Saturday, February 25, 2012

Don't Break the Product Rule or you'll be Slathered in Butter and Grilled!

My Pre-algebra class is reaching the end of their exponent unit.  I blogged about this unit before, but it was one of my first posts so I'm sorry about the roughness (I was also a little scribd happy because I was excited by how easy it was to throw documents right into my post.  It's magical!) I still feel like this unit is really important because really internalizing the exponent rules makes for a much smoother transition into algebra 2 and beyond, but after reading around a lot on other people's blogs, I'm not sure that stomping along through all the rules in order is the best way to teach them.  I also am aware that in the real world students will never have to simplify these ridiculous exponent problems. Though I still think that understanding these rules is necessary in creating a foundation for high school math and is a good way to introduce the logical system of math, I'm a little uncomfortable with how hard it is to tie them to the real world.  I ended the unit with exponential growth and decay and scientific notation which use the exponent rules in context, but I still wish I had a more concrete way to make these rules relevant to students.

I've been trying to think of a way to review what we learned throughout the whole unit.  Last year, I had the students just do a poster project where they had to neatly and creatively demonstrate all the rules, but I think that was just a desperate attempt on my part to have them review the material without adding a whole bunch more prep work on my part because I was swamped.  This year I created this review activity for them:

Exponent Unit 'Going on Vacation' Review Project After reading around so many blogs and seeing what the larger teaching community is doing, I've realized that even the activities I'm most proud of are lacking the real world context that has been stressed by so many other bloggers.  I've stressed teaching the logic of mathematics to my students because that is what is beautiful about math to me, but my students probably need more context driven activities and examples.  The problem is that my education in mathematics has been entirely traditional (i.e. contextless) and I don't think about applications and I don't interpret the "real world" through math.  I don't know how to see math in the world around me.  Yet I guess, just as we tend to repeat our parent's mistakes, it's so easy to teach the way I was taught and to focus strictly on logic.  I've realized that this is a grave deficiency in my teaching that I need to learn to correct, but changing the way I think about mathematics is going take a lot of time.  At least I'm pretty good at making fun and silly math assignments, even if they're not tied to context.  Two months ago when I came up with the idea for this project I was pretty proud of it.  Now I realize that it doesn't give students any deeper insights into math.  It will be silly and engaging I think, but I need to do better.

Sunday, February 19, 2012

Gifted and Talented

I was listening again to an old Radiolab episode and though I know it's a bit out of date and the conversation about this episode has probably long dried up, it just bothered me so much that I felt like I needed to rant about it somewhere.  My husband says that's what blogs are for, so here's the episode:

In it, Malcolm Gladwell talks about why he doesn't like gifted and talented education.  I've been entrenched in the gifted and talented debate in schools for my entire life because my older brother was identified as exceptionally gifted.  My younger brother and I were also identified as gifted, but I had a severe deficiency in reading that helped me integrate more into a public school environment because I wasn't yards ahead of my classmates.  My older brother though, was exceptional all around.  My parents trusted to the public school system to meet his needs, but like many very bright students he was bored.  So bored in fact that when the classes started to get legitimately challenging, he had trouble keeping up as he had never learned study skills because he had never needed to study.  My mother, throughout my brother's school career campaigned for gifted and talented education because the needs of her child were not met.  So I guess this all makes me ridiculously biased, but as a teacher, it has helped me work with my students because it taught me a very valuable lesson- school has to meet students where they are and it has to set expectations tailored to each student's abilities.

 Maybe I'll deal point by point with my disagreements with both Robert Krulwich and Malcolm Gladwell's comments.  I love Radiolab and Robert Krulwich, but I feel that in this instance, they didn't really know what they were talking about.

About a minute into the show, Krulwich says, "As we all know, in America, there's a real hunt on from a very early age to find the gifted and talented children.  And we have programs all over the country trying to identify exceptional kids."  Perhaps it's just Oregon, but here, Gifted and Talented programs have all but been eliminated from the public school system.  My mother has been fighting for the last 25 years to force the Oregon government to adhere to laws about TAG (Talented and Gifted) students in our state because these programs keep getting cut despite regulation meant to ensure that TAG students' needs are met.  She and many other TAG parents sued the Portland school district for noncompliance and won.  TAG programs are just another expense that school districts are reluctant to pay for, especially once No Child Left Behind was passed because NCLB forced schools to expend resources for the students at the bottom of the achievement spectrum, students at the top could be ignored.  The thought that there's a hunt out there to help these bright students succeed is laughable.  Portland just tried to close the highest performing school in the district because it was a "brain drain". They thought that those students, when redistributed among the other high schools, would help those high schools meet AYP. I know that perhaps he meant any gifted and talented students- those gifted at sports, art, or music.  Not necessarily those who are identified as Gifted and Talented academically by state tests.  These academically gifted students though, because their needs are not being met, make fewer and fewer gains as they move up through school and often end up having very poor grades or dropping out because their entire experience with school has been boring and frustrating.  Unfortunately, I have a terrible memory, and this is a rant, not an essay or an article so I won't dredge up statistics for you.  My mom could.  But how often have we teachers bemoaned the poor grades of our brightest students?  I know I and my colleagues have.

About a minute and a half into the show, Malcolm Gladwell, after Krulwich comments that Gladwell hates Gifted and Talented education says "It's ridiculous... We identify a child and call that child gifted because of their performance at the age of, whatever, nine or ten or eleven years old.  Why do we care particularly how well a child performs at nine or ten or eleven years old?  They're nine or ten or eleven.  They're a good 25 years from making any kind of substantial contribution to the world.  Why don't we wait?  What's the hurry?... So one child learns to read at 4 and one child learns to read at 2 and a half. So what?  Why does it matter?  Are the things that are being read between 2 and a half and four are of such incalculable..." And here's where Krulwich cuts in.  I get to what Krulwich's response is in a minute.  I can't even begin to express how much Gladwell misses the point with this comment.  He thinks that we identify early readers because we're interested in finding the next American author?  His comment makes it seem that education is about grooming students to serve society later in life.  This is an entirely valid perspective on education at a macro level.  Yes, education is about raising a new generation of thoughtful, intelligent and creative people to carry our civilization into the future.  But I as an educator am not interested in creating the next Obama, the next Gladwell or the next Bill Gates.  I am interested in helping my students learn to love learning.  I am interested in helping them learn to navigate the world so that they can explore and work towards goals that will make them happy.  I'm interested in helping them become happy, as well as successful human beings.  I am interested in the micro.  Gifted education is not about us selfishly selecting the brightest students to serve society in 30 years ala Ender's Game (If you haven't read this book, you should).  Gifted education is about helping students navigate school successfully without getting bullied and without being bored.  Gifted education is about challenging students so that they don't get lazy, as bright students are apt to do, so that they learn important study skills, expand their knowledge, interact positively with the world around them rather than disengaging as soon as they can no longer coast by.

Now to Krulwich's response: "No no no, it's just a normal parent's response to 'oh, if he's reading at two and a half, think of the things he'll do..' and it's just an extrapolation."  This response makes it sound like parents' only interest in their child's early skill is for boasting purposes or to satisfy a parent's pride.  What about those parents who recognize that their student will go to school and will be told by a friend, or a teacher "you're not supposed to read yet."  Or "it's nice that you can read, but the other children can't yet so you'll just need to sit here until they catch up."  These are comments that are made to young children when they can do things other students can't and it cuts down their morale, makes them feel freakish, tells them that what they do naturally is not ok, or that school is about sitting around being bored.  Parents become frustrated when their children are unhappy, and parents of gifted students are very frustrated.  My mother gets calls and e-mails daily from parents torn about what to do when their children come home from school demoralized, unmotivated and sad. (To make another literature reference- Wrinkle in Time has a great example of how gifted students can get treated at school.)

Now I know that this is not what Gladwell and Krulwich were focusing on.  There are parents out there that push their children to succeed out of selfish motives.  In many cases, there's no need to separate high performing students from average students.  But there are students out there who have special needs- just like deaf students, or a students with dyslexia.  Exceptionally gifted students need special programs with teachers who understand them in order to become successful, well adjusted adults.  I had dyslexia and I was accelerated in math.  Both of my needs were met.  I was challenged in both reading and math and being challenged taught me the beauty that could be found through learning and education.  I have a good work ethic because I was constantly challenged to keep trying, keep pushing, keep engaging with the world.  All students need this challenge, all students should have programs and teachers that work with them from where they are to where they want to go.  Even naturally academically gifted students need guidance and support. Krulwich and Gladwell have forgotten that education is also about students, not just about society at large.

Thursday, February 16, 2012

Perfect Squares and Perfect Cubes

My algebra 2 students are starting their unit on radicals.  This unit includes a review of square root arithmetic, the quadratic formula, higher order roots and fractional exponents.  It's kind of a dry unit.  I was trying to think of a way to spice it up and also, I was thinking about how in years past, my high school students have not been able to recognize perfect squares or perfect cubes at all because elementary schools around here don't emphasize the multiplication tables.  Then I remembered a comment on a blog (I think it was f(t) but I can't find the actual comment) from a guy who said he got students to recognize perfect squares by coding the alphabet with perfect squares from 1^2 through 26^2 and putting messages up on his board in code.  I thought this was a pretty awesome idea, so I went through and coded messages for my students for each day of the unit.  Because they're in algebra 2 though, I went beyond perfect squares and made more interesting codes.  I'm just awarding candy to those who solve the codes, but I'm also thinking about making it a competition.  The person who decodes the most messages correctly gets some prize.  I did my first one yesterday and my second today and so far, my students are really into it.  I thought I'd share the coded messages here for anyone who was interested.  I got all the messages by googling around for math jokes or math quotes.

Day 1:

“169-1-400-64  81-361  324-1-16-81-9-1-144”

Day 2:

81-400  81-361  196-225-400  400-64-25  100-225-4  225-36  169-1-400-64-25-169-1-400-81-9-1-196-361  400-225  16-225
9-225-324-324-25-9-400  1-324-81-400-64-169-25-400-81-9.   81-400  81-361  400-64-25  100-225-4  225-36  4-1-196-121 

Day 3:

676  196-676-49-361-484-196-676-49-324-676-169 324-64 676 529-484-25-324-576-484 441-144-81  49-36-81-169-324-169-400  576-144-441-441-484-484  324-169-49-144  49-361-484-144-81-484-196-64

Day 4:

676-289-64-361-49-144-16-361-64-4  64-324  1-16-64-169-36  676-1-121-16  361-196  4-196-400-169-361  400-225  361-196  361-484-16-169-361-576  484-64-361-49-196-400-361  361-676-100-64-169-36  196-25-25  576-196-400-289  324-49-196-16-324

Day 5:

4-144-1-9-121  64-225-144-25-361  324-25-361-441-144-400  36-324-225-169  49-225-16  16-81-484-81-16-81-196-49  400-64-25  441-196-81-484-25-324-361-25  4-625  676-25-324-225

Day 6:

441-324-25-484  144-36-49  144-441  441-144-36-81  121-484-144-121-225-484  361-676-25-484  49-81-144-36-625-225-484  16-324-49-361  441-81-676-576-49-324-144-169-64

Day 7:

3375-1728-64  2197-1-8000-512-125-2197-1-8000-729-27-729-1-2744-6859  2477-125-10648-125-5832  64-729-125  8000-512-125-15625  1000-9261-6859-8000  1728-3375-6859-125  6859-3375-2197-125  3375-216  8000-512-125-729-5832  216-9261-2744-27-8000-729-3375-2744-6859

Day 8:

8191-2-2187-256-32-8192-2-2187-512-8-729  512-729  27-243-9-19683-512-3-128  2187-256-32  8192-9-729-2187  9-4-19683-512-9-6561-729  2187-256-512-3-128  512-3  2197-256-32  4096-32-2-729-2187  9-4-19683-512-9-3561-729  59049-2-531441

Day 9:

3  1296-7776-3-7776-256-7776-256-27-256-3-625  27-3-625  64-3-49-243  64-256-1296  64-243-3-81  256-625  3-625  3125-49-243-625  3-625-81  64-256-1296  4-243-243-7776  256-625  256-27-243  3-625-81  64-243  343-256-25-25  1296-3-16807   7776-64-3-7776 3125-625  3-49-243-216-3-16-243  64-243  4-243-243-25-1296   4-256-625-243

I hope my students learn to recognize perfect squares and perfect cubes after doing these codes.  At the very least they're fun and they don't take away any class time because I'll pass them out on slips of paper at the end of class and students will have to decode in their own time.

If you want an answer key, just leave me a comment.

Saturday, February 11, 2012

Parallel Lines and Transversals

I wanted to share this document I made because I used this lesson on Tuesday with my geometry class and it worked really nicely.  I did it with them on a doc camera and shared their answers over the doc camera as well.  We all really enjoyed especially the last problem which was written about a student.  He really enjoyed the problem even though it poked fun at him.  I did a terrible job after this lesson though with reinforcing all the angle relationships and their names.  I just went over all the vocab- alternate interior/exterior etc. and had them do problems out of the book.  In my defense, I just didn't have time to do anything more exciting.  But at least we had a good intro to the topic I think.

The pictures came from  and   By the way, I'm still pretty new to this so is it best to cite pictures as I did above, in fine print below the picture, or should I try to restrict myself to only using pictures I myself have taken?
  Parallel Lines and Transversals Worksheet

Tuesday, February 7, 2012

Quadratic Functions Performance Assessment

My algebra 2 students just turned in a performance assessment over graphing quadratic functions and it turned out really nicely, so I thought I would share it.

First, I asked students to graph several functions I'd put together.  When graphed they form a man with wings.  I then asked them to make their own picture out of functions.  They then traded papers and tried to graph each other's pictures.  Finally, they colored in their pictures and we had an art show.

This project helped them reinforce function transformation rules, especially vertical dilation and some students even created their own functions that we hadn't studied to help them create their picture.  (One student taught himself how to flip an absolute value graph on its side and shift it, then another student had to figure out what he'd done to graph it.)   The best part of the project was when a student thanked me for having fun homework. And of course today was a lot of fun when we got to hang up their pictures, admire them, vote on the best one and eat candy.  The whole project took two and a half periods and I think a lot of learning got done.

Here's my original hand out:

*Oops, there's a typo.  Function 11 should be p(x)=x^2+(y-5)^2.  The squared should be on the outside of the parentheses.  I throw these sheets together too quickly...
  Quadratic Functions Performance Assessment Here are pictures of the students admiring each other's work and voting on the best piece of art:

Here are the pictures they created:

I do have more than 7 students, but two were absent today (I know, 9 students is still an awesome student teacher ratio.)
Here's the third place winner

Here's second place:

And here's first place

Saturday, February 4, 2012

Polynomial Division?

I'm trying to put together lesson plans for my Algebra 2 students' unit on polynomials and rational expressions.  The curriculum we're using doesn't have any lessons on factoring polynomials beyond quadratics, but our state standards do require factoring higher order polynomials.  I'm kind of torn as to how to teach this because I can't think of very many compelling reasons students would want to factor higher order polynomials without spending a lot more time on this topic than I want to.  I feel like it's a pretty useful skill for students to have- I know at least my calculus student has encountered polynomial division a few times this year, but I just feel a little tepid on the topic.  I don't know how important the broader math world regards this topic.  Is it worth spending more than a day on in Algebra 2?  Is there any way to make it fun and or applicable?  I've been searching around and I can't find anything at all that other teachers have put together other than drill and kill worksheets.  Anyway, here are two worksheets I've thrown together to help students factor higher order polynomials.  I think they should provide two relatively straight forward lessons albeit a little boring.  I would love any advice on spicing this topic up, or maybe compelling reasons to either go more in depth into the topic or to drop it all together.

Factoring With Pascal's Triangle Polynomial Division Exploratory Ws

One other note: Maybe this is something novices do, but I am a little torn about font choice.  I know in the larger scheme of things font matters so little, but I've heard so many people making comments about Comic Sans MS being unprofessional yet my students love it.  My first year teaching I was just playing with fonts, switching back and forth and not really caring, but my algebra 1 students who have had terrible experiences with math latched onto the assignments I printed up in Comic Sans as being friendlier and easier to understand.  They said it made math less scary.  Since then I've been using exclusively Comic Sans.  Another educator told me to stop using it because it was unprofessional, but when I polled all my students, they insisted that I stick to Comic Sans font.  Notice I did decide to switch fonts for the second worksheet.  I tried to find something that was equally friendly but didn't have the same stigma.  Sadly it's tiny.  What a silly thing to obsess over yet shouldn't I listen to my students?

Wednesday, February 1, 2012

Absolute Value Warm-up

I always try to plan every lesson to death.  I don't plan out minute by minute, but I do try to figure out exactly what I'm going to say, how I'm going to introduce an activity, exactly how the students will be organized for this or that problem, but sometimes I forget that the best activities or lessons can spring organically out of our collective energy.  I had an idea for a warm-up problem today about 2 minutes before class was going to begin, and I decided to run with it.

Our school is in Sheridan- which is a very small town, so most of the teachers and the students live in McMinnville which is a much larger town 20 minutes away.  The topic I was planning to introduce was graphing absolute value functions as the culmination of our algebra 1 unit on linear relationships.

When the students came in I asked who I should make part of the warm-up problem and everyone volunteered.  I picked two students- one who is 15 and one who is 13.  I said that the 15 year old just got his license but because he's so nervous, he drives to McMinnville at an agonizingly slow 10 mph.  Sheridan and McMinnville are about 20 miles away from each other.  At this point, my students- on their own I might add- started making x-y tables to figure out how far the two students are from McMinnville at any given time.  We made a simple table on the board and realized that it took the two students two hours to get to McMinnville from Sheridan.  At this point the students start poking fun at the 15 year old because he's driving so slowly, and he starts defending himself saying he doesn't want to kill anyone and however long it takes doesn't matter.  I asked the students if this relationship is linear and if we can model it with the equation of a line and immediately they start finding the equation for the line.  At this point I interjected with a new piece of information.  The 13 year old navigator wasn't paying attention and forgot to tell the 15 year old to stop and that they made it to McMinnville safely.  My 13 year old student says that actually, she fell asleep.  So the 15 year old driver drives right on through McMinnville and out the other side.  So we continue the table for distance vs. time, but now they're getting further and further away from McMinnville.  So here's what our table looked like:
           Hours driving                         0            1         2        3        4
           Distance from McMinnville     20        10        0        10       20
It made perfect sense to the students why we didn't go into the negatives, why the values started to "bounce" back up.  The students were having a lot of fun poking fun at the 15 year old and the 13 year old for driving so slowly and for spacing out and missing McMinnville.  One student even made his own table asserting that, since McMinnville is 4 miles across, the second half of the table should go 6, then 14 instead of 10 then 20.

After this warm up problem I had no trouble at all showing the students that an absolute value equation "bounces" but that on either side of the vertex, we have linear relationships.  I had complete engagement with the rest of the lesson.

I think maybe why it worked so well was that the problem was almost a "make your own adventure".  The students were able to add details for fun or change the numbers to fit in with their understandings of the real world.  I always start lessons with a warm-up that tries to pull the material we're learning that day into a real-world context to see what kinds of intuition the students already have for the subject material, but rarely does it work so well.  Sometimes no planning is the best planning- but I wish there were a linear relationship between how much time I spend on a lesson and how well it goes because then every day would be awesome.