Algebra II

Absolute Value

So I taught absolute value equations in Algebra II. And so far I think things have gone fairly well. I read Kate Nowak’s post on how she did absolute values, and I thought I would change my more traditional introduction to them… but I didn’t. I realized that the way Kate was motivating it (with the distance on the numberline model) was great, but I felt I could still get deep conceptual understanding with the traditional way she eschewed in her post.

So I stuck with that.

I used exit cards to see how they could do… and they were okay.

But after learning how to solve $|2x-3|=5$ or $|2x-3|=-10$ , I asked kids to solve things like $2-5|5x+6|=5$ or something similar. Many students said on their home enjoyment:

$2-5(5x+6)=5$ or $2-5(5x+6)=-5$ .

It is unsurprising to me, and yet, it makes me want to throw up. Because what’s coming more and more into focus, and I’m sure you’re going to hear me complain about this more and more in the coming months, is how reliant students are on “coming up with rules” and “applying rules” — without thinking. They desperately want unthinking rules. And this year, because I can’t handle throwing up all the time, I’m vowing to really not give rules to them.

I really got to the heart of this “I LIKE PROCEDURES” thing with them with a true-false activity that I did, using my poor man clickers. I think this exercise highlighted how dependent my kids are on procedures and coming up with simple rules that help them in the short term… but that can hurt them in the long term… It’s a bunch of True-False questions. And when we talked about each one of them, my class saw concretely how reliant they were on misconceptions and false rules. EVERY SINGLE QUESTION led to a great short discussion.

So here they are, for you to use. Sadly, I don’t have the blank slides to share with you, because my school laptop is not with me at home now.

These were great for asking “so who wants to justify their answer?”

School Store & Matrices

I spent a day on matrices and then we had winter vacation. Two weeks off. We came back and it took us two days to polish them off. In Algebra II, all we do is teach students some basics. I go over how to add, subtract, and multiply matrices. I remind students about multiplicative inverses. Then I introduce the identity matrix — so that we can talk about how $[A][A]^{-1}=I$ . And finally we write systems of equations in matrix form, and use our calculators to solve the systems.

Early on when introducing matrices, I threw the following two slides on the board:

And then I asked, without students doing calculations, which grade took in the most money? We took a poll. Then I asked how we might figure it out. A student answered “well we take the number of sweatshirts and multiply it by the cost of each sweatshirt and add it to the…” and I said “hmmm, this sound like you’re doing a lot of multiplying and adding… we just did a lot of multiplying and adding in this funny way.” MATRICES!

So we were able to figure this out using matrices (and I showed them how to use their calculators to do this). Turns out that no student guessed the 10th grade (which was the right answer). They were so enamored by the sweatshirts that they ignored the socks! (Next year I might have them do a ranking — who made the most to who made the least.)

The next day, before we embarked on using matrices to solve systems of equations, I threw the following on the board as a do now:

FIND THE PRICES OF THE ITEMS! They just sort of sat there blankly. Well, a few said “I remember how much things cost from yesterday” but I said the school store was under a new regime of leadership and the prices have changed. I told my kids to guess and check or try anything they wanted. Most just sat there dumbfounded. We left it.

We went through class as normal, going over home enjoyment and solving systems (which is not easy to teach, btw, because you have to talk about how matrix multiplication is not commutative, how there isn’t matrix multiplication, how you need to have an inverse matrix, and how there is something called the identity matrix and how it acts like the number “1”). At the end of class I threw up the same slide.

Most kids knew what to do. They saw the system of equations, and how matrices could help them solve it.

I don’t know if I’ll keep the ordering of these problems the same — in terms of when I introduce them in class. I don’t think I gave them due deference. But for some reason, I really enjoyed them. Although it doesn’t really answer why we do matrix multiplication the way we do it, the first day slides really show them that there is some logic to wanting to multiply and add, multiply and add…. The second day’s slide really highlights how intractable some problems might be at first glance, and how powerful matrices are to get us out of a seemingly impossible quandary.

Square Roots and Cube Roots

I’ve posted a lot about Calculus this year, and a bit about Multivariable Calculus too. But I’m not saying too much about Algebra II. Sorry. This year something is off, and the students aren’t as successful as in years past. I’m not exactly sure what to do. I’ve asked the student-led tutoring program to lead an Algebra II study group (we’ll see if anyone signs up). I also might change my teaching practice to allow more time for students to work problems in class — because I need to see more of them working and catch their errors in thinking earlier — before they go and practice the material wrongly at home. We’ll get through less curriculum though if I do that, and that itself is a problem, since we’ve pared down the curriculum so much.

Anyway, that’s generally where I am with Algebra II.

Specifically, I just wanted advice on how you guys teach cube roots (and fourth roots and fifth roots, etc.).

My ordering usually goes:

1. Turn to your partner, and explain to them what $\sqrt{5}$ means to someone who doesn’t know anything about square roots.

Students generally say that it’s the number that when multiplied by itself will give you 5. I then say “if the person doesn’t know anything about square roots, you might want to give them an easier example, like $\sqrt{25}$ … and explain how that is 5. But that $\sqrt{5}$ isn’t a perfect square so you’d get some number between 2 and 3. Yadda Yadda. I also then talk about the geometric interpretation (the side length of a square with area 5). Then I go back to the “it’s the number that when multiplied by itself will give you 5.”

I do not talk about there being two answers to “the number when multiplied by itself gives you 5” and the principal square root business. Because I want to use this to capitalize on their understanding of cube roots.

2. Then I put up $\sqrt[3]{8}$ and say this is 2. And to think about what this funny $\sqrt[3]{}$ symbol means. They get it. I put up a bunch more, and they usually can solve them. I put some negatives under the cube root symbol too.

3. I then ask them what $\sqrt[3]{}$ means, and they say “the number that when multiplied by itself three times gives you the value under the cubed root sign.”

4. I then throw up a bunch of problems, and three of these include $\sqrt{49}$ and $\sqrt{-49}$ and $-\sqrt{49}$ .

This is where the trouble comes in.

Some students now say $\sqrt{49}$ is $\pm 7$ . Because 7 and -7 are numbers when multiplied by itself which equals 49.

Here’s where I use the whole: “Don’t lose what you already know! Would you say $\sqrt{49}$ is -7 ten minutes ago? No. You’re right, that there are two numbers which, when multiplied by themselves, give you 49. So we can tell them apart, we say $\sqrt{49}$ is the positive one and $-\sqrt{49}$ is the negative one. So don’t lose what you know. When you see a radical sign, it just represents a single number. If there’s a negative in front of it, it represents a negative number. If not, it’s a positive number. Just like what you’ve always known.”

Okay, now I know the idea of “principal square roots” and all that. And I honestly don’t want to have this whole discussion about principal square roots with them, because every time I do, they come out more confused.

So here’s my question.

How do you introduce cube (and higher) roots? How do you engage with this idea of principle square roots so that students don’t leave confused? I just can’t get it totally right.

And just so I am being clear, I know the properties of square roots and cube roots and all that. I’m not looking for someone to explain that to me. I want a way to teach my KIDS these without confusing them all up. And I bet crowdsourcing is a good way to get ideas for next year.

Two Worksheets

ONE

On Thursday, I’m going to be introducing absolute value inequalities. Last year I used the picture below as motivation.

I then tried to work backwards to show kids absolute value inequalities. It wasn’t too hot a success. Certainly the “application” wasn’t a motivator, and working backwards just confused things.

This year, I’ve decided to start with a warm up. Without them knowing anything, I’m going to ask them to do this for the first 7 minutes of class with their partners.

View this document on Scribd

I already can see the great questioning and discussion that this simple worksheet will generate between partners. And then, when we come together: WHAM! powerful! It’s a simple thing, but Oh! So! Delicious!

After that, after we see some patterns and make some conclusions… then, then I can throw up the picture of the bag, and talk about it meaningfully. And have kids work backwards from their own conclusions to finding a way to express that region mathematically, using absolute value inequalities.

TWO

I’m introducing limits tomorrow. I pretty much have carte blanche in what I do. Last year what I did was sad. Like SAD. Like: “Here’s what a limit is. Get it?” This year, I’m stealing pretty much from CalcDave wholesale. Here’s his calculus questionairre. And here’s what I made.

View this document on Scribd

Pretty much the same thing. Then I’d like to somehow have them start thinking about how to get velocity from a position versus time graph. Haven’t quite figured that out yet. Either that, or Zeno.

Histograms, Standard Deviations, and Digital Cameras

Our last unit in Algebra II was statistics — and it was a hurried unit. (As last units always are.)

One of the topics I was covering was histogram basics. And I wanted to make it somewhat interesting. So I went online, and came across a page which explained how to understand histograms that your digital camera produces. You know what I’m talking about, right?

That’s the one. How do you get it on your camera? Heck if I know. I just pushed a lot of buttons and eventually the histogram appeared.

Because I had about 20 minutes, I just lectured my kids on how this histogram worked.The histogram has 256 columns (numbers 0 to 255). Each pixel on your camera is assigned a number from 0 (representing pure darkness) to 255 (representing pure lightness). Then the height of each bar represents the number of pixels with that particular level of darkness/lightness.

By that one little piece of information, you can start telling a lot about a photo. Such as when it is over-exposed and under-exposed, and when there is too much or too little contrast. You might wonder how photo editing software can increase the contrast or correct for a photo being over/under-exposed. One you learn about this, the answer is pretty simple. The program reassigns each pixel with a different brightness.

See examples that I cribbed from the website on my smartboard. Pay special attention to how the over-under exposed histograms differ from the “ideal” histogram (and similarly for the too high/too little contrast):

View this document on Scribd

I really enjoyed learning about this, and sharing what I learned with my students. But next year, I want to do something more. I want students to take photos and play with them in some image editing software — and see what happens to the data as they modify the image in certain ways. What does brightness mean? Will things change if the image goes from color to black and white? What does sharpening the image do to the histogram? I want them to talk about mean, median, and mode — and how they change. I want them to talk about standard deviation — and how it changes. I want them to talk about range and shape — and how they change. I want them to make a short writeup explaining their findings.

Look at what Picasa (free) offers:

You get the histogram (bottom left)! You also get all these ways to modify the picture!

And the histogram changes as you modify it! In REAL TIME as you slide sliders!

I don’t know quite yet how to make this rigorous or ways to ensure they’re learning. It’s kinda bad, because I just want to play around with this and discover what all these things do myself, not knowing what I want them to get out of it. I just want to explore. I’m not thinking backwards. But I suspect a good short bit on the shape of data can be made from this. (Alternative reading: I wouldn’t begrudge any of you if you, say, went out and made a short unit based on this and sent it to me.)

Some of my Algebra II class on Friday

I enter my Algebra II classroom two minutes before class, open my computer and plug it into the SmartBoard. By the time it powers up, most of my students have entered the room and are sitting down and chatting. I pull up the day’s SmartBoard and I get started. The day before was exhausting, and I was in a cranky mood then. (My Algebra II kids didn’t see this, because I gave them a test that day.) I tell my kids we all have bad days, but that when I was thinking “argh, bad day!” I started thinking of all the good that I have, and I thought of my wonderful Algebra II class. (Which they are.) So I wanted to let them know that. They liked hearing that. I liked saying that. It was a nice 30 seconds.

I then pointed to the SmartBoard

And we got started. I talked about how we’ve done so much algebraic manipulation and solving so far. Absolute value equations, exponent rules, radical equation, inequalities. And we’ve done some baby graphing (lines, crazy functions which we used our calculator to graph). But today, I said, was going to be a turning point in our course — and graphing would be the emphasis.

I introduce the discriminant, $b^2-4ac$ and we talk about where we’ve seen that (answer: in the quadratic formula). I tell them they will soon see the use of it. But first we should get familiar with it. [1] We calculate it for a few quadratics. And then I asked them “so what? what does this thing tell you?”

(Silence.)

I move on and say, “Okay, we just calculated a discriminant of -11 for a quadratic equation. Tell me something.”

I didn’t have them talk in partners, and when I got more silence, I highlighted the discriminant in the quadratic formula:

And then I asked “what is something mathematical you can tell me about the quadratic if the discriminant is -11?”

A few hands went up, and then I should have had them talk in partners. But I didn’t. I called on one, who said “there will be $i$ .” “What do you mean?” “The solutions will have imaginary numbers.” “Right!”

I then go on to explain it in more detail to those who still don’t see it. And then I explain how the two zeros are going to be complex (because they have a real part and an imaginary part). I see nods. I feel comfortable moving on.

I then ask “what happens if the discriminant is 10?”

I call on a random student whose hand is not raised, who answers “they will be real.” I ask for clarification, and they said “the solutions will be real.”

So I go to the next SmartBoard page and I start codifying our conclusions:

I’m hankering for someone to ask the obvious next question, and indeed, a student does. “What happens if the discriminant is 0?”

And we discuss, and realize there will only be one real solution. This gets added to the chart.

I then ask them to spend 15 seconds thinking about this — what they just learned. To see if it makes sense, or if they have any questions. Just some time.

I’m not surprised (in fact, I’m delighted) when a student asks: “Can you ever have a discriminant equal zero?”

I suddenly realized that for some of my kids, we’re now in the land of abstraction. There is this new thingamabob with a weird name, the discriminant, and the students don’t know what it’s for or why we use it. We’ve been talking about $a$ s, $b$ s, and $c$ s and even though we’ve done a few examples, it isn’t “there” for the kids yet.

I throw $x^2+2x+1$ on the board. He nods approvingly. Then I ask what the solution or solutions are for that equation, and they find the one real solution. Which gets repeated twice when we factor.

I then give them 5 minutes to check themselves by asking them to do the following 3 problems:

I walk around. Two students are actually doing the quadratic formula. So I go up to the board and underline the things in blue — and ask “do you need the full force of the quadratic formula to answer THIS question?” (Secretly I grimace, because who the heck cares if they use the QF or use the discriminant to answer the question? But if I’m teaching something, I want my kids to practice it.)

When we all come back together, I project the answers

And I get called out (rightfully so!) on improper mathematical language (imaginary vs. complex). So I fix that. I’m feeling slightly guilty about asking the two students to use the discriminant instead of the QF to answer the question, because who cares!, and so I tell the class that the discriminant is just a short way to tell the number and nature of the solutions, but don’t worry if you forget it, because you can always pull out the big guns: the quadratic formula. Which will not only tell you the number and nature of the solutions, but also what the solutions are!

I have them make a new heading in their notes

And I have them work with a desk partner to solve three quadratic equations using any method they like (they only know factoring, the quadratic formula, and completing the square).

They get the right answers, for the most part. The ones who aren’t getting it right are having trouble using their calculator to enter in their quadratic formula result. I want to move on, because of time, so I tell them that we can go over calculator questions in the next class but I want them to put those aside so we can see the bigger picture now.

They then are asked to graph the following three equations on a standard window:

We also talk about the difference between the two things they are working with:

We then look at the graph:

At this point, I haven’t pointed out the x-intercepts, but I asked students to see if they can relate the two questions. I grow a bit impatient, and I point to one x-intercept, and then the other, for the $x^2-5x-3=y$ equation. Hm. They start to see it. We then look at the $x^2+2x+1=y$ equation. Finally, the $x^2+x+2=y$ equation. Which doesn’t have any x-intercepts. Which confused some.

Someone said “that’s because the solutions are complex.” I pointed and said something like “yeah!” and then tried to explain. Some students got confused, because we did plot complex numbers on a complex plane, so they were like “you can plot complex solutions too!” I tried to address their concerns, by saying that the x-intercepts show us the solution when $y=0$ . But the x-axis is a REAL number line, not a complex number line. I don’t think they all got what I was saying.

We codify what we know:

This all took about 30 or 35 minutes.

I somehow totally forgot to do something key: bring the discussion back to discriminants. I didn’t ask them “so what can a discriminant tell you about a graph of a quadratic?” It might be obvious to us, but I guarantee you that only a few kids would actually be able to answer that after our lesson.

We spend the remaining 15 or 20 minutes on graphing quadratics of the form $y=x^2+bx+c$ and $y=-x^2+bx+c$ by hand. The students were working in pairs. Then at the end we make some observations as a class.

Class ended, and then I had more work to do.

The point of this post is two fold:

1) I’m in a teacher funk. You can see it in this class. I didn’t work backwards. I gave them what they needed to know (the discriminant), and then motivated it second. I did lots of teacher centered things. I rarely let them discuss things with each other. Blah. Especially for something conceptual, not good. Not terrible, not good.

2) Teaching is exhausting. Anyone who teaches knows that even in a non-interesting lesson like this, a teacher has to constantly be thinking “what do my kids get?” and “do I need to say that again and reword it?” and “do I address the calculator issue of 2 kids when 15 seem to be okay?” Basically, every 10 seconds is a choice that needs to be made, a thought about how to adapt, where to go, what to do.

[1] Honestly, personally, I think the whole idea of the discriminant is stupid and I would have no problem doing away with it. It’s a term with very little meaning and almost no use. But I am asked to teach it, so I do.

Aiming for Understanding…

On Thursday, in Algebra 2, I aimed for understanding… and came up short. It was one of those lessons I went in excited to teach, because I knew I could get my kids to “get” it. And then, I didn’t. It wasn’t a bad lesson, and I don’t think my kids are the worse for it. But it just wasn’t that killer lesson I had hoped for.

The constraints:

I had 50 minutes to teach absolute value inequalities. I definitely had to get through “less than” absolute value inequalities (like $|2x-1| \leq 15$ , but I wanted to at least introduce “greater than” absolute value inequalities (like $|-x+15| \geq 2$ ).

The plan:

1. Start off lesson with a fundamental question raised by an image (found at 360).

2. Generate a basic understanding of what a “less than” absolute value inequality is, using a simple question to illustrate why there are many solutions. We would then fill in a number line from our solutions and then talk specifically about why our answer is not [-12,2], but actually (-13,3) (for the question below).

3. Point out the basic geometric interpretation, and tie that back to the initial question we wanted to answer.

4. Show how we can use this geometric interpretation, and a compound inequality, to help us come up with a method to solve these sorts of inequalities. (Work backwards.)

5. Formalize our method of solution.

6. Practice, practice, practice.

The outcome:

I only was able to cover “less than” absolute value inequalities, and I doubt that my students have a good understanding of why our solution method works. I do think my students will be able to follow the procedure though.

Where exactly did I fail? I failed in part 3 and part 4 of my plan. For part 3, I should have found a better way to explain the geometric meaning. My students didn’t “get” it totally. You can see part 4 of my plan executed here on my SMARTBoard slide:

As you can see, I tried to start from the compound inequalities and work our way to the absolute value inequality. At the end, there is this “ta da” moment which was actually more like “ta WHA?” They didn’t get what I was trying to show them.

And I don’t blame them.

A huge part of me doesn’t want to teach something without proving — or at least deriving by example — why something works. I feel like a fraud, like I’m teaching ’em magic instead of math, when I teach a method of solution first and then show where it comes from. But in this case, it would have gone over so much better if I had shown the method of solution, and then after practicing it a few times, took a moment to look at our work backwards to see why it worked. Talking about what each step means — algebraically and geometrically — backwards might have clarified things a bit.

I could have also designed the lesson in a totally different way. I could have worked off of our understanding of absolute value equations (e.g. the equation $|2x-1|=5$ ). Then we could have had a great discussion on how to find solutions to $|2x-1|<5$ , focusing on why we use open dots at the solutions to the equality, and why we shade inbetween those dots. Now that I’m thinking about it, maybe I should remember to try this out next year.