The students are still wobbly in their performance of completing the square, but it’s obvious that much of this can be chalked up to testing anxiety (diagnostic tip: when the students gasp, hold their breath until their faces turn blue, and pass out in horror when you give them a quiz you declare will be ungraded, then you can probably attribute their performance problems to testing anxiety).

Fortunately, they are all experienced performers (as in the performing arts), so we will be discussing and journaling on test anxiety starting tomorrow as well.

In addition, I’ve made up a bunch of new guided practice worksheets — basically, they are “completing the square with training wheels first” practice sheets. They need what one piano professor of mine used to call “opportunities to kill off those nerve endings.” I’m confident that within a week or so, they will all be turning in 4 out of 4 performances.

If the beta test of these goes well, I’ll post links for anyone who wishes to take them out for a spin.

In the meantime, on to the next batch of skills and concepts! And thank you again for sharing your materials and experience.

Elizabeth (aka @cheesemonkeysf on Twitter)

http://

to the

.doc

portion of the file name. There are 4 files in total.

- Elizabeth

This was a gap I was glad I caught early. Their previous coverage of this material was too quick and unstructured to consolidate their understanding. From watching them flail around at the start, it seemed that you can’t just tell a group of 9th-graders something once and expect them to remember it. Moreover, if you teach them something half-assedly, any understanding they do achieve will quickly degrade into a distorted version of what you taught. So the fact that they had learned only that the square root of negative one is simply “i” was already festering away and mutating in their understanding.

I learned that I had to get them to know in their bones that the square root of anything is always going to be positive-or-negative something. This fact about the dual nature of square roots is what grounded the class for Day 2 of my lesson — as well as for future algebraic understanding. It’s what is keeping them from coming up with partially understood — or misunderstood — solutions.

Day 1 of my three-day-lesson accomplished this grounding, and I felt glad about that. My students are starting to really know this fact cold. If one of them forgets, another one will remind them of it. Throw a square root problem at them now, and they’ll give you both the positive and negative solutions. They do not seem to wobble off their foundation. Even if you throw an “i” at them, they know it could have either a positive or a negative coefficient in front of it.

They can no longer be fooled. I like that in a student.

This was another aspect of the beauty and simplicity of Sam’s lesson materials. It integrated great patience, building both concepts and skills up one thin layer at a time. It did not confuse or skip steps.

For this reason — and because the ones I made up worked so beautifully as reinforcement, next time I will add in a sequence of warm-up problems for each day that spirals back on these same concepts they’ve just seen and worked with— one easy problem involving the square root of a negative number, one problem that leads to a messy-looking fraction, and one problem that leads to a genuinely complicated fractional expression — probably something with a binomial in the numerator and an integer in the denominator. Then I would have them do a problem with a squared binomial that equals a perfect square. And finally, a problem that requires them to complete the square.

In fact, I decided to write up those problems and worksheets while they were still fresh in my mind. And so, in a spirit of gratitude for great generosity of spirit, here are the downloadable versions of the materials I adapted and used. These include a Read Me about how I used the lesson, the Student Packet of materials, the Pop Quiz, and a set of equation cards that can be used with Kate Nowak’s “Speed Dating” structure (see http://function-of-time.blogspot.com for details on how to set this up).

Hope these are helpful.

- Elizabeth (aka @cheesemonkeysf on Twitter)

http://www.elizabethstatmore.com/Z-LESSON-PLANNING/COMPLETE THE SQUARE/1-READ ME-COMPLETING THE SQUARE- LESSON PLANNING GUIDANCE.doc

http://www.elizabethstatmore.com/Z-LESSON-PLANNING/COMPLETE THE SQUARE/2-ACTUAL-Student Packet 20-Nov-10.doc

http://www.elizabethstatmore.com/Z-LESSON-PLANNING/COMPLETE THE SQUARE/3-completing-the-square-pop-quiz.doc

http://www.elizabethstatmore.com/Z-LESSON-PLANNING/COMPLETE THE SQUARE/4-Equation Cards-Complete-the-Square.doc

But of course, as we all know, the simplest and clearest lessons are the hardest to come up with. Lucky me to be able to draft off of Sam Shah! :-)

It’s always fascinating to discover which technique different people will glom onto. One of our students has been clinging to the quadratic formula, even when it means going through all kinds of contortions to solve the problem. Today I could see the exact moment when the light bulb about completing the square went on over this student’s head. There were sparks! :-)

Other students really liked setting up a little “data box” for themselves off to the side, finding the value of b, and then calculating the value of b/2 and (b/2)^2. With those values in hand, it was only a hop, skip, and a jump to completing the square and solving for x.

They still need a bit more practice before I give them the “Pop Quiz,” so I made up a batch of equation cards for Day 3 so we can use Kate Nowak’s “Speed Dating” practice structure. I’ve got hard, medium, and easy problems set up.

If there is interest, I’ll put the equation cards online on Box.net with the many other fine samples for this practice structure.

- Elizabeth (aka @cheesemonkeysf on Twitter)

Problem #9 —
I changed this problem to x^2 -4x=21 because I was getting a brain aneurysm trying to figure out how to explain your original problem ( x^2-8x=-12 ) to the aforementioned Algebra 2s;

Problem #10 —
For the same reason, I changed #10 to x^2-8x=-15 (original was x^2-8x=12 ) and also to mix things up a little bit (three problems involving x^2 and 8x and 12 were starting to look suspicious to me).

We’re on the block schedule system, so I only meet with these kids three times a week. That leaves plenty of room for misunderstandings and distorted memories of procedures to blossom.

Here are several reasons why I think it was a particularly successful lesson:

1) Right from the outset, it was obvious that at *LEAST* half of the class had only half-understood previous material on square roots and imaginary numbers. In fact, their square root-finding was a little wobbly too, but not as fatally so as their confusion about imaginary numbers.

I assigned problems #1-5 as a warm-up. So #1-3 gave allowed me to diagnose and remediate these misunderstandings right off the bat.

Students who felt ashamed of not understanding quickly got their sea legs back and were soon jumping in and volunteering to work out the next problem on the board. My skillful remediation was entirely due to your skillfully constructed series of problems to work through. Kudos and thanks for sharing this insight.

2) The next set of problems (#4 and 5) took them to the next level, shifting to a squared variable with an integer coefficient. They could have dislocated an elbow, what with all the back-patting for knowing what to do with the constant term and how to deal with the integer coefficient. Very wily, Mr. Shah. You helped me to lure them right into your trap.

3) As soon as they started working on these problems, they caught a glimpse of problem #6, and there was very nearly a whole group freak-out. But I assured them, Fear not, we will work through the dreaded problem #6 together in a minute, so just concentrate your pretty little heads on #4 and 5.

And the miraculous thing was that they did exactly that.

4) Problem #6 could have been a dastardly affair, had the previous problems not been sequenced so cleverly. So after we’d gone through #4 and 5, we did a guided run-through on #6 (you tell me what to do, I write it on the board and ask questions). When they divided through by 5, there was an audible gasp in the room. X-squared equals seven-fifths? WTF? Are you kidding me?

Once again, I just calmly questioned them through the process and the problem. When we reached the final answer, I asked them if I was done. A tidal wave of cognitive dissonance hit the room. Positive or negative the square root of a fraction? They were dumbfounded. But it’s not a neat integer answer. How could this be?

This led to an insightful discussion about reasons why this might — or might not — be an acceptable answer. I realized that nobody has ever really explained to them that they have permission to find the correct answer exactly as they find it.

We made a list of pros and cons as to why this might or might not be an acceptable answer, then we took a poll by a show of hands.

By this point, most of the students were beginning to have the courage of their convictions. They honestly couldn’t think of another humanly possible way there could be a better answer than this. A few more hands went up.

And then I confirmed their hypotheses that, Yes, this seemingly messy-looking thing — a positive-or-negative-square-root-of-an-irreducible-improper-fraction — is, indeed, the answer that you seek. At least it is in this problem.

I cannot tell you how relieved their faces looked after this. The release of tension in the air was palpable. All those years, every time they’ve come up with a messy-looking answer to a problem, they have doubted themselves. They’ve gone back and erased. They’re redone all their computations. They’ve disbelieved the arithmetic. They have struggled with shame. They’ve snuck their cell phone calculators out under the desk and poked in numbers to try and find a better answer.

And here it was, staring them in the face all this time. And they figured it out themselves.

5) After that, the class started attacking the rest of the problems on the worksheet with great vigor. We didn’t get through as much as I had hoped originally, but their understanding of the skills and concepts was so much sturdier than it’s ever been that I really didn’t care.

I project that it will take us three periods to get through the whole lesson, rather than the two I had hoped for, but I can see that they are already starting to sense what it feels like to taste success in math.

Thanks again for some fantastically well-thought-out teaching tools. I’ll keep you posted on developments!

- Elizabeth ( @cheesemonkeysf on Twitter)