Teaching Tricks I Have Learned This Year

I'm always experimenting with different ways to explain things, and each year, I am happy if I can just find one or two "biggies" that really make a big difference for my students. Here are the gems I gathered this year. There are a lot of them this year, since I taught all the same classes as I did the year before. It has been a good year to refine my teaching techniques a bit...


1. Visualizing solving equations as "unwrapping the onion" to get back to x. The kids who practiced this were awesome at re-arranging equations in terms of variables, even if the equation looks like this and they're asked to solve for Q:

R = S + 3Q - 5W
           T


2. Drilling equations of the form ax + b = c on mini whiteboards until every Grade 7 kid can do it in their sleep (including equations with fractional solutions). This really helped in the long run, because afterwards, when we moved on to much more complex equations, once they got down to an equation of this form, they could essentially do the rest while sleeping, and they wouldn't make any careless mistakes as their attention started to drift from the problem. It also helped them drill integer skills in algebra context, in the early stages of equation-solving.

Incidentally, I always read an equation like 5x = 3 as "5 copies of x is worth 3 in total. How much is each x worth?" I think this has helped my kids stay away from being confused about what to divide by what. If they can sound out the equation in their heads, the same way that I do, then it's not hard...


3. Always pronouncing fraction multiplication as "of" instead of "multiplied by", when reading an equation or expression out loud. For example, if the kids are simplifying (1/3)(x + 12), I say, "what is one-third of the quantity 'x plus 12'? Let's do it in steps. What is one-third of x? What is one-third of 12?" This helped to reinforce both the meaning of fractions and its connection to the multiplication operation.


4. Similarly, never let the kids get away with telling you that they don't know how to find a non-unit fraction, such as four-fifths of something. Always rephrase the question as, "What is one-fifth of ________? Then, how would you find four-fifths?" Push them, push them to do more in their heads. Don't let them think that fractions are harder than they are, or let them think that not knowing the meaning of fractions is OK.


5. Highlighting matching descriptions or units inside a proportion. Worked superbly to help kids set up proportions correct, consistently!


6. Teaching log by slowing down and focusing on its definition. This is a tried-and-true method for me this year; my students never had any weird issues at any point with logs this year, and I was able to repeat the same success with different students (some new transfers, some from other grades) at a later point. They were calm, independent, and their work all made sense from the start to finish during the entire unit. This had never happened to me before while teaching logs in any other way!!


7. Teaching sequences by making kids make a table of values (index vs. actual value) every single time, prior to setting up any equations. For some reason, this really slowed them down to thinking about what the word problem is giving them to work with, and they were consistently successful at tackling a variety of problems without getting confused.


8. Teaching Calculus by making kids sketch f'(x), f''(x), or f(x) graphs, given related graphs. They must do this consistently at the start of each class before moving on to work on anything else. The graphical understanding will underpin their entire algebraic understanding of Calculus, and help to bring everything together.


9. As soon as the kids differentiate a function via algebra, they must write down next to f(x) and f'(x) some word descriptions, such as "Height" for f(x) and "Gradient" for f'(x). This will build their independence in choosing the correct function to plug x-values into, and free them from having to ask you what to do at the next step of their analysis. Nag them while supervising/going over every problem, to write down these descriptions. Eventually, they won't need this anymore and they can visualize the descriptions in their heads. But, this builds their independence -- fast.


10. Repetitive quiz practice, on a complex topic, until you feel that it is quiz-worthy. This builds their confidence, while focusing their attention on a key skill, integrated with other skills they've seen before or that are nice to have.


I'm not done with the school year yet (still doing things that I'm pretty excited about, for the last few weeks of school), but I think that these are the little things that have made the most impact on my students' achievements/understanding this year. I hope that they will help you as well as they have helped me!! 

Mythical Form

My 7th-graders have been doing some lovely exploration and estimation activities on circles. It took a few days, but I think it was well worth our while, as it helped the abstract formulas make sense to them.

My students today were boggled by the fact that if pi has different digits that go on forever, that means that either the diameter or the circumference is a quantity with also digits that go on forever. That means that we have a "measurable" (ie. finite) quantity that is, in fact, not truly measurable. Trippy, eh? For a moment there, I felt the beauty of abstract math peek its head into our Grade 7 class. The kids now think the circle is a mythical, awe-inspiring form.

Last Week of IB Test Prep 2013

It's full-on test-prep season, and this year I feel very satisfied with how the test prep went for my Grade 12s (who are off on their study leave this week and have requested for just one last voluntary class session with me on Friday), as well as for my Grade 11s (who are starting their mock IB exams tomorrow).

Some things that I've done throughout the year that I found helpful:

• Sequence of repetitive quiz prep/practice, building up to a fairly complex quiz. I did this with my Grade 11s throughout the year, and I found it immensely helpful in repetitively drilling into them ways to think about incorporating graphical analysis into algebraic processes flexibly. I also did this with my Grade 12s regularly throughout this year, in order to go back and fill in some of their procedural gaps from last year. The Grade 12s have said to me that these quizzes have been very helpful, and more importantly, as they began to do mixed review this spring, I didn't feel like they had really any major gaps from last year yet to be filled or reviewed.
• Review packets organized by topic for Grade 12s, spiralling back through topics from last year. This year, instead of waiting until the spring to do review for old topics, I started handing out monthly review packets in August and giving detailed written feedback as the packets were handed in to me. I felt that these packets were very useful for me to have a written dialogue with each kid to get them thinking just a little bit further on each studied topic, and the threat of contacting their parents when they laxed on the completion meant that the kids were responding and at least doing some amount of review during the year instead of waiting until April to think about those old concepts.
• Weekly lunch time review sessions for Grade 12s starting in January, where they just did full-length old exams. Each week, I would pass out either a new calculator exam or a new non-calculator exam paper (I alternated which type to give them), and we would go over the previous week's exam paper problem-by-problem. The effect of this was that the motivated kids had a chance to try mixed problems on a regular basis, well before we finished learning all the topics in the IB syllabus. So, they got used to looking at full-length papers and feeling that sense of anxiety/uncertainty in their stomach during February, instead of during April. This was immensely helpful in building the confidence of those motivated kids over time.
• During the final weeks of concentrated old-exam practice during class, I asked the Grade 12s to identify orally at the start of each class the most common mistakes they tend to make within each topic. (ie. in circle sector problems, not using the correct radian mode; or in solving equations, forgetting that you can solve a complicated equation by simply graphing for intersection) This list helped to provide them with some mental focus even as they sat down for a mixed-problem practice session. 
• Skimming over/discussing the last semester's mock exam problem-by-problem with my Grade 11s, right before the end of our last class before their new mock exams. Although we had gone over these problems immediately after January, they were more focused now that the stakes were up again. Taking a fresh look at old problems after a few months helped them to focus on thinking about access points into each old problem that they had struggled with, in order to encourage them 1. to go back and revisit the last semester's mock exam and topics during their review 2. to think strategically and flexibly about how to approach each problem type during the test 3. to see how far along they have come in building confidence within those old topics.

I am EXCITED!!!!! I won't find out until July how my Grade 12s will do, but I feel very encouraged by their positive efforts, their confidence, and their general outlook. I also am excited to find out how my Grade 11s will improve from last time. Even a new girl in our class (who has been with us only for about a month, after transferring over from another teacher's class, and who has been seeing me a few times at lunch for help to fill in basic gaps) is shooting for a significant growth from her last test in January. Keeping fingers crossed all around!

How I Do IB Test Prep

I don't really like teaching the IB, because I feel as though there are a lot of topics to get through and it's hard to build all the concepts from the ground up (given the time constraints), so you really have to rely on the kids having a pretty strong foundation in the fundamentals prior to arriving in your class.

HOWEVER, I am definitely getting better at it. Kids are pretty happy in my class because they understand the topics and even the convoluted IB problems are becoming more accessible to them over time.

I was thinking about what made the big difference between last year and this year, and one thing is that I am giving consistent practice quizzes at the start of class on a non-trivial skill, through the course of 4 or 5 consecutive class meetings. Following that as a sort of long Do Now, we pick up with "regular" instruction for the rest of the period, on a usually unrelated topic that is concurrent. The quizzes and the instruction run parallel, but they're not usually on the same topic. After 4 or so such practice quizzes, I give a graded quiz on those skills. (Which may or may not be around the time when I give a coherent unit test on the concurrent topic. The kids don't seem to mind that there are always two topics going on at once, because they understand that the quizzes are only to build up specific test-prep skills, and the unit-based learning we do is more coherent and also much more time-consuming.) The key here is that the practice quizzes have to be sufficiently complex, yet similar each time in format and content. The first time they see a practice quiz, they usually cannot do it and they require me to explain the problem to them thoroughly after they try it. But if they're attentive and they keep good notes and they actually try all the practice quizzes earnestly, they can get close to 100% by the real quiz even as I add slightly more complicated parts on the final quiz. This is very important, because 1. it builds their confidence with complex problems, 2. it gives me an opportunity to reinforce the integration of multiple concepts.

For example, recently we did a set of practice quizzes in Grade 11 that looked like this (while we are learning logarithms and log / exponential functions in our "regular" instruction):

"For the function f(x) = 3x^2 - 2kx + k + 1, find all value(s) of k that would...
a.) Cause f to have two x-intercepts.
b.) Cause f to have one x-intercept.
c.) Cause f to have no x-intercept."

As part of this problem, the kids would define the discriminant function in terms of k:
D=(-2k)^2 - 4(3)(k+1), graph it in the calculator, and then use the graph to analyze when the discriminant value is zero (k = -0.791, k = 3.79 when rounded to 3 sig figs). They will then conclude that when k is those two approximated values, f has one x-intercept. They would then test it, by substituting k back into the equation for f... f(x) = 3x^2 - 2(-0.791)x - 0.791 + 1 and f(x) = 3x^2 - 2(3.79)x + 3.79 + 1 both look like they have only 1 x-intercept.

They will then look back at the same discriminant function they had set up, D, and then visually conclude that the discriminant function has a positive value when k < -0.791 or k > 3.79 (the discriminant graph is above the horizontal axis, which means that it has a positive value). And in order to test their hypothesis that k < -0.791 or k > 3.79 would cause f to have two x-intercepts, they will then replace nice integer values k = -1 and k = 4 into the equation for f, to verify that f(x) = 3x^2 - 2(-1)x - 1 + 1 and f(x) = 3x^2 - 2(4)x + 4 + 1 both have two x-intercepts. (And they do.)

Eventually, they repeat this process for the hypothesis that -0.791 < k < 3.79 will cause the discriminant function to be negative (below the horizontal axis), which will cause f to have no x-intercept. They can test it with a nice k value such as 0, 1, 2, or 3.

Well, what is the point of all of this?

1. Notice that I base the problem-solving around graphical analysis, even though we know that we can solve quadratic equations and inequalities without a graph. Research shows that the more you emphasize graphical analysis on the calculator, the better the kids get at visualizing equations and inequalities as graphs. In the long run, when you take the calculator away, they will still be willing to manually simplify their discriminant function, to sketch the graph, and use it to aid their analysis of variable k and its relationship, ultimately, to function f. And that ability to visualize is a powerful tool to have.

2. Specific to this topic, the notion that k and x are related but not the same can be often confusing to students who are not used to doing such sophisticated algebra analysis. A lot of times last year, my (former) 11th-graders used to ask me, "But why can k have two values when they're asking for there to be only one x-intercept for the function f?" Breaking down the process as we did above addresses that specifically, because when the kids substitute the values of k back into the original equation, they can see that the graph they're shooting for works BECAUSE the parameter k has taken on appropriate values. It helps to separate the meaning of k from the meaning of x.

3. For your weaker students, repetition breeds familiarity, which then breeds confidence. They also get to hear me explain the same concept 4 or 5 times, over the course of 4 or 5 classes, which is very helpful for them. They also get to use their notes on all the practice quizzes following the first one, which gradually forces/allows them to be independent while building up to the real quiz. You cannot achieve this type of comfort level by practicing/drilling completely different-looking problems everyday, even if the problems are essentially on the same core topic. Less is more, I think. Once they build an in-depth understanding of a single problem type of sufficient "juice" and complexity, I believe that they can then more easily transfer that understanding to other problems.

I know this method works, because not only did my 11th-graders do quite well on the mid-year mock exam this year, but generally speaking, when I think back on the topics that we have learned this year, I feel that they don't have collective holes/gaps as a class. We've gone through and filled them all in, using this consistent quiz policy that runs alongside our regular instruction of new topics.

I also use the same practice-practice-practice-practice-then-quiz policy to help my 12th-graders do spiral review from last year. The difference is that I try to integrate even more topics for them. On one set of practice quizzes, for example, they needed to: write a sine function from a graph; describe the step-by-step transformations from y=sin(x) to that graph; graph another quadratic formula within the same grid, labeling all important info; shade the enclosed area between the two curves; write an integral that represents the area between the two curves; evaluate the integral by calculator; then show how they can get the same integrated result by hand. I gave them 5 or so practice quizzes leading up to the quiz on this one, because there were so many different skills involved that they needed to practice. This is how I pull it altogether for them, using a combination of mixed concepts and (still, much needed) repetition.

I feel that we (meaning, my Grade 12s) are in much better shape this year, going into the review period, as a result of this quiz policy that I adopted. We had started the spiraling quizzes on last year's topics, all the way back in August. By now, I have gone through and drilled most of the particular weaknesses that I felt that they needed to see again. This method is systematic, more focused, and much better than just giving them random mixed practice! I feel that, even though we're not yet done learning all the topics (we still have one more to go, which is Vectors), my kids are already pretty OK now with doing mixed IB practice on their own and needing only moderate support from me. If only they can keep up the stamina for doing extra practice on their own during the remaining weeks, then I know that they'll be in great shape by the end of April!

I thought I would share this, you know, because I really think it has made all the difference in my IB classes. In fact, I have been trying to do the same in my MS classes, by spreading out test practice over the course of a week or two, leading up to the test, instead of just giving a single practice test. It feels kind of like a huge waste of time, because the practice quizzes take about 20 minutes each class. But, in the end, you're actually saving time because you're increasing mastery by providing more regular feedback and multiple opportunities to self-assess. This year, I didn't have to spend any extra time reviewing equations in Grade 7 after I taught that unit, because the kids all had the algebra skills down pat. So, yea, do try it if you're not already doing this in your class, and I hope that it helps to address a variety of learning and mastery issues!

They Like Logs!

I don't know if it's a coincidence, or if there are other forces at play. But today, during class, I noticed that all of my 11th-graders are solving all kinds of log and exponential equations fluidly without accepting any help from me. They were even completely comfortable finding inverse equations given a function like f(x) = a*b^(cx - d) all by themselves.

Amazeballs. This is the first time that I think my students as a whole really understand logs!

I still believe that the credit goes to this. Sometimes, it just pays to slooow theeem dooowwwn.

Pencast as My Substitute

I am going away for a conference on Friday. In order to give my 12th-graders the maximum chance for success, I have tried to create pencasts for all the problems in their packet, so that they can try them (hopefully really try them, OMG senioritis!), and then the substitute will just hit Play to go over the answers.

Like this one.

I am really impressed that PDF has the built in feature to support animations and wav files. wow. Yay to a hopefully still productive day in my absence. Yay to our new Livescribe toy, from which I can see many possibilities for the kids who just need a little more TLC...

What just MIGHT be (for me) the Secret of Teaching Logarithms

I have been teaching logarithm for a few years now. Each year, no matter how I approach it and how exploratory I make the whole thing to be, I find that my students are fairly unsuccessful at putting everything together, and they always get confused at some point. Last year, I finally had the idea of going back to basic definitions. The whole problem, I think, with kids getting confused with logs all the time is because they simply cannot remember, in the end, what the hell log even means after I make them derive all those rules. So, this year, I started with the definition very firmly, and every time the kids are doing a new problem, I repeat the hell out of that definition until they want to rip me into pieces. And, guess what! I don't care if they want to rip me up. It has worked like a charm. NO ONE is getting confused yet this year by the notion of logs. (I've skipped the exploratory stuff this year, in order to really keep their focus on what's important.)

This is the definition I taught them:

Log is just a way to ask a specific question.
log_a(b) asks the question: "What exponent is required to go from a base of a in order to reach a value of b?"

That's IT! We go over that with an example.

For example,
log₂(8) means "What exponent is required to go from a base of 2 to reach a value of 8?"
So, log₂(8) = ??

The kids said, "3!" (...OK, maybe first they said 4. I cannot remember now. But anyhow, they understood why it would be 3. Either they self-corrected or I corrected them.)

Then, we did some more simple numerical examples, as you always would do before kids start to get confused with logs. In each case, instead of just letting them be robots and following the previous numerical pattern mindlessly, I slowed them down and hammered into them the meaning of log. They had to say it OUT LOUD for every example:

log₃(81) means "What exponent is required to go from a base of 3 to a value of 81?" and that's why it's 4.
log₅(5)  means "What exponent is required to go from a base of 5 to a value of 5?" and that's why it's 1.
log₄(16) means "What exponent is required to go from a base of 4 to a value of 16?" and that's why it's 2.

etc. And then we went over the change of base formula, log_a(b) = log(b)/log(a). I am sorry, but I didn't try to make them discover it this year. Derivation is nice if the kids are already getting the basic concept, but else it obfuscates what's already a fairly tricky topic for a majority of kids. We practiced finding some decimal log results using the calculator, and testing them (as exponents) to make sure that they did give approximately the correct values that we desired, starting from the base.


And then we jumped right into solving equations! And the kids did brilliantly. I didn't even make a worksheet, I just started writing things on the board, a couple of simple problems at a time. Each time they got stuck, I just said, "Go back to your definition. What question does log help us ask? How can we use that?"

Each time they worked on a new type of problem and they needed help, they had to laboriously say out loud what the question is that log is asking. "What exponent is required to go from base of ___ to reach a value of ___?" and they then had to identify, based on the equation given, whether that question being posed had already been answered or not. Once they said all of this out loud, they were able to figure out on their own what x was fairly easily, without any help from me.

3^x =10  --> "What exponent is required to go from base 3 to reach a value of 10? That hasn't been answered yet." so, log is going to help us ask that of the calculator: log₃(10) = x

log₄(x) = 3  --> "What exponent is required to go from base 4 to reach a value of x? That has been answered already, 3." So, 4³ = x.

log_x(36) = 2  --> "What exponent is required to go from base x to reach a value of 36? That has been answered already, 2." So, x² = 36. For this one, it led us into a brief discussion of why x could not be -6, and of limitations on log inputs.

I was really shocked by how well the kids received this. I even tried after a few problems to introduce to them the memory trick from Amy Gruen, and they looked at me like, "Why would we need this?" (which I can assure you, was not the response I had gotten in the previous year.) I really, truly believe that going back to the definition of logs is the way to teach this often confusing concept.

Shortly after, they were able to do problems such as:

log₅(1/5) = ??

log₇(7^k) = ??  --> "log asks the question, what exponent is required to go from a base of 7 to reach a value of 7^k? The answer is, well, k!"

log₇(7^2n-3) = ??

So, being very pleased by their ability to recite and apply log definition, I started to put up some questions of multiple-step equations on the board, again just to let the kids try them first. (They needed a bit of hints only in the beginning, but for the most part they were pretty OK doing them by themselves.)

2*5^x =80  --> here was my hint. "Well, log does NOT ask the question, what exponent is required so that when I raise the base of 5 to it AND THEN MULTIPLY BY 2, the final value is 80. So, clearly the 2 here is a bit problematic..." and therefore the kids figured out that it needs to go away first.

-4^x =-73 --> here I helped them visualize order of operations by circling the x with the 4, and then circling the negative sign on an outside layer. I use this 7th-grade trick now even with my 11th and 12th graders to help them visualize how to peel away layers of the onion when solving for something.

3*6^x - 7 = 20

10^2x-9 =1098

So, this was all things that happened during our previous class. Today, after they returned, they were still very successful at transferring the log definition onto more complicated equations such as:

 6^x = 36^x-3     (which I realize, yes, they can easily solve in the future as a "change of base" problem, but since we're on the topic of introducing logs, I just wanted them to see how to apply the log definition to this problem.)

So, this is the question they decided to ask: "What exponent is required in order to go from a base of 6 to a value of  36^x-3 ?"  And they decided that the answer to that question has already been provided, as x.

So, log₆(36^x-3 ) = x

Now they apply a simple log rule of dropping the exponents in the front, which makes:

(x - 3) log₆(36) = x

And clearly since they know what log means, they can immediately simplify it now as:
(x - 3)(2) = x

and then just solve the rest as a linear equation. Tada!

Easy breezy. I'm going to always teach logs using definitions from now on. My little logarithm ninjas can even solve exponential equations for x in terms of other variables, and they can also tell me that log₆(6^{m^3}) should equal m³. YEAH. Not bad for being only two days into logs, I'd say.

If kids understand the definition of logs as something that asks a certain question, then down the road they won't be so confused when we discuss that 2^log2(k) = k, because the log part simply asks the right question, and the rest of the expression actually CARRIES OUT the instruction implied by that question. I find that when the situation looks complicated, I always go back to thinking about the definition of log in my own head. So, I have every reason to be hopeful that my kids, with consistent reinforcement from me, will create the same frame of reference in their little heads.

Ant on a Wheel

The hook for circular functions I had envisioned turned out to be pretty great, even better than I had imagined. The problem about the ant on the wheel was a hit, and really brought out some nice misconceptions from the kids. When I passed out the worksheet, I told them that they needed to estimate intelligently (not randomly guess) for the values in the table in #1, and that it should take a bit of time to complete if they were doing it correctly. They took their guesses, and most of them wrote things like

t = 0 --> h = 60
t = 0.5 --> h = 40
t = 1 --> h = 20
t = 1.5 --> h = 0
t = 2 --> h = 20
t = 2.5 --> h = 40
t = 3 --> h = 60
t = 3.5 --> h = 40
t = 4 --> h = 20
t = 4.5 --> h = 0


Then, when we discussed as a class, I drew a big wheel on the board and a vertical scale next to it to show height up to 60. I asked the kids where the ant was at 0 seconds, and everyone pointed to the top of the wheel, so I labeled it 0 sec. And then I asked them where the ant was at 3 seconds, and at 1.5 seconds, and we labeled those quickly as well since those were "obvious" to the kids. Then, less obviously, I asked them where on the wheel the ant was at 1 second and 2 seconds. To do this, they figured out that you have to divide the wheel up into thirds. And then we can further divide this wheel up to see where the ant is at 0.5, 2.5, 3.5, ... seconds.

By the end of our discussion, we got a diagram that looks something like this (on the board and also on their papers):

Each time, we referred horizontally over to the height scale that I had sketched on the board, and we estimated how high up the ant actually is. In doing so, the students noticed that they had to change the heights in their table. See below:

t = 0 --> h = 60
t = 0.5 --> h = 45
t = 1 --> h =15
t = 1.5 --> h = 0
t = 2 --> h =15
t = 2.5 --> h = 45
t = 3 --> h = 60
t = 3.5 --> h = 45
t = 4 --> h =15
t = 4.5 --> h = 0

One kid said, "But that doesn't make sense! I had divided it into equal parts before and that made sense." So, we had to discuss as a class that the ant is moving mostly vertically between certain parts of the wheel (ie. between 0.5 and 1 second, or between 2 and 2.5 seconds), and mostly horizontally between other parts of the wheel (ie. between 1 and 1.5 second, or from 1.5 to 2 seconds). Finally this kid is convinced that the height change is not a steady rate at all times.

So, with this consensus, I asked the kids to take out their graphing calcs and to create a scatterplot with this data (continuing the pattern all the way to 9 seconds, by 0.5 second increments, in order to reinforce their understanding of this circular pattern around the wheel). We looked at the graphs, and tada! It looks like a wave. Some kids were able to see it immediately, while others needed me to draw the scatterplot on the board and to connect it for them in order for them to see it.

Then, I asked them what type of function this was, and they were able to vaguely say sine or cosine (but they weren't sure which). We didn't get too far, but we started finding the equation of this function by hand for both sine and cosine functions and discussing the meaning of each part of the wave equation, which we will verify afterwards using the calculator's sine regression functionality. (Good time for the kids to practice all kinds of tech skills on the calculator, which they may or may not need for their internal assessment this year, depending on their internal assessment topics.)

Anyway, just thought I'd throw it out there. I didn't do super exploratory/introductory stuff this year in introducing waves, since the kids already have seen these equations the year before (in Grade 10), but I thought this lesson hook was really nice for re-introducing waves to them.

Addendum: I have decided that I'll be starting the next class by asking the kids where on the wheel the ant will likely raise two of his legs and say, "whee!" and then we draw it on the board on the wheel as well as on the wave graph, in order to connect this to rollercoasters and an itty bitty physics. On the other side, when the ant is rising rapidly, we can draw the ant hanging on barely, with two of his legs swinging in the wind.

Challenging Exponents/Log Problem

Fun/challenging problem from my colleague's test for his HL students (ouch!):

2(5^(x+1)) = 1 + 3/(5^x), solve for x in exact form "a + log(b)/log(5)", where a and b are both integers.

A student asked me after school for help with this, but I couldn't quickly figure it out on the spot. Afterwards, I thought over this and found it really fun to think of a variety of ways to try approaching this problem. (Too bad they mostly didn't work.) Over dinner, I thought of one way and it worked!

I am curious how you'd solve this. Please share!

Addendum: In hindsight, it's pretty foolish of me that I never looked at the graph even though I had suggested the graphical approach to that student. It's quite a simpler problem if you consider graphical approach as aid to the algebra. I was over-complicating things by doing it completely manually. (Still possible, but takes more work, obviously.) Silly silly me. These are the kinds of automatic-habit things I'd get better at, I guess, if I started coaching ISMTF competition teams....

Intro to Waves...?

I am just going to try introducing waves like this this year in Grade 11. I hope it doesn't suck.

PS. They've "seen" sine and cosine functions before in Grade 10. Else I don't think you can introduce it this way and expect them to associate it with sine and cosine automagically. So I guess the proper title of this post would have been "Re-Intro to Waves...?"

IB Internal Assessment 2014: Choosing a Topic

My students and I have been brainstorming topics appropriate for the new 2014 IB internal assessments. Our department created a timeline that would allow kids to submit topic choices by early November, write an outline and then some drafts in the winter / spring, followed by final submission in May. One of my colleagues recently went to an IB Category 3 training, which focused on the new internal assessments. After we sat down as a grade-level team to discuss the issues that came up at this conference, I now feel fairly confident that I understand what is expected (as much as is possible at this early stage of implementation of the new format), so here is what I gather. I hope it is helpful to you and/or your students.

The new format entails two parts: research and your own application. The math topic selected should be one that goes beyond the IB curriculum (quadratics is too easy, for example). The student is supposed to research and read literature on the math content beyond the course related to their topic, and then break down the math literature in their own words. Then, they need to basically show "engagement" by applying the learned knowledge to a situation of their own creation.

For example, one student of mine wishes to study a certain card game. I said that is OK, and that the research would entail this student explaining step-by-step how to calculate all of the probabilities in the standard rules of the game. And then, as part of their project, they could create a second game similar to the one that they have studied, but with varied rules, and then they would re-do all calculations to show that they had made the understanding uniquely their own. This kid can also run actual game simulations to compare with their theoretical results, in order to further reflect upon their understanding.

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If your kids are lost (as mine were) about exploration topics to pursue, I find that the Numbers Guy from WSJ is a good place to start to think about the connection between math and real-world issues. If they are familiar with GeoGebra, there are also possibilities of studying various GeoGebra animation-based problems that are comfortably within the realm of the SL topics. Or, any extension from 2-D to 3-D is always interesting. Besides that, Vi Hart has some unusual connections between math and various art forms, that might also jog some ideas. The kids can basically study anything, or even model data, so long as the data is sufficiently complicated. For me, a good rule of thumb is that if I can already imagine without any research what the data / result will likely look like, then the topic chosen is already too simple and not sufficiently "juicy" to get all the upper marks. In that case, the kid would need to add a twist to it to make it more interesting / complex, or abandon that idea and look for a new one.

Happy hunting for good topics! What an exciting time! Let me know if you find other sources of inspiration for you and your students. I find in my own class that if a kid picks a topic as a starting point, and they just take the time to start talking to me about it, usually we can brainstorm together until we find something that is sufficiently interesting/complex. The resulting direction may have nothing to do with what we started with, but just the process of talking a kid through a topic can jog my mind through related ideas that I have come across/read before.

Making "Effort" Transparent

So, one of the challenges that we face in teaching IB Standard Level Math in Germany is that German universities do not accept any lower level math as valid high-school graduation requirement, even though Mathematical Studies exists as a lower-level option within the IB program. So, as a result, a lot of kids who have a weak math background still want to take up SL Math to keep their university options open. As a school, we sympathize with kids and parents who have such goals, but we still need to make sure that we examine realistically whether a kid is correctly placed into SL Math and to either 1. realize that we will make an exception for this kid, even if it might mean that they get a 2 or 3 on the math IB exam, or 2. we communicate clearly to the parents that the kid is in jeopardy of failing the entire IB diploma if they continue to stay in this class (since a very low score on one IB exam will affect their cumulative score, and that can be a problem if the kid is already borderline passing in their other classes). It's tricky, because for all students the entry into an IB Diploma program can be a choppy transition (especially if they are coming from a different school), and so you cannot simply take into account a kid's current abilities, but also their motivation and commitment to improvement. Even in the two years that I have been teaching IB, I have already seen kids who came in being very weak, showing tremendous progress within the program.

So, one advantage that I have discovered with my 5-minute drills this year is that they help me to objectively determine who is committed to keeping pace with the class, and who isn't. Thus far in Grade 11 SL Math, we have had two quizzes after several days of consistent 5-minute practice. The first quiz was on finding equations of parallel or perpendicular lines through a given point (definitely a prerequisite skill to the IB), and about 1/4 of my class couldn't do it even after several days of practice. So, what does that mean? I gave back the quiz to the students and said simply that I cannot accept less than a perfect score on this quiz from any of them, because it is such a basic skill compared to the fast-moving curriculum. So, if they did not get full credit (4 out of 4) on this quiz, they must come see me within the week to get help, re-quiz, and get themselves off of my worrying radar. Immediately after that day, a girl went to change the class because she felt that the material was too advanced for her. Within the next week, 2 students came in for a re-quiz and one of them had to try three times to get a perfect score. A third student has not yet come in, so I emailed his parent and informed them that this is not a good sign to me in terms of his commitment to staying and keeping pace with SL Math. Simply, there are no excuses because I am always available during lunch, no appointment needed. A kid is either determined to catch up and seeing me for help/re-quiz, or they are not putting their action where their words might be.

The second quiz was on turning a standard-form quadratic equation into both vertex and factored form, and then sketching a graph using all info from all three equation forms. Again, a fundamental skill in IB math, independent of the application to webbed IB question formats. Some kids were able to do it correctly following our week-and-a-half of practicing during class, and others will be expected to come in for more help or a re-quiz on their own time.

I find it very simple and straight-forward this way. If a kid falls behind, they need to take responsibility to create extra contact time with me, in order to catch up. If they don't, and this develops into a consistent pattern, then they simply are not showing either sufficient understanding or a sufficient motivation to be giving SL Math a proper shot. At that point (still early in Grade 11), we would initiate the conversation with the parents to move the kid out of the class, with the understanding that it will impact their university applications. In the end, the decision will be clear as a combination of both their effort and their understanding, and we will be making the best decision to save the kid's overall IB Diploma result... This is much more rational and clear than in the past, when I had just feelings about which kid was doing something extra at home and which kid was not, in order to make up their existing gaps. This way, the expectation is clear and so is the result / necessary follow-up.

IB 2014 Investigation Introduction

This presentation might be helpful if you are an IB teacher looking to introduce the new internal assessment format. I made the presentation after poring over IB docs from the OCC, looking through their samples and FAQs, and talking to my departmental colleagues. I think I have a pretty good general feel of what is expected, and so I pasted over some excerpts from the official docs into this presentation and walked the kids briefly through the motivation behind internal assessments, why it is done in this new format this year, and what they should be considering as they try to brainstorm possible topics / narrow down to a single topic. We also looked briefly through some of the annotated online samples from OCC, but did not hand out any copies of those samples (to avoid even tempting future plagiarism), and as well examined some MYP task prompts that are formulated in the structure of an investigation. We still need to do one day in class when we just sit down with laptops and brainstorm possible topics together and individually, but I think today was a very clear intro. It's making me a bit less stressed out about everything.

5-Minute Drills

I am doing daily 5-minute drills this year in grades 7, 8, 11, and 12. It started off with my frustration that kids cannot remember unit circles, even though we had worked on them, explained them, practiced applying them. I was just fed up with them not memorizing the circles. So, instead of feeling frustrated, I decided that during Grade 12 we'd do daily drills of the unit circle at the start of the year. On Day 1, I asked them to take out a piece of scrap paper and to fill out the coordinates of Quadrant I of the unit circle. They failed miserably, so we went over again the hand trick for remembering the coordinates quickly, and I said that at the end of class I'd ask them to do it again. By the end of class, there was much more success (maybe half of the kids were able to get the coordinates correct). I think the immediate feedback helped to motivate them. And then I told them that we'd do the exercise again the next class. And we did, at the beginning and at the end of the next lesson, tagging on to the unit circle basic equations to solve within the range of 0 to 360 degrees. I said that the next time I see them, this'd be a quiz collected for a grade.

In Grade 11 we're doing something similar, but primarily to review older prerequisite skills (such as writing line equations) that I think the kids should already know, and that I only wish to brush up on. We would do the same skill at the start of class, end of class, and next start of class. And then soon we'd have a mini quiz on it also at the start of a class, collected for a grade.

So, a long time ago when I taught middle school for the first time, our school implemented daily quizzes. I kind of hated them, because it was so much grading, even though it was a good practice for the kids. I really like my new 5-minute drills, because I think they are the best of both worlds. The kids still feel the time pressure and the need to be correct, but they're not graded that often and it's less work for me. For my Grade 7 and Grade 8 students, I let them do two problems a day on mini whiteboards. (I got lucky and was gifted a class set, along with markers and erasers, when I sent out a request asking to borrow them.) This is important because in Grades 7 and 8 we are working on basic skills like fractions, percents, equations, etc. The boards are a nice way to quickly check in with all kids on a daily basis, and I can see who is sure of themselves and who is not. Now that I have taught with mini whiteboards, I really don't think I can go back! I love that  kids also write their normally snarky comments on the whiteboards instead of calling them out, so that only I can "hear" those comments. Cuteness. I saw one kid write "DUH!" on his board when another kid made an obvious observation. One day I kept them over the class accidentally (since our school does not have bells), and a kid raised his board that said, "Class ended!" So, they're great for classroom management as well as daily assessment.

I'm still trying all kinds of things this year, but working close to 60 hours this week is taking its toll. Grade 9 is my baby, because this again is a very low class, and this year a bigger group. I will be doing all kinds of experimental things with them, and if it works, I'll share the strategies with y'all. So far, so good. The kids are able to graph linear functions by making tables, and they're able to write linear equations from a table. Not a bad first week for kids who couldn't graph points on Day 1!!! They also go around and check off each other's answers, which I think is so awesome because they need to be building confidence alongside content knowledge.

So, 60 hours-ish this week. A bit rough. But I am loving it!!! I also really love sharing classrooms this year. I teach in about 5 different rooms, and I just love it. Even though I have to carry my supplies everywhere and it's a pain, I am all in other people's spaces and talking to them regularly as a result. It's really nice, because it forces me out of being in the workaholic zone. Anyway, I am hoping that things will calm down soon on the department chair side, so that I can get back to a normal work schedule and re-gain work-life balance.

I hope your years are off to a wonderful start! :) Hi web, goodbye web. See you soon, hopefully.

Introducing Calculus in the IB

In light of my recent reflections on my students' weakness in connecting application problems to algebra concepts (see this post), I started my new Calculus unit a different way than I did the last time. This is only the second time I have taught Calculus (in the IB, it's one of the topics and not an entire course), but the first time I did it I had started with the video that introduced the idea of instantaneous rates and worked my way through the idea of limits, then worked through the various traditional differentiation techniques, before we arrived finally at the applications. That was pretty ineffective, I found. The kids, by the time they got to the applications, struggled with putting them together with the algebra skills, and they found the entire topic to be very challenging and were daunted by the complexity and variety of problems/situations presented on different IB exams.

This year, I did something entirely different.

Day 1, I gave the kids a polynomial function and asked them to use the graphing calculator to find the dy/dx derivative values at specified places on the graph. They sketched the graph, wrote down the derivative values next to the appropriate parts of the graph, and then as a class we hypothesized what the derivative values meant. This was good for two reasons: 1. They're expected on the IB to know how to find derivative values on a graph using a GCD, 2. they got to go from concrete numerical examples to a more abstract definition / generalization of the meaning of a derivative, which helped me reach those concrete thinkers. Then, I gave them another sine equation, and they had to find all the places on the graph (within the standard viewing window) where the derivative was zero. Again, we discussed how they did this, and why that made sense based on their previously generated definition of the derivative.

Day 1 was very successful because by the end, the kids had conceptualized the meaning of a derivative value and were looking at me with these "duh! this is so easy!" looks. They learned the notations dy/dx and f'(x), and we went over how to find the derivative function of a polynomial function (justifying it by showing how, graphically, a cubic function "flattens" to a quadratic if you roughly sketch out its derivative values on the same graph).

Day 2 was also very successful. Instead of introducing more derivative rules, I introduced the idea of f"(x), and we linked f(x), f'(x), and f"(x) to physics. I told briefly the legend of Newton inventing Calculus to support physics, and together they figured out that if f(x) tells you the position at time x, then f'(x), its rate function, must be called the speed function, and f"(x) must be called the acceleration function. Great! With this itty-bitty bit of Calculus that they now "know", I threw them headlong into two 15-point application questions from old IB exams on differential Calculus. They struggled, of course. Two word problems took them about 60 minutes, including discussions as a class. But, it was very productive struggle, and in the end it had fully reinforced their understanding of the meaning of f(x), f'(x), and f"(x). Many of them figured out by themselves that if you need to find out the time that an object comes to a stop, you would set f'(x) = 0 and then get x, and then with that x value and the original f(x) formula, you can find the stopping position of the object.

It was lovely, because instead of the traditional approach of stuffing all the differential Calculus algebra skills down their throats at once, we slowed down enough for them to first digest the meaning, and zoomed out of the algebra skills just enough for them to see the bigger point of it before continuing with more detailed derivative rules.

On Day 3, we learned about the derivatives of sine, cosine, and also about the Chain Rule. It was pretty smooth. I had introduced the derivative of sine by asking them to sketch a sine wave, and from that, asked them to determine/sketch the shape of the derivative function graphically by first sketching the places where f'(x) = 0 and then thinking about what happens between those points. This was challenging, but some of them were able to figure it out. We referred back to the IB formula sheet afterwards to confirm our graphical intuition that f(x) = sin(x) --> f'(x) = cos(x), and we did some examples of the Chain Rule together before I threw them into another 3 old IB problems that required some resourcefulness and that involved the new rules learned on this day.

I am going to keep trying this approach in IB, breaking a big algebra concept into smaller and smaller chunks and integrating the end-to-end process very early, to see where the kids are getting stuck and to adjust instruction accordingly. I'll keep you posted on how effective this is, but my gut feeling is that it will increase their overall confidence with trying new problems on their own, because essentially they will be already doing this all the time with me as part of regular instruction.

Pancakes! (and Function Compositions)

I was reviewing composition of functions today with some of my students as part of a bigger review on function basics. The example question I used, a simple one, specified that
f(x) = 2x - 1 and g(x) = sqrt(x) and asked them to write the equations for (f ○ g)(x), (f ○ f)(x), and (g ○ f)(x).

I showed them how to re-write the composition notation like this:

(f ○ g)(x) = f(g(x))

and then I asked, "Is this composition primarily an f function, or primarily a g function? If you could figure that out, then writing the equation will be really easy..."

The kids stared at me in silence.

I tried hinting at it, "Let's think about which function occurs first and which occurs last."

"G occurs first, followed by f taking in the output of g as its input," so said the kids. But they still weren't able to state what the primary function is. The hint was too esoteric.

I tried again with a silly analogy: "OK, let's say you first make someone really fat, and then you smoosh them down and flatten them out like a pancake, then are they primarily fat or primarily a pancake??"

Kids giggled and mumbled more or less in unison, "Primarily a pancake!"

"OK then. If g occurs first and then followed by f, what's your primary function here? Which one is the pancake?"

Kids mumbled, "F is the pancake."

"Are you sure? Which is the primary function here?"

"F," they said. They're still not sure how this helps them write the formula.

So, together we copied down the equation for f, since we know that this is going to be the basic form/framework for our composed function:

f(x) = 2x - 1

But since we know that f is no longer just taking in a simple x value but taking in the entire function output from g, we can replace the bold parts above with g(x) and its associated formula as shown below:

f(g(x)) = 2*sqrt(x) - 1

Easy breezy. (On the board, by the way, I made sure to circle the bold parts and to draw arrows between the equations to indicate which x's got replaced with what.) The whole thing took less than 5 minutes to teach and all the kids said they now understand both how to do it and the concept behind it. (I gave them two more questions to try, just to be sure. They had to do
(f ○ f)(x) and (g ○ f)(x) on their own, and they did them both quickly, correctly, and without any doubts.)

I think review sessions for small groups of struggling students are helping me to become a more efficient explainer altogether. I find myself thinking of different ways of explaining things than how I normally structure it in class.

So yeah, pancakes! They are the magical ingredient to teaching function compositions, apparently.

MYP Grades

I know this seems obvious, but I am a big proponent of giving very realistic grades and very specific feedback. In the MYP, final class grades are given on a scale of 7. To me, this is how I view different levels:

MYP levels	How I see the student
level 7	A student has demonstrated both conceptual and technical fluency in all topic areas. They can approach new situations with independence and confidence. For a student in level 7, I pinpoint areas where they can still create further goals for themselves, so that they understand that a grade of 7 is not an end-all.
level 6	A student has demonstrated technical fluency in all skills learned during the semester, but sometimes cannot see the bigger picture and therefore misapplies skills. The difference between a level-6 and a level-7 student can also be their level of independence in approaching new situations.
level 5	A student demonstrates a "normal" achievement in their grade, with an ability to independently perform maybe 75% to 80% of all of the skills and a reasonable ability to articulate the big ideas.
level 4	A student has working knowledge of a vast majority of the topics but needs frequent help to get through the nitty gritty algebra of those problems. A level 4 student demonstrates procedural issues mostly, coupled with small conceptual issues.
level 3	A student has serious gaps/misconceptions in one third or one half of the key areas. Typically, a level 3 student has not taken the responsibility to seek help outside of class to address those major areas of weakness.
level 2	A student only demonstrate some small pockets of effort, very inconsistently, throughout the semester, and has also poor mastery across the board with the content topics.
level 1	A student basically sat around and did practically nothing (even after various conversations with students and parents), did not turn in most assignments, and has a poor understanding across the board with all topics.

For me, thinking about grades like this has been very liberating. I don't feel the need to nickel and dime kids on particular assignments, although I still record and look at all of their grades when making this determination.

In this sense, the grading encompasses both their learning habits and their academic performance, which I believe go hand-in-hand to indicate a child's success. In the MYP, by the way, this breaks down further into 4 different grading criteria: Knowledge and Understanding (for example, quizzes and test grades); Communication in Mathematics (for example, clarity of explanations in writing, and appropriate use of symbols and sufficiency of work shown); Modeling in Mathematics (for example, labs and patterns investigations that take a student from data to equation to predictions/generalization to explanations); and Reflections in Mathematics (for example, does the student complete test corrections regularly to reflect on and learn from their mistakes? After a lab, do they provide a complete error analysis to reflect on sources of inaccuracies in their data?). Those are criteria we look at when we determine whether a student is a 1 or a 7, but in general, their final grades still need to reflect where they stand, both academically and as a student who is still learning to learn.

It is important that you do not inflate grades in the middle school, because if you do (which unfortunately happens sometimes, because some teachers want to keep middle school chummy-feely like elementary school), you end up sending home the wrong signal that the student is doing "OK", and they end up in high school with both sub-par skills and study habits. Yikes. I find that when I have a clear framework of what different MYP grades mean to me (as I outlined above), I can more confidently assign grades and not have to feel bad one way or another for possibly inflating/deflating.

I also think that it is very important in an MYP program to have the same teacher for two-year rotations, because the goal of giving holistic grades is to provide opportunities for improvement. If you look at my impressions of the grade boundaries above, it is quite "easy" for the kids to move from one boundary to the next with some motivation and work. Transitioning students from teacher to teacher every year can be very disruptive in that process, and confusing in their attempt to grow as learners. Following a two-year rotation, it is my personal belief that going to a different teacher becomes beneficial, as the student learns to not rely too heavily on a single teacher's teaching style, and instead becomes more self-reliant.

Three-Variable Charts

In the context of preparing my students for their IB portfolios, I realized that it is absurd/funny that in secondary math we don't often deal with three-variable relationships, for example when both volume and temperature will affect pressure. In science, I am pretty sure that at the secondary schooling level you'd discuss things like when you change only one variable to see its effects, and afterwards you hold that one as constant and change another variable, and in the end you try to aggregate their overall impact on the outcome. In secondary math, we (ironically?) shield the kids from this. So, for much of their schooling, kids only know how to make and analyze two-column t-charts involving two variables.

In IB, the curriculum certainly expects a bit more, at least in its portfolio investigations. The IB portfolio tasks go straight into the investigation of the individual and aggregate effects of two variables on a third variable. This notion is unfamiliar to the students, so I was thinking about how to introduce them to this idea of observing and representing three-variable data.

In thinking about how to broach this, I was reminded of the sizing charts I often see on the back of the commercial packaging of tights/stockings. If you have never bought a pair of tights before from a certain brand, you often need to read the sizing chart to check the correct size to purchase. And what are the TWO factors that determine your size of tights? --Height and weight, you say. (Or, at least I hope you do. One of my girls immediately replied today, "Height and color!" sigh. Is it too much to hope that they like math over fashion?)



The boys in my class giggled when I brought this up as an "everyday" example of visualizing the relationship between three variables. (One of the boys said, "These IB tasks are sexist.") Two variables go on the outside of the table (ie. weight across and height vertically down), and the contents of the cells -- ie. the size in this case -- would represent the third variable. In mathematical terms, when you organize something in a table of this form, you can more easily find the formula f(a, b), where the "key" or input to function f is a tuple of two variables, rather than just a single variable.

Again, it's funny to think that this type of relationship must be very common in "the real world", and yet we discuss so little of it in school. Can you think of other everyday examples (ones that won't make my high-school boys giggle)?

My First Screen-Capture Vodcast!

Since I was not able to find a video introducing the kids to Type I tasks for the IB, I thought that I should try to make my own.

Here it is. I had to record it a few times because:

1. The screen capture program kept canceling out because Escape is one of the hot keys for canceling and I hadn't realized. So every time I hit Escape from the Prezi full screen, my screen capture video would disappear without an error message. I had thought for a while that I was maxing out the recording time limit, but eventually I figured out what was happening.

2. When I finally successfully taped one, I found myself "huhing" and "umming" too much. It was annoying, so I had to re-do it.

In the end, it's not perfect, but it's alright. I cannot tell if it's just me, but the audio and the video got a bit out of sync towards the end, but I think it's still something I'll save for the future, because it's always better to give the kids something they can watch at home in case they have questions while completing the project. And I am pretty amazed by how easy it was to do! I used "Free Screen Recorder" and the built-in microphone to my laptop, if you're curious. The program is free and the AVI pops up automatically after you save the file.

Addendum February 8, 2012: My students thought it was very helpful to look at the video side-by-side with looking at an actual project sample from the past. I walked them through the infinite surds task in order to illustrate the different stages of the mathematical investigation.

Intro to IB Type 1 Task

I pulled together a Prezi for introducing to my students the necessary elements of the IB internal assessment Type 1 task. In the IB program, the kids have to do two portfolio assignments, each of a very different nature. I personally enjoy looking over the tasks, because they involve doing very rich mathematics. What I don't enjoy is that the kids have to do these difficult tasks alone at home and that similar tasks are not built into the rest of the IB Math curriculum.

Anyway, different from Type 2 tasks (which involve modeling a real-world set of data with different function types, choosing the most appropriate function based on end behavior, and justifying the choice based on the context of the data), Type 1 tasks involve looking for and generalizing patterns. Last time I was able to find some great resources (vodcasts) for introducing the kids to Type 2 tasks, but this time I did not find parallel resources for Type 1 tasks. So, I pulled together a Prezi and when I have time over the summer, I'll flesh out another presentation in similar format, going through one specific example task.

Here is the Prezi, if you are an IB teacher and are interested in possibly using it in your class. To walk myself through the steps of the IB Task 1, I have looked at 4 different recent tasks: the stellar numbers, the infinite surds, logarithm bases (if log(X)/log(a) = c, log(X)/log(b) = d, what is log(X)/log(ab) in terms of c and d?), and a problem involving matrices. All four are drastically different in terms of the mathematical content they use, but I think they are all excellent applications of IB topics, just at a fairly sophisticated level. I have to really think about how to incorporate pieces of them better into my classroom in order to enrich student learning consistently and to reinforce their algebra skills!