Sep 072016

The struggle to understand why we teach K-12 mathematics in the order we do, and the content we do is real. I have wondered about this for a long time, and really have never found a good answer.

I threw out the idea of teaching y=mx+b as the only way to write lines (even though the district materials at the time said it was all we needed). I took a lot of heat for that decision from some people. I was told I was completely wrong; by teachers. I stuck to my guns because y=mx+b is a stupid way to teach lines. And in the end, I was told by other teachers that I influenced them to change too.

But really, K-12 mathematics education is nothing like this:

Mathematics as human pursuit

Think of Lockhart’s Lament.  You read Paul’s words, and you are hit by the poetry he sees in math. It is also 25 pages long. I read somewhere that Lockhart’s Lament is the the most powerful and often cited mathematics education document that is never acted upon. What does that say about us, as educators, who cite it?

Lockhart is passionate about math education, and he feels that the current state (in 2002) of math education is in trouble. His words may be as apt today as it was then. On page 2 he writes,

In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.

How much impact has Lockhart had on mathematics education? Often cited, rarely used or implemented. And yet, my Twitter feed and Facebook still have things like this pop up regularly.

Mathmatician is like a painter G H Hardy

What beautiful words representing fantastic ideals. Are you starting to see the cognitive dissonance I am feeling today? Too bad none of these ideals are found in our textbooks or our standards (and don’t get me wrong, I am not hating on the CCSS-M here). In fact, much of school mathematics is exactly how Seymour Papert described it here.

Papert - outwitting teachers as school goal

It is mindless, repetitive, and dissociated.

So as I was thinking of the question of “Why?”, I stumbled upon this article. Why We Learn Math Lessons That Date Back 500 Years? on NPR. To find out the answer is pretty much, “Because we always have,” is sad, disappointing and frustrating. We have taught it this way for the last 500 years, so we will continue to teach it this way for the foreseeable future.

I call B.S.

Seriously. We need to rethink how we teach math in a substantive manner.

We are part of a system that is not allowing learners to find the joy of mathematics, but the drudgery of mathematics and of learning. And this is not new. Not by any means. Edward Cubberly, Dean of the Stanford University School of Education around 1900) said,

Our schools are, in a sense, factories, in which the raw products (children) are to be shaped and fashioned into products to meet the various demands of life. The specifications for manufacturing come from the demands of twentieth-century civilization, and it is the business of the school to build its pupils according to the specifications laid down.

The fact that the specifications of education haven’t changed in hundreds of years is a problem (see the NPR article). It may even be THE problem. I am not so confident to claim that for sure, but it is definitely A problem.

At what point do we, as teacher leaders, rise up and demand this change. We see the damage. We see the issues. We must start demanding the curriculum be changed to meet the needs of our learners. I am not sure that the CCSS-M is that change. It seems like it is codifying the 500 year old problems that we are currently doing.

But it doesn’t not have to.

The Modeling Standard is gold. It is also 1 single page in the entire document.

I will just end this rant with that thought. Oh, and this thought. No more broccoli flavored ice cream.

textbook math is like broccoli ice cream

Jun 102015


I begin the class with a “what do you notice?” “what do you wonder?” session. This is probably the 5th or 6th day of class, and sets the stage for the entire rest of year. What do you notice? What do you wonder? I document all the noticings and wonderings, and then we discuss the mathematical questions.

Every year, the question of “I wonder how the 2 and the 1/2,” “I wonder how the 3 and the 1/3,” are related is asked. The best two questions that are always asked are, “Why are they all the same?” and “What changes when we change the exponent from 1 to 6?” I always say that I will answer every question by the end of the year; I will never lie to them and tell them something is impossible when it isn’t, but that some of their questions may need to be addressed in a future math class and not this math class. That honesty goes a long way.

I spend an entire period exploring the different functions with them, showing graphs on Desmos, asking for values to put in for a, h, and k. I ask questions like, “what do you predict the h will do?” and “Did your prediction come true?” The learners who are the typical aggressive type A learners hate it because they want the answers and want it now, but they will come around and start developing ideas on their own.

I start with lines for the (h,k) form because I think this form shows some reasons why to use the form, the benefits of using the (h,k) form over y=mx+b, as well as a simple function to cut our teeth on vocab.

I introduce this form first thing in the year as we get started. Fully Explaining & Understanding functions blank (double sided .docx file). I print off hundreds of this form, and we use it regularly. Some days I have the learners write the functions in their notebooks when I don’t have the forms, but I try to have a stack on hand always.

explaining This is what it looks like. There is A LOT of info asked for, and I start with lines so we can establish the understand of what the different elements are.

It always bugged me that we rarely talk about domain and range of lines. Why not? Why start introducing that idea with absolute value? Just because that is where it changes from all real numbers for both to only one, does not mean we shouldn’t introduce it earlier. Same thing with asymptotes and even/odd functions.

If I can get learners to identify the x and y intercepts on the line, and then connect those points with the standard form, so much the better.

Same thing with intercept and (h,k) form. Cut the teeth on a line, that is familiar and safe, so that as we move forward with quadratics, cubics, cube roots, etc, the learners can see the vocabulary does not change. What changes is the shape of the parent function.

I make sure every learner has one copy of this that is complete, pristine, written clearly and fully in their notes for every single function. When the learner puts them all side by side, they can see there is only one math, one set of ideas through the entire year. What changes is the amount of effort needed to get the intercepts for a cubic vs intercepts for a line? Why?

Another rich focus of questioning is “What makes the line unique?” “What makes the quadratic or cubic unique?” Some answers I have received are, “Only the quadratic always has a vertex form. The cubics can have that form, but usually not,” or “Every point on the line is a critical point, but we can’t always use every point for other graphs.”

Or can we? Hmmm. Leave it at that. Don’t tell them. Plant the seed and let it grow on its own.

This is a big picture post. Philosophy of teaching, approach to the topics, etc. No details yet. Just a pouring out of my thoughts on how I start. I will go more in depth. Notice that there is not enough room to work on the page. Only the results go there. The work is separate.


Jun 042015

My last post was about the three rules I use in my classroom. I developed the how and why in that post. In this post, I will explore some detailed “how I use them” in the classroom. I am careful to never say the word “rule” except for these three. We have exponent shortcuts, log shortcuts & properties, but never rules.

To do this right, I am going to use my Surface and do a lot of handwriting and posting of screenshots. If you are wondering, this is all done in OneNote with a Surface Pro 3. The bad handwriting is my own.

What really drove this point home for me, and made me codify it as something that needed to be talked about every day in class is the fact that if you do the “why?” for every step, you write down a “1” or a “0” for almost every step every time. Sure, you write down “I distributed,”  “I found the value to complete the square,” or “I factored” but why you do the next step is almost always a 1 or a 0.

See what I mean below.

To see how  I connect the rules, let’s start with an expression. It is not a very complex expression, but it sits solidly in the Alg 2 curriculum and throws learners for a loop often.

expression1 You can see that the expression is changed through the use of multiplying by one, and the convenient value we selected to use is a value that gets us a 1 when added. Why add? Because of the properties of exponents when we are multiplying bases.

Compare this to the rational expression of adding:


Here, we take the adding expression, and multiply both terms by 1, but a different one each time. Why? because we select the ones to use based upon the convenient terms to accomplish a common denominator.

Let’s move into some solving. Here is a straightforward quadratic that is in vertex form.

quadratic1 Yup, look at the bracket of zeros and ones.

How about turning a quadratic into vertex form?

Quadratic2 Here I explicitly used the first rule (I used it implicitly above as well, and I have the extra “I chose the 9 because it completes the square” step. Of course, these are bracketed by a zero and a one.

Finally, a log solving equation. I have just one, although I can do more. I chose an moderately ugly equation to solve, so it could not be solved any other way.

log1 [OOPS! I just realized I switched the 3 and 5 in my bcs statement. Damn dyscalculia. Sorry about that.] Here we have a double whammy. I conveniently chose to use Log base 5 to do to both sides. Why? Because Log base 5 of 5 equals 1! From experience, I know that taking the log of the more complex side reduces the number of steps. I don’t tell my learners that. They play and figure it out by doing the one problem both ways.  We also used Rule 1, do unto both sides.

In looking at the commonality between all of these problems, you can see the connection of “1” and “0” throughout. I stress this all year long, and have the learners write it all year long. This is the minimum requirement of writing I ask of my learners as they progress. We start writing much more, but I demand they write it. It reinforces the identities of addition and multiplication over and over again all year long. As the year goes on, they write less, but still write it.

Also, I almost never write a radical symbol until the final answers. All radicals are transferred to fractional exponents immediately all year. This helps explain why cubes are inverses of cube roots, and we don’t need to worry about notation. This is a big deal when dealing with some money problems and the exponent is 377 or some such nonsense  and we are solving for “r”. The “you just raise both sides to the power of 1/377 because the exponent will be 1” is automatic at that point.

I hope this gives a better understanding of what I mean by “zero’s” and “ones”. Please leave me questions here or on Twitter; @gwaddellnvhs.

Jun 022015

In Algebra 2, I start with my 3 rules. They really are not “my” rules they are just restatements of the multiplicative identity, additive identity and balancing equation. I believe how I use them to set the tone and stage for the entire year is different, however.

On the second day of class (the first day I usually do a problem solving activity, cell phones, or other type of activity) I introduce the “rules”.

2014-08-10 15.43.10 You can zoom in, or you can grab the files from here. I had a long discussion with several teachers about the wording, and Meg Craig made up the files once we settled on the phrasing. Can they be tweaked more? Absolutely. I would love to improve them. Leave the suggestions in the comments.

Back to how I use them, and why I think I use them differently. I will say with 100% assurance that I use these rules differently than I have seen others, and I absolutely use them differently than I used to.

This grew out of the frustration of having a nice, simple equation to solve like, y=5x+2, and needing to solve it for x. I really wondered why learners would mess up the math on such a simple equation so frequently. And don’t even get me started if the equation looked like v=ba-d, because that was impossible.

And I realized that although I was teaching the idea of inverses and identities, I was not connecting learners with or building the idea that these things are used.

So, I turned to the “SADMEP” idea. (this link is a google search for SADMEP. That is sad, huh.) But as I worked with this a year or two, I realized that the SA always made a zero, and the DM always made a 1, so I added that to my SADMEP poster (sadly, there are no pictures of this, I threw them all away several years ago).

Which lead me to the idea that it is NOT subtraction or addition that is important, it is the ZERO! Same thing with the ONE, those are the important ideas. Those are the identities. Why do we subtract 2 from both sides of the equation above? Because 2 – 2 = 0. No magic. We can actually subtract ANY number, but we chose to subtract 2 because that is the most convenient way to reach zero in one step.

And then I stumbled upon another magic word that goes hand in hand here; convenient. Why do we chose the values we chose to add, subtract, multiply, or divide? Because those values are convenient ways to make a zero or one the fastest.

Back to the ugly equation above: v=ba-d. Solve it for ‘a’. Add ‘d’. Why? because -d+d=0. Do we care what d is, or represents? No, we know how zero works. Same goes with multiplying by 1/b.

Then, show this video: up to the point where he goes into transcendental numbers (approximately minute 6).

I reach for my physics and chemistry books right about now, and find some ugly equations. These in fact. This is the file I start with to get the learners thinking about solving.

And then I go nuts. Put the formula sheet for AP Physics under the elmo. Do the same with AP Chem. Pick one. Pick another. Solve for any variable. Then solve the same equation for a different variable. For every single function / formula, the only thing they can write to justify their steps are “blah because it makes a zero” and “bleep because it makes a one.”

And we discuss that every single problem I can possible give them is solved simply by using these three simple rules. I make a huge deal of this in the log unit because they learn a NEW way to make a 1 in that unit. That is exciting.

All year long, my learners are shouting out, “because it makes a one” when we are working with exponent procedures (note they are not exponent ‘rules’) because that is how math works. Why do square roots and squares “cancel?” Well, they don’t. 1/2 exponents raised to the power of 2 means 1/2 times 2 which equals 1.

That’s it. I use it all year long. I rarely write a radical symbol, just fractional exponents. It just makes sense.

This is a couple of days of work, and I really think pulling from physics and chem texts helps. I have never had such success with solving and literal equations (in fact, they stop thinking literals are any different) as I have had the last two years.

There is a reason they are framed so nicely at the front of my room. They matter, and they solve every problem we encounter.


Some of the CCSS standards this idea hits:





May 302015

I tried to do a 180 blog, and made it to 90. I really don’t know how people like Justin Aion and Sam Shah do it. It is very difficult to find something to day for 180 days without it sounding boring and forced. They pull it off though. That is amazing.

Knowing I can’t pull of the 180 thing isn’t bad, however. I know I can do topics, and I have a topic I really want to crystallize for myself (as well as others.) I have really been toying with the idea of “one maths” the last three years, and I convinced / forced one of my fellow teachers in my building to start doing it as well. The results are amazing. The connections between the different topics are astounding, and the learners see them, are motivated by them, and create further connections as well. To see why the connections are so important, one just needs to read this “Math with Bad Drawings” post. The connections are vital.

Some tools I will use regularly in class.

1. The Three Essential Rules – from day one, these are the only “rules” I will ever talk about. Log “rules”? Nope, don’t have them. Those are shortcuts to understanding why the properties of logs work. Exponent rules? Nope, nothing more than shortcuts. The only rules we will ever explicitly say are these three: Additive Identity, Multiplicative Identity, and balancing equations. How I implemented them can be found here.

2. – This is the first website I load every morning as I get ready for my day. It is essential to visualizing and discussing function families. The main difficulty I have with desmos is I have so many ‘files’ created it is hard to find them all! That is a great problem to have I think.

3. My structure of functions: This is how I organize the entire year. We move from topic to topic, but as we move, the connection to the prior topic is constantly referred to and stressed.Functions
This list is the core of the connections I want to explore and develop this summer.

Some things I want to make explicit for myself.

1. How to connect this list to the CCSS standards and Essential Understandings explicitly.

2. How to connect each step to prior knowledge in a stronger way.

3. How to connect each step with the breadth of knowledge required (for example, quadratics have many ways to solve).

4. Finally, why in the first place! It seems odd to put the why at the end, but I think it is easier to think about the why once it is all laid out. Does this curriculum have an advantage over the standard “textbook” curriculum? Anecdotal evidence suggests yes, but it needs to be better explained before others can weigh in.

It is a large project, but well worth doing. I think it will really make me understand the mathematics better, and enhance my teaching tremendously.


I better not slack off. Lisa and Meg both called me out. and Stay focused Glenn!

Jan 172015

Nothing annoys me more in teaching math than a bunch of rules to memorize, and rational function come with their own complete set of rules to memorize. I really find that annoying, and I have been on a personal quest to make sense of algebra through a combined set of understandings that will bring comprehension, not rule following.

I have found that in large part through the (h,k) form of the algebraic functions (and here too). Not just a little, but the (h,k) form now drives my entire instruction to the point where my learners are asking me first “how do we undo this” instead of “what chapter is this” as we are learning the math.

So, rational functions. How do the “rules” of horizontal asymptotes fit for rational functions. I really struggled with this the first year I was working on the translations and (h,k) ideas, but this year it all fell into place.

Lets take two functions, f(x) and g(x) where the highest degree is m for the numerator and n for the denominator (just keeping things in alphabetical order).

The rules that everyone knows and hates:

If m=n, then horizontal asymptote is: y=a/b where a and b are the leading coefficients of the numerator and denominator.
If m>n, then there is no H asymptote [or some books say if m=n+1 then there is a slant asymptote]
if m<n, then H asymptote is: y=0.

Okay, I hate these. I really wanted to understand why, and I fully understood when I explored how to get any rational function into the (h,k) form. How do you do that, you ask? Simple. You do the long division and rewrite the equation in the new form.

First off, though, we need some functions to explore. I have a Desmos file with 1600 different possible rational functions:
 Seriously, 1600 possible functions. 40 for numerator and same 40 for denominator.

I tried typing it all out, but failed, so I wrote it out and took a picture:

2015-01-17 16.05.06

What we see is that the ‘k’ value is always the horizontal asymptote. What we also see, is that there is ALWAYS an asymptote when m>n, and sometimes it is a linear slant. It also, can be a quadratic slant, or cubic slant. What is important is that the horizontal asymptote is a way to discuss the END BEHAVIOR of the curve. If we have a slant asymptote, what is happening is the original function is approaching the value of another function instead of a constant.

Rock my world.

So, 2x^4 +3x^3-2x^2 + 5 divided by 2x^2+4x-2 gives us a ‘k’ of x^2 -.5x +1. The “slant” asymptote is a quadratic function.

2015-01-17 17.14.21Here is the math:
 and the Desmos file.

What is amazing here is the long division and putting the function into (h,k) form means you do not have to remember ANY rules with rational functions. It also means there is a reason to teach long division of functions as well.

If our goal is to create a unified, sense-making structure in algebra, this is how it is done.

Let me know if I have made a mistake somewhere or there are flaws in my thinking. This is one piece of the larger structure I am seeing with this approach to algebra, and I really want to push the envelop and limits of of the method.
At this point, what I see is that the “rules” of horizontal asymptotes are nothing more than tricks. The math is the long division and rewriting the function into the (h,k) form to show the translations, and reflection.

In addition, if you look at the functions I used in the explanation above (the first picture I used), you will see that only when the function is put in (h,k) for does the reason for the reflection show up. If the function is left in standard form, the reflection is hidden.

Nix the Tricks! This is the reason.

Nov 072014

This post is a branch of yesterday’s post on polynomials. As were are teaching this year, I am giving them some tough, torturous problems that are about incorporating all the math from previous units into one problem each time.

Yes, I want every problem to be a review of the previous year’s materials. And it is working so far. Another teacher and were looking at this problem set at lunch and we were discussing whether or not we had to have pairs of parenthesis that were conjugates.


I mean, I did give them pairs that were conjugates in each of these, but do I have to?

No. And not just no, but why should I?

Let’s think of a polynomial differently. A polynomials is nothing more that a series of “lines” that are multiplied. Those “lines” can be real lines  (x+2) or irrational lines (x+2sqrt(3)) or even imaginary “lines” (x+2i). When we take these “lines” and multiply them together we get a polynomial function.

cubic1 or cubic2

or even! cubic3. Yes, I had to do this in the Nspire software because this last graph breaks Desmos. Desmos cannot graph the imaginary numbers of the intercept form of a polynomial. #sadness

But notice that in each of these graphs I used the conjugates. Did I have to?



Not at all. Here is a polynomial function, perfectly formed, that is composed of three lines multiplied together. Can I do it with three different imaginary numbers? No. That is where we MUST use conjugates. Only in that one, special case.

So the question is why do some teachers think that conjugates are required in ALL cases of polynomials? The reason is very simple. Only when we have conjugates can we form a standard form function that can be solved with the quadratic formula.

That’s it. That is the only reason.

I was pleasantly surprised to find this series of old posts by Mr. Chase this week as well:

Do Irrational Roots come in Pairs Part 1

Do Irrational Roots come in Pairs Part 2

Do Irrational Roots come in Pairs Part 3

He approaches it from the standard form side, but I think it is far easier to see why it is perfectly okay from the intercept form side of a polynomial.

The real issue is do you have to have conjugates of imaginary numbers in a polynomial. I think, and I do emphasize the THINK, that we do. If we have an imaginary number in the standard form polynomial then we can not graph it on the real plane.  I have been trying to think of a work around (reminiscent of what I did with the quadratics here and here.) Still wrapping my head around it. Not sure if I can see a way out.


How does this relate to my class? Well, my learners are doing problems like the fourth picture. That counts.

Nov 062014

I haven’t posted in a while, mainly because I am just so happy with how my classes are going. I will focus on Alg 2 here, because these awesome learners are just knocking my socks off.

I am in the polynomial unit, knee deep in graphing, and increasing, decreasing, relative mins, relative max’s, absolute mins, etc. This is the problem set we were working on today in class:


Here are the questions I ask (docx format) for every single graph, from lines all the way through sin & cos at the end of the year.

Yes, some of these are going to be Does Not Exist. That is okay. Just because we don’t need to think about asymptotes with cubics does not mean we shouldn’t ask about them.

A little back story before I say something about my learners. I used to teach the textbook. I admit it. I sucked, horribly. My learners did not connect anything with anything and they did not see how to connect topic from one unit to the next. I was frustrated. So I first came up with my list of functions in (h,k) form, wrote it on my board and changed how I approached algebra.

2014-08-10 15.43.46

That was a win. But, then I was frustrated because every time I changed the graph, added an exponent, I had to teach a new set of vocab, but everything was the same; so why was I teaching new stuff? Why couldn’t I teach all the vocab up front, and then just explore the heck out of each function family?

Short answer was, I could. So, I did. That is where the form above came from. I introduced it last last year, and used it and modified it and tweaked it and the learners responded.

Enter this year, this class. I have everything set on day 1. We entered the year thinking about connections and planning our math and discussing end behaviors of lines (wow, that was easy, hey, they are always the same!, etc). Then quadratics, and we completed the square to get vertex forms, and we factored, and saw how intercept, standard and vertex forms were all the same function, and and and.

Enter polynomials.

We have done them from standard form, and done the division to get intercept form, we have broken these guys down every which way. I have tossed them fifth degree and fourth degree polynomials, they didn’t even blink. “Oh, so this just adds a hump to it.” I have explored more in polynomials this year than ever before.

And, since it is a constant review of prior material (“If this works with quartics, will it work with quadratics too? Yes”) I am constantly cycling and eliminating the mistakes my learners made in previous sections and on previous exams.

Which brings us to the problem set above. That is a killer set. The 4th and 5th are tricky, and they struggled. Until one of the class members said, “Don’t all we have to do is distribute them and so it is just a bigger distribution problem?”

Done. And. Done.

Now, of course there is a nicer way to do it. Substitute “u” or some other variable in for (x-3) in the fourth problem so you are multiplying binomials first. It saves time. BUT, it was not necessary to show it. They know distributing, so distributing is what works and they rocked the socks of of it.

So, why have I not been posting much? Because I have been enjoying the heck out of teaching. These learners are taking these ideas and running with them.  And I love it and them.

Sep 232014


That’s right, they got greedy, and lost. Well, everyone gained, knowledge and skills that is.  Today (and tomorrow in one period) in AP Stats we are playing The game of Greed. This is a great game, that challenges the learners in the end to make box plots and comparison statements about the created data.

You end up with some great data to use in class.

2014-09-23 10.46.52     2014-09-23 14.10.14


What is especially great is the right picture, period 5. Notice the big, fat zero? Yes, a female in the class purposefully took zero points. This was a VERY high scoring game as well, the die was very generous to them, and that zero affected everything. I just laughed when she said she was going to purposefully take a zero. It is well within the rules.

This achieved one goal of getting learners talking about the math, at least. 1 thing accomplished today for sure. They also learned more about comparing distributions and using boxplots. 2 and more things accomplished.


Algebra 2

This class was awesome today. I gave them a quiz. They had 1 quadratic function in vertex form, 1 in intercept form, and 1 in standard form. They had to turn the one they chose into the other two forms, and then answer all the questions about the function.

Oh, did I mention that if you choose the vertex form, the max points you can earn is 80% of the points? Intercept was worth 90% and standard form worth 100% of the points. They could choose 2 to do, but I would only grade the ONE they told me to.

As I walked around, I saw lots and lots of little mistakes. Silly mistakes. They would be losing, as a class, a ton of points because of not checking signs, and other silly things. I didn’t want that to happen, they knew better, but they were being inattentive to details. So, with 15 minutes of class I told them they could ask anyone in the room any question they wanted to, but they could not ask me.

They figured out pretty quickly they were being silly. Tomorrow’s quiz for real will go differently. Same set-up. Different equations.

Sep 182014

All models are wrong, but some are useful. George E.P. Box

AP Statistics

I was able to use this quote today in class. I was happy.

My learners were happy too, well, mostly happy. Well, okay, not happy at all at first. At first they hated me. They were struggling with learning how to do 1-variable stats, boxplots and histograms on their calculators in AP Stats. To force the issue of “you must do this, quickly and accurately” I gave them the following handout.

Ch 4 Box Plot Histogram 5 number summary INB 2013

5 data sets, all real, all crazy, none of them particularly easy. The golf data set is just weird.  These are clearly not data sets made up to look like something legit. They are data sets chosen to make them question whether or not their window is set right, whether they entered the data correctly. It forces discussion.

Then, they had this as homework.

Ch 4 – Tomato plant experiment

Yea, I am a demanding. They have until Monday, so I am not worried about the time it takes. But if they can’t make a graph, this is an impossible handout. If they try to get summary statistics by hand, they are in trouble.

I am interested to see what happens on Monday.


Algebra 2

Whew. This class started out brutal, but by the end of class they were ripping quadratic equations in standard form into (h, k) form in seconds. y=2x^2, or 3x^2, no problem. They were able to factor out the coefficient and jam on it. I was really happy about it. They struggled at first, but they were helping each other and they all had it by the end of class.

The assignment was to take 3 functions and put them all in the other two forms. Yes, the form for the second one requires the intercepts be written with complex  numbers. Are all functions factorable? Yes. Are all functions easily factorable? No.  Graphing will get them the intercepts? No. Graphing will get them the vertices at least? Yes, but (1,18) and (-3,-22) are two of the vertices. Not easy at all.

Sneaky Waddell, sneaky.

3 functions