Algebra Algebra Algebra

Phew! That's what seems to be consuming my thoughts these days. I like teaching it for the 2nd year, and also I like that I know what's coming up on the horizon in the higher level math classes for the kids, so I can know what skills to focus on. For example, we just covered solving proportions, and I had some problems with the variable in the denominator. Well. The kids learned from 8th grade that you can just change the equation by taking the reciprocal of both sides and then solve.

So 12/5 = 7/x becomes 5/12 = x/7.

This is all well and good, and sometimes that's the simplest way to solve, but what if in algebra 2 or above you come across:

3x/(x+1) = (3x-2)/x.

Then "flipping" does not get you any closer to the answer.

Oh. Then someone mentioned "cross multiplying". Well, I didn't even want to go there because of all the misuse of that I've seen later on: OH! magically any time I see 2 fractions together involving variables whether or not there's an equal sign, I'm going to cross multiply without knowing why it works or even IF it works. Voila!

Anyway .... tons of algebra fun. I did have a discussion with my coworker, and she had this brilliant idea that I tried with solving absolute value equations ... but it can work with solving any equation. Write down the equation. On top of it, write numbers in circles in the order of operation of what would be done to x if you were to plug in a number. Then on the side I listed PEMDAS and the circled numbers in order and wrote in words what that was:
1. multiply by 3
2. subtract 4
3. take absolute value
4. add 10

Then under that sidebar, I wrote "to solve: undo in backwards order SADMEP"
4. subtract 10
3. 2 cases
2. add 4
1. divide by 3

Then the kids had a road map of what to do, and they didn't do weird things like get rid of stuff inside the absolute value symbols before taking care of them.

Literal Equations

We've moved on to solving literal equations for a variable. I like this transition because it reinforces the same skills they've been working on forEVER. One thing came up in my first of 3 classes to teach it this year that made me change tactics for the next 2 classes.

We were all about "isolating the variable" and "undoing what's done to the variable you're solving for" and such. Then as I'm walking around, lo and behold, a student was actually moving the variable. She wanted to get it to the other side. What was she doing?! Don't touch that variable!

That led to the following in my next 2 classes. Suppose the problem is
Solve 3A = 2w + 4p for w.
I first made them get out another colored pen/pencil, and then identify the variable they were solving for. Then they had to write that variable in a DIFFERENT color:

3A = 2w + 4p

Then I made them draw an arrow to the w and write in their notes, "DON'T TOUCH THIS" while humming the M.C. Hammer song of the same name. We continued on, and most everyone successfully colored the "solve for" variables and left them alone in the remaining problems.

"Subtracting or Negative"

In the course of getting my algebra 1 kids up to speed on solving equations and inequalities, they have to combine like terms, and I'm getting the above question too often for comfort. Sometimes after they combine the terms, they'll squish them all together and instead of something like 6x - 5, it will be 6x-5, where that's a teeny tiny negative sign and not a subtraction sign. I hadn't clued into this until a kid wrote "7x4" instead of "7x + 4" because in her mind, it was a positive 4 that remained after combining the like terms, and not an adding of the 4.

So I'm trying to be careful to say things like: you're keeping track of all the steps you're doing, for example subtracting 5 and then adding 18, so at the end of the day, you haven't seen all the intermediate steps, and if you had just done it in one step, you might as well have just added 13 (and not "positive 13").

Also, in my continual attempt to bring in problems in context, I had interesting conversations with the kids about this problem I gave them:

1. Two cell phone plans Verizon offers are as follows. You can have a monthly fee of $40 and pay $0.20 per text message, or you can have a select plan costing $60 and unlimited texting. Consider the inequality
40 + 0.20t > 60
a. What does the t represent (give units)?
b. What is the person trying to find out by solving this inequality?
c. Solve for t and explain what this means in the context of the problem.
d. Cell Phone Sally has tons of friends and wants to see if her texting habits would be too expensive under the 1st plan. If she texts about 10 messages per day, what would her bill be per month under the 1st plan mentioned?
e. How many texts do you send per day? What would that be per month?
f. What would your monthly bill be under the 1st plan?

I had gone online to the Verizon website, and got that accurate information. I was very careful to have them do only a couple of problems and stop them. Invariably they answered "text messages" for a. Then we had to have a discussion about "what about the text messages? their length? their time? what?".

Then for problem b, practically everyone got it wrong. They said: they're trying to find out which plan is cheaper. So I asked them, okay what is your answer going to look like when you solve the inequality, and THAT'S going to tell you which one is cheaper? We held off on answering b until they did c.

Now note, my coworker and I decided to clump topics: solving equations and solving single variable inequalities one after another, because it would give the kids a chance to practice the same skills. Also note that we have not discussed linear equations and graphs and rates of change yet in the sense that they would not know how to set up the 40 + 0.20t if just the words were given to them.

So, my students solved c. And the answer is t > 100. So we had to have a discussion about what this means. Some kids had $100, some kids said "this means the left plan is more expensive", etc. We finally got to: if someone sends more than 100 text messages, then the first plan is more expensive. I asked what the t values were the solution to this problem, and they said "all t greater than 100", so I asked, would t = 101.3 work? and they said, no, it has to be an integer in this case. good.

d was also a problem that started discussion. There were kids that just started working it without asking me how many days in a month, so that was a good check of their careful reading.

And the last two were GREAT big eye-openers for me. I don't have a cell phone, so silly me, I knew "10 messages a day" was low, but I didn't know HOW low. Some of my students reported out that they sent anywhere from 100 - 500 - in one case 1000 text messages a day? Hmmmmm, I had to ask how many hours a day they were doing this, and we had to do the math to see if this was even physically possible. Holy Moly.

Duties and Conversations

Most likely, just like other schools, I have extra duties and guilt that sometimes makes me volunteer for EXTRA one time duties. And, just like other teachers, there's probably a fair bit of grumbling under my breath about how I don't have TIME for this and I need my time to prepare and on and on.

On the other hand, there are some nice perks that go along with my morning duty. The 2 other teachers I share it with teach different subjects and different grade levels, and I never see them other places. So my 15 minute morning duty allows me to chat with and get to know them and feel more connected to our school's "family".

Along the same lines, I volunteered to give up my lunch time to help sell t-shirts the other day. Not so bad, since I could eat my lunch at the same time, and I have the next period off, so I didn't feel too stressed. At this duty I got a chance to talk with yet a 3rd teacher I never see (new also this year) that teaches a foreign language. This conversation was super helpful to both of us. I shared with her some of the snippy behavior and struggles I'm having with a few of my students, and she mentioned that the same was happening with her. We talked about another student that's not doing so well in either of our classes, and I think we both left with the feeling of, "Whew! it's not just me.".

It's so easy to feel isolated and stressed and feel like the ONLY one that may be having certain troubles, that it's nice to commiserate with others that are going through the same thing ... not that I'd wish it on them, but you know...

Regrading Issues

Yeesh! I thought I was careful, brought on by bad experiences and lots of practice, but today ....

Way back when in the early days of teaching, when I'd grade a test/quiz/homework, and something was missing (a problem not done, a justification, some work), I learned pretty quickly to make sure to look carefully at ALL the blank space around it, and if there was, say a blank side of the page, I'd draw a diagonal mark through it to indicate I'd noted it, and also to prevent the kiddies from later on, after they'd got the paper back, ADDING something to that blank space and then claim that it'd been there all along, and then demand/request/beg more points because it was MY mistake. Ditto for blank parts of pages and such.

I know I make mistakes ... still happens ... daily ... so this marking and covering up of blank space cut down these instances dramatically. It also cut down my 2nd guessing myself that I'd missed something.

Well, we had a ton of swine flu absences a few weeks ago, and many students missed many days, and some are still making up old tests and homework. One student had come in last week to make up a geometry test. I had a meeting after school, so I put her in a windowed smaller room that I glanced in on periodically and had her work. She took forever, and it looked as if she had not studied. I think she was assuming I'd give her the same version as everyone else and not a make-up version, and she had used a friend's returned test to study the answers instead of studying concepts. She finally handed it in, and I graded it. 68/100. Ewww. On one unchanged problem, she MAGICALLY produced the answer and none of her work supported it. On another, she showed some work, but then guessed and checked an answer .... but didn't check the geometry constraints.

On a third problem, the one I had "regrading issues" with, I had made a table where students were supposed to fill in 7 cells. She left a whole column blank - 4 cells (logic symbols). I circled the whole column and put a question mark through it and graded on. Today she came to talk to me. Amongst other things, she showed me that she had written those missing answers in small letters in another column (this was a logic section, and the answers were like " p->q " and such). Hmmmmm, I didn't outright accuse her of lying, but I questioned her as to why she didn't put those answers in the right column. She had some story about how she had written them there to help her with another problem, and she didn't know what I was expecting in this problem's column titled "symbols".

I know I'm overly cynical about such things at times, and about 50% of the time I'm wrong, and I'd HATE to accuse someone of outright lying if it weren't true (it happened to me in high school with a teacher and I still remember it to this day). So, I didn't say more and gave her 1/2 the points back, but now I'll have to remember this and be more careful in the future.

Linear Word Problems

Whew! In algebra 1 I've finished the initial introducing of solving a variety of linear equations: one-step, two-step, multi-step, weird distributive action, variables on both sides. Now it's just a matter of having them practice their hearts out until most/all of them are successful. Yesterday, I wanted to have them see how these problems could be used in real life, and after scanning through books and such, I saw that a lot of people were fascinated with how many coins someone has, or how much of a certain type of mixture to use, or which 2 consecutive numbers add to another number.

Then, whew! I scanned throught he Hughes-Hallett book of "Functions Model Change" and adapted some of their ideas that were written in table form to use as linear equation problems. There's one problems about carbon-14 dating which I know is not linear, but, boom, call it a "model", and it can become a linear situation. Then there was one about weight of a person vs. calories burned doing various exercises. Finally there was one about the years since 1970 vs population of a town.

I liked my adaptations because I've altered the scales on the variables, so that the kids have to think about what the numbers mean in terms of units. Also, these are in-context types of problems, not "math world" type. Also, the kids didn't need a calculator, since I made the numbers "doable". We had a discussion about estimating. For example, for the fossils, t represented time the tree had been dead in 1000's of years. An answer came up as t=3 & 2/11, and I asked them what that meant. We discussed approximating 2/11 by 2/10 and having t=3.2 and they finally got to the point to see that was 3,200 years. I also liked that in 2c below, they had to think to put in w=1.4 instead of 140, for example.

Here's an example of one problem I adapted:

2. Exercise physiologists tested many people, and have come up with an equation that shows the number of calories used per minute as a function of body weight for various activities. For walking they calculated the equation
b – 4.6 = 3(w – 1.7)
where b represents the calories burned in one minute, and w is the weight in 100’s of pounds of the person.

a. If w is found to be 1.6, what does that person weigh? (don’t solve; interpret w=1.6)

b. If b is found to be 5.4, what does that mean? (don’t solve; interpret b=5.4)

c. Suppose a person weighs 140 pounds, how many calories did they burn walking in one minute? In 30 minutes?

d. Suppose someone burns 5.2 calories a minute, how much do they weigh?

Studying for Math Tests

Blach! I'm having an inner cursing and outer cursing with lots of hand gesturing and bad facial expressions weekend as I grade my Algebra 1 tests. This was their first true test on algebra (the other one was on topics they've seen since the womb: positive and negative numbers, fractions, simple graphing). During Friday I'd started grading my 1st set of exams (with inner grumbling and outer professionalism while my 2nd set of kiddies took the exam).

Then before my 3rd class started their exam, I had a quick informal vote: how many people did problems from scratch to study for this test (thumbs up or thumbs down)? How many people started studying before last night? How many people read through their notes? Hmmmm, the number of thumbs down was heart breaking.

Maybe it's an experience they have to go through: horrible test grades to see that they actually have to study. On a positive note, the way their grades are weighted with homework and tests, it doesn't HORRIBLY bring down their grades, but it does lower it (sometimes up to 6% points depending).

With a hope of helping them in the future, I've made up a document that I'm going to hand out to them a week before their next test. It walks through the steps of how to study for a math test. Hopefully, this will work.