Feb 092015
 

Besides the usual quote on the board today, I also have this math pickup line: How can I know hundreds of digits of pi and not know your phone number?  I am featuring a new math love / pickup line each day this week (some days will have more than one). If you want the list, Math & Multimedia is the source.

But anyway, I hate to even call this a #180 blog posting, because I gave up on that at the semester. I just was not focused enough to maintain. I don’t know how people do it. But I do want to share some of the Central Limit Theorem Love I just did.

The exercise is not my own. I stole it from Josh Tabor and I credit him fully with the idea. What you need for this exercise are pennies, chart paper, and some fun dots. That’s it. You need a lot of pennies though. By a lot I would estimate I have approximately 2500 pennies in a bucket. I don’t know exactly how many, but it is a huge number. I emailed the staff at the school and asked for pennies and they delivered. Each year I ask for more, and they deliver more. It is terrific.

Okay, on to the set up. When the learners walked in the room, they saw this:

2015-02-06 10.00.15

The instructions, the left chart paper for x’s, the middle for xbar’s and the right for p-hats. Yes, the scale is completely wrong on the p-hat chart. It should be from zero to 1. I fixed that.

Then, the learners pulled their coins, found the means, the proportion greater than 1985, and we graphed using stickers for the x’s, writing xbar and phat for the other two. At this point, we ended up with some good looking graphs. We discussed if we could tell the mean of the dates from the x graph, we decided we could not, so we OBVIOUSLY needed more data.

Do it again.

After two rounds, we ended up with these graphs:

2015-02-09 12.23.42 2015-02-09 12.23.54 2015-02-09 12.24.02

 

I did change the 1985 to 1995 by the time I took these pictures from my 3rd period of Stats. The newer pennies the staff gave me pulled the mean up.

I actually tore the “Actual Values” graph down and threw it on the ground because it was so useless. That was the point of that graph. I loved how the other two graphs were so clearly unimodal and symmetric. They fit the idea of the CLT perfectly. The fact they matched was just icing on the cake!

–http://onlinestatbook.com/stat_sim/sampling_dist/ Using this simulation for the CLT, we then looked at what happens when sample sizes are changed, whether the shape of the population matters, etc. It was very eye-opening.

Then we discussed the reason why, how, and what conditions must occur for one sample to then represent the population. The notes I used are here in pdf format. I am trying something for the end of the year where I post the notes before hand and they are required to read them as homework. I HATE going over the notes in class. So far it is a good experiment.

Next up are some in class problems.

This is the third time I have done this exercise, but only the second time I have used xbars and phats. It is very useful to have those there so the formulas make more sense.

The fact that the formula reads “the population mean is identical to the calculated mean of the sample” is very useful when the learners keep the population mean and the sample mean separate.

Jan 172015
 
2015-01-17 16.05.06

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 112014
 

On this bright,cheery Veteran’s day I took some time to clean up my reader, delete a bunch of feeds that I don’t read any more, organize the math teacher and other teacher feeds a bit and catch up on a couple of posts that I saved but didn’t have a chance to read yet. Be aware that the following is not a happy one, but a frustrated one. You can skip to the bottom to see the conclusion that is positive if you like.

 

—- Really, I am setting up an argument here in the beginning and middle, the end has a positive message. Totally okay if you skip the argument. —

 

What got me started writing was a statement by Dan Meyer that he followed Peg Cagle because, “she understands the concerns of Internet-enabled math teachers and she also understand the politics that concern the NCTM board of directors.” (via)

I read the link about “understanding concerns” which led me to think about the organizations I belong to and send money to each year. And let me be upfront about this. I am a member of the NCTM and have been continuously since I was in grad school getting my teaching credentials. I am actually subscribed to more than one journal, and have attended a national conference, a couple of regionals, and a couple of institutes. I have had district funding for some of this, but the majority have been paid for by me with my own money. I am critiquing the organization from the inside, not throwing bricks from the outside.

Okay, with that out of the way. I looked up the NCTM and found that they follow only 226 people / organizations. That’s it. Some of them are classroom teachers, but the teachers are vastly the minority. They follow mainly groups, college professors, and reporters.

nctm

But they are a large national organization. They can’t spend the time to read all the chatter from practicing math teachers. Let’s give them the benefit of the doubt. What about my local groups? Well the Northern NV Math Council does not even HAVE a Twitter presence. None. Scratch them off the list. Is it wrong that they don’t have a presence? No, that is their choice. I joined them in the past and suggested it. They refused because, “teachers don’t have the time for it.” Oh well.

What about the Southern NV Math Council? They do have a Twitter feed. They must follow lots of math teachers and spread the wonderfulness that is our math classrooms? No.

snmath

They follow 83 accounts, very few of which are math teachers. They have only 25 followers. I guess they are only slightly more engaged with the math teachers in Las Vegas than the NNMC is in Reno. But the state organization is doing better, right?

nvmath

No. They follow 83 accounts as well, virtually none of which are actual math teachers. I guess NV is a write off as far as math teacher engagement. I posted this frustration on Twitter, and Lisa Henry shared her state organization.

ohiomath  and I looked up the CA Math Council camath

Seriously folks. If the NCTM is wondering why math teachers are leaving and thinking it is not relevant, these screenshots encapsulate it pretty easily. These are organizations that are pushing to us, but not engaging with us. I only looked at number accounts they followed. Look at their tweet counts. The Nevada Math Council has 36 tweets? The CA Math Council has 552? They have  100,000 math teachers in CA, the birthplace of Twitter and they have only tweeted 552 times? The NCTM has 27,000 followers and have tweeted only 4500 times? Most of which are plugging their upcoming conferences?

But I did say I would look at all the organizations of which I am a professional member. Here they are; the National Council of Supervisors of Mathematics and the American Statistical Association.

ncsm

asa

The NCSM even follows me!? They follow MORE people than the EVERY organization listed above COMBINED! Now we know that following someone does not mean engaging with someone, but it is certainly true that you cannot engage if you do not follow. The chance of the NCSM engaging with math teachers (and look at who they follow, there are many math teachers in the list) is much higher than the NCTM, Ohio NCTM, CA NCTM, and both NV NCTM groups.

The ASA? I joined them to get some additional AP Stats materials through their magazines. They are a specialty group, but they still do a better job than the state general organizations. The general organizations that should be closer to us as math teachers.

 

—- Okay ,the positive that exists after the complaining. —

What really makes this important to me is we are forming a State Chapter of the NCSM. I volunteered to be a member of the committee. I did that a week ago before I thought of doing this comparison, but now I am firmly on the side of participating strongly. NV needs an organization that will engage and lead math teachers. We certainly are not getting either from the NNMC, the NMC or the SNMC. We get ignored and told, but not engaged.

What is odd is the leadership of these groups ARE math teachers. The leaders are people I know (at least in the NNMC) and yet they do not engage? This is a frustrating situation. How many other teachers are frustrated with their local group too?

I am going to send this post to the organizer of the NCSM so she knows where I stand and what I feel the problems are in NV. It appears this is not a NV problem, but a nationwide problem. It is not a problem of “Math teachers don’t have the time” like I was told previously. It is a problem of organizations that are supposed to be leading us, instead are ignoring us.

The new NV Chapter of the NCSM really does need to be a leader. The National NCSM clearly is trying to have more of a leader role than the other groups. I need to do my part locally to make this happen here too.

 

As a counterpoint to my complaining, here is my profile and a couple random profiles. Ilana Horn is a professor of mathematics education (I recommend following her if you don’t.) Dave is a teacher of high school math as well. I just picked them randomly out of my feed. Teachers do have the time. In the vacuum of leadership, we are constructing meaning on our own and further marginalizing the institutions that created the vacuum.

glennilana

dave

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.

problemset

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?

No.

cubic4

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.

Polynomials in Alg 2

 Alg 2, Personal  Comments Off
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:

problemset

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.

Oct 222014
 

I just sent this email to my department today. The subject was “If you still assign drill and kill problems”.

I am posting the text of the email without comment. It was spurred by a conversation on Twitter with @DDmeyer and @JStevens009.

Text is below:

Good morning all,

Hate to be the bearer of unwelcome news this morning, but if you are not keeping up with the world of math technology, it is reaching a tipping point of changing what we do. Check out the site: https://photomath.net/ and think about what it means to our profession and our classrooms.

Only around 30% of all kids have iOS devices, but in 2015 when it hits Android and the entire market is open, nearly 100% of our learners will have access to this. For free.

Kuta worksheets? Textbook problems? This type of software will render this type of homework obsolete. Desmos and TI-calculators have nearly done so already.

We have talked about the purpose and need to rethink what homework is for, and the tipping point is rapidly approaching where we really need to make some changes to what and how we handle it.

We need to think deeply about how we can create an environment of learning both in and outside the classroom, because the technology is making outside the classroom a moot point unless we make some changes long term.

Oct 212014
 

My last post was in September and was entitled “I am tired so tired.” That ended up prophetic because with the fall break I ended up not posting for three weeks.  Whew. I needed that break to get back on track and then caught up. I am glad I have my head on right now and am refocused.

So, on to new posts.

Today, Eli Luberof posts on Twitter:

 

Yay! This was amazeballs! You should try it. If you don’t want to try it, I took a screencap of what Stats looks like in Desmos. Desmos 2     

 

This just made my day. Now I can do data entry, just make sure to use the subscripts on the data points and the variables. Notice that you can do different forms of the equations! You can do yhat = a + bx or you can do yhat = mx + b. Either format gives you the r value and the residuals.

Go residual plots! That is awesome in and of itself. I am in love. Right now, the only thing that kind of saddens me is that I can not use “height” and “time” as variables. Desmos needs the x_1 and y_1 format to work. That is sad, because in stats we try to use words as variables. Oh well. At least this gives me a clean, nice, clear way to teach this topic to all classes.

Eli, two thumbs way up for this addition. [And just as an aside to the rest of the #MTBoS to show how responsive Eli is to us. He did a small focus session at #TMC14 with several stats teachers and asked us how we would want to do this. He told us he had not thought about residuals at that time, so to see residuals so easily pop up here was very exciting. Desmos is truly responsive to teachers needs.]  

Edit: Okay, so I kept playing and tried to build a lesson with it on residuals. Found some interesting things that I like and don’t like. I tried to calculate the predicted values, the yhat. Desmos didn’t like merging the algebraic with  the regression. Not at all. This is what I got when I tried: Desmos 3                          

 

 

 

 

 

 

 

I would like to have the ability to show not just the actual value, but also the predicted value. That way I can show where the residual actually comes from. Beggars can’t be choosers though.                      

Edit again: Desmos comes through like a champ. They tweeted this out to me last night.  

 

Which led me to do this:

desmos4Click on it and look at it large. You can see how I used a function per Desmos’ advice to then calculate the residuals instead of just using what is given. This pretty clearly shows where those points at the bottom come from.

I am teaching residuals right now in AP Stats, and I will use this as a demonstration (if it is up long enough) to show what the calculators are doing. Too often the learners don’t think beyond the buttons and just mechanically find the resid plot without thinking about what is going on.

Sep 302014
 

Today wiped me out. I had all 3 sections of AP Stats today, and I heard the same comment from all 3 classes. It went something like this, “I never understood the z score when we did it in Alg 2. My teacher just said, memorize the formula and get find the number.”

Sigh.

That kind of statement just grates on me. We are required to teach it, but some teachers don’t take the time to teach it well, and some just give it a cursory glance and turn learners off.

I am starting from zero with the idea, and building slowly and carefully. It is exhausting though. I never asked who their teachers were (although 3 minutes with the computer will tell me) and I won’t ask. I respect my colleagues too much to think less of them for the destruction of the stats unit.

At least I have learners from my Alg 2 class last year who cheered when I said the phrase “z score”. That made me happy.

Okay, repeat after me: The z score is the number of standard deviations from the mean.

Why over complicate such a simple idea?

Okay, back to grading hell.

Sep 292014
 

I tortured my learners with a game, a game that was awesome and they all agreed was worth while. We played a Stats Pictionary!

I used this document.  Ch 5 – various distributions- Pictionary   I created these distributions using the Illuminations Applet called plopit.   http://www.shodor.org/interactivate/activities/PlopIt/

Here are my rules:

1. Each pair gets one distribution.

2. You have to write your SOCS (Shape, Outlier, Center and Spread) so clearly, using values and descriptive words so that the other learner can duplicate the distribution without asking any questions.

3. Once the SOCS are written, trade papers, and then try to re-create the distributions from the descriptions only. DO NOT SHOW the original.

4. Once the distributions have been done, show the distributions and compare.

5. Repeat.

 

That’s it. Very simple. I did model one for one class. They were struggling with the idea. Once I modeled one, they were fine.

Big takeaways: They realized their SOCS sucked. The figured out what they needed to do to make them not suck, however. Also, the first round went poorly, but they quickly modified their SOCS statements to be clearer.  Finally, Spread was the one thing they still struggle with. They are getting better, but trying to estimate from a graph is hard.

We ended up doing around 4 to 5 graphs in the 35 minutes I allowed for it. It was a great experience I think.

 

I was asked to show my notes. This is the ppt I am using for all of 1 variable quantitative stats. I don’t think it is anything special, but I AM trying to be more creative and thoughtful with it.

I can’t get away from all the notes. I don’t know if it is me, or the material. I do know this is about 14 days worth of notes. I have not done a whole day. A few slides. Stop. Do activities. More notes tomorrow. More activities.  Check out slide 64. :)

Categorical & 1 variable Quantitative

 

——————-

Sorry to be silent last week. It was crazy and I was in a spiral of grading hell. I am not out of the grading hell, but I am out of the depression that results from the spiral. Now I am focused and getting caught up.

Sep 232014
 

gottoogreedy

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.