I'm so excited about this and hope it works.

Precalc Final Exam Review Blog 2009

If any of you drop by and want to help the kids out in the comments, please feel free. (They haven't done anything yet as of this posting.) If you want to make suggestions to me about setting it up/layout/etc, please comment HERE.

Mad props to the trailblazer.

## Alert!

**Hello, reader! If you intend to post a link to this blog on Twitter, be aware that for utterly mysterious reasons, Twitter thinks this blog is spam, and will prevent you from linking to it. Here's a workaround: change the .com in the address to .ca. I call it the "Maple Leaf Loophole." And thanks for sharing!**## Thursday, May 28, 2009

## Tuesday, May 26, 2009

### Productive Talk

I struggle, daily, with this (somewhat internal, somewhat not) dialog. Er, monologue. "Stop talking!" "Wait, I mean talk to each other!" "But about math!" "Oh, forget it, stop talking!" Really, my students talk, quite a bit. About math. In fact, trying to squash all talking seems to

Anyway, this made me giggle. AD was at its best when it let us gently laugh at ourselves by laughing at its flawed but lovable characters. Enjoy. (Also, I didn't know until today Hulu let you share a user-defined clip, which is rad.)

*promote*the off-task talking. But I still have to work to make it happen. It's not always organic, per se, to my lessons.Anyway, this made me giggle. AD was at its best when it let us gently laugh at ourselves by laughing at its flawed but lovable characters. Enjoy. (Also, I didn't know until today Hulu let you share a user-defined clip, which is rad.)

## Tuesday, May 19, 2009

### Wait! What?

There is a 75% chance of rain on any given day in the next week. What is the probability that it will rain on at least 5 of the 7 days?

There's a little surprise in there. I won't spoil the fun. (Until I do.~~Probably tomorrow.~~)

There's a little surprise in there. I won't spoil the fun. (Until I do.

**(also in comments): The***Update***weird**thing, that you might notice in the course of solving, if you calculate P(5), P(6), and P(7) separately and then add them together, is that P(5) and P(6) come out to the same thing. The probability it will rain 5 out of the 7 days is the SAME as the probability it will rain 6 out of the 7 days. Isn't that weird? If you write it out with fractions and the nCr formula you can how it happens, but the result is still counterintuitive to me. Combinations sometimes surprise me like that.## Monday, May 18, 2009

### Guest Post on Miss Calcul8's Classroom Management Series

Miss Calcul8 hasn't started a full-time teaching job yet, but she's already an experienced blogger and routinely offering sage reflections and observations from substitute teaching. I was very happy to contribute a guest post, and as a bonus, I really like what I wrote. Head over there and check it out, and don't leave without adding her to your reader.

## Friday, May 15, 2009

### The Stats Project

This is a summative assessment for the Algebra 1 Statistics unit. (I know that one-variable statistics is not algebra...insert grumbling about overloaded curriculum here.) The students are expected to be able to solve problems involving measures of central tendency, construct/interpret a box and whisker plot, construct/interpret a frequency table, frequency histogram and cumulative frequency histogram, and construct/interpret a circle graph. While learning the unit they practice constructing the graphs by hand. For the project, I teach them how to construct them with a computer and calculator and expect them to do so.

The whole evolution takes 4-5 days of class time. On the first day, we just collect data. Every student answers 24 questions. The questions are divided into 8 sections (A-H), consisting of 1 each of qualitative data ("What is your favorite color?"), integer quantities ("How many first cousins do you have?"), and non-integer quantities ("How long is your shoe in centimeters?"). I collect their work and, overnight, separate the 10 sections. The next day, each group of students gets everyone's responses to a section of three questions.

I give them explicit instructions about what graphs to create for each data set. We discuss why some graphs are more appropriate for different kinds of data, but if I let them pick, they would try to make a circle graph (or a "bar graph", which is not even an option) for everything. Two to three days are spent with the computers, calculators, poster board, etc, creating graphs, and writing up responses to analysis prompts. On the final day, each group presents their findings to the class.

Sometimes it seems they are all about making their graphs as ugly as possible:

There are some things I would change if I could. If I had a month for this unit, instead of 10 instruction days, I would want them to decide what data to collect, and give them a long period of time to do so. I would explore numerous ways to display data in informative and enlightening ways, and let them try whatever type of infographic they desired. As it is, I have 10 days to get them conversant with four specific types of graphs. So I make that process as engaging as I can.

Go here for Data Collection Sheets, Writing Prompts, Technology Instructions, and Scoring Rubric.

## Wednesday, May 6, 2009

### What Can You Do With This: Demon Mathematics

I'm going to go all Dan up in here and post this for a day or two without comment. (I have no idea if more than a scant few regular commenters are interested in participating...let's look at this as a little experiment.)

*Edit: I only made it about 7 hours without commenting. Everything I have to say is in the comments. Let's move on, shall we?*## Tuesday, May 5, 2009

### Solve Crumple Toss

I'm convinced that whoever is doing the work in the classroom is the one doing the learning. This means I try to minimize the time I spend doing dramatic performances of math problems, and maximize the time the students are working. My goal for every lesson is to expect the class to sit and listen to me for less than 20 minutes, leaving at least 25 minutes for them to work.

So we face a situation at odds with itself: kids need to practice doing math problems, but sitting and doing math problems is boring. To resolve the situation, I have collected an arsenal of structures that are a step up. Logarithm Wars is an example. Solve-Crumple-Toss came from a colleague, who is a master at thinking of these structures (thanks, Jen).

You need 6-8 problems that have somewhat lengthy solutions, not quick ones. Today I used this for proving and solving trig equations. Each problem gets its own 1/2 sheet of paper. Make copies and chop them in half. I usually allow 3-4 problems per student.

To kick things off, pass out 1 problem per student, and place the remaining problems where students can easily pick them up. When they have completed a problem, they bring it to you for checking. If there are errors, you give them some hints and send them back to their seat to fix it. If it's correct, the student crumples the paper, stands behind a line on the floor, and throws it at the garbage can for a bonus point. I actually use two targets: the recycling bin for 1 point, and the garbage can (which is smaller, and farther away) for 2 points.

It works pretty well. The athletic kids get to show off their skills, it requires students to get up and move around a little, and there's an incentive for getting a few done, and done correctly.

If you are lucky, you will have a student ask if they can get a running start and take off from the line. This can be very entertaining, and I love having my Flip for these moments:

So we face a situation at odds with itself: kids need to practice doing math problems, but sitting and doing math problems is boring. To resolve the situation, I have collected an arsenal of structures that are a step up. Logarithm Wars is an example. Solve-Crumple-Toss came from a colleague, who is a master at thinking of these structures (thanks, Jen).

You need 6-8 problems that have somewhat lengthy solutions, not quick ones. Today I used this for proving and solving trig equations. Each problem gets its own 1/2 sheet of paper. Make copies and chop them in half. I usually allow 3-4 problems per student.

To kick things off, pass out 1 problem per student, and place the remaining problems where students can easily pick them up. When they have completed a problem, they bring it to you for checking. If there are errors, you give them some hints and send them back to their seat to fix it. If it's correct, the student crumples the paper, stands behind a line on the floor, and throws it at the garbage can for a bonus point. I actually use two targets: the recycling bin for 1 point, and the garbage can (which is smaller, and farther away) for 2 points.

It works pretty well. The athletic kids get to show off their skills, it requires students to get up and move around a little, and there's an incentive for getting a few done, and done correctly.

If you are lucky, you will have a student ask if they can get a running start and take off from the line. This can be very entertaining, and I love having my Flip for these moments:

## Friday, May 1, 2009

### Imagine a Perfect Cube

This post grew from a comment I was going to write on my post Pi Whats, but as the comment grew and grew I felt it deserved its own post. It's about measurement and precision, and how whether you think they matter in mathematics depends on your perspective.

Do you think a string can be pi inches long?

You need to ask yourself this question:

Do you think a string can be 1 inch long? I mean EXACTLY 1 inch long?

If your answer is yes, a string can be exactly 1 inch long, then you automatically think it can be exactly pi inches long. Get a sphere whose diameter is exactly one inch, wrap the string around its circumference one time and cut it the exact length of the circumference, and the string is exactly pi inches long. If you are going to try to argue with me on this particular conclusion, then you are wrong. Pi is not some ill-defined Mysterioso of a number. It's very precisely defined. It's the circumference of any circle divided by its diameter. It's the circumference (in

If your answer is no, a string can't be exactly 1 inch long, then I'm okay with you saying it can't be exactly pi inches long, either.

The larger philosophical concept is that mathematics deals with ideal, perfect, abstract constructs that can really only exist in our minds. I can imagine a line segment of unit length. Cutting a physical piece of string with scissors to be exactly 1 inch long is much more treacherous terrain.

Scientists understand this, that's why they invented significant figures, and their calculations incorporate expected percent error. Whenever you measure something, you are necessarily rounding your measurement to the nearest level of precision on your measuring device. If you use a metric ruler with millimeter markings, and you measure something and say it's 2.6 centimeters, that means the thing you measured was closest to the marking at 2.6. It could be anywhere from 2.55 to 2.65 centimeters long.

Therefore, if I am thinking like a scientist or engineer (possibly

But in math we don't often discuss these problems of measurement. Because in math we deal in abstract perfection that can really only exist in our minds. We don't stop every time we give a measurement in a math problem and talk about the imprecision inherent in measuring something. In the physical realm there is no perfect square. No perfect right angle. No perfect circle. No length that is the square root of 2. But in my mind, the string can be exactly one inch long because I say so. These things can exist in our imaginations. In my mind, I can picture a square with sides of unit length, and draw in a diagonal, and I have created a length that is exactly the square root of two. In my mind, I can reason that somewhere between a string that is clearly less than one inch long and a string that is clearly greater than one inch long, there is a place I could theoretically cut that would be exactly one inch long. So as a mathematician, I say the string

I'm reminded of this litmus test from Out of the Labyrinth: Setting Mathematics Free:

Do you think a string can be pi inches long?

You need to ask yourself this question:

Do you think a string can be 1 inch long? I mean EXACTLY 1 inch long?

If your answer is yes, a string can be exactly 1 inch long, then you automatically think it can be exactly pi inches long. Get a sphere whose diameter is exactly one inch, wrap the string around its circumference one time and cut it the exact length of the circumference, and the string is exactly pi inches long. If you are going to try to argue with me on this particular conclusion, then you are wrong. Pi is not some ill-defined Mysterioso of a number. It's very precisely defined. It's the circumference of any circle divided by its diameter. It's the circumference (in

*u*units) of a circle whose diameter is 1*u*.If your answer is no, a string can't be exactly 1 inch long, then I'm okay with you saying it can't be exactly pi inches long, either.

The larger philosophical concept is that mathematics deals with ideal, perfect, abstract constructs that can really only exist in our minds. I can imagine a line segment of unit length. Cutting a physical piece of string with scissors to be exactly 1 inch long is much more treacherous terrain.

Scientists understand this, that's why they invented significant figures, and their calculations incorporate expected percent error. Whenever you measure something, you are necessarily rounding your measurement to the nearest level of precision on your measuring device. If you use a metric ruler with millimeter markings, and you measure something and say it's 2.6 centimeters, that means the thing you measured was closest to the marking at 2.6. It could be anywhere from 2.55 to 2.65 centimeters long.

Therefore, if I am thinking like a scientist or engineer (possibly

*using*math but not*thinking like a mathematician*) dealing with only objects I can hold in my hand, I don't think you can point at a piece of string and say "that string is exactly one inch long".But in math we don't often discuss these problems of measurement. Because in math we deal in abstract perfection that can really only exist in our minds. We don't stop every time we give a measurement in a math problem and talk about the imprecision inherent in measuring something. In the physical realm there is no perfect square. No perfect right angle. No perfect circle. No length that is the square root of 2. But in my mind, the string can be exactly one inch long because I say so. These things can exist in our imaginations. In my mind, I can picture a square with sides of unit length, and draw in a diagonal, and I have created a length that is exactly the square root of two. In my mind, I can reason that somewhere between a string that is clearly less than one inch long and a string that is clearly greater than one inch long, there is a place I could theoretically cut that would be exactly one inch long. So as a mathematician, I say the string

*can*be one inch long.I'm reminded of this litmus test from Out of the Labyrinth: Setting Mathematics Free:

*Close your eyes and imagine a perfect wooden cube. Now saw this cube three times, along each of its three axes.*

Question: did you see sawdust?

If so, you're no mathematician.Question: did you see sawdust?

If so, you're no mathematician.

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