## It Took Me 2 Years to Get This Approach to Imaginary Numbers

This past week the NCTM annual conference was held in Boston, and what an enriching epxerience! What made it so special this time around was meeting and hearing from my PLC of Twitter friends, many of whom I had admired from afar for some time. I’ll discuss the power of the MTBoS (Math-Twitter-Blog O’Spehere) in a later post.  Today I want to focus on a powerful session I attended in Boston, and how a new persepective developed – even after a 2 year delay.

The story starts 2 years ago at Twitter Math Camp in Philadelphia.  At that conference, I participated in an Algebra 2 small group, facilitated by the super-creative Max Ray, from the Math Forum. Splitting into smaller groups, I worked with a team to think about rational expressions – a unit which is often dry as sand in Alg 2 courses, and where I thought we could make some head-way. While we worked on our slightly-less dry, yet safe lessons, Max and a small group were discussing complex numbers on the board. There were mysterious circles, transformations, and discussions I didn’t understand.  I suppose I was taught about complex numbers the “traditional” way – we need them to solve certain quadratics and memrize some wierd rules about their behavior. We perform strange operations on them, and we definitely don’t ask why. I suppose I could have simply wandered over to the group and found out more, but the mathematical intimidation factor was high – I’m sometimes too proud to admit what I don’t know.

Fast forward 2 years, and I see Max is presenting a session with Michael Pershan. This is a must-attend. Two engaging speakers whom I appreciate for their ability to use students’ natural curiosity to facilitate math conversations.

Here’s the set-up: Michael finds a handful of volunteers to stand at the front of the room, standing on a hypothetical number line (Max stands at zero). The participants are then asked to consider the following transformations to their value, and move accordingly, returning after each move to their original position.

• Add 2 to your value – participants all move to the right 2 spaces.
• Multiply your value by 3 – participants all move to the left or right accordingly, depending on whether their original value is positive or negative.
• Multiply your value by -1. OK, now the plot thickens.  While we can find our new position, Michael does a materful job in having participatins reflect upon the transformation. The first two moves required left and right shifts; here we need to consider a rotation about the origin. This rotation provides a rule for multiplication by a negative.

The table has been set, the silverware polished. and now we need some new volunteers. We have a new number line, and some new transformations to think about.  BUT this time around we want to complete our movement by using the same transformation twice.  Let’s roll!

• First, add 4 by using the same transformation twice.  This is a nice appetizer – let’s move 2, then 2 more.
• Next, multiply by 9. This is a little trickier, as some folks almost crashed into the next presentation room. But two multiply by 3 moves do the job.
• Now, multiply by 5. Oooh….we have an entry point into radicals. Some quick discussion, and we two moves – multiplying by a little mroe than 2 each time.
• Finally, multiply by -1…in two moves…..

WAIT!  This is the stuff Max was talking about 2 years ago that I didn’t get.  The bulbs have gone off.  I GET this now!  We do a 180 degree rotation do perform a multiplication by -1, so now we need two 90 degree rotations.  And now we have an entry point into imaginary numbers, without the scary-sounding term.

What I appreciate most here is that we don’t need to wait until deep into algebra 2 to think about the imaginary unit.  These concepts are accessible to younger students, and we have a responsibility to achieve some conceptual buy-in before just thrusting abstract ideas in front of our students. You can find Michael and Max’s shared files here on their Teaching Complex Numbers page.

I get it now…I think….and I’m not ashamed to say it took me 2 years.

UPDATE: You need to immediately run to check out the fun summary Ashli has provided of this session. Her notebook sketches are unreal (in the non-numbr sense)!

## Statistics Arts and Crafts

The Chi-Squared chapter in AP Statistics provides a welcome diversion from the means and proportions tests which dominate hypothesis test conversations. After a few tweets last week about a clay die activity I use, there were many requests for this post – and I don’t like to disappoint my stats friends! I first heard of this activity from Beth Benzing, who is part of our local PASTA (Philly Area Stats Teachers) group, and who shares her many professional development sessions on her school website. I’ve added a few wrinkles, but the concept is all Beth’s.

ACTIVITY SUMMARY: students make their own clay dice, then roll their dice to assess the “fairness” of the die. The chi-squared statistic is introduced and used to assess fairness.

You’ll need to go out to your local arts and crafts store and buy a tub of air-dry clay. The day before this activity, my students took their two-sample hypothesis tests.  As they completed the test, I gave each a hunk of clay and instructions to make a die – reminding them that opposite sides of a die sum to 7. Completed dice are placed on index cards with the students names and left to dry. Overnight is sufficient drying time for nice, solid dice, and the die farm was shared in a tweet, which led to some stats jealousy:

The next day, students were handed this Clay Dice worksheet to record data in our die rolling experiment.

In part 1, students rolled their die 60 times (ideal for computing expected counts), recorded their rolls and computed the chi-squared statistic by hand / formula. This was our first experience with this new statistic, and it was easy to see how larger deviations from the expected cause this statistic to grow, and also the property that chi-squared must always be postivie (or, in rare instances, zero).

Students then contributed their chi-squared statistic to a class graph. I keep bingo daubers around my classroom to make these quick graphs. After all students shared their point, I asked students to think about how much evidence would cause one to think a die was NOT fair – just how big does that chi-squared number need to be? I was thrilled that students volunteered numbers like 11,12,13….they have generated a “feel” for significance. With 5 degrees of freedom, the critical value is 11.07, which I did not share on the graph here until afterwards.

In part 2, I wanted students to experience the same statistic through a truly “random” die. Using the RandInt feature on our calculators, students generated 60 random rolls, computed the chi-squared statistic, and shared their findings on a new dotplot.  The results were striking:

In stats, variability is everywhere, and activities don’t often provide the results we hope will occur. This is one of those rare occasions where things fell nicely into place. None of the RandInt dice exceeded the critical value, and we had a number of clay dice which clearly need to go back to the die factory.

## Introducing Discovery Hour with Codebreaking

Our school has been on a semester block schedule for over 20 years, with some tweaks made to accomodate building size, AP courses and electives. But this year brought a major schedule change, and an opportunity to think about how we use time to engage students. After periods 1 and 2 (each 75 minutes), all students move into something called HATS period. The acronym stands for Hatters Achieving Targeted Success, and during the period students have a lunch period, along with assigned time with teachers. It’s a great mid-day block for students to touch base with activities and clubs, seek help, make up work, and our RTII team has utilized the time to meet formally with students and facilitate individual help sessions.

I saw an opportunity to engage students in meaningful activities during this time, and have started Hatters Discovery Hour – modeled after the Genius Hour concept many elementary and middle schools offer. My thought is that so many of our teachers have awesome ideas to share which don’t quite fit class time. Also, it’s an opportunity for students to experience teachers they may not cross paths with during their high school career. Let’s build more connections!

The past 2 months have seen some fascinating offerings.  Our No Place for Hate Team has used Discovery Hour to facilitate open discussions on race relations.  Meanwhile, a science teacher shared his experiences working as an EMT in a medical diagnosis session. Juggling was the fun focus of one session, and Discovery Hours on memory systems, photography and meditation are in the works.

THE REAL IMITATION GAME – CRYPTOGRAPHY

For my Discovery Hour session, I shared many of my collected activities on codebreaking. With Oscar season just passed and some simmering interest in the Imitation Game, it was a perfect time to talk about the role of codebreaking through history. Even better, my principal and district curriculum director (and my former boss) were on hand to join in the fun:

I was ambitious, trying to fit 4 codebreaking challenges into the hour. In the end, we had just enough time to keep things moving and hold some fun discussions in these 4 areas. Scroll below to download the handouts.

CRYPTOGRAMS – We started with a basic letter-to-letter cipher. I used a long quote from Bill Gates, which almost turned out to be too long – as I felt a time crunch hitting early. But longer quotes allow more entry points, and I couldn’t pull my principla away from the challenge!

CAESAR SHIFTS – Here we used an online applet to explore shifts, and this provided an entry point for modular arithmetic, which few of the students had encountered before.

HILL CIPHER – By now we had established that the first two coding procedures did not seem too secure. I have shared Hill Cipher with students in my classes before during matrix units, and again a cryptography website was helpful in providing some easy codebreaking trials. When I have done these in class, I often develop problems which get around the modular arithmetic issue (it takes longer to discuss than I often have time for) but we were able to squeeze in a 5-minute mod primer.  See below for other Hill Cipher problems I have used.

THE ENIGMA – The cherry on the sundae, and where many students were stunned by the complexity. This online Enigma simulator is one of my favorites – I love the visual of the wiring. So many good questions concerning inverses, how codebooks were traded and how the British broke the code. I left enough time to show Numberphile’s Enigma video, which capped off the hour nicely.

Looking forward to sharing more of what I know in later Discovery Hour sessions, and thrilled so many of my colleagues are buying into the idea.

## Inverse Function Partner Share

We’re working through functions in my college-prep pre-calculus class; meaning a more rigorous treatment of domain, range, and composition  ideas than what students experienced in earlier courses. As I was about to start inverses last week, I sought an activity which would provide some discovery, some personalization, and less of me rambling on.

These are the times when searching the MTBoS (math-twitter-blog o’sphere) leads to some exciting leads, and the search for inverse functions ideas didn’t disappoint – leading me to Sam Shah’s blog, and an awesome discussion of inverse functions which I turned into a sharing activity. A great list of blogs and MTBoS folks appears on this Weebly site.

To start, I wrote a function on the board, and asked students to think about the sequence of steps needed to evaluate the function:

$f(x)=3x^{2}+1$

The class was easily able to generate, and agree upon a list of steps:

1. Square the input
2. Multiply by three
3. Add 1

From here, I asked the class to divide into teams of 2. Each partnership was then given two functions on printed slips (shown below) to examine: list the steps of the function, and provide 3 ordered pairs which satisfy the function.

THE FUNCTIONS:

Notice that the functions are arranged so that A and B in each set are inverses.  Partners were given two different functions, but never an inverse pair. So a team could get 2A and 4B, but not 3A and 3B.

My plan was to complete this entire activity in one class period, BUT weather took hold. They day we started we had a two-hour delay, and the next two days were lost due to snow, then a weekend. SO, the best-laid intentions of activity, sharing and resolution became activity…..then 5 days later.

As we started the next class day, I asked students to review their given functions (and re-familiarize themselves), then seek out the teams who had the other half of the function pair and share information. So a team which had 2B sought out 2A, and so on.

After the sharing, a classwide discussion of the pairs was then seamless. Students clearly saw the relationships beteen the inverse pairs and the idea of “undoing” steps, and we could now apply formal definitions and procedures with an enhanced understanding. Also, by sharing ordered pairs, students saw the domain-range relationship between functions and their inverses, and this made graphing tasks much easier. I’m definitely doing this again!

Finally, notice that pair 2A / 2B features a quadratic / square root. While we didn’t dive right in at the time, this set the trap for a discussion of one-to-one fucntions and the horizontal line test the next day.

## Desmos + Statistics = Happiness

Sunday – a quiet evening before President’s Day – checking out twitter – not looking for trouble – and then,

Wait..what’s this?  Standard Deviation?  It was my birthday this past Saturday, and the Desmos folks knew exactly what to get me as a present.  Abandon all plans, it’s time to play.  A lesson I picked up from Daren Starnes (of The Practice of Statistics fame) is a favorite of mine when looking at scatterplots.  In the past, Fathom had been the tool of choice, but now it’s time to fly with Desmos.  There are a few nuggets from AP Statistics here, and efforts to build conceptual understanding.

CORRELATION, LSRL’S AND STANDARD DEVIATION

Click the icon to the right to open a Desmos document, which contains a table of data from The Practice of Statistics.  In you are playing along at home, this data set comes from page 194 of TPS5e and shows the body mess and resting metabolic rate of 12 adult female subjects. One of the points is “moveable” – find the ghosted point, give it a drag, and observe the change in the LSRL (least-squares regression line) – explore changes and think about what it means to be an “influential” point.

Next, click the “Means” folder to activate it.  Here, we are given a vertical line and horizontal line, representing the means of the explanatory (x) and response (y) variables. Note the intersection of these lines.  Having AP students buy into the importance of the (x-bar, y-bar) point in regression beyond a memorized fact is tricky in this unit.  Drag the point, play, and hopefully we can develop the idea that this landmark point always lies on the LSRL.

Another “fact” from this unit which can easily wind up in the “just memorize it” bin is this formula which brings together slope, correlation, and standard deviation:

$b_{1}=r\frac{s_{y}}{s_{x}}$

The formula is given on the exam, with b1 acting as the slope, so even memorizing it isn’t required, but we can develop a “feel” for the formula by looking at its components.

Click the “Means plus Std Devs” Folder and two new lines appear. we have moved one standard deviation in each direction for the x and y variables. Note that the intersection of these new lines is no longer on the LSRL. But it’s pretty close…seems like there is something going on here.

Ask students to play with the moveable point, and observe how close the rise comes to the intersection point. Can it ever reach the intersection? Can we ever over-shoot it? In the “Rise Over Run” folder, we can then verify that the slope of the LSRL can be found by taking a “rise” of one standard deviation of y, dividing by a “run” of one standard deviation of x, and multiplying by the correlation coefficient, r.

There’s other great stuff happening in the Desmos universe as well.

1.  This summer brings the 4th edition of Twitter Math Camp, to be held at Harvey Mudd College in California. I’m thrilled to have latched onto a team leading a morning session on Desmos. Consider coming out for the free PD event, and join myself, Michael Fenton, Jed Butler, and Glenn Waddell for what promise to be awesome mornings. To be honest, I feel the Ringo of this crew….

2. Can’t make it to the west coast this summer? Join me at the ISTE conference in Philadelphia, where I will present a learning session: “Rethink Math Class with the Desmos Graphing Calculator“. Bring your own device and join in the fun!

3. Are you new to the world of Desmos? Michael Fenton has organized an outstanding series of challenges, with 3 difficulty levels, to help you learn by doing. Try them out – they promise to get you think about how you and your students approach relationships.

4. Merry GIFSmos everybody!  The team at Desmos has developed GIFSmos to let you build your own animated gifs from Desmos files. EDIT – as Eli noted in the comments, credit for GIFSmos goes to Chris Lusto.  Thanks for being so awesome, Chris!

## Odds and Ends from a New Semester

A new semester for me started last week.  Lots of excitement, new faces, new classes, and much going on professionally. So many feelings coming from all directions….

I’M INSPIRED BY THE MATH FORUM FOLKS

Last fall, I participated in an Ingite session with the awesome folks from the Math Forum at the Association of Math Teachers of New Jersey conference.  Thanks to Suzanne for her kind invitation to share with a group of math leaders I have admired from afar.  The videos from the Ignite are finally available. Enjoy my flailing arms in a talk related to a blog post from long ago on math phrases I’d like to expunge.

While you are on YouTube, check out talks from the rest of the panel especially Max Ray and Annie Fetter. I marvel at Max’s ability to weave a math story cleanly and effectively about a theme. And Annie always provides ideas I need to try the next day in my math class.

I’M FRAZZLED BY NEW CLASSES

This new semester brings me 3 new classes, which are 3 completely different preps: AP Statistics, Academic Prob/Stat (which is co-taught) and CP Pre-Calculus. Are there 3 classes which could be any more different? I feel troubled when I have a schedule like this, since I feel like none of the classes get the attention they deserves, and I spend most of my time chasing my tail. Also, CP Pre-Calc is a course I have never taught before. Does anyone else have trouble becoming invested in developing inquiry lessons for a course, when you know you may never teach it again?  Tough days….

I’M AWASH IN ANTICIPATION…

For my new niece, who may be born any second now!

I’M ENCOURAGED BY NEW DISCOVERY SESSIONS

Our school adjusted the bell schedule this year; the result being structured time in the middle of the day where all students have lunch and/or study time. This semester, I am organizing Hatters’ Discovery Time – an opportunity for teachers to share their passions with students – modeled after the Genius Hour concept.  So far, our No Place for Hate team has facilitated an open discussion on race – a well-attended event with productive, positive discussions. This month, one of our science teachers who is also an EMT will share lessons on medical diagnoses, while a fellow math teacher offers an introduction to juggling. My session at the end of February will feature codebreaking, inspired the movie The Imitation Game, and just in time for Oscar season. Sessions on medidation and woodworking are also in the hopper. Such great ideas from my colleagues and hope students enjoy the offerings!

I’M EXCITED TO SHARE MORE…BUT…

The last month has brought a flurry of professional plans for the summer and beyond. I’ve been invited to attend the AP Statistics reading for a 4th year, which is one of the highlight of my year…it’s summer camp for the Stats Kids!  Late in June, I have been accepted to share Desmos ideas for the classroom at the ISTE conference in my hometown of Philadelphia. And plans for Twitter Math Camp in July are taking shape: a team of 3 math professionals whose work I all admire and I will team to present morning sessions on “Next Steps with Desmos”, while I have been accepted to present a session on tackling those “tricky statistics concepts”.  Should be a blast in LA.

But, today also brought rejection from NCTM, as my session on Standards-Based Grading in Algebra 1 was rejected for 2 regionals. Last year, I was rejected for a Desmos session, and the year before for a session on encouraging writing skills in math class. At this point, I really think I am being punked.  But congrats to all of my friends and colleagues who have been accepted, I’ll be curled up at home with a box of cookies while you all enjoy the conference……

All is not lost though. I think it’s time to put the SBG talk away, as I have done it at PASCD and a local conference now. I never got around to editing the video of this talk from our fall ATMOPAV conference, but I’ll share it now for old time’s sake.  Enjoy!

## Adding Distributions of Simulated Data

The current chapter on expected value and combining distributions in AP Statistics is one of my favorites for a number of reasons.  First, we have the opportunity to play games and analyze them…if you can’t make this fun, you are doing something wrong. Second, it often feels like the first time in the course we are doing some heavy lifting. Until now, we have discussed ideas like sampling, scatterplots and describing distributions – nothing really “new”, though we are certainly taking a much deeper dive.

The section on combining distributions contains a number of “major league” ideas; non-negotiable concepts which help build the engine for hypothesis testing later.  The activity I’m sharing today will focus on these facts:

• The variance of the sum of independent, random variables is the sum of their variances.
• The variance of the difference of independent, random variables is the sum of their variances.
• The sum of normal variables is also normal.

First, we need to have student “buy in” that variances add. Then we have the strange second fact: how can it be that we ADD variances, when we are subtracting random variables? In this activity, we’ll look at large samples, and what happens when we add and subtract these samples. Since many students taking AP Stats have the SAT on their brain, and there is a natural need to add and subtract these variables, we have a meaningful context for exploration.

SIMULATING SAT MATH AND VERBAL SCORES

The printable classroom instructions for this activity are given at the bottom of this post.

To begin, students use their graphing calculator to generate 200 simulated SAT math scores, using the “randnorm” feature on their TI calculators, and using the fact that section scores have an approximately normal distribution with mean 500 and standard deviation 110. Note – some older, non-silver edition TI-84’s won’t be happy with this, and a few students had to downgrade and use a sample of size 50 instead. There are a few issues with realism here: SAT section scores are always multiples of 10, which randnorm doesn’t “know”, and occasionally we will get a score below 200 or above 800, which are outside the possible range of scores. Also, there is a clear dependence on SAT section scores (higher math scores are associated with higher verbal scores, and vice versa), and here we are treating them independently.  But since our intent is to observe behavior of distributions, and not reach conclusions about actual SAT scores, we can live with this. In my class, no student questioned this as problematic.

Repeat the simulation in another column to simulate verbal scores. Then, for both columns, compute and record the sample mean and standard deviation. For my simulated data, we have the following:

It’s time to pause and make sure all students are clear on what we are simulating. We now have 200 students with paired data – the math and verbal score for each. Like most students, our simulated students would like to know their overall score, so adding math and verbal scores is natural. I help students write this command in a new column, then let them loose with the remaining instructions on both sides of the paper.

Students had little trouble finding the sum of the math and verbal scores, and computing the summary statistics. For my sample data, we have:

$\large&space;\overline{x}_{T}=1000.82,&space;s_{T}=165.16$

As students work through this, I want to make sure they are making connections to the notes they have already taken on combining distributions. I visited each student group (my students sit in groups of 4) to discuss their findings. Most groups could quickly identify that the means add, but what about those standard deviations? By now, if my students have taken good notes, they know that standard deviations don’t add, and that variances should. I leave groups with the task of verifying that the variances add.

Here’s the beautiful thing: students who immediately tell me that they “checked” the variances and verified the addition get the evil eye from me. In this simuation, students should find that the variances are “close” to adding, but not quite.  At the end of the acitivity, I ask students to conjecture why the addition is a “not quite” – even after I have beat into them that variances add.  There are two main reasons for this, and I was happy that a number of students sniffed these out.

1. We are dealing with samples, not populations. There is inherent variability in the samples which causes the sample variances to not behave nicely.
2. Variances add – but only if distributions are independent. Here, even though we created large random samples, there is still some small dependence. And while we don’t specifically cover the formula for dependent distributions in AP Stats, it’s worth discussing.

Next, it’s time to look at the differences.  Here’s students are asked to subtract math and verbal scores, compute the summary statistics, and compare the sum and differences. This was a nice way to go back and re-visit center, shape and spread.

CENTER: Sums are centered around 1000, while differences are centered around zero.

SHAPE: Both distributions appear approximately normal.

SPREAD: The sum and difference distributions appear to have similar variability.

And this idea that the spread, and standard deviation, will be similar for both the sum and difference, can be also be explained by looking at the range of each population distribution.

• For the sums, the max score is 1600 (800 M and 800 V), with a min of 400 (200 each)
• For the differences, the max score is +600 (800M and 200 V), with a min of -600 (200M and 800 V).

Here, we can see that both distributiuons has the same range.

From start to finish, this exploration took about 30-40 minutes, and was worthwhile for verifying and developing understanding of the facts for combining distributions.  The student instructions and video notes students take beforehand are given below.  Enjoy!