Sep 092016

To my last post, “No more broccoli ice creamDavid Griswold challenged me with a very serious and thoughtful reply.

The phrase, “No more broccoli ice cream” came from this meme that I saved. I collect these memes, just because the provide interesting fodder for conversations about math in class.

textbook math is like broccoli ice cream

So who is Denise Gaskins? She is a home school parent who specializes in K-6 math (I am inferring this from her website and the books / content she talks about. I could be wrong about the grade levels.) She tweets and has a FaceBook page under the name “Lets Play Math.” It is clear she has a focus on making math fun, interesting, and engaging. At that age, my experience is that learners are very much into mathematics. I saw this bulletin board in a hallway last year.

2015-09-28 12.56.29

If you zoom in, you will see that almost every single one of those 4th graders said their favorite subject was math, or they enjoyed math, or they were good at math. 2 out of the 15 had no positive mention of math. A bulletin board next door to it had a similar proportion. When I saw this board, I wrote a post called “Where does the joy go?” This issue is one that I have been struggling with for a while. Why are young children excited about math, but junior or high school learners typically are not?

I believe it is because at some point, I and my fellow teachers stop thinking about math as joyful, and start thinking about it as “serious work.” We can’t have fun solving these equations, this is “serious business.” But that is true of all subjects in high school. It isn’t just math teachers, but English teachers, history teachers, and other teachers. We turn our subjects into these “serious business” topics that must be “mastered” and “assessed.” If you don’t pass the classes, then you can’t graduate, you can’t be successful in life without knowing “algebra.”

[yes, I used a lot of scare quotes in that paragraph because I do not want anyone to infer that there is an agreed upon meaning of those terms.]

Here is what David said in his questioning of my post:

I’m not sure I completely agree with this, or Lockhart for that matter. There are a lot of people who find joy and beauty in the curriculum, and there are lots of ways to encourage and celebrate that joyousness without throwing too much out. Will some people hate it? Sure. I didn’t like AP US History very much, though I liked my teacher. But I had friends who thrived there. And I’m okay with that.

Personally, I don’t think the ice cream metaphor is realistic. Math isn’t ice cream. Nothing is ice cream. No field or subject is as universally loved and delicious as ice cream, certainly nothing with any practical application. Math isn’t ice cream, it’s vegetables! So maybe “textbook math” is steamed broccoli and it’s up to us to add peas and roasted cauliflower and sweet potatoes (maybe even with some marshmallows on top) and even pickles, but the fact is some people don’t like ANY vegetables and some people like simple steamed broccoli the best and some people like ALL vegetables and, importantly, all of them are part of a well balanced diet. So our job is to be a math nutritionist.

The first paragraph I will not reply to, because it is his personal feelings and I don’t think there is anything there to discuss. It is real.

The second paragraph is the challenge. “No field or subject is as universally loved and delicious as ice cream.” But … I don’t like ice cream. I eat it maybe one time every four or five months, because my wife wants to share something.

And, guess what? Yes, math is as loved as ice cream at the lower grades. I have observed 4th and 5th grade classrooms where the learners are excited, joyful, and enthusiastic about math. The bulletin board above is anecdotal evidence.

I think we need to stop saying it is the subject that is like vegetables, and accept the fact that it is the way we teach the subject that turns it into vegetables.

Watch this video (it is 5 minutes) of these middle school learners struggle and succeed in math.

There is honest to goodness joy there.

They ate some yummy ice cream in that lesson. Why can’t we do that every day?

To answer David. Is math like vegetables? I think it can be. Is math like ice cream? I think it can be. The choice is mine.

If I get to choose whether math is more like vegetables or like ice cream in my classroom, I will choose ice cream (even though I don’t like ice cream).

I choose this not for myself, but for my learners. And David is right. Not everyone will love every subject. I am okay with that. But if I choose to present math like brussel sprouts instead of chocolate fudge peanut butter ripple, then I have denied some learners even the ability to choose whether or not they enjoy math.

And then, how do I make my pre-service teachers understand that it is a choice they can make too? [Wow, that is a whole different can of worms.]

So, is math like ice cream? For my classroom, for my pre-service teachers, the answer must be Yes.

David responded on Twitter with these series of tweets. I think they add a great deal to the conversation.


Thank you David for making me think.

Aug 062016

Keeping up the #BlAugust posts! Yay. Weekends are harder to do.


Yesterdays post where I questioned my skills and abilities in questioning created two comments. Both of them helped me directly shape the focus of what I am doing in this class. First, @Druinok suggested I get a hold of some of the AVID materials, check out Making Thinking Visible, and think about Socratic Seminars as a structure to the course.

Bust of Socrates from wikimedia

I love that idea! Socratic Seminars (SS) give a great structure and fit neatly with the ideas and goals I have for the course. But, some resources to fall back on and refer to regularly would be helpful. With a little google work, I found these that I plan on using in one form or another.

Facing History ( has a good page on SS, the rational, the process, etc. I like this page for its simplicity of thought. It is clear and directs me to the info I need.

Next up, from, a 4 page PDF on how to begin, manage, and work within the structure of SS. This is a detailed document, but not so detailed that it can’t be used as a quick reference in the middle of the semester when I will need it.

Next up, a 31 page PDF from AVID*on how to run Socratic Seminars! Wow. If you need some detailed instruction in SS, this is the document for you. I found a lot of valuable tips and techniques in it. This document has teacher advice, learner advice, black line masters, etc.

Finally, a 1 page poster PDF for learners on the rules of SS. I am not sure if I will actually use this info yet. I may just discuss it, and not hand it out.

The other comment from Andrew reminded me that even though the skills of teaching math are not transferable, the PROCESS I used to learn how to teach math is still in play, and that process is what I will need to focus on over the next couple of weeks.

Thank you Andrew and DruinOK. You both helped me tremendously. I appreciate it greatly!

*The star is because Avid doesn’t like to share. I received the DCMA request (they didn’t even have the courtesy to leave a comment or communicate with me) to remove the file.

I did. They suck. Here is their lawyer speak for everyone to read.

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Aug 022016

Since I committed to BlAugust (see the #MTBoSBlaugust hashtag) and I am in a course design hole developing an Educational Theory course for my program I decided to merge the two ideas together.

I think it is important for teachers to regularly revisit the theories we learned in ed courses, because I know that when I looked at a particular theory after teaching for several years I saw the theory in a new light and had a different understanding of the approach. With that in mind, over the next month I am going to lay out the structure, the readings, the justification for the readings, and some of the questions I plan on asking.

This course is specifically for math and science majors who are also double majoring in education. At that point, I can make several assumptions about my learners.

  1. These learners are all comfortable with math up through (and some beyond) calculus. Even if they are in the precalc courses at the start of the semester, they are required to take calculus, so the ideas of algebra and infinitesimals are will not be foreign to them.
  2. All of the examples, questions, and discussions are to be centered around what it means to know math or science [facts, processes, methods, ideas, etc], and / or how to learn those same set of ‘things’.
  3. Finally, I can assume that these learners are committed to the course and the program. This is the third course in the program, not the first. These learners have already been in an elementary and middle school classroom teaching lessons they have written themselves. They have a small amount of experience, they have had frustrating lessons, great lessons, and they have written their own lessons.

With those three assumptions made, I will start outlining the thinking that is going into the lessons. I have a calendar already mapped out for each class period in the semester. I have the assignments identified, and questions.  No, I am not starting from scratch, but using another professor’s template who teaches this course at another University in a similar program.

With this in mind, my goal of the “talking it through” here is to make sure that I am connecting the first day of class to the last day of class. Is each class period started and ended with intentionality. What I want to avoid is the feeling of randomness that can occur the first time a course is taught. And this IS the first time this course will have been taught at my university and by me.

No pressure.

Just do it right the first time.

Tomorrow’s topic: Week 1: Intro’s and Assessment.

Happy BlAugust!

Feb 142016

This is a class assignment. Not to blog about it, but to write a paper about it.  The “it” is critical pedagogy/theory. Do you know what that means? I didn’t (and probably still don’t) either. I realize that I acted in ways that inched towards critical pedagogy, but I didn’t understand the theory. When I say, “I inched towards,” I mean it. I think I was heading in a direction that was taking me towards being a critical educator. I was not there.

When I first read this quote, I thought “Females are the half being held back. Right, I have to teach to all learners, not just the male learners.” But, as I developed ideas of critical pedagogy (again, before I knew what it was) I thought about my learners who didn’t like math. Was I holding them back? What about my Hispanic learners? Or the gay learners? Or what if they are gay, Hispanic, and female!

The next question I started asking myself is, “If I am not teaching to Hispanics or females, am I holding them back? After all, If I am teaching ‘neutrally’ then isn’t their problem to learn, not mine?” I have heard a version of that question from many teachers: I teach, learners learn, that is the way it goes. I once told a teacher, “A teacher who says I taught it, but they didn’t learn it is the same as a salesperson who says I sold it, but they didn’t buy it.” (No, really, I told that to the teacher’s face. They were … not happy with me.)

If I am not teaching TO the female learners, or TO the Hispanic learners or TO the low SES learners, than I AM holding them back. Purposefully, with foreknowledge, and now with malicious intent. I am using my power as an educator to purposefully hold back some learners over others.

I can’t do that. I will not be that teacher who doesn’t teach to ALL my learners, to get ALL of the learners to their maximum potential. I will not be the teacher who abuses their power.

But does that mean I am a critical educator?

The short answer is No.

The long answer is also No.

Because I didn’t create a classroom environment where I challenged the learners to engage and change the world. I changed my classroom for them, but what did I encourage them to change?

That is the difference between being an aware, a reflective, educator and a Critical Educator. And the more I learn, the more I believe every teacher SHOULD be involved in critical theory / pedagogy. It should not be an option to opt out. Being neutral on this topic does harm to learners.

Being neutral on the topic of power is wrong There is no such thing as being neutral. Yes, those are pretty strong words. I believe them. I will act on them.

I will put the rest of this “paper” below the fold, so it is not taking up tons of space. However, I encourage you to read on. I am going to try to become explicit in understanding what critical pedagogy is. I won’t apologize for that. It will be technical. And yet, I don’t see what I write below as optional practice in the classroom.

Critical Pedagogy / Theory: as I see it today, February 2016

Continue reading »

Oct 222015

As I was observing my students teaching I stood in an elementary school hallway and saw this display.

2015-09-28 12.56.29

This was on both sides of the hallway, 15 on one wall, 15 on the other. So you don’t have to blow it up to see, I will explain it. Each page says, “Who am I” and below that says, “My favorite: book, subject, pet, food, hobby, tv show, I’m Good at, When I grow up, I would like to be” on the left with blanks to fill in.

Here is the thing that really made me smile, and then get angry. Between the two boards, over half of the students  said “My Favorite Subject is Math” or “I’m good at Math.”

No joke. This is a Title 1 elementary school, and in the sample of these two classrooms, these learners said they enjoy or they were good at math.

I was so happy.

Then I thought about high school math and I got angry.

Where does this joy go?

At what point in the education trajectory of learners does the joy disappear to be replaced by frustration, anger and dislike?

And then the bigger question of Why? What changed? The learners didn’t change? They progress through the classes, learning, enjoying, and being good at math.

My conclusion was that WE, teachers, the adults, change how we approach the math. I can only speak to high school, but I know I would have many discussions about math in PLC’s, and trying to steer the conversation to the learners is tough with some teachers. Why was this hard? It should be the standard.

It is not about content, it is about learners; people, human beings with needs and desires. Are we showing them through interesting problems they need the math? Why not?

Dan Meyer has been asking frequently, if xxxx is the headache, how is yyyyyy the aspirin? This is the right question we, as upper level K-12 teachers, need to be asking. Over and over. How are we fulfilling the needs of our learners? It isn’t with “it is on the test.”

I don’t have any answers to questions in this post. I really needed to share the picture. A picture of a group of learners who truly enjoyed math, and the emotional response I had to it. It shook me to the core to realize that as a math teacher, I was and am part of the problem.

I will be part of the solution too.

Just to end on a happy note, one of my learners from last year tweeted me and made me smile. People. I teach people. Not content.

Jun 102015


I begin the class with a “what do you notice?” “what do you wonder?” session. This is probably the 5th or 6th day of class, and sets the stage for the entire rest of year. What do you notice? What do you wonder? I document all the noticings and wonderings, and then we discuss the mathematical questions.

Every year, the question of “I wonder how the 2 and the 1/2,” “I wonder how the 3 and the 1/3,” are related is asked. The best two questions that are always asked are, “Why are they all the same?” and “What changes when we change the exponent from 1 to 6?” I always say that I will answer every question by the end of the year; I will never lie to them and tell them something is impossible when it isn’t, but that some of their questions may need to be addressed in a future math class and not this math class. That honesty goes a long way.

I spend an entire period exploring the different functions with them, showing graphs on Desmos, asking for values to put in for a, h, and k. I ask questions like, “what do you predict the h will do?” and “Did your prediction come true?” The learners who are the typical aggressive type A learners hate it because they want the answers and want it now, but they will come around and start developing ideas on their own.

I start with lines for the (h,k) form because I think this form shows some reasons why to use the form, the benefits of using the (h,k) form over y=mx+b, as well as a simple function to cut our teeth on vocab.

I introduce this form first thing in the year as we get started. Fully Explaining & Understanding functions blank (double sided .docx file). I print off hundreds of this form, and we use it regularly. Some days I have the learners write the functions in their notebooks when I don’t have the forms, but I try to have a stack on hand always.

explaining This is what it looks like. There is A LOT of info asked for, and I start with lines so we can establish the understand of what the different elements are.

It always bugged me that we rarely talk about domain and range of lines. Why not? Why start introducing that idea with absolute value? Just because that is where it changes from all real numbers for both to only one, does not mean we shouldn’t introduce it earlier. Same thing with asymptotes and even/odd functions.

If I can get learners to identify the x and y intercepts on the line, and then connect those points with the standard form, so much the better.

Same thing with intercept and (h,k) form. Cut the teeth on a line, that is familiar and safe, so that as we move forward with quadratics, cubics, cube roots, etc, the learners can see the vocabulary does not change. What changes is the shape of the parent function.

I make sure every learner has one copy of this that is complete, pristine, written clearly and fully in their notes for every single function. When the learner puts them all side by side, they can see there is only one math, one set of ideas through the entire year. What changes is the amount of effort needed to get the intercepts for a cubic vs intercepts for a line? Why?

Another rich focus of questioning is “What makes the line unique?” “What makes the quadratic or cubic unique?” Some answers I have received are, “Only the quadratic always has a vertex form. The cubics can have that form, but usually not,” or “Every point on the line is a critical point, but we can’t always use every point for other graphs.”

Or can we? Hmmm. Leave it at that. Don’t tell them. Plant the seed and let it grow on its own.

This is a big picture post. Philosophy of teaching, approach to the topics, etc. No details yet. Just a pouring out of my thoughts on how I start. I will go more in depth. Notice that there is not enough room to work on the page. Only the results go there. The work is separate.


Jun 042015

My last post was about the three rules I use in my classroom. I developed the how and why in that post. In this post, I will explore some detailed “how I use them” in the classroom. I am careful to never say the word “rule” except for these three. We have exponent shortcuts, log shortcuts & properties, but never rules.

To do this right, I am going to use my Surface and do a lot of handwriting and posting of screenshots. If you are wondering, this is all done in OneNote with a Surface Pro 3. The bad handwriting is my own.

What really drove this point home for me, and made me codify it as something that needed to be talked about every day in class is the fact that if you do the “why?” for every step, you write down a “1” or a “0” for almost every step every time. Sure, you write down “I distributed,”  “I found the value to complete the square,” or “I factored” but why you do the next step is almost always a 1 or a 0.

See what I mean below.

To see how  I connect the rules, let’s start with an expression. It is not a very complex expression, but it sits solidly in the Alg 2 curriculum and throws learners for a loop often.

expression1 You can see that the expression is changed through the use of multiplying by one, and the convenient value we selected to use is a value that gets us a 1 when added. Why add? Because of the properties of exponents when we are multiplying bases.

Compare this to the rational expression of adding:


Here, we take the adding expression, and multiply both terms by 1, but a different one each time. Why? because we select the ones to use based upon the convenient terms to accomplish a common denominator.

Let’s move into some solving. Here is a straightforward quadratic that is in vertex form.

quadratic1 Yup, look at the bracket of zeros and ones.

How about turning a quadratic into vertex form?

Quadratic2 Here I explicitly used the first rule (I used it implicitly above as well, and I have the extra “I chose the 9 because it completes the square” step. Of course, these are bracketed by a zero and a one.

Finally, a log solving equation. I have just one, although I can do more. I chose an moderately ugly equation to solve, so it could not be solved any other way.

log1 [OOPS! I just realized I switched the 3 and 5 in my bcs statement. Damn dyscalculia. Sorry about that.] Here we have a double whammy. I conveniently chose to use Log base 5 to do to both sides. Why? Because Log base 5 of 5 equals 1! From experience, I know that taking the log of the more complex side reduces the number of steps. I don’t tell my learners that. They play and figure it out by doing the one problem both ways.  We also used Rule 1, do unto both sides.

In looking at the commonality between all of these problems, you can see the connection of “1” and “0” throughout. I stress this all year long, and have the learners write it all year long. This is the minimum requirement of writing I ask of my learners as they progress. We start writing much more, but I demand they write it. It reinforces the identities of addition and multiplication over and over again all year long. As the year goes on, they write less, but still write it.

Also, I almost never write a radical symbol until the final answers. All radicals are transferred to fractional exponents immediately all year. This helps explain why cubes are inverses of cube roots, and we don’t need to worry about notation. This is a big deal when dealing with some money problems and the exponent is 377 or some such nonsense  and we are solving for “r”. The “you just raise both sides to the power of 1/377 because the exponent will be 1” is automatic at that point.

I hope this gives a better understanding of what I mean by “zero’s” and “ones”. Please leave me questions here or on Twitter; @gwaddellnvhs.

Aug 032014

At TMC14 (Twitter Math Camp 2014) this year I did not attend many sessions, because I was the co-lead or lead in several blocks of time. It was great, and the comments I received were very complementary. I think the teachers telling me that were just being nice a little bit, but I hope they did receive some benefit from attending. With that said, the first thing I want to do in my TMC Recap posts is communicate some of what occurred in the sessions.

First up, Algebra 2. I co-led these sessions (there were 3 days of 2 hours each) with Jonathon Claydon (@rawrdimas) who blogs over at InfiniteSums. The 3 days were split into the following structure. Day 1 was about how to teach algebra 2 with some structure and form so that you can connect all the disparate topics of Alg 2. Day 2 was about a different way of cycling through the topics to allow for constant review and building of knowledge (pivot algebra), while day 3 as all about modeling.

Day 1 started off with the question, “How do you currently teach alg 2?” We had several answer. Graphing all the parent functions and creating a hook to hang the rest of the year that way (Family of Functions), or solving the equations and connecting the graphs later (equations first), going through the textbook units and color coding them, and then I introduced my (h, k) format. There was great interest in the (h, k) structure so we spent the rest of the time on that method.

What is that method, you ask? Well, on my board under the heading of Algebra 2, I have the following forms written down:


First off, what do you notice and wonder about all these forms? Yes, I do ask that and spend some class time on the noticings and wonderings about this list. I actually have a “You are Here” note that moves from one to the next to the next as we go through Alg 2 and I make a big deal about that move.

The really nice thing about organizing the class in this way is that clearly the learners are learning ONE set of math operations, not 12. The amazing similarity between all of these forms encourages the learners to actually look at the math and ask “what is the same, what is different” and STOP thinking “all of this is different each time.” It takes some work, but the learners figure out that my 3 rules (the ONLY 3 rules I allow them to use/ write/ or say in class) are how ALL of these functions are solved. [make sure you read the comments too]

Also, shown (but not handed out) during the session was how I consolidate all of the maths for all of the functions and what I expect for every single function listed. It looks like this:


All of the links for the handouts and materials are on the TwitterMathCamp Wiki site. If you want this handout or any other handouts from TMC, please feel free to download them.

My goal with this process is getting the learners to think of math as ONE body of knowledge and not a segmented series of things we memorize. We LEARN how to factor, how to graph, how to identify points on a graph, and we USE that same knowledge over and over again.

I have had some success with this last year and I am looking forward to doing it again and blogging about it as I go. Yes this means I am planning on blogging more. That is one goal I have for the year. It was created because of this article on the secret to writing. (hint, there isn’t one.)


Jun 292014

This post is really for a friend of mine who is very much math aware and capable, but does not teach math. He has twin sons who are absolutely amazeballs smart, and through talking with him one day I espoused my approach to algebra. He was intrigued and wanted more info. This is my attempt at more info.  I figured that others may find value in it, so I am publically posting it for all to see (and read, hopefully.) Please rip it to shreds if you feel I am in error or made a mistake. I want to do better, and I can’t if I am doing something wrong.

So let me begin with where this came from. At the NCTM Las Vegas regional conference in 2013 I was introduced by an elementary teacher to what she called 13 Rules that Expire.

13 rules that expire

There are some definite Algebra concepts on here like number 2) Use keywords to solve word problems and number 3) you can’t take a bigger number from a smaller number or what really kills me is number 8) multiply everything inside the parenthesis by the number outside the parenthesis.

Yea, right. Try that with f(x)=2x+5.


And from there we have Nix the Tricks. A more rigorous treatment of the stupid things we as math teachers do to mess up learners in the guise of teaching them to get an answer instead of understand the mathematics.

So I sat back and tried to come up with some rules that did not expire. Some essential rules that always work, that always build understanding and not destroy it. I ended up with 3.

1. When solving an equation, you can do absolutely anything you want, as long as you do it to all terms of the equation.

2. When working with expressions, you can only change it by adding or subtracting 0 (by using additive inverses), or when solving equations, you create zeros by adding or subtracting 0 (by using additive inverses).

3. When working with expressions, you can only change it by multiplying by 1 (in any form), or when solving equations, you create 1 by using the inverse functions/operations.

That’s it. When I am teaching math, I stress the idea of creating a zero or using a zero. When demanding written explanations, I demand they say that 5+-5=0 in their explanation. So, a nice short example.

Find the function that is the inverse of y = 1/3x – 4

Work                             Why did I do what I did?

x = 1/3y – 4                    Because the first step in finding the inverse is switching the x & y

+ 4        +4                     Add 4 to both sides because –4 + 4 = 0 (additive inverses = identity)

x + 4 = 1/3y                    result

3(x+4) = 3(1/3)y              Mult by 3 because 3(1/3) = 1, (multiplicative inverses = identity)

3(x+4) = y                       finished, but should check it by …….

So there is an example of the work I require. I want them to be using the language of inverses and identity. Why does the square and the square root cancel each other? Because the exponent of 1/2 (the square root) and the exponent of 2 (the square) when multiplied equals 1, the identity.

Along with this, there are some forbidden words and phrases in my classroom. One is above, “Cancel”. I do not allow my learners to use it. At all. Ever.

Why? Because I have seen all of the following described by the word:  -5 + 5 cancels to make 0. 3/3 cancels to make 1. Log10^4 cancels to make 4, sqrt(5^2) cancels to make 5 and on and on and on.

If those all “cancel” then that word means nothing, and it does not mean anything at all. It is just a word used to hid the mathematical knowledge of inverses and identities.

Another phrase that I will not allow is the common answer to the question, “Why did we add 4 in the first step above?” They typical answer I get is, “Because we want the y by itself.”

Huh?  That is not why we added 4. We could do anything we want in the world. We could have added 6, or subtracted 3, or taken both left and right side and made them exponents with a base of 7.5. We added 4 because –4 + 4 = 0. That zero is important. I also write it down. I think most teachers do not. Or they draw a line through the –4 and 4. What does that line mean?  I tell them it means a 1, and did we make a one? No, we made a zero.

It takes some time to unlearn the bad habits, but this is a Nix the Tricks kind of endeavor. In the end, I think I am helping the learners understand math better and more deeply. At least, I hope I am. Only time will tell.

Jul 012013

Adding variances is the Pythagorean Theorem of stats, we all probably say that to the learners, but it is ONLY true in one special instance: When the two variances are independent of one another.

That pesky “independence or n < 10%” condition check plays a role here, because if we fail to check it, then we can’t add the variances.

Or can we?

Well, I won’t jump to the end yet. I want to explore a classroom activity on this first.

Combining Random Variables: Speed Dating

To save time and money, many single people have decided to try speed dating. At a speed dating event, women sit in a circle and men spend about 5 minutes getting to know a woman before moving on to the next one. Suppose that the height M of male speed daters follows a Normal distribution with a mean of 69.5 inches and a standard deviation of 4 inches and the height F of female speed daters follows a Normal distribution with a mean of 65 inches and a standard deviation of 3 inches. What is the probability that a randomly selected male speed dater is taller than the randomly selected female speed dater he is paired with?

Pair the class off, and have the boys do randomnormal (69.5,4) and the girls do randnormal(65,3). Make them compare and raise hands. Once they do it, shift and do it again. Do this for a sample of 20 or so. Just enough so they can see sometimes there will be taller girls, and sometimes taller boys, and just doing this won’t answer the question.

Then use lists and put 100 males into list 1, and 100 females into list 2 using the store command on the board calculator up front. Go through the list one by one. Who is taller, boy or girl? Who is taller, boy or girl. Etc. Wait for someone in the class to say “Couldn’t we just subtract the two?” bingo. Let them come up with it though.

Calculate the SD of the L1, L2, and L3. What do they notice? Hmmm, 3, 4, 5…. Wait for it. Wait for it. If you need, question, but don’t give it away. Let them reach back and think of Pythagorean Theorem.

Now they discovered the rule instead of you telling them the rule, and it should stick a little better.


But back to the question above. If the two samples are not independent, then we can’t add the variances, right? Wrong. Here is where the Law of Cosines plays a role.

Think back to geometry or trigonometry and we had the following situation:


When the triangle was a right angle, we had the top situation, the nice simple Pythagorean Theorem. But if the two sides are not orthogonal, then we don’t have a right triangle, and then we use the law of cosines to find the length of the third side.

Guess what happens in stats when we do not have orthogonal (or independent) samples? You are right!


Where rho is the correlation coefficient between the two samples. When the two samples are independent, rho is zero, and the term goes away. It turns out in AP Stats, we are teaching a special case! If we have non-independent samples we can still add the variances, as long as we adjust for the correlation!

Isn’t it awesome when you learn where it goes so we can teach the curriculum better? This kind of stuff truly makes me a better teacher.