Feb 202016
 

As I have been examining my practice through the lens of Critical Theory, I asked myself how would I teach differently now than I did even a year ago? Great question.

If-you-change-the-way__quotes-by-Wayne-Dyer-40  It is time for me to look at AP Stats differently.

The last year I taught AP Statistics, I created great connections through the entire year on each topic, how each piece fit together, and how the end results started from the beginning topics. I carefully planned it so that every element of the year connected. And then, after the AP Exam, we had 4 weeks where I challenged the learners to come up with a question, do the research, and answer the question. Topics ranged from bullying, treatment of gays in stores, to trash on the side of the road. A huge, broad range of topics.

But, we did not do anything about those topics. We didn’t share them with the community. We didn’t have time. We collected great information, but we did not ACT on it.

If I teach it again, the first week is answering the following questions.

  1. What problem in our community do you wish to solve?
  2. Is this problem something on which we can we collect data?
  3. What data do we need to collect before we can formulate a solution?
  4. Share with class.
  5. Are there any similarities in problems?
  6. Can we consolidate any of the ideas?
  7. Discuss.
  8. Revisit 1 – 7 until we can not do 6 any further.
  9. List the topics for the class.
  10. Form groups for each topic based on your own interest and your own passion.

After that first week process is over I would think we have between 2 and 7 different projects in each section. I would have to be flexible and let the class drive the number and type of projects. The only thing I can think of why to reject a project is if we would have to deal with FERPA violations, incredibly sensitive topics like rape or incest, or other legally sensitive issues.

This is the truly difficult part of the teacher’s role, is playing the gatekeeper. I would want the learners to make decisions on what they want to study, but I know that there are some topics that are not researchable by high school learners. We don’t have an IRB to do experiments on people, for example. But we want some groups to do experiments. So I would need a committee of people at the school willing to be the final Yes / No on some topics. This is actually true to the real practice of research.

After we have decided on the specific topics, then we start into the process of answering the following questions:

  1. What types of data are there in your question?
  2. How do we display those types of data?
  3. How do we collect the data in the most scientific manner?
  4. etc, etc, etc.

These are questions that come out of the AP curriculum word for word. The only difference is that I, as the teacher, will be phrasing the lessons in the context of their projects. We will be learning from the different groups why we need to know about categorical and quantitative data. We will be learning from the different groups why bar graphs work for one type, but not for another type, and we will have to dive deeply into cluster, stratified and every other type of sampling in order to come up with the BEST way of collecting data for each project.

The goal now is to dive into the AP Stats curriculum deeply. We won’t need to come up for air because we will be inhaling the vapors of our excitement for our project. (wow, that metaphor was tortured, wasn’t it?)

  • What if a learner wants to switch? I don’t think there is a problem with that. Let them choose their enthusiasm.
  • What if an entire group decides they are more passionate about different other projects? Great, then we dissolve that research group and form a new one.
  • What if they decide to start over with a new question 1/2 way through the year? If they really want to go backwards, and redo all the work they have done on experimental design, research design, question creating, data analysis, and all of the rest of the topics then why stop them from relearning the material in a different context? Granted it is a ton of work, but they are learning, relearning, and taking charge of their education on a topic they are interested in. Why block them artificially?

Second semester is about finishing probability so the learners can moving into confidence intervals and tests. This is where the decision making comes into play, and as the learners become confident in this area, they will be making decisions on their topics.

Those last four weeks of school when I used to do projects would now be turned into “Action.”

  • Meet with the administration or counselors of the school about the data collected and share the statistics and conclusions. Work with the them to come up with a plan to solve problems, or at least come up with a plan to work on solving problems.
  • Write letters to the newspapers and media.
  • Write letters and meet with politicians.

The end goal is to allow the learners to drive the content of the class. They would be much more engaged in their own questions than any question I could come up with.

They would still be learning 100% of the AP Statistics curriculum, but now they would be more engaged and see the purpose for each “module” of the curriculum in a more solid, substantial way. This should help with AP scores (but I have no data to support this).

And in the end, hopefully it would make the community (however the learners bounded this) better.

What it would take from me is a huge willingness to give the learners power over their own education. They would have the ability to make decisions, and be allowed to follow through on those decisions. Some of those decisions will not turn out with positive (statistically speaking) results. They will get negative results. That is real life.

It would take time to plan, to organize content around their projects, and to think deeper about the connections. It would take time to connect with admin and parents to explain why I am doing this. It would require the admin of the school to be willing to allow learners to have the power.

It would absolutely weaken the oppression of the learners done by curriculum designers.

I want to do this. I am not in a classroom any more to do it.

Is anyone willing to partner? I will help. I will support. I will do everything in my power to make your life easier while doing this.

I think it is worth doing.

Nov 122015
 

I have been in Elementary school classrooms this semester observing my learners teach lessons. They are amazing, and the UTeach model of teacher education is one with which I am completely on board. My learners will have spent so much time in the classroom being observed and getting feedback that they will have no choice but to be amazing teachers. Add in the fact that my math teachers will only be taught to use interactive and engaging methods like the 5E model, and you have a home run.

BUT, as I have been in 3rd through 5th grade classrooms, I have noticed a very disturbing trend. Like this board I saw in a 5th grade classroom.

2015-11-02 12.20.48 (2)

Notice that the objective here was to “Reacquaint yourself” with the math terms by designing a city. OMG. Seriously. This was in the 5th freaking grade. No wonder geometry is such a difficult class to teach in HS, the learners are bored stiff and resentful the teacher is lecturing them on something they have spent time on already.

Next up, a 4th grade classroom. The terms I heard LEARNERS using today were; expression, equation, identity, and inverse.

No joke. 4th grade. The learners were using the terms correctly, and identifying the difference between an expression and equation and using inverses to construct identities while solving equations.

This was not a Gifted and Talented classroom, this was an at risk, high needs, pretty normal, typical classroom.

If I were to summarize what I have learned this semester as a teacher of teachers, it is this. High School teachers, we need to seriously up our game. We need to realize that the reason our learners look bored and apathetic is because we are rehashing what they already know.

We are NOT connecting to what they already know (even if we think we are.)

We are NOT challenging them to reach for deeper understanding (even if we think we are.)

And, we are NOT realizing the learners are entering our classrooms with a great deal of prior math experience and love. Connect with it. Pull it out. Create engagement.

My eyes are open, and it scares me to death what I have done in the past to my learners. The CCSS standards are working. The shifts in mathematics education are working. We must be leaders and take advantage of it.

Go spend time in elementary school classrooms. It will shock you what the learners are doing today. What are we doing?

Sep 062015
 

In my Feedly this morning popped up the article by Larry Ferlazzo called, “Disappointing NY Times Article On Teachers & ‘A Sharing Economy’.” Okay, let me be more blunt. I am not disappointed in the NYT, I am frustrated and a little ticked off. It stems from this article in the NYT: A Sharing Economy where Teachers Win by Natasha Singer.

Read the article. I call foul AND shenanigans. How much did TeachersPayTeachers pay for this fluff piece that was nothing more than an advertisement for teachers selling out other teachers.

youblewit

Maybe it is because I am active and love the #MTBoS (that is the MathTwitterBlogo’Sphere, if you are not familiar with it.) I embrace the sharing, the collaboration and the freely giving of resources that the math teachers do on Twitter, their blogs and the internet in general.

The article should have been titled, “A sharing economy where teachers win, but collaboration dies.” Sure, some teacher just made $1000 by selling her lesson plans to a 1000 different teachers for a buck. She won, but collaboration died. Is she seeking feedback from people who have used her lessons? Is she improving them by discussing and talking about how others have used them? Probably not. It is in a store, and people are buying it. There is no reason or need to improve it.

Meanwhile, in the #MTBoS, teachers are making, sharing, improving and resharing lessons all the time. They are coming together to make better lessons. And then, they talk about these lessons, which spawn more, better lessons. This is a collaborative community where ALL teachers win, and more importantly, our learners win. And our learners continue to win. Over and over again.

Seriously, look at the amount of resources freely created and given away.

First up, websites created by teachers collaborating:

  • Let’s start with the MTBoS Directory. No one claims this is an exhaustive list. It requires teachers to add their names to it, but there are currently 344 teachers in the list, all with an online presence, and all sharing things.
  • Nixthetricks.com – created by Tina Cardone and teachers all over the #MTBoS who contributed tricks. You can download the most excellent book for free.
  • Fawn Nguyen’s Visual Patterns and Math Talks. Both are excellent sites. I have used the Visual Patterns site frequently in my high school classroom, and am working on learning more about Math Talks and implementing them in the college classroom where I am now.
  • Would you Rather Math is a site I used regularly in my teaching as well. Great questions, created by and curated by John Stevens.
  • Michael Pershan’s Math Mistakes. See an interesting math mistake? Submit it to this site and have a discussion on the thinking the learner made while making the mistake. We can learn more from mistakes than we can from correct work.
  • Dan Meyer’s Google spreadsheet of 3 Acts lessons. More on this to come. I am working on an idea taking shape out of my current position as a Master Teacher with a UTeach model school.
  • Mary Bourassa’s Which One Doesn’t Belong. So Mary saw Christopher Danielson’s great shapes idea, and realized that there was some amazing math thinking that could be done. BOOM, another collaborative website created.
  • Open Middle Dan Meyer introduced the idea, Nanette Johnson, Robert Kaplinsky and Bryan Anderson ran with and created the platform.
  • Desmos Activity Bank A site created by Jed Butler out of the need to share Desmos files, first showed at TMC15 at Harvey Mudd College.
  • MTBoS Activity Bank created by John Stevens (second time his name is on the list) to collect and curate some of the awesome materials created. Anyone can submit their own, and searching is easy.
  • The MTBoS Blog Search also created by John Stevens (I don’t think he sleeps). This site allows you search the blogs of a long list of math teachers for lessons, content, whatever you are looking for.
  • Robert Kaplinsky has a Problem Based Search Engine, to find those specialized lessons that are, you guessed it, problem based!
  • The Welcome to the MathTwitterBlogoSphere website has a further collection of collaborative efforts that includes some of the above but is even larger.

But that isn’t even all of it. There are teachers who are collecting curriculum, links or materials and sharing it all back out; lock, stock and barrel. These teachers have “Virtual Filing Cabinets” full of lessons that have been tried and tested, re-written and shared back out. Some call their pages VFC’s, some are just curated sites of materials.

And then there are great organizations giving away curriculum:

  • Illustrative Mathematics, free ever-more-complete curriculum that is CCSS aligned and incredibly high quality.
  • Shells Center/Mathematics Assessment Project, good as lessons, problems or assessments. I forget about this site until I am desperate, and then kick myself because it is just so good and thorough.
  • Mathalicious has free lessons and paid lessons. I have used them in class. They are worth paying for!
  • Igor Kokcharov has an international effort in APlusClick. Lots of great problems and lessons.

And this list is FAR from complete. It is what I pulled together in 15 minutes of thought. And this list does not even begin to talk about the 180 blogs

So, NY Times and Natasha Singer. You blew it. You didn’t show teachers winning, you showed teachers selling out. If you want to see winning teachers, click on any link above and read their sites.

The above are all winning teachers. TeachersPayTeachers is an example of teachers losing out on this kind of collaboration.

Jun 102015
 

Functions

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:

expression2

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.

Jun 022015
 

In Algebra 2, I start with my 3 rules. They really are not “my” rules they are just restatements of the multiplicative identity, additive identity and balancing equation. I believe how I use them to set the tone and stage for the entire year is different, however.

On the second day of class (the first day I usually do a problem solving activity, cell phones, or other type of activity) I introduce the “rules”.

2014-08-10 15.43.10 You can zoom in, or you can grab the files from here. I had a long discussion with several teachers about the wording, and Meg Craig made up the files once we settled on the phrasing. Can they be tweaked more? Absolutely. I would love to improve them. Leave the suggestions in the comments.

Back to how I use them, and why I think I use them differently. I will say with 100% assurance that I use these rules differently than I have seen others, and I absolutely use them differently than I used to.

This grew out of the frustration of having a nice, simple equation to solve like, y=5x+2, and needing to solve it for x. I really wondered why learners would mess up the math on such a simple equation so frequently. And don’t even get me started if the equation looked like v=ba-d, because that was impossible.

And I realized that although I was teaching the idea of inverses and identities, I was not connecting learners with or building the idea that these things are used.

So, I turned to the “SADMEP” idea. (this link is a google search for SADMEP. That is sad, huh.) But as I worked with this a year or two, I realized that the SA always made a zero, and the DM always made a 1, so I added that to my SADMEP poster (sadly, there are no pictures of this, I threw them all away several years ago).

Which lead me to the idea that it is NOT subtraction or addition that is important, it is the ZERO! Same thing with the ONE, those are the important ideas. Those are the identities. Why do we subtract 2 from both sides of the equation above? Because 2 – 2 = 0. No magic. We can actually subtract ANY number, but we chose to subtract 2 because that is the most convenient way to reach zero in one step.

And then I stumbled upon another magic word that goes hand in hand here; convenient. Why do we chose the values we chose to add, subtract, multiply, or divide? Because those values are convenient ways to make a zero or one the fastest.

Back to the ugly equation above: v=ba-d. Solve it for ‘a’. Add ‘d’. Why? because -d+d=0. Do we care what d is, or represents? No, we know how zero works. Same goes with multiplying by 1/b.

Then, show this video: http://www.youtube.com/watch?v=seUU2bZtfgM up to the point where he goes into transcendental numbers (approximately minute 6).

I reach for my physics and chemistry books right about now, and find some ugly equations. These in fact. This is the file I start with to get the learners thinking about solving.

And then I go nuts. Put the formula sheet for AP Physics under the elmo. Do the same with AP Chem. Pick one. Pick another. Solve for any variable. Then solve the same equation for a different variable. For every single function / formula, the only thing they can write to justify their steps are “blah because it makes a zero” and “bleep because it makes a one.”

And we discuss that every single problem I can possible give them is solved simply by using these three simple rules. I make a huge deal of this in the log unit because they learn a NEW way to make a 1 in that unit. That is exciting.

All year long, my learners are shouting out, “because it makes a one” when we are working with exponent procedures (note they are not exponent ‘rules’) because that is how math works. Why do square roots and squares “cancel?” Well, they don’t. 1/2 exponents raised to the power of 2 means 1/2 times 2 which equals 1.

That’s it. I use it all year long. I rarely write a radical symbol, just fractional exponents. It just makes sense.

This is a couple of days of work, and I really think pulling from physics and chem texts helps. I have never had such success with solving and literal equations (in fact, they stop thinking literals are any different) as I have had the last two years.

There is a reason they are framed so nicely at the front of my room. They matter, and they solve every problem we encounter.

 

Some of the CCSS standards this idea hits:

A.SSE.1-4

A.REI.1-4a

A.REI.5-7

 

May 302015
 

I tried to do a 180 blog, and made it to 90. I really don’t know how people like Justin Aion and Sam Shah do it. It is very difficult to find something to day for 180 days without it sounding boring and forced. They pull it off though. That is amazing.

Knowing I can’t pull of the 180 thing isn’t bad, however. I know I can do topics, and I have a topic I really want to crystallize for myself (as well as others.) I have really been toying with the idea of “one maths” the last three years, and I convinced / forced one of my fellow teachers in my building to start doing it as well. The results are amazing. The connections between the different topics are astounding, and the learners see them, are motivated by them, and create further connections as well. To see why the connections are so important, one just needs to read this “Math with Bad Drawings” post. The connections are vital.

Some tools I will use regularly in class.

1. The Three Essential Rules – from day one, these are the only “rules” I will ever talk about. Log “rules”? Nope, don’t have them. Those are shortcuts to understanding why the properties of logs work. Exponent rules? Nope, nothing more than shortcuts. The only rules we will ever explicitly say are these three: Additive Identity, Multiplicative Identity, and balancing equations. How I implemented them can be found here.

2. Desmos.com – This is the first website I load every morning as I get ready for my day. It is essential to visualizing and discussing function families. The main difficulty I have with desmos is I have so many ‘files’ created it is hard to find them all! That is a great problem to have I think.

3. My structure of functions: This is how I organize the entire year. We move from topic to topic, but as we move, the connection to the prior topic is constantly referred to and stressed.Functions
This list is the core of the connections I want to explore and develop this summer.

Some things I want to make explicit for myself.

1. How to connect this list to the CCSS standards and Essential Understandings explicitly.

2. How to connect each step to prior knowledge in a stronger way.

3. How to connect each step with the breadth of knowledge required (for example, quadratics have many ways to solve).

4. Finally, why in the first place! It seems odd to put the why at the end, but I think it is easier to think about the why once it is all laid out. Does this curriculum have an advantage over the standard “textbook” curriculum? Anecdotal evidence suggests yes, but it needs to be better explained before others can weigh in.

It is a large project, but well worth doing. I think it will really make me understand the mathematics better, and enhance my teaching tremendously.

edit:

I better not slack off. Lisa and Meg both called me out. http://www.teachesmath.com/?p=765 and http://www.megcraig.org/?p=394. Stay focused Glenn!

Aug 142014
 

2014-08-10 15.43.10

My 3 rules for Algebra. Meg Craig made them pretty, I framed them, and went over them in Alg 2 Honors class last period.  Today we moved into Literal Equations. That is how I teach the rules and reinforce the rules, with literals.

Typically, these are one of the hardest things learners to grasp and wrap their head around. I am taking my physics book from college to the board and writing down formulas from it and from chemistry. Go.

And it was tough. Learners who can tell you that 25-25 is zero in a heartbeat will balk at v – v is zero. So there is a lot of coaching an patience as they work through the ideas of “The Rules” and the differences in how addition and multiplication act with respect to distribution.

It was difficult, fun, and frustrating but they are getting the concept of solving equations, all equations, no matter what they look like.  I will spend one more day, so that means 2 and a partial period on this. I think it is time well spent.

 

AP Statistics

The 6 W’s a H:

Who, What, When, Where, Why, by Whom and How. This is the introduction to thinking and reading scientifically. ScienceDaily.com is my friend during these couple of days, and the biggest stumbling blocks were finding the variables (the What) and the population of interest (the Who). I have added a 6th W, “by Whom”. I have found that helps with the learners wanting to put the author’s name in the “Who” spot. If they have a place for them up front, they can’t be part of the “who”.

Yesterday’s assignment was to grab one article from ScienceDaily and detail the W’s and H.I don’t think one learner did it fully correctly, though. I know that every single learner I helped one on one did not (and that was most!).

To help, in class we did a couple of problems from the textbook, which are … okay. Nothing earth shattering; they are short, and the learners can use keywords to figure out the answers quickly. Then we did 1 article from ScienceDaily on the board. I projected it from the web and we went through it, finding all the information. That helped too.

Finally, I handed each group of 2 an article and had them discuss the W’s and H together. I circulated and answered questions. They then swapped articles and did it again.

At this point, they have read 4 to 5 different articles from ScienceDaily. That is 4 to 5 more than they have read in the past. Reading these articles is a different skill than reading the fiction that English teachers have them read, and I think this is useful to getting the learners to be in a Stats frame of mind. I hope so at least.

The assignment for next class is to redo last night’s homework. Cross it out, start over, and do it right.

Feb 242014
 

I am trying to come up with ways to connect the ideas of Statistics and Calculus down through the curriculum in all levels, even into Algebra 1. I think I have a way to do it that makes partial sense and can be done with reasonable effectiveness, but I have an interpretation problem.  If anyone can see a way out of it at the end, please let me know in the comments.

First off, I started with http://graphingstories.com and selected the water volume exercise. My reasons are focused and simple. 1. It is linear, so it works well in the algebra 1 course during the first semester. 2. Finding the slope of the line is straightforward, and requires some transfer of skills, but not a lot of transfer. 3. The area part of it is just a triangle, so the math is not complex.

I am trying to build from simple through the more complex, so starting off simple is helping me wrap my brain around it.

So, I start with this video:

watervolume

You end up with a graph that looks like this (click to enlarge):

watervolumepic

Great, now for some math. Find the rate of change of the line: Best guess: (610ml – 0ml)/(15 sec-0sec) = 40.67 ml/sec.

What does that mean? As the seconds increase by 1, the water volume increases by 40.67 ml.

The equation of the line is milliliters= 40.67 ml/sec * seconds + 0.

So far so good. The interpretation of everything so far makes complete sense. It is attainable for algebra 1 or geometry, and has meaning in context of the video.

So I want to inject a little geometry into the problem and I find the area under the curve. No big deal, it is a triangle, so it is 1/2*610 ml * 15sec = 4575 ml*sec.

What does that even mean? Does it have a meaning? Am I doing something that should not be done? What does a value of 4575 ml*sec even mean? It is not the total volume, that is 610 ml. It is not the sum from 1 to 610, that is 186,355 according to wolframalpha.

In calculus, we find the areas under curves all the time, but they are specific types of curves.  We find the area and it is the displacement, or it is the total distance. But does this area have a context or meaning?

I really would like to figure out a way to make sense of this, but if there is not a way, then I will have to go back to the drawing board.

Edit: The word for what I am looking for is called absement. Links to resources are:
http://nqtpi.blogspot.com/2012/09/the-area-under-distance-time-graph.html
http://thespectrumofriemannium.wordpress.com/tag/absement/
http://wearcam.org/absement/examples.htm
http://forums.xkcd.com/viewtopic.php?f=18&t=34744 [read all the way to the bottom]

And a pic that makes the idea of absement more clear: http://ow.ly/i/4HEwh

Jul 032013
 

This is a two part lesson, an experiment and simulation that together meets the CCSSM S.IC.B.4-6 and explains AP Stats concepts as well:

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This exercise is one we did at the SilverState AP Institute with Josh Tabor, but the nice this is that if you do the first part it fits in Alg 2 nicely, but you can extend it to AP Stats as well.

It is LONG, with lots of pictures to explain how to use the applet so I will put the post after a break:

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