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:

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:

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.

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.

or

or even! . 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.

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.

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.

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.

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?

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

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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.

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.

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.

Today was a mixed bag of, well, weirdness and frustration with some awesomeness.

I will start with the awesomeness. This morning at 7:15am, a learner from last year walked in and asked for help with his college math class. Loved it. I worked with him for about an hour (into the first class of the day) and he left feeling much better about his class and caught up. I felt really good about being able to continue to help my learners even after they have graduated and moved on.

The frustration was that classes can be so very very different. My one period is chatty. And by chatty I mean so talkative they actually miss out on some of the lesson because they just won’t learn to listen. They are great learners, but the social aspect is killing them. Meanwhile my other two classes have exactly the opposite problem. They are so un-chatty that they sit there in silence waiting for someone to speak up.

The classes were so polar opposite today, and I was completely flummoxed by it. I need to get the one class to talk about the math, and the other classes I need to get to talk about the math! Well, at least it is a common problem.

Finally, the weirdness. I gave my AP Stats class this lesson. Ch 4 – Tomato plant experiment But today we were learning how to calculate the fences for outliers and we applied the calculation to data set E.

The NSpire says the data point 20.2 is an outlier, as does the TI-84. But using the 5 number summary and doing 1.5(IQR) + Q3 and Q1-1.5(IQR) we get a fence of 19.95.

The value o 20.2 is not an outlier, but the graphing calculators call it an outlier. That is weird.  I used JMP hoping it would give me different values for the 5 number summary. Nope. Same as the graphing calculators.

This is some weirdness I can’t easily explain, but it did hammer home the idea we should not trust the calculator.

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.

One thing I am really working on in AP Stats is the amount of notes, the lack of notes, and the engagement of my learners. AP Stats is one of those courses where the amount of vocab to assimilate is so huge, that it cannot all be done by activities. I have found that a mixture of activities and notes, and assignments and cycling back again helps tremendously.

I have the one slide from my notes today above. The literal, not figurative, brick wall between the two ideas of mean & standard deviation and median & IQR was very well communicated this year. The learners told me they understood. The formative checks I did supported that.

I still am not confident. Too many learners mess up this idea every year for me to take the face value word on it. I will be giving some questions over the next couple of days to make sure.

The re-writing of my slides to be word minimal, picture heavy, and discussion focused has changed how the class goes when I am doing notes, at least. I am happy with that aspect, and the learners I have asked directly about the notes have told me they are very useful and not boring.

That is something at least!

——————————

PhD spillover

As an aside, the class on non-parametric statistics has taught me one thing that has impacted my AP Class. The structure I used last year as far as how I teach the content is right on the money.

In the PhD level class, we look at every problem first from the perspective of “is it categorical or quantitative” and then “how many variables”. So far, we have limited the decision to just categorical, non-normal problems (hence the non-parametric! label of the course.)

For Inference section, the course will be divided up into a. quantitative 1 sample, a1. confidence interval, a2 hypothesis testing; b. quantitative 2 sample b1. confidence interval, b2. hypothesis testing, etc. I think this structure leads better to the advanced level stats if they take a next class.

It is also the exact opposite of what our textbook does. Oh well. I didn’t use the textbook structure for 2nd semester anyway for the last 3 years. This just reinforces that decision as a good one.

——————————

Finally, some lesson ideas I am working on.

That’s right. Funky dice!

On the left we have odd shaped, non-standard dice. Awesome. Are they fair? Not sure. On the right we have, yes, for reals, 5 sided, 7 sided and up dice. No joke. I once argued that a 5 sided fair die could not exist. Is it fair? Not sure. I am writing some lessons for expected value to take advantage of both of these.

I also received word from Robert at http://thedicelab.com/ that my order of weighted dice is coming soon.

Heh heh heh. That’s right. Real, honest to goodness (well, dishonest to goodness) weighted dice.

Expected value here we come! More later on this idea.

You are here! That is my AP Stats objectives board for the next few weeks. Today and yesterday we finished up Categorical data analysis with Relay Cards. It was very successful. I had many learners telling me they understood what they were doing, and they were saying this even though they were making mistakes in the reading of the problems.

I like the fact they were happy with the content and realize that making mistakes in reading did not mean they were not understanding. I need to figure out a way to make sure they realize that.  This is an issue I need to think on tonight and figure out a way to pull it together for them to think on as well.

I wish I had a magic phrase that everyone would hear and just go, “Aha.  I understand that making mistakes does not mean I don’t understand, it just means I made a mistake.”

I have RADICALLY revamped the notes I am doing as well.

This is the old PPT from the book. I am ashamed to say I used this for several years.

Here is my notes for this year, same topic. Yes, the quote is from Dr. Who. I will see how many learners pick that up.

Yes, there is still text on the slide, but less. And more of a story instead of regurgitating stupid words.

I am trying to do more of this type of thing with my notes instead of the “The definition of a relative frequency histogram is” blah blah blah. So far, the learners are telling me my notes are not horrible. They read less, they write less, and they are learning more and being much more quick in doing problems and asking better questions.

So far, success on that front.

Ch 3 – Relay Cards (made by Shelli Temple)

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Algebra 2

Whew, but Alg 2 is brutal.

We are working our way through a series of Quadratics. Today I introduced completing the square and justified it by needing the vertex form. All of the quadratics we have done are found here:

I started them off in vertex form, they had to provide intercept and  standard form. Now I am giving them standard form, and they provide vertex and intercept form (among all the other information found on the exploration sheet.)

They are hating me right now, but it is getting easier. The idea that ALL quadratics are factorable, is stressing them out. Some are easily factorable, some require the quadratic formula, but ALL are factorable.

Ouch.