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.

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.

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.

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.

Yup, look at the bracket of zeros and ones.

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.

[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.

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. Desmos.com – 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.
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.

edit:

I better not slack off. Lisa and Meg both called me out. http://www.teachesmath.com/?p=765 and http://www.megcraig.org/?p=394. Stay focused Glenn!

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:

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.

Here 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.

Vocab is killer in stats class. If there was one single thing that determines success or failure in stats class it is mastery of the vocabulary. I am really trying to make that a smoother transition, and I think so far it has been successful.

One new thing I tried is Shelli Temple’s (@druinok) Quiz-Quiz Trade exercise.

They are 32 flash cards with a sampling situation on the front and the answer on the back. Nothing truly spectacular in difficulty setting up, but really, how much of what we do is super complicated.

I gave one class about 30 minutes to do in class, and the others 15, but all of them had great questions about why this was cluster, but that was stratified, etc.

On the quiz they are getting, the question I am most looking forward to the answers is: “Explain the difference between cluster and stratified sampling.” I hope this helps them see the differences in a sharper, more focused way.

And then there is tonight. This is a short post because of tonight.

I leave school in 15 minutes for UNR, where at 4:00 pm I have my first grad school class to begin my Ph.D. process. I am a little nervous, but a LOT excited.

I am not clear in my mind if I am going to use this blog as a platform for reflecting on my studies, or if I should keep them separate and start a new one.

Any guidance from the readers? All 2 of you? 🙂

Anyway, off to a class in Qualitative Research in Education.

Grant Wiggins threw down a challenge, and it has taken me so long in thinking about it that there have been responses and followups already. I hope I am not too late to the discussion.

I have a different idea of “Big Ideas” than Mr. Honner or Mr. Wiggins, and I think it fits Mr. Wiggin’s idea of what a big idea should look like.

#### My first big idea is Inverses & Identities.

I want to start with a statement Mr. Wiggins makes in his response,  “I am looking for those ideas that are big – powerful and fecund – for both novice and expert.”

Does the circular idea of Inverses and Identities meet this criteria? I think it does.

In elementary school, learners focus on the most basic inverses, addition and subtraction, before moving on to the still basic, multiplication and division. In Algebra 1, these inverses are used to solve equations through the application of identities.

Every equation solving is about the application of inverses to get the identity, whether it is just adding and multiplying or using roots (exponent of 1/2 times exponent of 2 = 1: how many learners never see this because of the radical sign!) or trig functions, all the way up to derivatives and integrals. All of solving can be viewed through the big idea of inverses and identities.

We can extend the idea of inverse and identity to functions as well. This explain WHY two functions are inverse if f(g(x)) and g(f(x)) both equal 1x . It isn’t the “x” that is important in the composition statements, it is the ONE!

Viewing algebra through the lens of Identity and Inverses allows us to see why we do mathematics the way we do, from arithmetic all the way through trigonometry and calculus and beyond. Abstract algebra makes use of these ideas as well in the structure and definition of operations even.

Novices and experts alike must use and understand just how powerful the concept of inverse and identity are in doing mathematics. I think this is a “big idea” in the spirit of Mr. Wiggins definition.

#### My second big idea is Transformations

I have been trying to make a lot of connections in Algebra class, and I think the idea of function transformations fits this definition as well. What do I mean by transformations?

These transformation rules are shocking to a beginner when they realize that all of the functions have the same properties for all transformations.

And even the expert makes use of the fact that there really is only one set of math “rules” through all of mathematics. The connection between them is demonstrated through the use of the transformation properties and can be taught through the properties. The underlying structure of mathematics that the expert uses to is still evident as well.

I am not sure if this completely fits the big ideas that Mr. Wiggins was asking for, but I know these are ideas that I have been working towards as big ideas for my classroom and my own thinking.

I know my learners have responded to the use of Identity and Inverse in solving and graphing, and the transformation rules concept has definitely had an impact on the learning of mathematics in my classroom.

Homework has been in discussion for a while in my circles. Is it useful? Why do we assign it? Etc. I came across one strong reason why we should really stop grading homework and start grading other activities that demonstrate learning.

That reason? It is Chegg.com. I used to think Chegg was just a place from which to rent textbooks, but that is just not true. They also SELL the complete solutions to every textbook we use in my district.

That’s right. They sell the solutions manual. This list of MATH textbooks they have available is 75 pages long if you wanted to print it. I checked. I found our Algebra 2 book, the Trig/Precalc book, and the AP Calc book on the list.

Now don’t get me wrong, I do not think Chegg is doing anything illegal. Immoral perhaps, but not illegal. They are just selling something that some learners will use to cheat. That is not Chegg’s problem, it is the learner’s problem. Other learners would use it as a tool for learning. Don’t blame the tool, focus on the learner, AND focus on my policies that encourage cheating.

After all, why would I make a large percent of a learners grade something called “homework” when the reality is that category really could be called, “that stuff you copy out of the book but don’t really understand but still get full credit.”

Having a service like Chegg available just makes it easier to justify moving to SBG and away from traditional “homework” type assignments.

Thank you Chegg!

I have had this long term struggle going on in my head this year that we really don’t do a good job making connections between material in our classes, and that lack of connection is one reason why “transfer” (ala. Grant Wiggins and Understanding by Design) doesn’t occur as  frequently as I would like.

Well, I am not going to talk about it any more. I have the beginnings of a plan to enact. There will be many steps to this plan, but I think the starting point needs to be simple to enact and creates some opportunity for connections to be made.

Every test in my department from Algebra 1 through Trig/Precalc must have a couple of different kinds of problems on it. This is step 1 I am implementing next year.

The first type is a literal equation. Of course, as a stats teacher my first thought was M=z*root(pq/n). Perhaps at the algebra 1 level we won’t start there, but we can select most of the formulas needed in geometry and use them as literal equations and every quiz and test solve for a different  variable of one of the formulas. And, here is the kicker, EVERY time, the learner must explain why they are doing the operation. Justification is mandatory. If we look at the Margin of Error formula above, there are 4 different questions to be asked. That is 4 quizzes or tests that one question can be used.

The goal is get learners to think of literal equations a part of algebra and the justifications as the same thing as every other problem. By the time they reach AP stats, they will have seen this equation repeatedly and know how to manipulate it as a literal, not just with numbers in it. We need to connect AP Stats to Algebra 1.

Next, every test at algebra 1 level must have some form of the following question:

Evaluate (x – (x+h))/x with x = 2 and h = 3. Yes, I know it reduces to h/x, but as we move forward with notation, it becomes:

Evaluate [f(x) – f(x+h)]/f(x)  with f(x) = 2x+5, x = 2 and h = 3.  As the years progress the function can be moved from linear to quadratics to absolute value to cubics or rationals.

Finally, truly stress and monitor that verbage “rate of change of” every time the word “slope” is used.  The learners need to hear and write over and over the “rate of change of” the line in algebra 1, geometry, and algebra 2.

The goal is to create a common language / strands through all math courses and chapters that lead to AP calculus and AP statistics. All learners need to be exposed to the language of statistics and calculus repeatedly throughout their education so it is not different at the upper levels.

So those are the three things I can and will implement next year, without fail.

What am I missing?

Any other language to implement? Any other formulas / concepts that can be used at the lower levels of math that lead directly to the upper levels?

Every year in AP stats for the last 4 years I have struggled with getting my learners to understand and use the all important conditions checks in Confidence Intervals and Hypothesis tests. This year I changed up how I taught it, and it has really made an impact. In fact, I can honestly say that all of my learners are using and completing the conditions checks with consistency.

Here is what I did.

First, there are two essential assumptions / conditions:

1. Randomization – is the data collected through some sort of random process (usually given)

2. Independence – what makes the data independent? Stop and think about the context.

Okay, those two generally are not the problem, because they are the same across all types. No mess, no fuss, stop, think and read and think again and you have those two. It is the next two that requires the real thought and memorization. I changed it up this year and put my own spin on them. Instead of naming them as the book, I realized that what they are trying to set is the maximum and minimum sample size that will give you an appropriate sample:

 Z CI or HT T CI or HT 3. Max Sample Size n < 10% of population of interest n < 10% of population of interest 4. Min Sample Size np & nq >= 10 3 parts to “nearly normal”

With this kind of setup, they are thinking about the context more. Why do they have to check the 10% condition? Because that gives you a ceiling on your sample size. Why the success / fail condition? It gives you the floor to the sample size.

My learners are much more focused and they are planning better this year on this topic. I am very pleased by how fluid and easily they are incorporating the CI’s and the HT’s into their language, whereas in prior years it was a struggle to get them to memorize what they were doing.

I think the fact they understand why is making a difference.

My AP Stat class has been frustrating me the last couple of weeks. They have been very blah. Not interested in being aggressive in their learning, not thinking about the formulas for stats, and in general willing to just sit there and expect the knowledge to jump into their heads with no effort.

It came to a head last week where we did a problem as a warm-up, then did two problems from homework, and when I handed out an AP FRQ, they all looked at me as if it was the first time they had ever seen the problem when in reality we had just did three of the identical problems prior.

Yea, I was frustrated.

But, instead of freaking out, I started asking them questions, mainly, “Why?” It turns out they were feeling overwhelmed and frustrated too. The formula sheet was a complete mystery to them, the concepts where going over their heads, and they wanted to freak out as well.

So I burned a day going over the formula sheet.

Guess what, no one has ever taught them how to understand what is on the formula sheet. That should not be a surprise to me, but it is. So here is a sample of what I did:

7 elements in the definition of mean are; xbar, equals, sigma, x, i, n, and divide by. I made them explain what each was, and the fact that the equals sign can mean different things was the toughest. So was the fact that the ‘1’s’ in standard deviation are different. We really discussed what these things meant.

Each formula / equation / definition has the number of elements indicated, and a “Hand or Calc?” statement next to it. I didn’t want them to understand the formula and then forget they are supposed to do standard deviation in their calc not by hand. But they need to understand the formula too.

No one has ever done this with them in any class before, this was the first time they have had to really understand a formula sheet. They hated it. They struggled, they discussed, they asked questions, they growled and shot me dirty looks, and EVERY SINGLE person said it helped them understand the formulas better. This, in turn, means they understand the class better.

Success is a beautiful thing when it works. This worked.

Here is the docx if you want it. It is formatted for use in Interactive Notebooks.