Aug 262012
 

Okay, all along I was promising a massive file upload for all the readers who want the Exeter materials. I will explain what each group of files are for as I go.

All files are in WORD or PDF format, and all are in a zipped folder. Downloading and unzipping the folders will speed up your access tremendously. All in all there are 44 megs of files here. That does not sound large, but word files and pdfs are incredibly small these days!

[The placement tests were posted on Exeter’s website, but they didn’t realize they were made public. Links to that page have been removed. If you are a teacher and would like the files, let me know.]

The progression at Exeter begins with a Placement test to determine what course the learner should be enrolled. These are released Placement tests from Exeter:

Released Placement tests
After being placed in the correct course, the learners then start in on the problem sets. I have 2 years archived, but I would love more if someone has them.

Problem Sets 2011-12

Problem Sets 2012-13
The current year is 2012-13, so the archive is an August download of the new materials, including the change logs. If someone has the change logs for the 2011-12 or the files for previous years, I would add them also. The live location for the current year’s materials can be found on their site.

I do have solutions to the 2009-10 problem sets (I was given these without the actual problem sets) and solutions to the 2011-12 problem sets. Will I post them? No. I know I would not be happy if a teacher posted solutions to all the problem sets I created. That is the one thing I won’t post.

During the class the learners are in, they will do hands on activities, and use Geometers Sketchpad to explore math. The Instructor of the Exeter sessions I attended was nice enough to share these. They are all written by Exeter teachers, so no poaching and claiming them for yourself. Please attribute them accordingly.

Hands On Activities 2011
(Both Word and PDF documents!)

GSP Document and Sketches
GeoSketchpad misc
If you are looking at these thinking, “Dang! That is a lot of material to go through!” You are absolutely right. The 2 docs for Alg 1 are 59 pages combined, the Geo doc is 62 pages, and the Alg 2 Hands on is 69 pages. Right there are enough docs to keep a person busy in class for a long time, and you would be learning terrific math as you go. In the GSP Document and Sketches folder, there is a document called “2011 gsp.doc” It is 101 pages of GSP constructions.

So the learners are working problem sets, they are working activities and extending their learning beyond the problems and being active with the math. Now it comes time for some assessments.

Math 1 tests
Math 2 tests 
Math 3 tests
These are all in word format, so you can edit and use them in your classroom if you like. These tests give you some idea of how Exeter assesses their learners. Something you should know is that every one of these assessments are open notes. Every problem set they have worked is available to them on the exam.

Finally, the year is over, the faculty get  together and evaluate the problem sets. What worked, what didn’t, what can be improved. And the writing committee collects all those comments and distills them down into a commentary on the problem sets for the rest of the staff. Then the rewrites happen, and the new problem sets are published, and the cycle starts all over again.

Commentary 2011-12  [if you would like the commentaries, and can demonstrate you are teacher, please email me or comment and  I can email them to you. The files have been removed at Exeter’s request.]

 

And there you have it. This is the cycle of development of the Exeter curriculum and materials. The vast majority of the work is done by the writing committee, compiling the commentaries and editing the problems. That is a huge task, and I would love to have a serious discussion with someone at Exeter just about that. Heck, I would spend a week with them just asking questions about the writing of the questions, let alone working and thinking about the problems themselves.

I hope this is of some help to other teachers out there.

Aug 112012
 

In this post I want to show Exeter’s problem solving strategy. This is important, because it is SO different from how a problem like this is typically approached.

First off, the problem I am going to model is M1:21:11 [Math 1, page 21, problem 11]

11. Alex was hired to unpack and clean 576 very small items of glassware, at five cents per piece successfully unpacked. For every item broken during the process, however, Alex had to pay $1.98. At the end of the job, Alex received $22.71. How many items did Alex break?

In a typical Algebra 1 class we would try to get the learner to see the equation is:

.05(576-x) + 1.98x = 22.71

In fact we try to get the learner to jump directly to the equation from the problem by deconstructing the sentences, and then solve the equation. x = 3, by the way.

Now, let’s see how Exeter expects and demands that ALL of the modeling problems are handled.

First off, we will be making a table. The headings in this table are mandatory and can not be short cut. The learners must label the table thoroughly so that it makes sense. Remember, this is the same problem as above. I am going to paste in my table all filled out, and then explain the essential elements.

Guess: # of broken bottles

# of unbroken

$ Paid for unbroken

$ subtracted for broken

Amount paid

Goal

Check

0

576-0=576

.05(576-0)=28.8

(0)(1.98)=0

28.8-0=28.8

22.71

no

5

576-5=571

.05(576-5)=28.55

(5)(1.98)=9.90

28.55-9.90=18.65

22.71

No

3

576-3=573

.05(576-3)=28.65

(3)(1.98)=5.94

28.65-5.94=22.71

22.71

YES!

B

576-B

.05(576-B)

1.98B

.05(576-B)-1.98B =

22.71

 

 

Okay, there we have. A decent example of what a modeling, problem solving solution would look like. At the beginning stages of Math 1, they would not demand the last row, the equation row. But quickly they would ask the learners to start generalizing their solution.

The guesses column are not set in stone. The guesses are going to be the learners guesses. They are going to guess whatever they want. I started with 0, because maybe he didn’t break any. Then I saw that was too high to my goal, so I figured Alex broke a few. Then I was too low, so I picked one in the middle.

Now, let’s examine what the columns mean. It is clear from the headings that each column has a very specific purpose and is clearly labeled. What are we guessing? We are guessing the number of glasses he broke. If he breaks 5, then he didn’t break 571. How do we get that, we subtract. Each column must have in it HOW they get the number, not just what the number is. And so on.

Notice that by the time the learner reaches the answer, they have worked several times the process, they know the multiplications, the subtractions, and they have the solution worked out. Where does the variable go? It goes into the spot where numbers change. What do we call the variable? Don’t care, use a letter that makes sense to the problem.

How do they start this process? The first problem that is a modeling / problem solving problem is M1:9:4. It looks like this:

image

Notice that they start by giving the table and even filling out the first row. The problem I worked above, didn’t have that level of detail. The learner had to provide it. That is the point.

EXETER MODELS AND LADDERS THE LEARNING UP TO THE LEVEL THEY WANT.

Yea, I shouted that. We have this impression that Exeter is so fabulous, that they don’t have to ladder or work with learners. We think that the learners just will magically go *poof* and be able to do all these things that we struggle with.

Guess, what, they struggle with similar things there as we do in our schools. It might be easier because of smaller class sizes, but the root problems are the same.

Okay, off my soap box.

The Algebra 1 activities have some problem solving activities, and they even are sneaky by giving a blank table with fewer columns than the learners need! The learner is pushed to make the table for themselves.

Think about this type of problem solving for special ed, or EL Learners. They have the numbers set up, they can see where the Letter for the Unknown goes, because it is the only number that changes when they are doing the problems. Wouldn’t this method help them out so much?!

Think about your average learner who struggles with parsing the language of the problem. If they work 10 or 20 of these as starters, as homework, as in class activities, do you really think they are going to stress about a word problem?

Nope, they are going to say, “Mr. Waddell, these are easy, can we move on to something harder?” And you know they will.

Think about the really advanced learner. They are going to resent the table after a short time, but they will go to the generalization much faster because of it.

Can you think of any downside to this method of problem solving? I can’t. I have done Algebraic Thinking’s “SOLVE” method, and other methods. None of them are as straightforward and easy to put together as this method. We could spend THOUSANDS of dollars on professional development on problem solving, and none of that money would come close to the success of just creating a table, labeling, and working it out step by step.

Guess and Check. That is what Exeter calls it. I call it just downright successful for every level of learner.

Aug 112012
 

Before I begin going through this problem I have selected, I want to link you to where the PDF’s of the documents can be found. Notice these are NOT the problem sets, these are activities that ARE used in class, but also are pulled together for teacher use at the Phillips Exeter Academy Summer Math Institute. Just so we are all on the same page on where these are coming from.

Also, at the end of this series of posts (and I have mapped out at least FIVE more of them) I will post these resources in WORD format instead of PDF found on Exeter’s website. I asked the instructor and he gave me permission to post them as long as credit is given to Exeter. Have I said how much I like the Exeter Academy, their curriculum and how much respect I have for their willingness to share?

Okay, so let’s jump into the problem. Before I begin, I must say that we spent 3 days working this problem I am presenting here. We did a bit of it one day, a bit more another, and finished it as a third. We did parts as a starter problem, parts as an activity and parts as a “could we do this with this problem?” extension. This translates very well to the high school curriculum because it allows for a stepped pacing, starts off slowly with multiple entry points for all learners to accomplish, and them moves them slowly up the ladder to high levels.  Oh, it is essentially the Common Core.

I am going to put this below the fold because it gets long. No, really, lots of pictures, and it is long. It is completely worth the read though, and you will see how this fits into algebra and geometry.

Continue reading »

Aug 102012
 

Before I get into some examples of what we did for the 4 days of Exeter training, I want to discuss what the overall philosophy of pedagogy that was modeled exhaustively. Never once did the instructor actually SAY, “This is how we designed and planned the problems.” Instead, he set up the situation where we worked problems, and through my experience at #TMC12 and paying attention to how he phrased things, and how he moved from one problem to the next I was able to work it out.

Then I asked him point blank if I was right. I was. We had a short (very short) discussion on it, and then we did more math.

In describing the system / plan / organizational structure below, I want to make it clear these are my words, my descriptions, and my labels about their essential methods.

I am going to use the following descriptors for my understanding of their system; the setup & modeling, naming, and extension. I will work through each one of these separately, and connect them to some problems found in the Exeter problem sets as examples. When I am talking about problems found in the sets, I will use the following notation: M1:1:3 means Math 1, Page 1, Problem 3. All problems can be found on Exeter’s site here.

  • The set up & modeling: M1:15:6, M1:17:3,4,5,6, M1:18:1,3,4 M1:19:2,3

All of the problems listed above have one thing in common; they all are slope problems and yet NEVER ONCE mention the word slope. They use “rise” and “run” or some other variant. They discuss the change of one thing and the change in another, they talk about stairs or setting up a table of values or walking at a continuous rate or … you get the picture.

All these problem (and more, I just pulled out a sample) model the idea of slope of a line in a REAL WORLD basis. It has the learner calculate slope 5 different ways, from 20 different types of constant change. It clearly equates the idea of slope and rate of change, and puts emphasis on units and context. Never once is slope mentioned. No definition, no definition of a line, no y=mx+b, nothing. Just; here is a situation, figure out the answer. And the figuring isn’t all that difficult. It just asks the learner to understand what the rate of change is for each real world problem.

  • The naming / defining: M1:19:4image

AND THEN they spring the definition of slope on the learner. It is kind of off handed, “hey, you know that thing you have figured out how to calculate in all those different situations, it has a name, it is slope btw. How cool is that, we now have a name for the idea we have been working with the last two weeks.”

I asked our instructor if he would ever say the word slope before they worked this problem, and the answer was no. Let’s think about that for a second. This problem is on page 19, which essentially means (but not necessarily) the 19th day of class. The learners in this class have been working with linear equations, problem solving with linear situations, and slope problems for about 10 days of the last 19 and the teacher JUST NOW UTTERED THE WORD SLOPE!

This is essentially 100% backwards from how we are taught to teach, and completely and diametrically opposite the textbook approach. We use the word, give the word a meaning, try to get the learner to memorize the meaning, create foldables to help them memorize the meaning, and then are frustrated when the learner forgets the meaning.

Exeter has the learner work with the meaning, solve problems with the meaning, completely understand the meaning, demonstrate they know the meaning through 15 different applications of the meaning AND then they say, “Oh, btw, that meaning has a name.”

Which method do YOU think is better? I know the answer for myself. Having tried and failed the last 5 years at getting learners to memorize meanings for words, it is a hell of a lot easier to get them to memorize a word when it is being attached to a meaning that is well understood.

  • The extension: M1:19:5,6,8; M1:20:9; M1:21:3,4,11

So now that the learner knows the word, the world opens up because instead of long problems that model real world situations the questions can become much more abstract. But they don’t! That is the point!

The problems just step up the level of thought required to a new level. Now the instructor can create activities that challenge the learner to think about slope and y-intercepts in a more thoughtful way. For instance, one hands on activity has the learners working with geoboards and thinking in depth on what it means to attach certain adjectives to the slope, or certain numbers to a slope.

This extension piece can not be stressed enough. It is not working problems 2-30 even from section 3.2. It is; explain what the slope of a line looks like as you take the value of the slope from 1/5 to 5/1 through all integer steps of both numerator and denominator. Then explain WHY your explanation in the first part makes sense.

This is the pedagogical pattern used at every level, and for every topic. Setup the topic simply, show how the topic models a real world situation and work with that topic for several days in several different ways, then define the topic in a very straightforward manner, and finally extend the topic to new, novel, and more complex situations.

It is important to understand this progression of understanding for the next post, which will take one very straightforward idea, and end up in a place where we can calculate the volume of a 3 dimensional derive the formula for calculating the volume of any 3 dimensional parallelogram. Yea, it really can be that easy.

Jul 292012
 

Many people ask me why I ride my motorcycle long distances in the summer. This summer I traveled from Reno, NV to St. Louis, MO. It was around 4000 miles, round trip, and brutally hot for a couple of states worth of riding.

But, that traveling allows me one single thing I rarely get. Time away from all distractions. It worked. I thought long and hard about the problem I talked about last post; A visual representation to imaginary solutions of quadratics. Somewhere in Wyoming I had the idea on how to prove it. By the time I hit Utah, I had the solution worked out in my head, and I needed to jot some notes. It honestly took me several hours to type up the solution, and without further ado, here it is.

graph5

The Goal:

To prove that in a general case, the circle that is created by reflecting a parabola with imaginary roots (the orange one) about its vertex (the black one) will have as its radius the value of the imaginary roots of the original.

We will begin with clip_image002[12] as our initial equation, with one requirement that the discriminant is negative;  clip_image004[10]. This will ensure that our initial quadratic equation has imaginary roots and the parabola exists above the x axis as shown.

 

Now, we need to reflect this equation around the vertex, but just adding a negative sign in front of the “a” will not do it. If we add that sign in and make it “-a” it will reflect around the x-axis, not the vertex. Therefore, we are going to need to complete the square, get the original equation in vertex form and then add the minus sign to reflect.

 

Given equation                                                                       clip_image002[13]

 

First, divide all terms by “a” and set the y = 0                              clip_image006 

This gives us a first coefficient of 1, which makes

Completing the square possible. Next, we will complete

the square by using clip_image008 and its square.                        clip_image010

 

now that the perfect square trinomial has been constructed       clip_image012

we can factor the trinomial into vertex form.

 

The center of the circle above can be clearly seen in this form, and is: clip_image014 We will need this later.

 

Now we need to solve the reflected parabola for x.               clip_image016

 

Add & Subtract the constant terms from both sides to get:          clip_image018

 

Move the negative sign from the right to the left side:                 clip_image020

 

Take the square root of both sides:                                       clip_image022

 

Finally subtract the constant term from both sides:                  clip_image024


Notice that we have essentially derived a version of the quadratic formula. It doesn’t look exactly like the standard version we all memorize, but it is the same, with one important difference. There is a sign change to the terms inside the radical sign! That will be very important.

 

This formula gives us where the reflected parabola crosses the x-axis, so we now have 2 points on the circle, the plus and minus, and the center of the circle.

 

The final step of the proof is to show that the radius of the circle, or to put it in another way, the distance from the center of the circle to one of the roots of the reflected parabola, is identical to the imaginary part of the solution / roots of the non-inverted parabola. So, onward to the distance formula.

 

We need to find the distance from clip_image014[1] to clip_image026.

 

Distance formula:                                            clip_image028

Insert the point values for x and y       clip_image030


Using just 1 of the 2 values for the + or -.

 

Simplify the subtractions:                                               clip_image032

 

Finally, square the inside term leaving the following:             clip_image034

 

This leaves us with a pseudo-determinant of:                     clip_image036

 

 

However, in setting up the problem initially, we stipulated that the determinant clip_image038 would be negative. If that is true, then the value of inside the radical sign in our last step must be positive!

 

[And yes, I am cheating. I am leaving it to the reader to show that the way it is written above in the last step as the “pseudo-determinant” and the regular determinant are essentially equivalent.]

 

Not only that, but the value of clip_image040 which is from our inverted quadratic, is the same value but opposite sign of the more familiar clip_image042 from the quadratic equation.  If clip_image038[1] is negative, our inverted quadratic will be positive with the same value (oh, and it works in reverse too!)

 

There, I now proved that the reflecting a parabola with imaginary roots around its vertex will allow you to calculate the imaginary part of the complex answer as the radius of a circle created by the reflection.

 

QED.

Feb 212012
 

I tried something very new in AP Stats this year. Okay, it may not be all that new, but it was new for me. Last year when teaching confidence intervals, I taught it as it shows in the book, first 1 prop z, then 2 prop z, then inference testing, then 1 sample t, then 2 sample t.

I ended up with a class that saw confidence intervals as 4 separate things, and never once (except for those few exceptional learners) connected the dots to see that all 4 intervals, 5 actually, because you can have 1 prop z and 1 sample z, were all the same, exact idea separated only by what kind of data you have.

This year, while working with a colleague in another state (thank you @druinok and your blog) I learned that while the curriculum to AP Stats is pretty set, the creativity to teach it better comes from me. So, I changed it up. Last year, my problem was that the learners did not see the intervals as the same thing.

This year, I decided to teach all 4 intervals at the same time. Slight exaggeration. I taught the 1 prop z interval, and the conditions for it, and how to interpret, and how to do them. Then, I offhandedly mentioned, “and you know, there are other types of intervals we will get to as well.” In the restaurant business, that is called planting the seed. English teachers call it foreshadowing. I call it darn good stuff.

The reaction from the learners was immediate. “What are they?” “Are they different?” “How are they different?” were some of the immediate questions. So I went into them. Next I covered the 1 sample t interval. Here are the conditions, here is where you use them, etc.

They were hooked. The class, as a whole immediately saw that the intervals really weren’t all that different. Next, I made a worksheet with 12 problems. Several from each type, but purposefully NOT 3 of each type. Actually, 3 of the 1 prop/sample z, 2 of the 2 prop z, 5 of the 1 sample t, and 2 of the 2 sample t. The learners cut out the 12 problems, and had to sort them. There were a couple of purposefully tricky examples, like:

Suppose the average height of randomly sampled 100 male students at University of Reno is 67.45 inches with a standard deviation of 2.93 inches. Find a 95% confidence interval estimating the height.

The class put this in the “t interval” category at first, and that categorization would probably not be wrong on an AP test. It fits better in the “z” category though. Why? This was tricky because it doesn’t say we actually DID take a sample of 100. It says “Suppose ….” Yup, this is Mr. Waddell being a jerk and trying to trick the learners. But they got that. It was the only question worded that way on purpose.

At the end of class, the learners had 4 stacks of problems. I worked 1 problem all the way out using PANIC (Parameter of interest, Assumptions check, Name the interval do the math, Interval in correct notation, Conclusion in context). They had to pick 1 from each stack and do the problem.

They left class in good spirits with some very complex problems. They left feeling like they understood something important. I walked away chalking the last 3 days up as a success. Now it is just up to them to recall it.

Jan 242012
 

My last post was about an idea to use old scantrons as a visual aid to build knowledge of the binomial probability formula before the learners actually were introduced to the formula.

Short post: it worked, I think.

Long post: I passed out the scantrons, which immediately brought forth a groan. We just had the final exam last week, and I was already giving them a quiz! Once we got over that part of it, I asked a question.

I was giving them a 1 question quiz. What is the probability they would get it right? All they had is a scantron, no other papers or anything else, so they asked if they were just supposed to guess at the answer. My response was “yes” and quickly they had in hand that the probability was 1/4. It always surprises me how long it takes to get to that point with some learners though.  It is not as quick as I would think. But we all got there within a minute or two, which is faster than normal.

Next, I told them they were taking a 5 question quiz. And I asked the question, “what is the probability you will get a passing grade on the quiz?”

Now the frustration started. They wanted initially to just say (1/4)(1/4)(1/4) = 1/64.  Not so. I killed that pretty quick. But before I did, I wrote it in exponent notation, so they would be comfortable with the idea of exponents having a meaning in the problem. It worked.

The class started guessing lots of things then. (1/4)^3(3/5). That one was creative, accounting for the 3 out of 5 questions. I did not tell them what was right, but we had in the end 5 options the class thought were possible. One of the options was (1/4)^3(3/4). Pretty close to the basic part of the formula, just missing the fact they missed 2 questions, not 1.

So I asked them which of the options written down actually referred to something bubbled on their scantron. After all, 1/4 means something physical related to their scantron. They quickly ID’d the right equation as meaning something consistent,  and very quickly said they needed an exponent on the 3/4.

The last step was asking them how many different ways to get 3 right out of 5. I did the standard counting and after drawing 3 different options someone said, “there must be an easier way.” The class suggested permutations and combinations, and we quickly settled on the combination of 5 C3.

Done. In 15 minutes, we constructed the Binomial Probability formula using nothing but a stack of old scantrons. I then wrote the formula down for them, and they explained what the pieces were for. They made the connection much quicker than before, and I was really pleased with how fast they could plug numbers into the formula.

I remember last year just screaming inside because they could not get the difference between the probability of the problem as a whole, and the p and q values in the formula. They understand that now.

Success, probably. I like it.  Below is a jpg of the notes I made while doing the exercise.

 click to enlarge.

May 302011
 

I am very much a fan of repurposing objects and creating my own solutions to tech problems. Often times, you can do for your self much cheaper and get better quality if you think outside the box and create your own solutions instead of buying a premade solution.

A long time ago, I wrote about how I used a KVM switch to connect multiple sources to my projector. After two years, I am very happy with it. I use it daily to connect my Elmo to my projector, along with my laptop and 2 empty ports all through a single wire. It makes me very flexible with my tech usage.

When my Elmo finally dies (it is a fairly old model, the precursor to the HV-110u Digital Visual Presenter that I bought on eBay for around $80) I think I will make my own instead of buying another. Don’t get me wrong, I love my Elmo. I use it often. I also won’t spend a couple of hundred dollars to buy what essentially is a webcam on a stick.

Speaking of webcam on a stick, guess what! Another math teacher figured out how to mount a web cam inside of an old light mount and accomplish the same thing the Elmo does!

Now that is awesome. When it comes time to replace my old trusty Elmo, I will be using a web cam, purchased on sale, some inexpensive LED lights from Target or Walmart, and a light arm scrounged from a garage sale. At that point, I have a $200 Elmo for a quarter of the cost. Even better, because I can upgrade the webcam and modify it as needed. It is more flexible and useful than the Elmo ever could be.

We are only limited by our imagination, once we give up the idea that things have to be made for us.

May 042011
 

The FRQ review that I did last class was fairly successful. The learners worked their problems, thought about how to teach that problem, and gave good presentations to the class. The one thing I was very disappointed in was the lack of questions on how to do the problems the class was teaching. They all said, “yup, I understand how to do it, that makes sense.”

Come on, as a teacher, I know that is b.s. They were just humoring their classmates giving the presentation on the problem.

Today I am giving them another MC practice exam. I am still mixing it up though. I numbered them off by 1,2’s. Now they all have an odd or an even number. The ones went to one side of the room, the evens to the other.  I gave them 5 minutes to work on as many problems as they can in pairs, then called “switch”. Then they worked more problems for 5 minutes before switching again. And again. And again.

When the class was half way over, I required them do a “switch different”. Now they were paired up 1-2 instead of 1-1, and they were teaching each other the problems they already did.

It is a modification of a think-pair-share but for AP Stats review.

First period went off terrifically! Second period also! At great day of reviewing multiple choice problems where everyone was engaged and focused on the problems at hand.

Mar 012011
 

In AP statistics I was having a problem getting the class to actually engage during class! I know! The whole purpose of being .. in class .. is to BE IN CLASS!

Thanks to my twitter PLN, I learned a terrific idea to get the class engaged. It works. Hardcore it works. It is called a “halfsheet”. I take one problem. Just one. It goes at the top of the page. Then you put the acronym of choice down the left side to remind the learners how to do inference problems (as an example). At this point you have one problem, the acronym (PHANTOM, or I use the books Think: H:, Think: C:, Show Tell, whatever you use), on one half a piece of paper.

Small, neat, and concise. You give this to groups of 2 to work on. They work. They struggle, they learn. Then they bring you the half sheet for you to grade. You point out any errors, suggest wording changes, and then send them back. They bring it back with the corrections and you either send them back or accept the sheet.

Once you accept the sheet, you give them problem number 2!

Repeat.

Problem 3!

Repeat.

It works. They are engaged. They are working, and you get to spend your time tweaking, suggesting, and helping instead of talking at them.

Amazing what a half sheet of paper can do.