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!

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