Teaching Repeat Students

February 3rd, 2010

I’ve taught 7th grade students Algebra – now I’m teaching high school students Algebra. There is definitely a world of difference there. Whereas I used to come up with challenge problems all the time to keep the students interested, I’m now struggling with the basics. I will admit, teaching is not as fun as it was – it’s actually fun when students care about the learning.

At any rate, I’m a little upset at the end of this first semester because one of my best classes has been taken from me. Even though they were all struggling students, and I had to work hard to make them master the basics, everyone in the class tried very hard, put in the effort, and managed to pass. You would think they would leave a class like that alone. I have never had a class like that – EVER – not even with my 7th graders, because even middle schoolers who are above average in math can be lazy because they know they’re smart enough to get by. Not only that, but this class that I had never once gave me a single problem in that regard. Nor did they ever give me problems with their behavior – I’m not exagerating when I say that I never had to ask anyone to stop talking, to stop disrupting the class, to remain seated, to put away their ipods or cell phones – an entire semester and not one single word of correction ever had to come out of my mouth. They took all my sweethearts away and replaced them with 34 first semester repeaters, some of whom were suspended every other week, some of whom are coming back from detention or long term suspensions, and plenty who are special ed. Sadly, there were several students in that “good” class that I know would not do very well if they were mixed with other students. I sure do hope I don’t see any of them repeating second semester next year.

Just wanted to vent.

Algebra FunSheets

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Teaching Laws of Exponents

November 6th, 2009

This year I made the change from teaching middle school to teaching high school. I still teach Algebra, and the difference is tremendous. All the methods I’ve used in the past really don’t apply anymore. I no longer look for ways to challenge students, but am now focused on how to get the students to understand the basics. It’s very discouraging.

Take teaching exponents. You either multiply, and therefore add the exponents; if you have a power, you multiply the power with the exponent(s); and if you divide, you subtract. Three simple “rules.” And, yes, I taught it that way, and regret doing that. I do have a handful of students that do get the laws of exponents – but most confuse all the different rules and do whichever operation they fancy at the moment with no rhyme or reason.

For students who don’t have a good handle on math, the above “rule” method is not a good one. I should have gone to the basic method of having them write it out: x^2(x^3) = x*x*x*x*x which is x^5. And (x^3)^2 = x^3*x^3 = x*x*x*x*x*x which is x^6. As for division, they can just cancel out top/bottom pairs after expanding them out. Even though this method is rather crude, it’s something the students can understand without relying on rules. And it’s probably true that those handful of students who got the “rules” would have seen the pattern of the rules after writing everything out anyways, and that self discovery of the rules would have been that much more meaningful to them.

My point is, you know the level of your students. If your students tend to struggle, then teach them to use the definition of exponents (i.e. writing everything out in expanded form) to evaluate exponential expressions. In fact, the next thing on my list is to create worksheets with smaller exponents so that expanding them out won’t be so torturous on the students. I think one of the reasons why I taught “using the rules” is because their workbook is full of problems with larger exponents like (x^12)^4. Who on earth wants to write out 48 x’s. You’d think after writing the first few, they’d get the point, but you can’t always assume those kinds of things.

So if you haven’t gotten to that unit yet, hopefully by the time you do, I will have several worksheets on expressions that have small enough exponents that are more ideal for writing out/expanding. I’m fairly fortunate in that I taught this unit to my students who are repeating second semester Algebra, and based on the mistake I made with them, I can make sure I don’t repeat that mistake with my one-year Algebra students. Towards the end of the year, I will reteach my second semester students this unit using expansion, since they probably won’t recall having done anything with exponents come March/April…

By the way, I had a student ask me why x + x is 2x and x * x is x^2. I explained that multiplication is “shorthand” for adding the same thing over and over again, or repeated addition. Instead of saying 7 + 7 + 7, it’s better to say 3 times 7. Likewise, exponents is “shorthand” for repeated multiplication. She did not get it. I used the visual, if you have an x in one hand, and an x in the other, all together you literally have two x’s, that’s what addition is. Adding does not change what you have, just how many of them you have. She didn’t get that. I expanded exponents to show her why only multiplication produces changes in the exponents, and she still didn’t get it. If you have a better way to explain this common error away, please let me know…

algebrafunsheets

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A Quadratic Project Idea

March 25th, 2009

I just made a worksheet in which students estimate the U.S. and world population for various years, an application of quadratic equations, and came up with an idea for a project. It integrates U.S. History and Algebra, so anything that is cross-curricula is a good thing, right?

Anyways, A good group (or individual) project idea is for students to create a time line (poster board, computer, etc.) of the United States. Using the quadratic equation that models U.S. population growth, have them select 20-25 (or however many they have time for) important dates (Civil War, World Wars, etc) and calculate and graph the approximate U.S. population for that year. So the y-axis would be U.S. population and on the x-axis, students will label the dates of important events. This way, students are graphing a relevant, meaningful quadratic equation instead of just a random equation with arbitrary values. And since they can relate a growing population with the growth of a quadratic curve, it adds meaning to graphing quadratic equations.

This would only work for the U.S. population since the equation for the world population is now more exponential than quadratic, and, if I remember correctly, involve natural logs. Of course, for higher level math, you can use the more complex equation that models world population growth and modify the project to incorporate important world-wide dates.

Just a thought.

www.algebrafunsheets.com

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Teaching Factoring

February 25th, 2009

I had written a post a while back on how to factor trinomials using the slide and divide method (or slip and slide, it has numerous names). However, I did not use that method this year when I taught this unit. The reason for that is because I don’t think it enables students to understand factoring, it was simply a memorized algorithm. That’s not always a bad thing, but they weren’t able to relate this skill to other types of factoring. First, there are the basic trinomials that have a leading coefficient of 1. Those are fairly easy to do quickly, once you understand the concept of finding factors of ac whose sum = b. Then, for trinomials with a leading coefficient other than 1, there was the slide and divide. Then they had to understand to factor out any GCF’s, which sounds simple enough, but most students want to jump right into the problem and don’t take the time. Then they get to factor by grouping, which doesn’t seem related to all the factoring of trinomials that they’ve been doing, so by the end of the unit, many students simply became confused.

This year, I went strictly with factoring by grouping. As strange as it might seem, I taught trinomials with coefficients other than 1 first. For the problem 2x^2 + 13x + 20, students multiplied 2(20) = 40. So they had to find factors of 40 that had a sum of 13. They used the numbers to rewrite the problem as 2x^2 + 8x + 5x + 20. Then they factored by grouping: 2x(x + 4) + 5(x + 4). So the two factors are (2x + 5)(x + 4).

Next I taught the basic trinomials (with leading coefficient of 1). These were a lot easier.
For the problem x^2 – 12x – 28, students still multiplied the leading coefficient with the constant, but since the leading coefficient = 1, then the number they arrive at is -28. So factors of -28 that had a sum of -12 are -14 and + 2. They rewrite the problem as x^2 – 14x + 2x – 28 and then factored by grouping. x(x – 14) + 2(x – 14). So the 2 factors are (x + 2)(x – 14). For some reason, when you add an extra step to a method, students complain about it being so hard. But when you go from “harder” to “easier,” the students love how it’s so much “easier.” By the way, it is at this point that I explain the relationship of multiplying 2 binomials and factoring, which is basically “undoing” multiplying 2 binomials, so that they understand what factoring is.

Now when you have trinomials with a GCF, it doesn’t matter. Take 4x^2 – 32x + 64. Students tend to forget pulling out the GCF, but with this method, it’s no big deal. The product of 4(64) = 256. (Okay, yes it makes for large numbers). So factors of 256 that have a sum of -32 are -16 and -16. They rewrite: 4x^2 – 16x – 16x + 64, then factor. 4x(x – 4) – 16(x – 4). So the 2 factors are (4x – 16)(x – 4). So there’s an added step. If a factor can be broken down further, then they must do that. So it ends up being 4(x – 4)(x – 4). All done, and you can see that it doesn’t matter if they forget to factor the GCF first. Of course, for many problems, they’ll want to since they’ll just end up with huge numbers. I tell them that if they end up with such a large number, that’s a clue that maybe they should factor out a GCF first.

Factoring special cases involves no extra steps and work as any other trinomials. Take x^2 – 100. Since the product of 1(-100) = -100, then they’re looking for factors of -100 that = 0. (since there is no x term, there are zero x’s and so b = 0) So it must be 10 and -10. Rewrite: x^2 – 10x + 10x – 100 then factor: x(x – 10) + 10(x – 10). (Many students will recognize by now the pattern and will understand that the 2 factors are (x + 10)(x – 10) without doing that middle step). What about those with a GCF? 3x^2 – 363 for example. Well, 3(-363) = -1089. Students might discover that the factors of -1089 that have a sum of zero are 33 and -33, so that 3x^2 + 33x – 33x – 363 = 3x(x + 11) – 33(x + 11). The factors would be (3x – 33)(x +11) but they must break the first factor down to 3(x – 11)(x + 11)) However, -1089 is such a huge number that most students will now be hoping that there is a GCF and there is: 3(x^2 – 121). So now they can continue factoring: 3(x^2 – 11x + 11x – 121) which factored leads to the same answer of 3(x – 11)(x + 11).

And finally, by the time they get to the factor by grouping method, they completely understand and find these easier than the rest. Take for example 4x^3 – 20x^2 + 3x – 15. They don’t have to multiply, search for factors or anything. It’s already set up for them: 4x^2(x – 5) + 3(x – 5). So the 2 factors are: (4x^2 + 3)(x – 5).

Also, keep in mind that 1 is still a factor. In the problem 2x^2 + 5x + 3, 2(3) = 6, so factors of 6 that have a sum of 5 are 3 and 2. Rewrite: 2x^2 + 3x + 2x + 3. Now the first half has a common factor of x. But the second half, which seemingly has no common factors, does have that common factor of 1 and since you MUST factor something, factor out the 1.
x(2x + 3) + 1(2x + 3). Which makes the factors (x + 1)(2x + 3)

So although slide and divide seems an easy enough algorithm, in the long run, the more connected each topic seems, the easier it will be on the students (and on you!). Hope this helps.

algebrafunsheets

Posted in Algebra Lesson Plans, Teaching Algebra | 7 Comments »

Coming soon: Funsheets4math

February 5th, 2009

DUE TO TECHNICAL REASONS, THE TWO WEBSITES HAD TO BE SEPARATED. SUBSCRIBERS ONLY HAVE ACCESS TO THE WEBSITE THEY SUBSCRIBED TO.

Another website has already been created: funsheets4math! Don’t try to access it quite yet … the code and gui are working, it’s just a matter of tying up a few loose ends. As soon as it is up, we’ll provide a link. It will be the same creative worksheets, just different content. This will be a sister site to algebrafunsheets, so if you’re already a subscriber, you will have unlimited access to both sites. (Good news for our members who teach elementary school or middle school, but probably not much use to those of you who teach high school … unless you teach high school Algebra AB, in which case the beginning algebra skills worksheets for pre-algebra might come in handy.)

As a middle school teacher, I found that the workbooks/textbooks did not have adequate resources for certain topics. And many computer generated worksheets I found online were also a little under par (one worksheet for area of irregular shapes kept repeating the same 5 problems on a 20 question worksheet). So, since I have a few worksheets for these particular areas, it seemed like a good idea to provide access for these resources as well.

It’ll be a slow process, though – don’t expect too much in the beginning. It’s taken me 6 months to create roughly 130 algebra worksheets, and I still need to create many more to cover those areas that are still lacking in resources. So the funsheets4math won’t be open for subscription for quite some time – only those with algebrafunsheets subscriptions will be able to access it during this time.

Math content for late elementary school to pre-algebra will be covered.

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Creating funsheets

January 25th, 2009

It’s been quite hectic, and quite draining to teach Algebra all day and then come home to even more Algebra in creating these funsheets. Even so, many units are still lacking a full range of adequate resources. I’ve jumped ahead to work on those units that aren’t typically taught until the end of the year, but I’ll add random worksheets in the other units, as they can be used for review. I’m also going to create an end of the year review type of unit, which will be good for those of you, like me, who need resources that prepare students for standardized tests. Anyways, I expect that by the end of the summer there should be a full range of activities for the entire curriculum.

I’ve learned quite a bit these past several months, as to which formats work and which don’t. Riddle worksheets seem to work best, so I’ve given up trying to simply stick with math jokes. Some of the last few math jokes have been real gropers, so I’ve opened up my options to all jokes. The students really enjoy “solving” the riddles , even if they are dumb.

Also, if you know you need a specific type of worksheet, please feel free to let me know. I’ll do my best to accommodate your needs within a couple of days.

www.algebrafunsheets.com

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Six Technology Tips For The Algebra Classroom

December 13th, 2008

Since I recently threw out some ideas for how to make Algebra fun without technology, I thought I’d follow it up with ways to make Algebra fun with technology. This one is far easier, in regards to coming up with ideas. After all, simply using technology in the class room captures the students attention since they do love their gadgets.

It’s not so simple if these resources are not available at your school. Maybe after a bit of begging, you can get your school to invest in a few of these resources. I’m very fortunate in that I work at a Technology Magnet school, and I have all of the resources that I list below. I will list them in the order of the most beneficial resource, to those that aren’t quite as important to have. You’ll see that, in my opinion, the TI family takes precedence in regards to importance in the Algebra classroom, and this almost seems like a sales page for the TI. Once you use it, though, you’ll understand completely.

1. The most important Algebra resource you can acquire is the TI Graphing Calculator, along with all it’s accessories. The TI Graphing calculator itself is a necessity, and as long as you have this, you can do a lot of activities with them. Most schools should already have this. However, if you use it simply as any other 4 function calculator, then you are missing most of what the TI is all about. (Sorry, I’m not a Casio fan, but I’m sure you can do the same thing with the Casios). It’s a great exploration tool, so long as the students are carefully guided through deliberate activities and not simply just told to graph such and so equations. The TI website itself provides a lot of resources in regards to this, so you don’t have to do a lot of work, just sift through the activities yourself and choose the best activities for your students. You can download a worksheet that guides your students through an activity, or you can download applications that guide students through an exploratory session right on the calculator. Of course, that means you’ll have to download the app to all the calculators, but it’s well worth it. Use these sessions as a supplement, though, don’t think it takes the place of your teaching.

2. Along with the calculators themselves, the TI Smartview is wonderful. One thing it allows you to do is drag and drop a snapshot of your calculator screen onto any document or power point, which is wonderful for creating worksheets or quizzes and tests. But most importantly, it projects a very large version of the TI itself onto your SmartBoard, which then can be used as an actual calculator. (You must have a projector and a SmartBoard for this to work). This makes things so much easier when students can actually see which keys you are using, as opposed to having to listen to things like, “push the button on the top left, the blue one that says 2nd, etc…” Actually, you still have to say those things, it just eliminates the need for you having to repeat it a hundred times, since the students are given a visual of your instructions. At any rate, aside from the calculators themselves, the SmartView is the second most important resource to have in the classroom. And since everything is done on the fly, so to speak, even mistakes open up great math discussions.

3. And finally, to complete the TI family, you need the TI Navigator. One of the best features about the Navigator is that it allows you to capture your students’ screens at any moment, which allows you to make sure all your students are on task. Keeping students on task is key in any class room. But in regards to fun, the Navigator opens the door to so many activities. One of our favorites is the golf ball activity, in which the students actually input equations that line up a golf ball with the tee (yes, that is where I came up with the golf ball worksheet, which isn’t quite as fun as doing it on the TI). Because the activities are so interactive, the entire class becomes involved in the same discussion (as opposed to a single student on a single computer). I think students learn much more when they dialogue with each other. For example, during the golf ball activity, a student will enter an equation with a positive slope, when the required equation to get the ball to the tee is clearly a decreasing line. Students will discuss that they at least know the slope must be negative, that the angle from the ball to tee is relatively flat, so it must have a slope < 1, etc.

4. Okay, so enough about the TI. If you don't have a TI, but you do have a SmartBoard, that itself can open up a lot of doors in regards to discussions. First of all, the SmartBoard itself does wonders in keeping the students engaged, they love the technology of it. They like seeing a digital projection of their own writing. They beg for a turn to do a problem on the SmartBoard. They actually take ownership of it and are very protective of it. If you have one in your room, you must know what I'm talking about. In regards to Algebra applications, though, it's also fairly interactive. For example, in regards to graphing lines, all you have to do is put a coordinate plane background and then draw a line with the drawing tool. The line can be manipulated by hand to be placed anywhere on the screen. There are quite a number of other interactive tools specific to each topic that can be downloaded from the SmartBoard gallery as well, too many to simply list here.

5. Fifth on my list is the interactive quizzing systems - especially if you don't have a Navigator. It does require a projector. And it also requires more planning, since every detail needs to planned in advance. It's simply multiple choice math problems projected on a screen, but students can input their answers simultaneously and both you and they get immediate feedback. If you use this, one thing that I've found to be very important is to ask each TYPE of question TWICE in a row. After the first question, you access the students' knowledge and discuss why the wrong answers are wrong, what mistakes the students made, etc. But if you leave it at that, and just continue, then you can't be sure the students have grasped the important ideas of the question. Ask it again, a different problem, but same idea, and make sure that the percentage of students answering correctly has gone up. You'll be surprised the number of times this does not happen, and the misconceptions you thought you had cleared up really hadn't been cleared up at all. In other words, it gives you a more accurate assessment of what the students do or do not understand.

6. Last on the list are the standard programs on any computer, such as power point. These resources aren't quite as invaluable as those above, in regards to Algebra itself. But they do offer their benefits. As far as I can tell, the best thing about these tools in the Algebra class room is that it gives students a much needed change from simply doing worksheets all the time. Explaining a problem on power point is more interesting to a student than simply writing several paragraphs about it on a sheet of paper. PhotoStory is not standard software, but if it is available, along with student digital cameras, then you can come up with great projects. One project idea is have students do the old car on a ramp lab (steal this from your science teacher, they've got the lab sheet and required accessories), as a supplement to slope, and the students can take pictures and record an audio file of their explanation of slope and it's effect on the speed of the car, etc. Students can record an original rap/poem/song they wrote to help remember the main ideas of slope, which along with images, makes for a wonderful final product. (I guess I'm stuck on linear equation ideas at the moment, because that is what we are currently working on in class). So even though these tools are not specific to Algebra itself, the list of ideas for these resources are endless and fun, so long as you lend a bit of creativity into them.

Get Algebra Worksheets at
www.algebrafunsheets.com

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Challenging students in Algebra

December 6th, 2008

The other day I was looking at some upper level algebra problems and I was wondering how many classes as a whole ever get to that level. I know that my classes typically don’t. Not the entire class at any rate. There is just way too much diversity in any single class. Take a simple problem like 2(x – 3); it’s enough of a challenge to get some students to remember that they need to distribute the 2 to the ‘caboose’ or last term, and not just to the first term of x. On top of that, it’s a battle trying to get those same students to realize the 3 is negative, therefore they should get -6 and not +6.

When you have so many classes that have students struggling with the basics, how do you ever get to the problems like:

1/4(17/3 – 18x) + 2(14/17x – 19) = 3/5(16x – 92) – (27x + 4)?

If your class is at that level, and you’re not teaching a TAG class of 7th graders, please let me in on your tactics. I sure could use them.

But one thing I have discovered over the past several years is that it doesn’t matter what kind of class you teach, there is always at least ONE student who is capable of being challenged. And in that case, it is worth it to push the students to that higher level. I’m not talking about handing the class a worksheet of 20 problems similar to the above, that would certainly seem overwhelming. But I have found a way to integrate those higher level problems into our lesson every now and then.

Not every single day of course, since the novelty would wear off. And I definitely vary the problems that I challenge the students with. But every now and then, regardless of what unit we’re in, I’ll give them a problem like the one above. Not necessarily a solve the equation problem, that was just a random example. Any type of challenge problems. Because the goal in challenging students is not for them to find the answer – sometimes, as with problems like the above, I actually give them the answer. The goal is for them to show each step in between. It’s certainly a challenge to the students – literally. They tend to compete with each other to see who can get all the steps correctly first. I didn’t set it up that way intentionally, but students who actually put forth the effort to solve these type of problems also have some intrinsic motivation that makes them equally competitive.

As for those students who struggle with the 2(x – 3) type problems, well, they have the rest of the period to concentrate on Algebra basics. But for those few students who actually are capable of doing more, well, this is their time. And my experience has been that they thoroughly enjoy it.

Find some good challenge problems for your students. A few of them will surprise you.

Get Algebra worksheets at
www.algebrafunsheets.com

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Online Activity for Slope and Slope-Intercept Linear Equations

November 22nd, 2008

The other day, my students spent the day in the computer lab and explored the myriad of web sites on line that let them explore linear equations. I directed them to some fairly decent tutorials, and various online tools that allowed them to physically move lines to explore the effect it had on slope and equations. It was the first time I took them to the lab this year, and thought they would thoroughly enjoy this hands on opportunity. But they blandly went through the tutorials, and some even complained that they would prefer next time to stay in the class room, because it’s more interesting when I explain things to them.

However, the last 15-20 minutes of the day, I directed them to the website below. The student loved it. Many students who struggled with rate of change/slope and writing slope-intercept equations were proficient by the end of class. Unlike most other math “games,” the graphics on this are excellent and students actually enjoy the concept of the game, which is rather strange (which is why they liked it). There are roaches that traverse the coordinate plane at a particular line, for which the students must identify the slope or slope-intercept equation that will exterminate the roaches. They can even choose the method of exterminating the roaches (fumigator, shoe …).

Linear equations game

If you don’t have the time to do this during a class period, show the students the site and they will go there on their own time. Many of my students asked me to send them the link at their home address so they could play at home!

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Slide and Divide Method of Factoring Trinomials

November 22nd, 2008

Someone asked about how to teach factoring trinomials where the leading coefficient is not 1, so I thought I would post it here as well.

The best method I’ve found is often referred to as the slide and divide method, for trinomials of the type ax^2 + bx + c

Anyways, take the trinomial 3x^2 + x – 10.
(Always ensure to pull out any common factors before continuing with this method).
First you slide the leading coefficient to the end and multiply it by the constant:
You get x^2 + x – 30
Then factor like normal
(x + 6)(x – 5)
You have to “put back” the number you slid out, so to speak. You do this by dividing the constant in each factor by the leading coefficient you “slid” out of the way.
(x + 6/3)(x – 5/3)
Simplify the fractional terms you end up with.
(x + 2)(x – 5/3)
Once it’s simplified, if there’s a fraction left, the denominator becomes the coefficient of the variable term.
ANSWER: (x + 2)(3x – 5)

Here’s another example without the explanations:

2x^2 – 7x + 5
x^2 – 7x + 10
(x – 5)(x – 2)
(x – 5/2)(x – 2/2)
(x – 5/2)(x – 1)
ANSWER: (2x – 5)(x – 1)

Finally, an example where you have to pull out a common factor first:
12x^10 + 42x^9 + 18x^8
6x^8(2x^2 + 7x + 3)
6x^8(x^2 + 7x + 6)
6x^8(x + 6)(x + 1)
6x^8(x + 6/2)(x + 1/2)
6x^8(x + 3)(x + 1/2)
6x^8(x + 3)(2x + 1)

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