I started this lesson with some theatrics. I asked them to simplify the fraction shown in the picture, and of course they all wanted to cancel the terms (as expected). I let them do it, and then changed the pretty pink heart into the fiery eruption you see here. I told them that those red slashes are like daggers through a math teacher's heart. I also told them that, when they go to college, I never ever want them to make the mistake of canceling out terms. Cancel factors, not terms! We spent a lot of time talking about the difference between factors and terms, and why this rule is true. We talked about why you can't add 5 and 5x, but you can cancel the 5's in 5/5x. I think this was time well spent, because this canceling problem is a persistent weed. From there, we practiced factoring and canceling. Pretty straightforward. In the following lesson, we multiplied and reduced products of polynomial fractions. There really were no new skills to learn, so after modeling one problem, I had them do independent practice work.
Continuing with the lessons, we learned to factor difference of squares expressions. I used a geometric approach to help make sense out of the pattern, and it has really helped some students figure out how to more easily factor the nasty ones like 25x^2 - 16y^4. A quick sketch of the squares, labeled with their side lengths, has proven quite useful.
It's been a while since I posted. The last week of February was our Junior Trip, in which we take all of our junior class on a 4-day-long trip around California to visit various CSU campuses. It's an incredibly important part of our program, because it is the time when our juniors really start to imagine themselves as college students. The tours, the student panels, seeing the dorms and classrooms, the admissions directors, and the DCP alumni all bring things into sharper focus for the 11th graders. We moved the trip earlier this year (it used to be in April) because kids come back inspired and ready to make positive changes, and so we wanted them to have more time to improve their grades before the end of the semester. It's also a great time for students and staff to bond and get to know each other in different ways. Needless to say, a 4-day, 3-night field trip with 80 high schoolers is tiring. We're all pretty much recovered now, and it's been back to business as usual. Time to catch up on some lesson postings.
In Algebra 2, we're nearing the end of the polynomials and factoring unit. I've been focusing on basic factoring techniques (look for the GCF first, then either use trinomial factoring or difference of squares, if possible). I'm still deciding whether to throw sum/difference of cubes into the mix this time around. I decided to bring simplifying and multiplying rational expressions into this unit (instead of waiting for the rationals unit) because it seemed like a good way to have them get more practice with factoring without repeating the same exact problems again and again. Plus, these questions are prominently featured on the STAR test.
One thing that has been helping students deal with factoring out the GCF is teaching them to write the prime factorization of each term in the polynomial, every time (including a -1 factor when there is a minus sign). Though it takes longer, this is pretty much a foolproof way of factoring out the GCF - many students have a lot of difficulty with the "what's the largest expression that divides into both" method.
Ok, so I guess it should really be titled Algebra 1, not 2. But my students always need to review/relearn this topic. We'll go easy for the first lesson - only problems where the GCF = 1 and where the leading coefficient is 1. I made a puzzle for them to put together so that it is more fun than just doing a worksheet. I did something like this in the past with my honors class (but with much harder polynomial equations) and they really enjoyed it. That puzzle, once assembled, instructed them to do push-ups to get some candy. This one only requires that they tell me a joke - I'll add them to my arsenal if they're any good.
I've been using Keynote this semester as an experiment, to see how it could work in my Numeracy class. So far, it's gone pretty well - especially after I bought a remote mouse so I could control it from anywhere in the room. Combined with the mini-whiteboards, it's been a really efficient way of getting students to do work. After presenting a concept, I can have them practice a few problems right away by showing the next slide, and having them work on their boards. There is no time wasted passing out worksheets. Also, I can make sure all students are focusing on a specific set of problems (versus on a worksheet, where they tend to start jumping around right away, based on what seems easiest). Then, I can show work/answers on the slide without having to pull out a transparency.
Since I've got the projector reserved and set up now, I can easily insert fun and interesting images, sounds, and video clips. I've recorded myself and other teachers singing little ditties (like the infamous "Don't add across"). I've started scouring YouTube for interesting stuff... though the ratio of total crap to interesting stuff is quite high, I've found a couple of gems. I even unearthed my old calculus professor from college, who recorded a "top ten algebra mistakes hit parade" as well as "all of calculus in 20 minutes".
So I'm in my fraction adding unit now, and we've been working with fraction circles to understand adding. Now, we're taking a break from that to do some work on prime factorization, reducing fractions by canceling common prime factors, and finding LCM. Once they get all this mastered, we can go back to adding fractions using common denominators. I hope they don't forget it all over spring break... I've always found it difficult to teach factors and multiples, and GCF and LCM because students confuse these concepts very easily. Part of the problem is their difficulty with the language of division. Just about every student I have says "divide 6 by 40" when they mean 40÷6. If I ask "does 3 go into 12?", they'll say yes. But they'll also say yes if I ask "does 12 go into 3?". (Aside: I think I'm going to devote an entire lesson to this issue - along with the whole "subtracted from"/"subtracted to" issue.)
In any case, I YouTubed LCM and GCF to see if there was anything interesting out there. I was surprised to find a method for finding both LCM and GCF at the same time using Venn Diagrams that I'd never seen before. It's mathematically equivalent to looking at the prime factorizations and picking the right factors, but it provides a nice structure for students to remember which is which. So I designed a lesson to practice finding factors and multiples, and then using this model to find LCM and GCF. It went quite well. I don't know how much will be retained over the weekend, but we'll practice more on Monday/Tuesday because I want them to have LCM down solid. Here are two of my slides, and then the original video I got the idea from.
This lesson will start with the new weekly quiz + notes check. If students' are unorganized or have incomplete notes, they will lose points. I just did this in my reglar algebra 2 class this morning, and only a handful of people got full credit on the notes check. While that is discouraging, it also shows how critical it is that they get continually assessed on short bursts of organization (they only needed to show me a binder with 2 days worth of notes, and an index with 2 entries).
After this, there will be some basic direct instruction on factoring polynomials with the difference and sum of cubes patterns (which I'm not exactly clear why this is worthy of a state standard, but there you go..). The only interesting part of this is that I will use the visual model shown in this post to help them see where the pattern comes from. You can think of a^3 - b^3 as the big cube's volume minus the small cube's volume. This volume is equal to the volumes of shapes I + II + III. You can easily get expressions for their volumes, and then factor out the common (a - b) factor from each term to derive the (a - b)(a^2 + ab + b^2) pattern.
That segues nicely into the next piece, which is factoring cubic polynomial by grouping. Students already learned this method for factoring quadratics, so this piece should go pretty smoothly.
I hope to get through the instruction piece fast enough to allow a good chunk of time for students to just practice using these techniques. They definitely have troubles applying the patterns to expressions like 125x^3 - 64y^3, or even worse, when you need to factor out a common monomial first. I'm trying to coach them to always write the problem in the form ( )^3 + ( )^3 first, so they can clearly see what the values for a and b are.
Today in class, we started with a mini-lesson on finding the powers of i, using the "multiplying by i = rotating by 90 degrees about the origin" metaphor. I misjudged - I made it a self-paced worksheet, but it was too complex (no pun intended) for them to handle on their own. When I went over it with them as a class, they started to understand it a lot better, but it ended up taking much more time than I had planned. The circle help students see why the pattern repeats in 4s. If you start at 1 on the real axis, and make a full circle, you will have multiplied by i 4 times. Therefore, any multiple of 4 in the exponent will cause an integral number of circles, landing you back at 1. Then, the remainder indicates how many more rotations you need and where to end up on the complex plane.
I'm at the point in the year where finals are looming and I feel pressed to rush through the curriculum to finish the unit. But what's the point, if they're not going to retain information as I charge ahead? I may need to scale back my learning objectives for this unit, or for the upcoming units.
After the powers of i, I wanted to do a quick review of factoring out the greatest common monomial factor from a polynomial, as well as factoring quadratic trinomials where a = 1. This is basic stuff that they did in algebra 1, but they had forgotten so much of it. By the end of class, most of them were starting to remember having done this before... so I go back to my earlier question about retention. I feel like this is a fundamental algebra 1 concept that should be retained from spring of this year (when they learned it in algebra 1) until now. Maybe going through it again now will help them get it into longer term memory... but how many times do they need to review the same topic until it sinks in?
Instead of giving a worksheet for practice, I wanted to show students how I create factorable trinomials, and for them to create their own and quiz each other. I think that this will be an effective way to help give them a deeper understanding. Unfortunately, we ran out of time because of the earlier problems. I am moving this into tomorrow's lesson, so hopefully it will work out.
I am a math teacher in the Bay Area, California. Previously, I taught for eleven years at Downtown College Prep charter high school in San Jose, where the target student is low-skilled and of the first generation to go to college. Currently, I am teaching at Capuchino high school in San Bruno.
On this blog, I will post strategies that I am trying, and I would love to see strategies that others are using to help bring students up to grade level and teach them real math skills.
If you want more information on any of my lessons, feel free to email me at yochanan AT gmail.com. Also, if you have ideas to share, please comment.
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