that shaded regions can form the base of 3D objects with a cross-sectional area of a common geometric shape. Students understand that shaded regions can serve as the profile of objects rotated about the x-axis or y-axis. Students can generate integral expression for the volume of a region for any of these scenarios.

Fear Not the Y-Axis

An early introduction to integrals can make volume expressions particularly smooth. The concept of an integral is not foreign, its representation as an area is well known at this point. The new idea is that much like integrating a velocity functions yields position, integrating an area function yields volume. The integral serves to move us "up" the relationship.

Students grasped the concept fairly quickly, what we spent most of our time on was the variety of scenarios. We discuss 3D objects with square, rectangular, "leg-on" right triangle, "hypotenuse-on" right triangle, equilateral triangle, or semi-circle cross sections set perpendicular to either the x-axis or y-axis. In my first run through Calculus I ignored the y-axis to my detriment. It doesn't have to be a big deal if you don't make it a big deal.

I hand out a set of regions.

For each region, I give them three types of cross sections to consider. The y-axis regions introduce an important concept: what do I do if a part of the region is uniform in size? Seeing that a volume expression can result from two integrals will become useful later.

As students will make all kinds of notes on these pictures while working through them, I have a second set of regions ready later in the week.

This set of regions is for introducing solids of revolution. After all the mixing of shapes before, students find it relatively simple to throw this in the mix. Again I present them with three scenarios to consider for each region, and they determine the appropriate volume expression. Multiple integrals may be required.

Full PDFs are here: Regions 1, Regions 2


To demonstrate understanding of volume expressions, students have be able to apply any scenario to any region. My goal here is to remove any fear of "y-axis" especially since it's not something they've been used to. Revolving a region about the y-axis should be no different than stacking a bunch of square cross sections along the x-axis.

Their (admittedly too lengthy) understanding task looked like this:

Six figures accompanied the task. None of the curves were labeled, I wanted to make a point about innate knowledge of parent functions. None of the given functions were in the appropriate form for y-axis use. I wanted to make a point about knowing how to invert something to suit your needs.

Full PDF: Questions, Regions

Despite the 2016 AP exam not offering a straight forward question on volume, the students exceeded my expectations on this topic. The y-axis was no big deal.

Total time spent was 5 class periods. The assessment took two 2 class periods.

AuthorJonathan Claydon

A continuing series of posts on how I implement topics in my classroom. Last year we looked at Pre-Calculus. Let's explore some Calculus, shall we?


Students understand the relationship between an original function, its derivative, and its second derivative. Students understand that a value of zero on one graph implies meaning about the equivalent location at steps "above" and "below" that graph.


I found great success with curve sketching, a subject I found difficult as a student, through the early introduction of integrals. I establish an "up" and "down" narrative about the role of integrals and derivatives. I find it helpful to introduce this with a bit of physics and introduce the connections between position, velocity, and acceleration.

Eventually, the "up" and "down" visual looks like this:

I do a little physics demonstration with a tennis ball, discussing its velocity and direction and various moving away or towards a table. It's a remarkable "OHHHHHHHHHHH" kind of moment that I wish your average physics class included.

Units are a critical part of the discussion. Later you need students to understand which operator is necessary based on interpreting a lot of language. Unit awareness helps with some of this. Being given information noted in meters/second with an answer requested in meters should be a huge flag that going "up" or integrating is appropriate. See: What Helps Me?


This takes a lot of repetition and discussion to get right. I find this Desmos graph helpful. I place an emphasis on values vs. behaviors, a tricky idea conceptual. For example: a piece of a function can have positive values yet negative behavior. Say that piece belonged to a graph labeled f'(x). What graph can we construct from those values? What graph depends on the behaviors? Catching that difference takes a minute.

Primarily we spend a lot of time drawing families. Students are given a starting graph at any point in the family.

Eventually we arrive at the idea that my pattern of behaviors at one level, say f'(x), should be the values of my graph at the f''(x) level. If we moved "up" the family, the relationship is inverted.


Curve sketching is great and all, but it means nothing if you can't interpret the drawings. Sequence wise, I go Riemann Sums > Curve Sketching > Integration > Curve Interpretation. At this point we do a lot of heavy vocabulary lifting: minimums, maximums, points of inflection. concavity, and justifications for the appearance of all these things. Justifications in Calculus are relatively simple, but teaching students to make simple arguments is strangely difficult.

At this point we've reached December and students are ready for a concluding task:

Going Further

Soon enough, I've introduced the concept of algebraic integration and the +C terms that follow. Students will start asking questions throughout curve sketching. When going "up" a level, many will ask how they know where to put the graph along the y-axis. I play a little dumb and say the answer is coming soon.

Once you've established +C as a class norm and they understand the idea of what that constant represents, you get into the numerous possibilities with what the graph one level "up" could look like, and bam, you're ready for slope fields and differential equations.

Early access to integrals saves so many headaches later. The idea has had time to marinate in their minds. It's hard to talk about slope fields if kids aren't square on the base concept on why there could be so many options in the first place.

AuthorJonathan Claydon

A continuing series of posts on how I implement topics in my classroom. Last year we looked at Pre-Calculus. Let's explore some Calculus, shall we?


Students have a fundamental understanding of integration in all its forms: approximate area through Riemann sums, abstract area through geometric breakdowns, and algebraically through antiderivatives.


My approach is unique, as far as I can tell. In all the Calc AB material I've seen, integration enters the picture really late, after the idea of curve relations (f, f prime, and f double prime). In my opinion this does students a disservice.


Discuss the Riemann sum methods: left, right, midpoint, and trapezoid.

Place emphasis on an integral as the approximation of area under the curve. At this point in time I will use functions defined only with a set of data. I show them definite integral notation and how our Riemann sum values are one option.

The biggest lesson of all needs to happen right here. WHY would a data table defined with x-units of seconds and y-units of meters/second have an integral value measured in meters? Don't walk away from the concept until they see that the dimensions used in your sum approximations have units attached. Multiplying those dimensions to create an area have consequences for the resulting units. This is how your April fortunes are won and lost.

Before moving on, take the opportunity to discuss antiderivatives. Keep the examples simple. Reverse power rules and definition based trig functions without limits of integration. The word "antiderivative" is the next key.


Return to the concept of a definite integral. Review the idea of a Riemann sum. With arbitrary data, we can make approximations of an integral's value. What if we had a function defined solely with a picture?

Knowing your final learning target matters here. Can students recognize when to interpret a function value as an area? In the graph shown here, what, exactly, is that integral symbol implying about the relationship between g(x) and f(t)?

This type of integration scenario immediately follows f, f prime, f double prime relations and the curve sketching exercises associated with the topic. If you're students know about antiderivatives in the midst of curve sketching, you'll be amazed at how quickly they can draw a curve family AND discuss the transition between curves using the concepts of derivation and integration.


Right before you break for winter, briefly return to the antiderivative idea mentioned in October. Using the same simple scenarios (reverse power rule, base trig functions), add limits of integration. Where have we seen those before? What do those imply? Are we still talking about area even though I have no picture? Could I make a picture? What does my algebra demonstrate on that picture?

If you find students struggle with implementing the fundamental theorem, see what happens if they know what an integral is months before you introduce it.


Pat yourself on the back because the heavy conceptual lifting is over. Now you can move to integrals that require chain rule concepts (or u-substitution if that's your thing, it's not mine). A huge problem for Calculus is battling student understanding of skills while at the same time realizing they have no conceptual idea of what's going on. April is the wrong time to realize that students might be able to parrot all the appropriate methods associated with integrals, but can't think their way around the concept at all.


Students need time with this stuff. Cramming the many many facets of integration (we didn't even talk about volume!) into 4 weeks is incredibly strenuous, and I think it breeds a student who can do everything you say but has no idea what they're doing.

Don't make your students become math teachers before they really get this stuff.

AuthorJonathan Claydon

This is a continuing series of posts about how I approach topics in Pre-Cal They represent a sample of what I get the most questions about which always seem to start "how do you..."


Students understand the unusual nature of a trigonometric equation. Students understand there is a base set of solutions on the domain 0 to 2pi, but an extension of that domain extends the solution set. Students understand that it is possible to make a statement that summarizes all the terms in the infinite series of solutions. Students understand that the frequency of the trig functions relates to the way solutions repeat along the domain.


We don't start with algebra. We start with the concept of an inverse trig function. For convenience, I stick with angles on the unit circle. My first request is that given a particular x or y coordinate from the unit circle, can you find all the angles associated with that coordinate? We condense that statement into math notation, sin^-1 x or arcsin x, for example. The idea that multiple answers are possible is interesting, and lays the foundation for the algebra later.

Next, we look at graphs. I hand out a set of trig equations.

To keep things simple I have them graph the even set or odd set on a restricted domain of 0 to 4.5pi, recording every intersection they notice with a sketch of what they saw. This continues a big algebra idea, every equation has a graph that validates the algebra. For more complex algebra, I always like to start with the graph.

For some of these equations there are many intersections, for others there is but one. Some comments on how weird tangent is, and questions on how to type sin^2 x.

Now the interesting question. Why so many answers? What happens if we widen the domain restriction? What if we removed it entirely? Many correctly guess that there are far more solutions than we could ever write down.

But is there something to this? Is there a way to condense a small infinity like this into something a little more manageable? We take a closer look at just what numbers appear as solutions. Are they random or regular? How might we predict the next number?

After the graphing exercise I graph another random trig equation. We focus on the first two positive solutions. I name them Solution Zero. We discuss the frequency of the function in question. I take our Solution Zero and add one repetition of the frequency to those numbers. Amazingly, the next number in the solution sequence comes out. Minds blown.

Eventually we discuss that an infinite set of solutions condenses to Solution Zero + n[frequency] both in radians and degrees. Finding Solution 1000 is an easy task now. It's fun to ask about how we would determine solutions to the left of zero, many determining that n = -1 would do the trick.

Lastly I demonstrate the algebra necessary to find these solutions without a graph, validating the need for an inverse trig function.

We build from there and talk about squared functions and how the scope of Solution Zero is much larger (base 4 instead of 2). All of it a lovely dance back and forth between algebra and graphs, avoiding impractical (and graphically dubious) equations used by textbooks to convince somebody (ANYBODY) that kooky trig identities are a thing real people use.


We discuss Trig Equations in the early parts of the second semester. I look for student understanding of both the graphical and algebra aspects of the solution sets. Students validate whether a graph accurately represents a trig equation and determine solutions beyond Solution Zero.

Test 11, Test 12

AuthorJonathan Claydon

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."


Students realize that the hypotenuse of a right triangle, the magnitude of a vector, and the r-value of a polar coordinate are three different names for the same quantity. Students realize that the direction of a vector and the theta-value of a polar coordinate are the same quantity. Students understand how to use polar equations to find r-values and how those r-values are plotted. Students understand how parameters in a polar equation affect the properties of its graph.


A couple years ago I moved straight from vectors to the polar coordinate system because the math is identical. With such recent exposure to the mechanics it just becomes a vocabulary lesson.

We start simple, taking rectangular coordinates and finding their polar equivalent. We draw some right triangle systems and define the hypotenuse as the length of the radius. The theta-value of the coordinate is just the angle relative to zero. It's the same math from vectors.

Next, what makes these things unique. Graph of some high-res polar grids. Have students make a table in 15 degree increments, 0 to 360. Give them some simple polar equations like r = 3 + 3 cos theta and r = 3 + 3 sin theta. Have them complete the table for all the values of thetas. Ignore the groans. Once finished, show them how to plot a point along the circles and then connect the dots.

To better understand the weird shapes, they do an exploration. Given a set of polar graphs and some equation frames (a+/-b cos/sin theta; m sin/cos k theta) students determine the equation associated with a graph and we have a discussion about properties. They have iPads to do the graphing.

Questions to ask: why does 5 + 3 sin theta appear the same as -5 + 3 sin theta? why do graphs have a favored orientation? what determines petal count in a rose curve? What induces a loop? is there anything odd about petal counts? how would you make a rose curve with 10 petals?

Next we get a little more technical. Students are given a second set of polar graphs. This time, no iPads to help. Come up with the equation. Using the equation you came up with, find r-values at 0º, 90º, 180º, and 270º. Use this information to make a connection between negative r-values and the appearance of loops. Discuss the angle separation of rose curve petals. Discuss how far off-axis a sin based rose curve is rotated from its cos counterpart.

One last set of polar equations. Having their fill of cardiods, limaçons, and roses, we discuss ellipses and circles. Pass out the iPads again and send them to a polar ellipse generator. Give them a set of ellipses and have them determine the appropriate parameters. Have a brief discussion about any effects they notice and give them some definitions: semi-major axis, eccentricity, and focus. The key part of this study is to get an idea for eccentricity and how it relates to the dimensions of a shape.

If you really want to keep going (and at this point it's mid-April and probably have the itch to start Calculus), you can explore parabolas and hyperbolas.


Great stuff here. After the initial discussion of equations that match graphs, the students get a chance to design their own. Then we draw them outside on the sidewalk.

When discussing polar conics, you can have some really fun discussion about the solar system. Polar ellipses are plotted using Kepler's equations, which nicely map the orbits of any celestial body you've heard about. After the exploration of making equations of arbitrary ellipses, send them to Wikipedia and lookup the semi-major axis and eccentricity for Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto, Eris, and Halley's Comet. Sticking those values into a Kepler equation gets you a lovely solar system, complete with Pluto's encroachment of Neptune.

There's a great video to show as a followup that discusses how bodies interact with one another in space.


I assessed the mechanics: coordinate conversion and appropriate equations for a graph. Then wandered into the conceptual with discussion questions about eccentricity and written descriptions of rose curves. It spanned three tests and 4 SBG topics.

Test 16, Test 17, Test 18

AuthorJonathan Claydon

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."


Students can define a vector as an arrow of varying length pointed in a particular direction. Student can compose a right triangle from a vector by finding the horizontal and vertical components of the vector. Students can combine vectors in component form or with given magnitudes and directions and finding a resulting magnitude and direction. Student can correct the results of an inverse tangent function based on the quadrant of the vector.


Vectors goes pretty quickly because all the math is a new spin on right triangle stuff. If you spent some time relating cos with horizontal measurements and sin with vertical measurements the breakdown of a vector into components is pretty easy. Some years I've talked about coming up with vector components to describe how to travel from one coordinate point to another. I skipped that this year.

I don't have any particularly interesting intros. It's a lot of drawing arrows with a known magnitude and direction and turning them into right triangles, then just showing them the mag * cos theta or mag * sin theta thing. You could have them draw some arbitrary arrows, do some measuring, and then work back to solving right triangles if you want.

They practice component breakdowns and combining vectors that are already in component form (I use the i, j, and k physics conventions). They figure that out on their own, it's easy.

Show some videos of airplanes landing in crosswinds.

Next we work backwards on finding magnitude and direction given only the components. I spend a moment talking about solving right triangles which will lead them to the use of an inverse tangent.  I always discuss angles as relative to zero (getting answers that range from 0-360), and we always work in degree mode. Calculator output needs to be interpreted here, because in quadrant II, III, and IV the calculator is solving a triangle, and the answer will always be less than 90. The answers are also relative to the horizontal, not zero. I emphasize drawing vector systems quite heavily to help with this idea. If the vector is located in quadrant III, but the calculator tells me the direction is 75º, what is that 75º really describing?

Through this discussion we come up with corrections: quadrant I is fine, quad II is ans+180, quad III is ans+180 and quad IV is ans+360. Students will usually fail to correct an angle for a quad III vector because the calculator outputs a positive number. Kids always seem to think positive = success.

Next we do the whole thing in three dimensions. It's limited though. They'll describe the appropriate octant for a 3D vector, and its magnitude. I don't have time to get into the subtleties of angles relative to the three axes. In the future when they're learning with holograms, sure.

Last, we talk about the angles between two given vectors. And no, I don't use the hokey formula with dot products. To start they draw the two vectors and literally measure it. Going further we talk about how to find relative angle measurements. If you're feeling really adventurous you can have them solve the arbitrary triangle created or find its area.

Vector problems are easy enough that a lot of the intro stuff I'll just make things up on the board. I have two typed up bits. Vector Intro, 3D Vectors


Three activities that crop up. A couple of them are going to change in the future due to changing time constraints.

To stress how a vector system is arranged, I have them create a few in three of the four quadrants. They demonstrate the components and the vector itself using straws cut to scale. An error I've made is rushing this activity before the students have a good idea what I'm talking about. It's also a bit messy and there's nothing necessary about the straws other than giving a physical component.

Since students are bad at visualizing 3D coordinates (they really are, I know you don't believe me), we do some walking around in mathematically defined 3D space. Take some yarn and make an x-y-z axis set in the room. Label the octants. Show the students around and discuss sign conventions. Then have them locate a big set of coordinates printed on labels (set of 60, Word, PDF) inside your 3D coordinate box.

Last, I have them construct their own 3D coordinate system using a bunch of straws and electrical tape. They label each octant and then give me an example of a point that belongs there. Only problem with this one is it is very time consuming (solid 80 minutes start to finish) and might be a little light on learning. I have some ideas about how to improve it next year.

Due to the nature of the components these will fall apart very quickly, some within hours of finishing. Students or are more dutiful with their taping have much better results.


The concepts in vectors span a long period of time even though the material is pretty simple. There are a lot of conceptual things to talk about here. The foundations of this unit come back when discussing polar coordinate systems, so the more time you spend on it the better.

The topics covered span four tests and strike a 50/50 balance between mechanics and concepts.

Test 13, Test 14, Test 15, Test 16

AuthorJonathan Claydon

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."


Students understand how values of the unit circle are periodic and are connected with the appearance of a sin and cos graph. Students understand the ideas of amplitude and period. Students understand that it is possible for an identical graph to be produced by a sin and cos function with appropriate use of transformations. Students understand that tan has a different period by default.


There's a lot to do here and I find this to be one of my favorite topics. We move from values on the unit circle to more generic methods for sketching the graph of sin and cos functions. We make some science connections with sound waves.

First grab some trig graph paper and have students get out a unit circle and some markers. Have them systematically plot the values of sin for the whole circle and connect the dots. Do the same for cos and then tan.

Define the terms amplitude and period. Some of this will be familiar from prior science classes. I interchange period with wavelength and frequency. As a further connection, we talk about the "speed" of the function and define this base form as "normal" speed.

Together, I show them how you can plot the graph of modified sin/cos/tan functions if you know something about how amplitude and period frame the picture. We plot a few examples together. Students often find it weird that sin 2x has a shorter period. There is a tendency to start talking about arbitrary opposites a la "oh well, we do the opposite of the modification, therefore 2x implies x / 2" hand waving. I found it really useful to continue that metaphor of speed. The function sin 2x completes its run two times "faster" than the "normal" we defined previously. For something to be faster, the elapsed time must be shorter. Therefore, one wave takes up less distance along the x-axis. This becomes really handy when you want to talk about sin 2x/3, where the whole opposite operation thing gets ugly to describe. Rather, at 2/3 the speed, we should expect waves to be longer. Throwing in stuff like -3 sin x or sin x + 2 doesn't cause much confusion. I don't do horizontal transformations just yet.

Next they work through two activities to get more practice. It's easy to go fast through this topic and wind up with lots of confusion. I made a concerted effort to slow down and offer lots of opportunities for students to discuss this with each other.

After the activities, they completed some individual practice.


First I have to make some connections between pictures and the equations associated with them. I hand a out a set of 16 unidentified graphs (Answer Key). Students are given an iPad and desmos to determine how the pictures were made.

Next students play a little game drawing randomly generated trig functions. Students are given whiteboards, markers, cards, and 1 die. Students flip a card which describes something about the amplitude, period, and/or transformation of a function. Students roll the die. A 1/2 means sin, 3/4 cos, and 5/6 tan. Students must draw the appropriate function that has the properties of the card. I wrote a more extensive description at the time.

Next students are given a chance to design some trig functions of their own, labeling the graphs appropriately. They design 3 sin and 3 cos functions. They must demonstrate amplitude changes, slower periods, and faster periods in whatever combination they choose. Students create the graphs with desmos on an iPad, print them out and annotate.

Last, we make some connections with sound waves. How much I spend with this depends on where I am in the semester. Last year we were right up against the end, so the science stuff was just a couple side discussions. You could flesh it out more as I have in the past, or start with this. I use a tone generator to talk about how we perceive amplitude and period with sound. I also have a set of blinking lights that are fun to talk about too. You can go really deep into harmonics and springs and things if you have time to fill in the physics background.


This traditionally hits at the end of the first semester. Assessment was pretty straightforward. I'm looking at their ability to sketch sin/cos/tan functions by interpreting amplitude and period correctly spread across two SBG topics. I ask some conceptual questions as a third topic.

Test 9, Test 10

AuthorJonathan Claydon

This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."


Students define sin, cos, tan, csc, sec, and cot as the ratio of sides in a right triangle. Students identify certain special ratios for 30º, 45º, and 60º angles and organize them in the unit circle. Students can determine unknown sides and angles in a right triangle. Students should see that the size of an angle has a direct or inverse relationship to the value of certain side ratios.


Other good resources on the topic come from Sam Shah (with a follow up) and Kate Nowak.

My approach and theirs have some similar messages. Students should understand that trig ratios mean something and that while 30/45/60 ratios are considered special, there's nothing particularly special about them other than reducing to nice-ish numbers.

First, I start with the "special" triangles. Have them draw a set of right triangles (4-5) of arbitrary size for 30/60/90, 60/30/90, and 45/45/90. Establish the definition of the opposite, adjacent, and hypotenuse. Measure the ratio of opposite/hypotenuse and adjacent/hypotenuse for your triangles and record them in a table.

They should see the ratios are the same for an angle regardless of triangle size. We use the three triangles and build a collection we call the Unit Circle where the hypotenuse is 1. I offer them things like root(2)/2 as the exact value of the ~0.71 number they see. We define opposite/hypotenuse as the sine of an angle, adjacent/hypotenuse as the cosine. It's important to take a moment and talk about opposite and adjacent as relative definitions. Some will catch that, others might not. They spend a week finding ratios on the unit circle and defining them as sin/cos/tan/csc/sec/cot.

Now we take a known angle and find the missing side of a right triangle. This is an exercise in vocabulary. Given the location of angle, what is the relative definition of the three sides? What trig ratio is relevant to the given information? This is the first time they compute sin/cos/tan values with a calculator. You could choose not to and save it for later.

Next, using a calculator I have them complete a table of values for sin/cos/tan of the angles 0 to 90 in increments of 5 (this is similar to Sam's large packet of triangles to build a database of ratio values). I give them a set of right triangles of arbitrary size and they use their table of ratios to approximate the angles. I avoid using the inverse trig functions on the calculator until we start talking about trig equations later on. I want them to make a connection between ratio value and angle value. I want them to have an idea of sin and cos trends to help when we graph those functions later.

Lastly, they create a set of right triangles, one in each quadrant, of arbitrary size where they demonstrate the values of the six trig functions for one interior angle and use those ratios to approximate the value of the interior angle.

They have an assignment with a set of right triangles where sides and angles are missing and they have to find the missing information using any of the methods we discussed.


Most of their work through this unit is a series of activities. Here's a picture of the arbitrary special right triangles and the table of ratios:

Here's a picture of the ratio table they complete with the calculator:

Here's a picture of the final assignment, an arbitrary set of right triangles with all ratios computed and angles approximated:

Another option is the classic "how tall is the tall thing?" activity. Take them somewhere where you have some tall objects. Have them measure 1 side and 1 angle. Use a relevant trig ratio to approximate the height of the object and compare results.


Triangles cover 5 topics in my SBG system. Student demonstrate knowledge similar to classwork and in a couple cases are asked to draw triangles oriented correctly in a quadrant, label sides based on a defined trig ratio, and determine the rest of the information.

Test 7, Test 8, Test 9

AuthorJonathan Claydon
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This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."


Students should be comfortable with function notation. Students should see the division of two functions as a ratio. Students should be able to discuss the difference between an asymptote and removable discontinuity. Students should be able to understand why they occur and how they appear on a graph.


There is a lot here. I talk a lot about algebra, a lot about graphs, and try to cycle between them frequently.

Start where we left off in composite functions. We discussed addition, subtraction, and multiplication but have ignored division. I specify division as the ratio of two independent function and coin the term rational as shorthand. Define some functions f(x) and g(x), both factorable quadratics, and such that the ratio will generate some errors. Have the students create a table of x values from -10 to 10. Have them evaluate f(x) and g(x) for the length of the table. Then evaluate the ratio of f(x) / g(x). They should see a g(x) value of 0 causes a problem. Glenn Waddell has a prepackaged alternative version of this in Desmos.

Graph the ratio of f(x) / g(x) with Desmos (either individually or just you) to provide some explanation for the errors. At this point I'll define asymptote and removable discontinuity.

Now we get into why these errors happen, why some are considered removable, and how you could find their location without a graph. I prepare a number of factorable quadratics as ratios and have them simplify them through factoring to see if any share a binomial in the denominator and numerator. It is tempting to say that if you get something like (x - 3)(x + 4) / (x + 4)(x - 7) that the ratio will fail at -4 and 7 because those are the opposites. It is much better to phrase it as "what value of x will make the binomial zero?" I'm careful to say divide out and not cancel out as well when defining x = -4 as a hole. 


Graph a Rational without a Calculator: present them with a ratio made of factorable quadratics, have them identify the location of any asymptotes or removable discontinuities. Use these locations as boundaries. Mark the asymptote with a dotted line and notch a spot on the axis where the hole should appear. Use test points on either side of the asymptote and hole to see if the graph is generally positive or negative. Draw curves that follow that behavior.

Identify a ratio from a graph: Provide some graphs of rational functions. Have them work backwards from the location of the asymptotes/holes to write out the possible denominator. Fill in what's possible based on the presence of a hole. Use the idea of a placeholder in the numerator to signify the lack of additional information. Ex: unknown A / (x + 3)(x - 9) for something with two asymptotes but no holes.

Design a Rational: make up two ratios that demonstrate the characteristics we have seen. Demonstrate a function with multiple asymptotes and one of them should have at least one asymptote and a hole. Label the graph with the corresponding ratio and explain why the features appear where they do.


Students are tested on mechanics: identifying the location of an asymptote or removable discontinuity from a ratio of factorable quadratics/cubics. Students are tested on context by being given a ratio and possible graph of that ratio and have to develop an argument as to whether or not that graph could represent the ratio. A few students did a fantastic job with those questions. This is tested as two SBG topics.

Test 5, Test 6

AuthorJonathan Claydon
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