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!

Quadratic Function Project Brainstorm

I'm brainstorming / laying out my end-of-year plans for my 8th-graders. After their end-of-year exam in late May, we will close grades, but we will still have about 3 or so weeks of instruction, which is enough time to do something very rich and not have to coordinate with other classes (since we use the May test to do placement for Grade 9). Last year, I used this extra time to let the 8th-graders define their own math projects, which were plenty of fun, but I wasn't entirely happy with the rigor of their mathematical results. This year, I'm toying with the idea of doing an exploratory quadratic functions unit. (Technically, quadratic FUNCTIONS are a Grade 9 topic for us, but previewing it in Grade 8 is always beneficial.)

I'm thinking of making it largely exploratory, since by then pacing won't be much of an issue and I can let them really take the time to develop their conceptual understanding of quadratic functions, which is the essential access point to a lot of higher-level algebra analysis down the road.... The timing is tight (as it was last year with my other end-of-year projects), but I think it's still doable and has a lot of potential!!!

Let me know what you think. Is it an OK approach for intro to quadratic functions / basic function transformations?? This is based on my rumination about a different way to think about flexible factorization of quadratic functions.

Day 1: Developing the understanding of how to graph y = x² + bx.

Plan - In pairs, kids will be given y = x² + 2x, y = x² + 5x, y = x²- 3x, y = x² - 7x. to graph on the calculator. They will sketch results in their notes, recording all intercepts in the form (x, y), and writing a one-sentence hypothesis about what the graph of y = x² + bx will look like.

Then, they will be reminded that in a function, we can solve for the x-intercept(s) by setting the height of the point, y, equal to zero. They will algebraically show that their hypothesis works for all b values.

Day 2:  Developing the understanding of how to graph y = ax² + bx, which is a more general version of the quadratic function.

Plan - In pairs, kids will be given y = x² + 6x,  y = 2x² + 6x, y = 3x² + 6x, y = 12x² + 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the graph. They will show how to solve for the x-intercepts using algebra only.

Day 3: Developing the understanding of how the graph is affected by the sign of its leading coefficient.

Plan - In pairs, kids will be given y = -x² + 6x,  y = -2x² + 6x, y = -3x² - 6x, y = -12x² - 6x. They will again sketch graphs, noting x-intercepts, and then make a hypothesis about the effect of the leading coefficient on the shape of the graph. They will show how to solve for the x-intercepts using algebra.

As part of Day 3, they will do some matching between equations and pictures of graphs and to justify their choices orally.

By the end of Day 3, they should also be able to explain in writing how to graph y = ax² + bx.

Day 4: Developing the understanding of the effect of the constant term c.

Plan - In pairs, the kids will put in a function like y = x², and look at its table in the calculator. They will be asked to generate a second function that would increase all y-values by 1. They will prove their new equation works, by showing the values of both functions side by side in the calculator (y₁ and y₂), and copying down the table values. Then, they will write down the formula for the new function.

They will then look at the graphs of the two functions to determine "what happened" visually to the original graph when the equation got changed that way.

They will keep playing around with this idea, translating upwards and downwards and checking both the table and the graph to observe/verify the effect of c. 

By the end of Day 4, they will be given a graph of two functions. One of the functions will have an accompanying formula, and they will be able to see visually what happened to the points on the graph. They will then need to "guess" at the equation of the other, vertically shifted function, and verify it in the calculator.

Day 5: Putting the algebra pieces altogether

Plan - One partner of the pair will have a set of sequenced instructions, to be given to their partner one step at a time. The first step will sound like, "Sketch a graph of y = x² - 9x, labeling x-intercepts with values." Then, after that has been successfully completed, the next instruction will be given: "Now sketch the result of shifting that graph vertically up 4 units, labeling the resulting images of those original points you knew." After that has been done, the partner gives the third instruction: "Now, write the formula for this new graph." Once the partner is finished, they verify their results using the graphing calculator's graphing and table features and write a brief explanation of how they checked their results. Then, they switch, and the new partner has instructions that has to do with a downwards facing function like y =  -x² - 6x, and repeat a similar sequence of instructions to generate a new graph, a new / related equation, and to verify all results against the calculator.

Both partners will then work together to complete problems starting with functions of the form
y = ax² + bx and translating those graphs vertically to get new graphs.

By the end of Day 5, they should be able to explain the connection between y = ax² + bx and
y = ax² + bx + c, and explain how to use this connection to graph any standard-form quadratic function quickly in under 1 minute.

Day 6: Practicing/drilling the connection between quadratic function equation and graphs

Plan - In pairs, they will start with a function y = x² - 9x + 1, highlight the first two terms, sketch that function using dashed lines, and then sketch in the "real" final function using solid line. They will repeat this a few times with different functions, until they can fluidly graph any y = ax² + bx + c function. On this day, they'll also learn to visualize the axis of symmetry and to write its equation by inspection of graph.

Day 7: Going backwards from a graph to an equation

Plan - In pairs, they will be given one quadratic graph with two "nice", symmetric integer points being emphasized on the graph, one of the points being on the y-axis. They will be asked to sketch using dashed lines what this function would look like if you shifted those two points down to the x-axis, and be asked to write the function equation of both graphs. They will practice this a few times.

At the end of Day 7, they will be given a quadratic graph whose two "nice", symmetric integer points are both not on the y-axis. This tests them to see if they can figure out that the translated graph would have an equation that looks like y = (x - m)(x - n) + p instead of y = x(x - n) + p

Day 8: Playing around with the idea of adjusting "a".

Plan - In pairs, they will import Dan Meyer's basketball photo into GeoGebra. We will discuss as a class the need to find a modeling equation in order to fully predict whether the ball will make it into the hoop. From there, they will choose two nice integer points, write the equation, and graph. If they notice that the curve goes through those two points but doesn't have the correct steepness desired in order to fit the photo, then they will create a slider value in GeoGebra and toggle the value of "a" until they get a good "fit" around the graph, and record their results.

As a class, we will then go over the idea of solving for "a" using an unused point (x, y) and link it to solving for the y-intercept in linear functions. They will solve for "a" this way to compare analytical results against the technology results.

Day 9: Modeling Individually

Plan - Following a discussion of examples of parabolic applications, each pair will find and import their own photos of "real-life" parabolic shapes from the web. They will then model the function in Geogebra both using technology and using algebraic analysis.

Day 10: Creating posters

Plan - Each pair will create two posters, one with the modeled functions overlaying the photos, and one poster explaining the general process of graphing y = ax² + bx + c and the general process of fitting an equation to a parabolic graph.

Day 11: Practice presentations

Day 12: Math fair for other classes / parents?!

Totally Silly but Works

I made up a totally silly call-and-response thing this year for practicing exponent rules (after we did the initial exploration, obviously, so that they could understand why the rules work). It's mad cheesy, but the kids totally remember the rules now!! The hardest part is keeping the clapping going, but I'm not sure if it's because of my students being totally off-rhythm in general or what (they're super suburban kids).

So we clap, step from side to side, and I say, "8B, are you ready?" and they chant, "Yeah, oh yeah!"

And then I call on a random kid, "Nora, are you ready?" and she chants, "Yeah, oh yeah!"

and then I call out one of the following: "Power times power", "Power to a power", or "Power over power" while holding up fingers in each hand (up to 5, obviously, in each hand) to represent the original exponents we're working with.

Depending on which one I call out, the kids need to reply with, "You gotta add them up!" "You gotta mul-ti-ply!"  or "You gotta can-cel out!" in a sing-song voice, and that kid I named would then have to say the answer (resulting exponent) immediately after. (For example, if I am holding up 3 fingers and 5 fingers, and it's "power times power, you gotta add them up!" then the kid would shout out "8!")

And then we'd resume with me calling on the next kid randomly. It's mad cheesy, but it works! Afterwards, they were all loose and happy when practicing exponent rules. Every practice problem I would put on the board, I'd ask them which rule can be applied first or next, and they'd say it back in that sing-song voice, "you gotta mul-ti-ply!"

Go kids for being good sports!! It helps to make a boring topic a little less tedious! Next year, I'll necessarily add dance moves to help our kinesthetic learners. (I already have them. I came up with them after we did the exercise.)

Yup... I've got little shame left. :)

Thinking About Factorization Flexibly

I was randomly thinking about this on the way home today and truly fascinated by the teaching possibilities:

Sketching a graph of f(x) = x² + 6x + 7

is the same as sketching f(x) = x(x + 6) + 7

which is the same as sketching g(x) = x(x + 6) and then shifting g up 7 units.

Since the points (0, 0) and (-6, 0) are on the graph of g, the points (0, 7), (-6, 7) must be on the graph of f. This allows us to quickly see the symmetry line at x = -3 without memorizing x=-b/(2a).

Another way of "partially" factoring f to get the middle term 6x is f(x) = (x + 1)(x + 5) + 2

So, if h(x) = (x + 1)(x + 5), then we can imagine the points (-1, 0) and (-5, 0) from h being translated up 2 units to get the points (-1, 2), (-5, 2) on f.

Another way of "partially" factoring f to get the middle term 6x is f(x) = (x + 2)(x + 4) - 1

So, if j(x) = (x + 2)(x + 4), by thinking about the relationship between j and f, we can deduce that f must have the points (-2, -1), (-4, -1).

Similarly, we can get the partial factorization f(x) = (x + 3)(x + 3) - 2.

If we assume m(x) = (x + 3)(x + 3) and consider the relationship between m and f, we can deduce that (-3, -2) must exist on the graph of f.

So, we can pull together all those points so far to get (0, 7), (-6, 7), (-1, 2), (-5, 2), (-2, -1), (-4, -1), and (-3, -2) as points that must be on f. This way of thinking about graphing quadratics ties together strongly the ideas of factorization and transformation. They're no longer two separate concepts but integrated as one. Since I've never seen this connection in a textbook before, I decided to call it flexible factorization.

One distinct advantage of flexible factorization is that as soon as you are given y = x² + kx + m, you can quickly factor it partially into y = x(x + k) + m, which allows you to quickly determine two points on the graph, (0, m) and (-k, m) and to find the axis of symmetry at x=-k/2. You can sketch the graph roughly in about 30 seconds for any standard quadratic function (this extends to y = ax² + bx + c, as it factors into y = x(ax + b) + c, which means that (0, c) and (-b/a, c) are two points on this graph and the parabola opens in the direction as indicated by the leading coefficient "a".)

Of course, this does not mean that the kids won't have to learn the standard analysis techniques, but I think being able to connect factorization with transformation gives them another tool when modeling and thinking about graphs.

I'm going to keep playing around with this idea, possibly turning it into an end-of-year project in Grade 8. Stay tuned!

Confidence

I keep circling in my head about this. To help your weak-ability students, the singular gift you can give them is the gift of confidence in math. Yes, it's important to make the lessons relevant. Yes, it's important to have open-ended questions. But, math is inherently fun when you feel confident when approaching a new problem and can see it as a challenge rather than an obstacle, and this is a gift that they can retain even after they leave your class.

Confidence. That's what it all boils down to, for me anyway, when working with weak-ability students. I love this time of the year when I can see the transformations that have occurred from August until now!

Wedding Dress Saga - in Hindsight

Since I only get married once (I hope), I think it's OK for me to recap the crazy journey that was to find a wedding dress!!! I really dislike German wedding dresses. I think most of them are over the top, unfortunately. Most of the dresses I tried on just overwhelmed me in appearance. I felt like a girl swimming in an ocean of overly heavy fabric and overdone details.

This was one that I liked, if I would have gotten married in a colder climate. My high school best friend loved this one! I liked the sense of movement in the fabric pattern, but I didn't like the fact that I'd have to wear a frame underneath in order to hold up the poofy shape.

Once I started looking at shorter dresses, this was the favorite one that my girls had sent me. It's simple and beautiful (with the gloves) and the cut looks like one of my existing dresses. I thought quite hard about getting a dress like this made from scratch, and even investigated costs associated with this. But, I was a little concerned that since I wasn't planning to wear a veil on the (windy) beach, that the rest of the dress would be overly casual.

Then when I found this dress, I liked it right away. There's something about the simple elegance of the dress that I loved. The cocktail style made me feel like I could walk through a party and play hostess without being bogged down by layers of fabric. But, I didn't love that the left breast was not covered in the same material as the rest of the dress. (It was of a bra material. Why?! German wedding fashion is so weird.) I also didn't cry like the girls in Say Yes to the Dress when they found their favorite dresses.

Unfortunately, when I went back to the store a month later, this dress was already sold. I bought one that was the same style, but 4 or 5 sizes too large. They had to rip it apart, re-make the whole thing to fit me, and also hand-make crinkled fabric in order to cover the left breast to match the rest of the dress.

I also liked the idea of mixing and matching short and long dresses, so I sent them this picture to ask them to make this detachable train for me from scratch. I wanted a detachable train for functionality and -- so that just in case I hated it in the end, I could still take it off.

When I picked up my finished dress (which costed only 500 Euros even including all the alterations), I was in love. (A few fittings had to occur, remember? It was too small after they resized it, and so they had to resize it again a couple of times.) I was thrilled by how fabulous my seamstress was; you literally couldn't tell that part of the crinkle on the dress was made by hand. I bought a hair accessory to match the modernness of my dress, and a single-pearl necklace to match its simplicity.

Then, because of all the health- and job-search related stress, I lost weight. I don't have a scale at home, but I estimate that I must have lost between 5 and 10 pounds from pure stress. So, I had to bring the dress back, etc etc. get it altered again so that it didn't look like a borrowed dress. In the end, it was such a relief that everything still worked out!! (Reposting final picture from an earlier post... Once I get the professional photos, maybe I'll post an update.) So, whew. If you're a bride-to-be, good luck!!! I hope you can get through the whole process with fewer fittings than me. I estimate that in all, I had to have about 8 or 9 fittings, which is well above the average for a bride-to-be and is a lot for a gal who goes shopping maybe twice or three times a year and who tends to shop like a man (in and out in 45 minutes with a lot of things purchased... Geoff shops way slower than me!).

Final Months in Berlin

Amazingly, time is flying by and we find ourselves sprinting towards the last months -- ready or not -- before another big life change.

1. Geoff's going back to Seattle after our pre- and post- wedding hanging out time in Berlin. This time, he's bringing with him most of our wall decorations that we wish to keep (paintings, cuckoo clock, vintage posters, marionette, and some other knick knacks from our travels). It's helping the reality of moving sink in...

2. I am in the process of actively looking for jobs in Seattle, with some positive/encouraging progress. In May, I hope to ask for a day off to spend a long weekend interviewing with schools in Seattle. (So far, one interview is for sure. Another one during the same weekend would be fabulous to have, if I can get the extra day off and rebook my flight into Seattle.) These are two really great schools and they both seem quite interested in continuing the conversation of hiring me, so I'm feeling overall pretty hopeful with the job-search prospects.

Coupled with this, I also need to look into the logistics of filming my class just for about 10 minutes. Two different Seattle schools have offered this to me as an alternative to teaching a demo lesson on site. If I can get this working, I could just send the link to other schools that request the same...

3. In other news, since I'm the department chair, I am simultaneously interviewing potential people to hire into my current department. There's something that feels pretty funny about interviewing other people and being interviewed all at the same time, especially when some of those people I'm interviewing are supposed to fill my spot. One person I've interviewed so far is a rock star, and I hope secretly that he'll take the offer so that I can leave my students in good hands.

4. Next week will be my final full week of instruction with my Grade 12s, and also the last week of instruction before the semesterly mock exams in Grade 11. I feel quite excited to see how they will do!!

5. In Grade 9, we're doing my favorite 3-D project again. The kids are making good progress so far -- they've already gotten their designs/dimensions checked off, and half of the groups also had their calculated volumes already checked off. Next week, we'll be working on the surface area calculations, drawing 2-D nets, and starting the construction during class. EXCITED!!!

6. The sun is rolling in, slowly but surely. It's my favorite time of the year in Berlin!!! In May, there will be the annual Karnival der Kulturen, and in June, we're going to try to work out one last trip -- to Kiev (where my friend has offered for us to stay at her condo) or Vienna or Bamberg (with their unique "smoked beer"/rauchbier local breweries) or kayaking through / camping by the beautiful lakes surrounding Berlin. Lots to choose from, but so little time left!! I'm sad just thinking about it.

Survey Project - I'm Back!

I recently gave a survey project to my 7th-graders, that involved them creating/administering a survey, creating circle graphs with a protractor, drawing conclusions from graphs, and making educated predictions for a larger population.

Here are some clear photos I managed to snap of a few of their posters. I am impressed by how informative they are, considering that the kids didn't turn in any rough drafts. (Note to self: Threatening to turn it into a full-blown writing assignment really helps to bring up the quality of submitted posters.)

See full-sized yellow poster here

See the complete blue poster here.

See the full-sized cream poster here.

PS. Yes, I got married!! Here are a few pics.

Belize looks like this:

Snorkeling with sea turtles!

 

Our beach ceremony (people sat in a spiral form). It was awesome to have 50 friends and family join us from as close as El Salvador and as far as Sydney and Shanghai!


Newly weds! (My dress worked out fine in the end. It was short, with a detachable train.)

A couple of days after the wedding, we had a sunset cruise with the guests who were still around. It was lovely!!!