Aug 122013

I tried something new this year. I was a little nervous trying it, but I did NOT want to do the whole “here is the syllabus, let me read it to you” shtick.

So, I made the first day very active and involved.

Did it work? What did I find went well, what didn’t?

In my Alg 2 STEM class, which is an honors level Alg 2 course, the seating randomly with cards and making the learners share something about themselves was rocky at first. I actually had to ask and model, “Did you share? And by sharing, did you actually do this?” walked up to learner, introduced myself, and said 3 things. The learners sat there in absolute silence until I did that, and then they talked, shared, and actually met their table mates.

Next,  the cell phone assignment. OMG, it was a blazing success.  I had 33 learners discussing rich mathematics, arguing about what kind of display to make, the benefit of bar graphs over line graphs over tables, what is important in choosing a plan, etc. The only thing I had to do was answer questions like, “Do you use a lot or a little data?”. The idea of rich problem solving and being less helpful was glorious and absolutely a positive in the classroom.

One learner did ask if I had a syllabus for them. I told them they would get it next week. This week they needed to establish some habits of conversation and working together. Several of them were shocked and amazed that I did not talk at them all period on the first day of class. Win.

My next class was Algebra 3, or Advanced Alg. It is a senior level class for learners who may or may not be going to college, but who need a 4th year of math and refuse or feel they are not capable of doing Trig/Precalc. This is a tough class, and can be challenging to teach because I get a lot of “I don’t get this” or “I am not good at math” etc.

This group I did the literal equation assignment and it was … rough. What was specifically rough was the idea of multiplying two binomials. I got a couple of “why can’t we just do this together?” questions, and on the second I stopped the entire class and gave the following reason why not. “During your freshman year, you did it together. And your sophomore year, you did it together, and your junior year, guess what, you … did it together. Who here thinks that me “doing it together” with you now will make a difference?”  Not one person said it would.

This was a tough assignment for them, and the conversation was heavy on the, “Well, if it was easy, you shouldn’t be in this class.” and “Why would I waste your time with something you already know?” This resonated with them, and made them realize that I was on their side, I was just challenging them. I felt some success, even though they didn’t finish the assignment at all.

Finally, AP Stats. We did the Gender Discrimination problem / simulation. It was a great lesson, where we got to use the vocabulary of simulation right from day one. This was a great thing, and I think it set the tone for a class where discussion and simulation are the most important elements.

Time will tell. This was only day one, but at least it was a thought provoking day for my learners and a successful day for me. I believe learning did occur.

Aug 102013

I am trying a new approach to my first week of classes. Instead of the usual first day where we hand out syllabi, and go over policies and procedures, and bore everyone, including me, to death I am trying something completely different.

First off, I am keeping my room set up in groups. I have 6 groups of 4 each and 2 groups of 5 each for a total of 34 desks. That is my largest class so far, so I will keep that arrangement. I did this last year for the first time, and it was successful. I am working on making it more so this year.

On each table (I think of each groups of desks as a table) is a paper that says:

Please fill out and share with the individuals at your table. Once everyone at the table has shared, raise your hands so Mr. Waddell can pick up this paper.

Table # ________________________

My name is: _______ I prefer to be called: ________

3 things that make me interesting are:

  1. ___________________________________
  2. ___________________________________
  3. ___________________________________

Last year I realized that even after a couple of weeks, some of the tables had not even really shared their names. I am forcing the issue with this file (in word docx format). There is space for 5 learners to write their names since I have 2 tables of 5.

Next, in my Algebra 2 class, they are getting a problem. Here it is, in its entirety.

I need new phones for my house, and we are currently not on contract so we can switch to any company.

I need 2 smartphones and a plan to go with them.

Your task:  In your groups, create a poster that lays out the 2 year cost of my 2 new phones and plan. Each person’s handwriting must be represented equally on the poster for everyone to receive full credit. Decide which plan I should get, and explain why clearly enough that my non-mathy significant other will understand. Graphical depictions of the costs would be helpful.

On the desks will be a poster paper, some markers, and that is all. Of course, I have more information for them, and when they write down a question that needs that information to be answered, I will give it to them. Not until they ask AND have a question written down and ready to be answered will I give it to them, however. If you want that file, it is also in docx format.

Finally, and this is the one I think I am most proud of as a first day lesson, is the literal equation solving review for my Algebra 3 course. I am also going to be using this in my Alg 2 class as a day 2 review.

Start off by using the first 6 minutes of this video:

Right, the “game” they propose is AWESOME. The fact they then also show how they are creating equations to solve is amazing, and if we push the game into the CCSS model and require the writing out of steps to accomplish, then we will end up with something REALLY cool.

So, I created this document (yup, in docx so you can edit it as well).

The first page is explanatory and the link. The second page is the “game” as it is presented by Numberphile, and the third page is 7 literal equations from the AP Physics formulas sheet.

My goal is to have the learners in 1 70 minute class period work together, write out the steps to the game in each case, and then apply those steps to something that learners normally have a HUGE difficulty with.

Why am I doing this? I want to set the tone for the year that the learners need to write, they need to discuss, and they need to ask questions. These are challenging tasks that require the learner to process and develop ideas, not simply puke up something that I have told them previously.

That is my goal.

Will it work? I don’t know. I will find out on Monday.

Feb 012013



Yea, that’s right. I just had 2 classes in a row teach themselves and others how to complete the square with circles and ellipses. How did I accomplish this miracle, because I really do consider it to be a miracle.

In my Advanced Algebra Class we have the following problem called, creatively enough, The Lost Hiker.

Suppose you need to find yet another lost hiker.  Fortunately you have information from 3 different radio transmitters.  From this information you know that he is:

25 miles from Transmitter A

5 miles from Transmitter B and

13 miles from Transmitter C

These radio transmitters are at the following coordinates:

Transmitter A – (25.5, 7.5)

Transmitter B – (-2.5, 3.5)

Transmitter C – (6.5, -11.5)

Use this information to find the coordinates of the lost hiker using the intersections of the theoretical circles.  Show your work below.

There are 4 major steps here; 1. Write the equation of 3 circles, 2. Expand the circle equations, 3. “Collapse” the circles through polynomial subtraction to 2 lines, and 4. Solve the system of lines.

We worked on these for 3 days. These problems are huge and complex, and one single minus sign incorrect means the whole thing works out wrong. But they persevered for three periods in class, until every single learner could do 3 unique problems on their own, and get the correct answer.

I have an excel spreadsheet programmed to calculate an infinite number of nice problems with solutions. The solutions are posted on the board w/ a magnet, and I walk around with a bunch of problems in my hand. Get one right, here is another.

Well, at the end of 3 days, I wrote the following equation on the board and asked them to turn it back into a circle.

x2+ y2 – 6x + 8y = 12

It took them all of 1 minute to have the entire class finished. One learner who failed Alg2 taught the rest how to do it because it was, “Easy, just take the -6, divide by two, because when you square it you have two of them.”

Then I wrote 5 problems on the board.

x2+ y2 – 4x + 12y = 10

x2+ y2 + 2x – 10y = -8

x2+ y2 – 4x + 7y = -20

2x2+ 3y2 + 6x + 15y = 0

4x2+ 5y2 + 16x – 100y = 27

The first two were just some more practice. The question was asked, “Can we have a decimal?” When the answer was yes, they didn’t bat an eye.

The last two did make them think, but they had the factoring done correctly on the left side, they just needed a little hinting for the right side of the equals.

Why did it work so nicely? I think because they really were engaged with the lost hiker problem and honestly worked them. And they worked. They agonized over why they didn’t get the correct answer. They gave me dirty looks, and when we found the minus sign they missed or the extra number they wrote in, they were angry at themselves.

But they persisted! It was a thing of beauty and I loved it.

When they did the completing the square on their own, with only some gentle nudging from me, I told them how proud of them I am. They needed to hear it.

Heck, they EARNED it.

Oct 132012

Grr. Really. I have some frustration I need to get off my chest, so this will be a “constructive venting” post. Constructive, because I really do have a point, and venting because I can’t believe the things that occur in math education.

First off, I have been really interested in how the curriculum of mathematics is designed and how that curriculum actually works against the learning of math as it is presented in our current textbooks. Honestly, I am really disappointed in textbooks right now. I have spent some time examining the vocabulary in different sections of our current books, and I realized that the books set up the topics of mathematics as if each chapter, each topic, is a disjoint Venn diagram. Chapter 2 has no relation to Chapter 3, and that is how math is done.

No wonder learners struggle with math and don’t make connections. We have DESIGNED math to be taught that way. Don’t agree with me? Just look at your textbooks and tell me if the linear vocabulary is identical to the quadratic vocabulary. Then ask yourself why not? 80% of the vocab should be IDENTICAL.

Let me show you what I mean. This is the current state of affairs:

Linear Equations Quadratic Equations
Standard Form Ax + By = C y = ax2 + bx + c
Equation forms y – y1 = m(x – x1) y = a(x – h)2 + k
y = mx + b y = (x – a)(x – b)
m = (y2 – y1)/(x2 – x1)
Vocabulary Rate of change, y- intercept, slope, rise over run, x – intercept x-intercepts, solutions, roots, axis of symmetry, reflection, translations, y – intercept

Hmmm, do you see ANYTHING that overlaps there? The only thing that is the same is the fact that the x-intercept and y-intercept are both used. However, in the quadratic unit, the y-intercept is rarely used, and the x-intercepts are normally called roots or solutions. Learners = 0, Notation = 1, point goes to the book.

Seriously, even look at the idea of the “Standard Form”. Why is it “Standard”? Because that is how we would normally want to write the equation. Lines get some weird, special form that relates to nothing else in polynomials, while quadratics and all other polynomials get something that makes sense and flows from one type to the next. Learners = 0, Complicated Notation = 2; the book is clearly leading.

Oh, that Point-Slope form looks pretty promising, though. It is very similar to the Vertex form of the quadratic. Well, kind of similar. It is the only form in the family of polynomials that uses subscripts, but that is okay, subscripts are super easy to learn. Oh, they aren’t? Subscripts confuse the heck out of learners? You mean there is a difference between a number in front of a variable, a number above and to the right of a variable, and a number below and to the right of a variable? Darn. And if the Point-Slope is supposed to come from the definition of slope, where did the subscripts of 2 go? Why ? I don’t get it. Why did things change? Learners = 0; Confusing notation = 3. Clearly the learners are getting the worst end of the deal here.

How can we make this better? Well first off, we need to realize that there is only ONE type of math we are doing; Polynomials. Start there, and build a set of math from scratch.

Damn. That sucks. That is a lot of work. That is akin to what Exeter has done or what Milton Academy has done. [Terrific article on Milton’s efforts here by the way.] And there in come the rub. Here we have the idea, but it takes a team of willing people who think alike to create. It takes time, energy, money, and effort. And then, it takes only one or two teachers to block the whole thing from implementation in a school or district.


Double Frick.

Here comes the venting. I need to move beyond this so I will get something done.

I didn’t even realize this bothered me so much until I started thinking about why I was so paralyzed for the last two weeks. Very hostile. Very frustrated and angry, and then I started asking myself why. I think I have it figured out. I was at a department lead meeting two weeks ago, and the leader of the meeting presented my idea for lines. (okay, not mine, but Exeter’s). And the department leaders from other schools at the meeting said, “Why?” “Why would we do that?” “Lines are y=mx+b”. “That is stupid, the textbooks don’t teach it this way.” etc.

I. Kept. My. Mouth. Shut. It took heroic efforts, but I did not rip into anyone. I was good. I mean, they didn’t come up with any good arguments why it doesn’t work, shouldn’t work, isn’t a good idea. They just said, “ThatisnotthewayIwastaughtthereforeitiswrong <breath> thebookteachesitonewayandwecannot/shouldnotchange.”

THEN! OMFG. The pain wasn’t over yet. I was speaking to our assessment director after the meeting, and she was just as frustrated as I was, if not more. She told me a teacher actually contacted her and told her she was doing a bad job because she left the section of the book that taught lines as y=mx+b out of the curriculum guide. Why was that bad, you ask? Because that is the ONLY way that teacher teaches lines. That teacher won’t teach it any other way because it is the easiest way.


Really? Sigh. You know that ONE teacher will end up on the next textbook committee, and that ONE teacher will end up railroading 4 other great teachers into buying a shitty textbook simply to shut that ONE teacher up. How do we fix that? How can we?

And that is my frustration? What can I do about that? It has eaten at me these last 2 weeks, and it shouldn’t. It won’t now that I have written about it. I have gotten it off my chest, and that helps.

How do I respond?

I push forward with my Exeter project. I write a paper and submit it to Mathematics Teacher. I stay focused on what I need to do to accomplish my goals.

And remember:

“Don’t say you don’t have enough time. You have exactly the same number of hours per day that were given to Helen Keller, Pasteur, Michelangelo, Mother Teresa, Leonardo da Vinci, Thomas Jefferson, and Albert Einstein.” – Life’s Little Instruction Book, compiled by H. Jackson Brown, Jr.

I have the time. I just need to use it better.

Jul 292012

Many people ask me why I ride my motorcycle long distances in the summer. This summer I traveled from Reno, NV to St. Louis, MO. It was around 4000 miles, round trip, and brutally hot for a couple of states worth of riding.

But, that traveling allows me one single thing I rarely get. Time away from all distractions. It worked. I thought long and hard about the problem I talked about last post; A visual representation to imaginary solutions of quadratics. Somewhere in Wyoming I had the idea on how to prove it. By the time I hit Utah, I had the solution worked out in my head, and I needed to jot some notes. It honestly took me several hours to type up the solution, and without further ado, here it is.


The Goal:

To prove that in a general case, the circle that is created by reflecting a parabola with imaginary roots (the orange one) about its vertex (the black one) will have as its radius the value of the imaginary roots of the original.

We will begin with clip_image002[12] as our initial equation, with one requirement that the discriminant is negative;  clip_image004[10]. This will ensure that our initial quadratic equation has imaginary roots and the parabola exists above the x axis as shown.


Now, we need to reflect this equation around the vertex, but just adding a negative sign in front of the “a” will not do it. If we add that sign in and make it “-a” it will reflect around the x-axis, not the vertex. Therefore, we are going to need to complete the square, get the original equation in vertex form and then add the minus sign to reflect.


Given equation                                                                       clip_image002[13]


First, divide all terms by “a” and set the y = 0                              clip_image006 

This gives us a first coefficient of 1, which makes

Completing the square possible. Next, we will complete

the square by using clip_image008 and its square.                        clip_image010


now that the perfect square trinomial has been constructed       clip_image012

we can factor the trinomial into vertex form.


The center of the circle above can be clearly seen in this form, and is: clip_image014 We will need this later.


Now we need to solve the reflected parabola for x.               clip_image016


Add & Subtract the constant terms from both sides to get:          clip_image018


Move the negative sign from the right to the left side:                 clip_image020


Take the square root of both sides:                                       clip_image022


Finally subtract the constant term from both sides:                  clip_image024

Notice that we have essentially derived a version of the quadratic formula. It doesn’t look exactly like the standard version we all memorize, but it is the same, with one important difference. There is a sign change to the terms inside the radical sign! That will be very important.


This formula gives us where the reflected parabola crosses the x-axis, so we now have 2 points on the circle, the plus and minus, and the center of the circle.


The final step of the proof is to show that the radius of the circle, or to put it in another way, the distance from the center of the circle to one of the roots of the reflected parabola, is identical to the imaginary part of the solution / roots of the non-inverted parabola. So, onward to the distance formula.


We need to find the distance from clip_image014[1] to clip_image026.


Distance formula:                                            clip_image028

Insert the point values for x and y       clip_image030

Using just 1 of the 2 values for the + or -.


Simplify the subtractions:                                               clip_image032


Finally, square the inside term leaving the following:             clip_image034


This leaves us with a pseudo-determinant of:                     clip_image036



However, in setting up the problem initially, we stipulated that the determinant clip_image038 would be negative. If that is true, then the value of inside the radical sign in our last step must be positive!


[And yes, I am cheating. I am leaving it to the reader to show that the way it is written above in the last step as the “pseudo-determinant” and the regular determinant are essentially equivalent.]


Not only that, but the value of clip_image040 which is from our inverted quadratic, is the same value but opposite sign of the more familiar clip_image042 from the quadratic equation.  If clip_image038[1] is negative, our inverted quadratic will be positive with the same value (oh, and it works in reverse too!)


There, I now proved that the reflecting a parabola with imaginary roots around its vertex will allow you to calculate the imaginary part of the complex answer as the radius of a circle created by the reflection.



Jul 272012

For today’s #myfavfriday I am presenting an idea that has been percolating in my head for a while. If you want to know what a #myfavfriday is, then see Druinok’s blog here.


Learners have a devil of a time with quadratics. Afterall, there can be 2 solutions, 1 solution, or no solutions in Algebra 1, and then in Algebra 2, we come at them with the fact that those equations with no solution really do have 2 solutions after all, they are just “imaginary” (could there be a worse name for them, really? Thanks a lot Descarte.”)

But I came across a picture on some site one day, and it has stuck with me. I never bookmarked it, or wrote down the site, so it is lost to me (and I have searched hard for it) but the work blew my mind, and as I have shown it to learners, they have at least gotten a sense that the “imaginary” really does have meaning.

Let’s begin with 2 equations and graphs that are simple, straight-forward and make sense. [all images are clickable to see full size]

graph1and graph2

The equations are y = x^2 – 4x + 3 and y = x^2 – 4x + 4.

A simple change of one number changes the number of solutions from 2 distinct to 2 repeating solutions, and learners don’t have a problem with that idea, generally. Then comes this bad boy.

graph3 y = x^2 –4x + 6

Now they have to do the whole Quadratic formula on it to get the solution, and the solution has those i thingies in it, which makes them all confused and irritated until they wrap their heads around it. And why does it still have 2 solutions? It doesn’t touch at all!

But wait! We can play a game with this quadratic function. What if we reflect the parabola around the vertex in the downward direction? Then we end up with something that looks like this:

graph4a To do this reflection, we first had to complete the square on the original equation to get y = (x-2)^2 + 2. Now, with this equation, we can put the – sign into the equation and get the reflection, y = -(x-2)^2 + 2.

But hold on, see those 2 points where it crosses the X-axis? And see the Axis of Symmetry that goes through both equations? If we use those three points as definitions for a circle, we get the following graph and equation.

graph5 (x-2)^2 + y^2 = 2

Guess what the solution to the quadratic equation y = x^2 –4x + 6 is. If you guessed 2 + root(2)i and 2 – root(2)i  then you are absolutely correct.

The real number part of the complex solution of a quadratic with two imaginary roots is the X value of the Axis of Symmetry, and the imaginary part of the solution is the radius of the circle created by the center and endpoints created when the inverted parabola crosses the X-Axis!

Okay, mind blown. Why? How could I prove this?

Aha! now come into play the hours I spend on a motorcycle every summer. How could I PROVE that this will always work? I have the proof. I am working it up, but it is a pain to type. That, I think, will be the focus of a future, #myfavfriday!

[And I really need to look in to a LaTex module for my blog if I am going to do math. The equations look horrible.]

Edit: 29 July 2012: I proved this assertion, at least to my satisfaction in a followup post:

Edit: 4 August 2012: I found, stuffed in the bottom of my backpack, a rumpled piece of paper with this link on it. I think I did this page justice with my treatment. I wish I had found the page before I spent hours thinking about how to prove it, it gives the suggestion right there at the bottom!

Edit: 18 Dec 2012: @Mythagon posted this picture on Twitter. It is a great visual of what is discussed above, and clearly shows why the rotation is so important.

From: Teaching Mathematics, 2nd edition by m. Sobel and E. Maletsky

Edit: 27 Sep 2015: Wow, a long time since the original post, however I still come back to this every year. Love it. Now, Luke Walsh, aka @LukeSelfwalker added this to the mix. Love it. Click it for the live Desmos file.

Aug 042011

I have done a very poor job of writing about advanced algebra, the course I helped co-author 4 years ago with 5 other teachers in my district. I would like to rectify that this year, and explain more about the course and honestly, get better ideas for the course.

The course is more project based, and it has four distinct, but overlapping sections. Quarter 1 is financial math, quarter 2 is math in art, quarter 3 is math in technology and quarter 4 is math in health / human body. If you go to the site you will see the basic structure and some of the lessons / sites I use to teach the course.

Starting off this year, I am going to do something different with the quarter 1. It always felt a little disjointed to me. We use the materials from NEFE to get us started, and then we jump in much much deeper than NEFE goes. We spend a lot of time on spreadsheets (which are nothing more than giant algebra problems using variables) polynomial equations and rational equations (another way to think of Pert and other compounding equations, ie. purchasing a car and annuities) as well as some basic ideas of personal finance.

This year, instead of just teaching each module as a standalone, I am going to tie all of the quarter 1 together with this outline.

At the end of this project, you will need to show the Banker your portfolio that demonstrates you have the financial knowledge and ability to purchase a house in the North Valleys.

Part 1: Setting S.M.A.R.T. Goals

1. Short term, medium term, & long term goals

2. Tracking your goals

3. Adjusting and rewriting your goals

Part 2: Savings & Compound Interest

1. Calculating compounding interest for n

2. Calculating compounding interest for continuously

3. Finding rate or time in both kinds of compounding (working backwards)

4. Knowing and demonstrating differences between APR and interest rate

Part 3: Career & Income

1. Identify 3 careers for yourself

2. Calculate lifetime earnings

3. Calculate $1,000,000 earnings timeline

4. Comparing 2 and 3 for your careers

Part 4: Purchasing a Car

1. Calculating your payment

2. Deciding on years of repayment

3. Comparing years to payment and making a good choice

Part 5: Investing & Credit

1. Risk vs. Reward

2. “Safe” vs. “Intermediate” vs. “Risky” investments

3. Annuities

4. Credit Cards

5. Credit Scores

Part 6: Budgeting your spending and savings

1. Creating a budget (will be working on all quarter)

2. Projecting income

3. Projecting expenses

Part 7: Buying your home

1. Put it all together for the Banker, and using the financial information gathered to justify to the banker that you are a good loan candidate

That’s right. The end goal and purpose of the portfolio will be to purchase a house. It is the biggest investment a person generally makes, but I also know of 4 learners who graduated within the last 3 years who are now homeowners. It is a reality they can achieve now, whereas 3 years ago it was out of reach.

The purpose of this structure is to create a buy-in. Now they see the end goal. This goes hand in hand with backwards design and Understanding by Design principles. I have emailed out a draft to the other Advanced Algebra teachers, and once I have it more fleshed out I will email out another copy. I also hope to get some feedback from them to see what they would add as well.

So, what do you think? Is this a viable way to put together a quarter on personal finance? Let me know in the comments.