Complex Numbers

I wanted to do something cool with complex numbers- basic operations are rather boring, especially since it was review from last year. So, I thought about fractals and decided to have students practice magnitude, adding and multiplying complex numbers in the context of the Mandelbrot set. It's a simple iteration that makes a pretty picture, seemed perfect to me!

I had a bit of a false start because I assumed students had seen iterations before or at least used subscripts to name terms in a sequence. False assumption! After some back tracking and translating, they understood the methodology. I did a few examples with them first, asking them to predict what types of results could occur.  After exploring some key points I let them in on the magnitude greater than 2 trick.  We also listened to the Mandelbrot song (beware- make sure to use a clean version) which both provides additional information and gives the formula in the form of a catchy tune!

Between the two classes they managed to check 81 points, which seems like a lot, but doesn't actually make the most spectacular picture. If I do this again I will assign a set number of points to each student (10?) and give them time to work in class but also expect some work at home. Then we would get a more filled in picture.

Or, I could have each student choose their own iterating function, playing around on a fractal app until they got a picture they liked. They could then check 10 points by hand to show what the different colors mean. If I used a color printer these could be nice posters.

Even if it wasn't beautiful math art this year, it gave some motivation for operations with complex numbers and they got in a good amount of practice.

Following fractals we solved quadratics using Sam's awesome scale of efficiency. I moved quickly to complex roots of quadratics and then into polynomials of higher degree when students showed they had mastery of quadratics. Students enjoyed making up challenging problems for each other to solve. Asking them to make up problems was a great way to figure out/reinforce that a polynomial with real coefficients will always have imaginary roots in conjugate pairs.

The final challenge was coming up with an assessment. I decided not to have students solve a quintic with complex roots on the test but instead had several questions which focused on one step of the process at a time. I'm still undecided if that was the right choice. Students generally did well on the exam and it was easier for me to pinpoint their errors with shorter problems. It also gave students the opportunity to show me that they knew how to do a later step even if they messed up an earlier one because they were separate problems. As I'm typing this I am realizing that the best thing to do would have been a scratch off hint bank! Then students could scratch off a hint if need be and continue the problem to show mastery of later steps. (The benefit of scratch off hints is I would know who used the hints.) So many good ideas for next year, I really hope I get to teach the course again and remember to look back at this post!

Fail Friday: Proof

I knew going in this lesson would be rough. I'm not sure if that makes the end result better or worse.  It's satisfying to be at the point where I know my students and curriculum well enough to anticipate the stumbling blocks.  However, it's frustrating that I spent several days thinking, writing, asking for help, researching and discussing this lesson, yet it still failed.

The SMART board had been unresponsive all morning and my students (being helpful problem solvers!) wanted to troubleshoot with me even though I'd already tried their ideas in the previous class.  I appreciate that they wanted to help, especially because most of the tricks I've learned on the SMART board have come from students, but the class wasn't settling down and getting to work on the do now.  By the time I finished checking homework only a couple kids had gotten started (usually this isn't such an issue, but we also usually start class with a quiz, a habit thrown off due to the blizzard- I hadn't see this class in 5 days!).  Once I focused my attention on getting them to work, they were engaged in trying to come up with the shortest possible definitions.

They did a nice job with this task.  Some students wanted to include properties in the definition, for example defining a rectangle as a shape with two pairs of congruent sides and four congruent angles.  I got the chance to differentiate between definition and property, as well as preview the rest of the lesson- proving properties.  By the time we finished this and going over the homework (which sparked some good debate over whether a rectangle is always/sometimes/never a rhombus) we were about half an hour into class.  Normally this would have been great, but I neglected to consider the early release schedule due to PD when planning the rest of the lesson.  Instead of 90 minute classes we only had 60, which I know is still a lot for many of you, but I only see them once more before vacation and wanted to test on quadrilaterals before all this knowledge escapes over break!

While I started distributing the proof cards, I bribed my students- any property they proved today I would provide on the test, the rest they would need to remember or figure out again on Friday.  This received comments from both ends of the emotional spectrum: proof evoked "is that the given thing? I can't do that!" while the potential to get extra information on the test got students to perk up.

I showed them the chart below and reviewed how some quads are subsets of others, so anything we prove for a parallelogram will hold true for all the other shapes "underneath" it.  Then I asked what property they wanted to start with.  One student suggested "a diagonal divides a parallelogram into congruent triangles" which sounded good to me since I knew many of the proofs depended on use of congruent triangles.  We started with the definition of a parallelogram (which they knew, but I didn't have a card for), then they stalled.  I prompted a bit and we got consecutive angles are supplementary.  This is a key property, so I ran over to the computer (since the SMART board was broken) to add this property to the slide- one thing proven!  On the way back to the board I glanced at the clock- panic! Not only would we run out of time before they proved many properties on their own, at this rate we might run out of time before finishing this proof!  As I stood at the board I was wondering what I could possibly do to get them through the rest of the steps without giving everything away.  I suggested we draw in the diagonal, look at their cards, look at particular color cards (triangles and lines are different colors), draw the triangles separately...  Lots of wait time in between as I desperately searched the eyes of my students to find one who had the spark of an idea I could pounce on.  In the end I gave some bold hints, did a lot of pointing, and they figured it out.  But by that point there were only 5 minutes left in the period.  Definitely not enough time to send them off to do some other proofs on their own, especially after the defeated looks most of them had about a minute into this process.  In the steps of that one proof we did prove three properties, so there was that redeeming fact.

As we started cleaning up I was feeling frustrated that I’d had to drag them through the process rather than the students being able to figure it out on their own; I didn’t think they had any investment in the problem. But then the student who had figured out the last steps yelled out “wait! Don’t erase mine yet- I want to post it on Instagram!” And then I stood up a bit straighter because if even one student was proud enough of the work she had done to want to share it with the world, it wasn’t a total failure of a lesson.  (By the way, this was my One Good Thing for the day, if you haven't seen that blog yet you need to subscribe, it's good for the mind and the soul.)



I think my first mistake was letting them choose the example.  I should have chosen a short but representative example ahead of time and structured this more carefully so that the proofs built up from previous properties.  Then I could have emphasized how once they've proven something, they can use it as a reason in the next proof.  If I'd set up a flow then I wouldn't even need to do an example as a whole class; just reminding them that the proof cards on lines are still useful would have been enough to get them going.  I'm struggling with when it's better to let them come up with the questions (that worked great for discovering properties) and when it's better for me to provide some guidance.  If we had unlimited time then it would have been fine for them to start a proof, realize they needed some other information, go do another proof first and jump around until finally they'd proven all the properties.  But between the time crunch (2 classes between blizzard and vacation! Must wrap up unit now!) and the defeatist attitude they already take with proof, this class just isn't there yet.  And I need to remember that's okay.  Sure, in my ideal class we sit around and play with ideas, teasing apart details until they all have eureka moments and see math as a thing of wonder and beauty.  But, the world isn't ideal, students have past histories with math, preconceived notions, are distracted by valentines day and vacation and snow.  That's why they need a teacher- I have to find the right amount of support and independence that allows them to engage in productive struggle.

What I'm hoping to hear from you is that you've done this exact lesson and can hand me the perfected version and that you've also perfected the next lesson- proving these statements are biconditional.  Barring that, what I really want to know is how you approach proof in geometry.  I hated proof when I took high school geometry because they were fill in the blank two column format.  I could never figure out where they were going, how that reason could possibly provide a helpful statement or what theorem number went with the statement.  I've tried to make changes that I think help, but students still have strong emotional reactions to the word proof, so I'm not there yet.

Quadrilateral Proof

I've been thinking about how in my PreCalc class we prove theorems and then use them to solve problems.  However, in Geometry we generally prove things that seem self evident after a few examples or that seem completely arbitrary and we never use them again.  To try to change this I want to spend class on Wednesday proving theorems.  Whatever theorems they prove, I will allow them to use on the test Friday! (Block schedule plus snow days means I only have two 90 minute periods with Geometry classes this week.)

So far we have found (and proven) the angle sum rule for all polygons, defined concave/convex and done the quadrilateral sort.  From there I had students make quadrilaterals on their geoboards (thanks so much to Mimi for inspiring me to use them!) and record at least 3 observations.  From those observations students proposed characteristics we should use to describe the shapes and then completed the entire row of this chart:

Characteristics of Quads by

This went great to start!  I was really impressed with how many properties they came up with, and the only instructions I provided were "measure sides, angles, diagonals and record observations."  However, there were too many properties for this activity to hold their attention.  I knew the good streak was over when I got the dreaded "When are we ever going to need to know this?" (side note: Seriously child? I know you're just looking for an excuse to quit, but we're sorting things by characteristics, that's an important life skill that crops up everywhere!)

So, now I need a follow up that's less tedious and proof is always iffy.  The proof cards are helpful, and I don't think I'll even need any new ones since most of these characteristics come from parallel lines and properties of triangles.  But somehow I need to get students to want to prove characteristics, because I'm pretty sure they're all totally content with having observed them.  Ideas?

p.s. After the test students get to work on this awesome worksheet Fawn put together (originally from Don) since we're studying area after vacation.

Fail Friday

Cross Posted from Productive Struggle:

I hereby declare Friday a day of reflection.  Druinok has been running My Favorite Friday for a few months, and now we would like to offer the complement, Fail Friday.  Word around the mathtwitterblogosphere is that we need to learn from our mistakes and challenge ourselves as professionals.  The message I heard loud and clear on twitter today is that we need a place to get advice when our lessons don’t go as expected.  Having your own blog is a great way to get that feedback, but it can be scary to post a lesson that failed.  Well, it’s not so scary any more because we’re all doing it!  You have the option of just posting here or cross posting from your own blog.  Either way, all you have to do is fill out this form.

My goal is to schedule the posts throughout the week, so if you check in daily there will always be a new conundrum to ponder and offer feedback on (it’s always a great idea to subscribe in a reader, but this blog will only thrive if you click through and comment!).  This means we just chose Fridays for the alliterative aspect, you can submit any day of the week.  To get this project started though, it would be awesome if tons of people posted a Fail Friday on their own blog February 15.  I can’t wait to hear the ideas you all have, because I know you’re doing awesome things.  I hope this blogs provides the outside perspective that allows you to figure out the tweak that will turn a lesson from fail to fabulous!

Inverse Functions

After completing the study of trigonometry, I have reached the section of Pre-Calculus that is a smattering of review, extensions and seemingly random concepts.  They aren't actually random, in truth, I sat down with the AP Calculus teacher (who is also teaching one section of Honors PreCalc with me) and we went through the textbook choosing topics that students are weak in when they reach Calculus.  We are still figuring out how to connect them into a coherent curriculum, but at least I can point to my Math Practice Standards posters as one constant throughout the year.

So, when I got to inverse functions I put out a call on twitter for interesting ways to study inverses.  If students had already learned about them last year I didn't want to go over everything as if they knew nothing, but also couldn't assume they remembered all about inverse functions.  Hedge to the rescue!  She came up with an awesome introduction to inverse functions that I tweaked slightly to fit my needs.

inverse review by

It turned out that students didn't remember (or learn?) that the graphs of inverses were reflections over the line y=x, so this was the perfect combination of review and new.  There were also many choices in how students wanted to complete the assignment.  The only definition I provided was that an inverse "undoes" a function, so outputs get turned back into inputs. (Someone want to add inverse to the vocab list with a better definition? Please?)  This meant that some students inverted the tables, graphed, then figured out the function while other students inverted the functions, made new tables and graphed.  The difference in method provided an excellent opportunity to start talking about restricting domain, the biggest challenge of this unit.

The second day we practiced taking inverses of functions using inverse operations.  Most students did well with this, except for a couple students who wanted to do everything in their head and inevitably mixed something up.  There really is a reason I ask students to show their work, and it isn't because I love to torture children.  Mixed in with their practice problems was the interesting case of f(x)=-x+3.  It is its own inverse!  So of course we discussed this anomaly and yet again alternated between graphic and algebraic representations until they were convinced that they had sufficiently described all possible cases where a function is its own inverse.

After introducing the term one-to-one and practicing a few domain restrictions we played a 'game.'  Each student received a number cube (and I told them not to call them dice, because then we must be gambling! and they laughed).  I projected 6 functions called f(x) and 6 functions called g(x).  Each function had an inverse on the screen, but they weren't matched up.  Some students immediately started matching functions (next time I'll make them harder to match), but the goal was to practice function composition as well, so I told them to keep that information secret.  Each person rolled their cube; the partner on the left determined f(x) while the partner on the right determined g(x).  Then the partner on the left found f(g(x)) while the partner on the right simplified g(f(x)).  Finally, the pair compared and if the functions are inverses they decided if a domain restriction is necessary.  You may have noticed it's not really a game at all, but it involved dice so students thought it was fun.  They did tired of it before completing all 36 pairs (shocking, right?) so we finally matched the pairs and listed all of the domain restrictions.

The best part of this unit was the result: 5 perfect scores out of less than 40 tests and 17 students aced at least one standard.  One of the students who didn't do well came after school to remediate and when I talked him through one example he exclaimed "This is easy! Why didn't I get this before?" and his friend said "Maybe because you were on your phone..."  We all laughed, then he proceeded to get a perfect score on the retake.  It's amazing what they can accomplish when they focus.

Writing Trig Equations

While most of my students are fine when I give them a trig function and ask them to graph it, they struggle with the converse- writing an equation from a graph.  For most of the quizzes and tests I give I only need to create one version of the retake, some students will need to retake a standard twice.  But for the standard that required students to write a sine or cosine equation from a graph?  I made four versions and more than one student needed them.  (Kudos to those kids for caring enough to put in the time and energy to master the standard.)  When students have an equation in front of them they do just fine reading off the coefficients and constants, then deciding what each one does.  The graph has each of those numbers hidden inside it, but there are four different things to check for (amplitude, period, horizontal shift, vertical shift - plus choosing sine or cosine) and students inevitably forgot one of them.

I introduced writing equations with the standard Ferris Wheel Problem (although I really need to check my numbers for accuracy, the original was in seconds so I know it wasn't based in reality).  Students had a very hard time with this problem, which surprised me since they all had graphing calculators.  I expected students to do plenty of guess and check until they found an equation that matched.  Instead, I got equations that were completely off, heard frustration from students (and parents, poor timing assigning this the night before conferences) and had a crew of kids stay after school to figure it out.  A couple students even attempted the bonus after struggling through the first section.  That was my first real insight to the dedication of some of these kids, it was awesome to watch how excited they were that they succeeded and hear them brag about staying after for two hours.

So, when I assigned the hours of daylight problem during midterm review week I didn't expect it to go perfectly.  It turned out, I didn't correctly anticipate any of what happened.  First, it took forever to set up the graphs.  This was Ashli's project (that she got from someone else, thanks mystery original author!) and she used excel, smart lady.  Second, students had great intuition about what the data meant, they have a deep understanding of daylight, the tilt of the earth and the effect latitude has.  Impressive!  Third, some students forgot the difference between sine and cosine graphs - I expected them to need reminders about amplitude or period, but to be mad at me for asking them to figure out the difference between sine and cosine (with notebooks, graphing calculators and unit circles at their fingertips) was surprising.  But, we survived it all.  I wish that I'd given this assignment when we had more time to examine it.  Since it took so long to graph the original data I just asked students to describe the shape of the graph of the changes.  Most students picked up on the fact that it was another curve, but I didn't get the chance to expound upon how strange and exciting that was.  Next year I'll plan things better.  I'm not sure yet what better will look like, but it will be better!

New Blog / PreCalc Midterm Analysis

The new blog I mentioned last week is live!  My goal for Productive Struggle is to get people thinking deeply about lessons or assignments that go awry.  I would love to see it become a place for conversations that push each of us to examine the decisions we make and offer our expertise (we are professionals, after all).  However, none of us is perfect.  To turn our failures into successes, we need to work together, ask good questions and share resources; the same work we do with our students.  I hope you will participate in the hard practice and subscribe.

Last week I attended a meeting on data analysis.  It motivated me to do some deeper analysis of my midterms.  I'm putting the overview here and the detailed parsing of problems on the new blog. We had scantron to tally and sort and make charts for geometry. For PreCalc I had the kids do the grunt work! I asked each student to go through their exam and list every question they got full credit on. From this list, they determined what topics they were good at. Then they did the same thing for the problems they earned no points on (ignoring problems with partial credit). Here are the compiled results:

Most students listed as a strength:
Circles (arc length, area of sector)
Radians (coterminal, converting to/from degrees)
Basic Trig (right triangle, unit circle)
Laws of sines and cosines
Area of triangles

Near even split between strength and weakness columns:
Graphing (several students split drawing the graph from writing an equation from a graph. Writing an equation was harder - post on this coming soon)
Inverse functions (not trig)

Most students listed as a weakness:
Inverse trig (no calculator)
Identities (several students specified PROOF).

I also analyzed the spread of the test. Five topics (circles, radians, area, inverse functions and identities) had two questions each. There were three problems on basic trig while the remaining three topics (graphing, inverse trig and laws) had four questions. Luckily the topics with extra questions are somewhat balanced between topics students did well with and topics they struggled with. The test as a whole was too long so I think next year we will aim for two questions per topic.

I hope you head over to Productive Struggle to help me sort through the questions students really struggled with.  I always have doubts whether the problem was written poorly or if the students truly didn't understand.  Your opinion would be greatly appreciated!

PreCalc Reflections

Some thoughts from students and parents about pre-calc so far.  (I started this post at the end of first quarter, but never published it.  Now it has a few second quarter additions to make it relevant.)

Many parents came to me during first quarter conferences to report that students find this class hard.  Some parents were happier about this than others, but students have gotten the lecture a few times "Honors PreCalculus is supposed to be challenging, but not impossible.  If you're stuck Ask For Help!  That said, you need to try things on your own and study, which may be a new concept for you."  I was surprised to discover that juniors needed to read the article on how to study, but shouldn't have been since I actually gave parents this warning at Meet the Teacher night:

Your kids are good at math.  They have excelled so far; for some math comes easily and for others they have had to work to reach honors pre-calculus.  If your child hasn't had to work hard in math yet, this will probably be the course where they have to start.  I expect them to find some stumbling blocks as I ask them to prove their rule works, rather than giving them the rule to memorize and use.  But I know that all of them are capable of the work and look forward to teaching them.

Just like I had, some parents forgot this warning.  But I reminded them, and by second quarter I wasn't hearing the same concerns at conferences, because the students did rise to the challenge!

Student reflections on Quarter 1:

"I just still need to get used to not working with numbers and seeing these problems for what they really are."

"I think the hardest thing is to apply algebra concepts to trig functions"

"I love dayback!" (our term for after school help)

"My favorite subject so far is probably doing the proofs because I like the feeling of coming across something sneaky and understanding."

"I learn everything separately and it's hard to put it together."

"I like every topic after I understand"

"I like this class even though its kinda hard"
My initial response to reading that is: Why aren't we supposed to like things that are challenging?  Why not "I like this class because it's hard."?

One response the the question: Anything else to share?
"Yes."

Reflections on Quarter 2:
Sadly, we made the midterm far too long, so there wasn't time to have students answer the reflection questions I had planned.  I'm realizing now that I should have asked them last week, but I was too excited about plotting the Mandelbrot Fractal to remember.  However, a few students had missed some journal assignments during the quarter, so I gave them the prompt "What did you learn this quarter?" rather than looking up all the questions.

"This is easily the best class in the whole world." "Along with all of these fantastic concepts, I've had a great time with my classmates and my fantastic teacher and I can't wait to learn more!"  Slight chance he was aiming for extra points, but he is a very enthusiastic student who is excelling in the class.  He has stayed until 5 pm on a few occasions, once going above and beyond on a project, so he definitely doesn't hate me or the course.

"Ok, to be honest I actually kind of enjoy pre-calculus but I hate the way I always feel afterwards." "helping me with fighting against terrible and horrible identities" "Without [Ms. Cardone's] near-endless help I EASILY would've failed" This student struggles; math doesn't come easily to him and as an over-committed senior he doesn't have the time to dedicate to the class to be as successful as he could be.  He amuses all of us during class, and since he's in the near-silent course his endless imagination is a welcome interlude.  

I look forward to hearing their thoughts on third and fourth quarters!  I just have to remember to ask...

Proof: algebraic vs geometric

Proof seems to be the dividing line in our school between honors, regular and fundamentals courses.  In geometry the fundamentals course does no formal proof (though they are required to conjecture and justify solutions), regular level does some proof and the honors group does quite a bit of formal proof.  In PreCalculus I'm trying to have the honors class prove every rule before they use it, in contrast to the other classes.  In the regular level the teacher occasionally presents a proof but students never derive anything independently (which is not to say they never justify their work or understand where a rule comes from, but they're not going through the process of proving every formula they use).

To begin identities we proved sin^2+cos^2=1 geometrically (using triangle trig).  I then asked students to prove algebraically or geometrically: 1+cot^2=csc^2 and tan^2+1=sec^2.  Some students missed the part that once we had proved an identity, we knew it was true and could use it.  They were confused how we could start with sin^2+cos^2=1 when we wanted to prove 1+cot^2=csc^2 and instead started over with the triangles just like we had proven the first equation.  So much of proof in geometry class is random- we don't use what we ask them to prove.  But in reality, the great thing about proving something is that we now know it is true!

At NCTM I learned how to prove sin(a+b) and cos(a+b) geometrically using paper folding, which was a novel approach.  I used the setup outlined in these powerpoint slides.  Again, we used the new equations to prove some other ones algebraically: sin(a-b), cos(a-b), sin(2a), cos(2b)

After the second set of proofs I asked if students preferred algebraic or geometric proofs:

F Block:
4 algebraic
8 geometric
1 no preference
/22

G Block:
8 algebraic
2 geometric
3 no preference
/17

Conclusion: good thing we do some of each! (and, my response rate wasn't very good in F Block)

The next set of rules (law of sines, law of cosines, Area=1/2absin(C)) we did both geometrically and algebraically.  For these, each proof starts with a diagram which is used to set up the first equation(s), then students manipulate terms algebraically to find the rule.  I am enjoying the way algebra and geometry really complement each other in pre-calculus.  The separation of concepts into the seemingly neat categories of algebra and geometry for 3 years presents a rather false notion of mathematics as being partitioned.  In truth, the different methods support each other and when a mathematician hits a dead end using one representation, they switch to another, which often provides a hidden insight.  I've convinced students (some more than others) to be flexible in their approach- going from the unit circle, to a graph, to a triangle, to identities as they work to solve trig problems.