# Monthly Archives: December 2014

## Class Opener – Day 74 – PolyGraphs

My 9th graders have only about 2 weeks left with me before their final exam. Most of them will move on to Algebra 2 next semester, so my strategy with them has been 2-pronged: ensure we are produtive with new material and put them in a “happy place” to make a seamless tranisiton to Algebra 2. With a unit review today, and a pre-holiday-break quiz Monday, this was a perfect time to test-drive the new FREE PolyGraph activity from my friends at Desmos, along with the awesome work of Dan Meyer and Christopher Danielson. The Parabola activity sounded perfect for my class, though there are also activities featuring linear functions, rational functions and hexagons.

My freshmen have limited understanding of quadratic functions. While we have encountered some useful vocabulary regarding parabolas in my class (intercepts, vertex, domain and range), these students have not had a formal unit in graphing them yet. I was curious if students could transfer what they already knew to a new scenario. I was tempted to do a quick review of vocabulary before sending kids to the lab, but thought better of it. I want gut reactions.

In the activity, one student acts as the “picker” and chooses 1 parabola from a set of 16. The “guesser” then asks yes or no questions to help narrow down which parabola was chosen. “I don’t know” is also available as an option, if the question is not clear.

Between games, students are given challenges to help guide their understanding of vocabualry and “good” leading questions.  I found these “intermission” questions to be extremely helpful, and noticed that the quality of the questions students asked improved after participating in them.

Some obeservations about my students in this activity, which we did for about 30 minutes.

• Students didn’t have vocabulary to describe parabolas which “open up” (a>0) versus those which “open down”. The question “is it a smiley face or a frowny face?” was used by more than one student and led to some side discussions of what this meant.
• Students also recognized that parabolas could have different widths, and describing the differences between these was more challenging. Questions like “Is it wide?” or “Is it narrow?” are helpful for identifying some extreme curves, but without a baseline for what a “regular” curve looks like, this leads to some confusion over which parabolas should be eliminated.
• In the first round, few student mentioned the x or y-axis in their questions. Later, I noticed these became valuable tools for elimination.
• Questions which attempted to use the vertex showed mixed success. “Is the vertex positive?” is unclear, but these attempts improved with more game plays. Similarly, attempts to describe domain or range often needed more work.
• Students can be sneaky, and mine are no exception. Some students attempted to bypass mathematical conversation by asking “Is it in the top row?”.  Nice try – until they realize the parabolas are mixed up. Also, sometimes students were assigned to play against students sitting right next to them. Not ideal, but workable.

Here’s where I would go with this, if my next unit was on a formal discussion of quadratics: copy the student-developed phrases like “smiley face” nto a document. As we encounter those ideas formally in graphs, develop more math-specific language, match them up with the student descriptors, and improve the document. I want students to take ownership of their descriptions, and allow for their self-generated language. Hopefully, this builds richer connections to the vocabulary.

At the end of each class, I had students complete a Google Form evaluation. I appreciate the feedback from students who took this task seriously!

• This activity was really really FUN!! I liked it because I was able to interact with my classmates. I had fun as well as learn.
• It was interesting to see the language people used to describe the graphs.
• i really liked that it was an interactive activity that we could do with our class mates. It really allowed us to think about math in a fun way!

How did this activity increase your understanding of parabolas?

• I learned that a “smiley face” is positive and a “frowny face” is negative. these math terms will probably be useful.
• It forced me to think mathematically and use many math terms to figure out the answer.

Down the road, I think it could be fun to have one class code and invite students from a number of schools to join in. Knowing that their partner was somewhere in the room caused some goofy behavior, and I wonder how much more focus they would have if their partner was from a different school. In the end, I appreciate this activity because it is fun, forces students to think mathematically, and has clear entry points for class discussion after leaving the computers. Finally, can we have students use their already-existing Desmos accounts for logging in? I like that the data from all students is shared with me, and would be even better if their activity data is all in one place.  Awesome job team!

## Class Opener – Day 73 – You Can’t…Because You Can’t

In the past few days, my 9th graders have worked through a chapter on polynomials: multiplying, factoring, solving, simplifying. There’s a lot of process here, and often my fear is that students attempt to memorize short-cuts (such as the old stand-by…FOIL) without fully understanding the reasons WHY procedures are valid. It’s an easy “out” to tell students they will need procedures for their next class – I drink from this well sometimes – but I need something more my students. I want them to be able to clearly articulate and verify, using precise vocabulary, the rationale for all steps they take in math class.

In today’s opener students were presented with two problems on the board I had “solved”, and were asked to comment on my procedures:

The problem on the left is one we had completed yesterday in class, and a number of students noticed that one of the solutions, zero, was missing. I asked students to identify reasons why my solution gave a different solution set:

Because you don’t get the right answer.

Because zero is supposed to be an answer.

We’re not quite getting to the heart of the matter….I asked students to look over my solution to the problem on the right and comment.

You have to subtract the 5 over first.

You need to set it equal to zero.

In each case, students were fixated more on what I should have done, rather than what was presented in front of them as a solution.  Time to re-direct the conversation – I asked students to think about each step I had done in the problems, and tell me specifically which step was in error. This is a much more uncomfortable experience.  For each problem, the steps “feel” right. In both of my classes, the breakthrough eventually came, with some coaxing:

Students: You set both factors equal to 5. You need to set them equal to zero.

Me: Why can’t I set them equal to 5? The equation equals five.

Students (eventually): Because if two things multiply to make zero, one of them must equal zero.

Now we are getting someplace. The zero-product property is often taken for granted in this unit, but it is a powerful little engine. Name two numbers which multiply to make a product of 5….is it guaranteed that one of the two numbers MUST be 5? Nope. Zero is the hero. Hoepfully, some new conenctions were made regarding the nature of zeroes here.

The problem on the left was a much tougher nut to crack. The conversation eventually focused on the “other” solution – zero – and the perils of dividing by zero. Definitely look for more “devil’s advocate” moments as we explore rational expressions further.

## Class Opener – Day 72 – Fermi Questions

Today’s class opener caused a student to string together a sequence of words I don’t think I’ve ever heard in this history of sentences:

Can we please just factor now?

Today I exposed my freshmen classes to Fermi Questions, a series of unusual estimation experiences, like the one we started with below:

The Fermi site here provides a slider where you can change the power of 10 to integer values. Some students had trouble wrapping their heads around the expectation, until some students summarized the ideas quite nicely:

• It has to be in the thousdands.
• I don’t think it could be in the millions.
• Think about how many it would take to go along the side and multiply.

Students really got into the estimates, and I enjoyed listening to them argue their position with neighbors while attempting to estimate unknown quantities. I facilitated the group-think by moving the slider based on loud “higher” or “lower” from the group, until it seemed we were satisfied.  The site then gives you a result and a score based on how close you were.  There are a few thousand questions on the site, and we got through about a dozen today before settling into class.

Some of my favorite questions are those which demand a negative exponent, such as this one:

Determine the diameter of a 22 caliber bullet divided by the length of the Nile river.

Do we think it is one-tenth the length? One hundredth? One millionth? This was a fun way to re-visit laws of exponents, especially negative exponents.

While most of the class was engaged in the discussion, a few shyed away, which led to the quote at the start of this post. Are this questions really so threatening to students that they would RATHER factor? It also plants the seeds for some potential stats data collection, down the road.

## Class Opener – Day 71 – Factoring Drills

What’s the first rule of factoring?

It’s a shame that few students get my Fight Club references anymore, because they’re GOLD!

I’m not sure there’s much in math class I look forward to less than factoring. There are many cool applications of functions and quadratics when we get to max/min problems or start to connect factors to quadratic graphs, but there’s also a lot of necessary drill time which really taxes my creative juices. Fortunately, the Nrich Maths site provided a fun opener which allowed my students to work collaboratively and strategically.

In the Finding Factors task, students are given a square grid of expressions to factor. Students then must determine which factor belows at the head of each row or column by strategically choosing expressions to factor.

Each group in my classes today was given one netbook to use, and worked together to factor expressions and discuss possible factor placements.

On the front board, a more challenging 6×6 problem board was projected, and interested students helped crowd-source a solution. This interactive applet allowed us to move from endless drill to a collaborative experience.

Thanks as always Nrich!

## Class Opener – Day 70 – A “Homer”ic Effort

I confess – I was a bad person today. And here’s why….

One of my recent reads is Simon’s Singh‘s fun book The Simpsons and Their Mathematical Secrets, where Simon shares math gems from Simpson’s episodes hidden craftily by the math-centric series writers.

I confess I haven’t been a regular watcher of the Simpson’s for many years. This book took me back to the fun of many of the early episodes – like the corny “R D R R” gag from season 1.

One of my favorite math memories from the Simpsons is the early “Treehouse of Horror” episodes, where Homer is sucked into a strange “3D” world. It’s from this episode which I shared a screen-grab from the strange “Tron-esque” world as today’s class opener:

My math radar always goes off some when I see math included in movies or TV. Who exactly is providing the math – do they know anything about what they are sharing?  Here, we are presented with an interesting equation:



Where the heck did they get that from? Is it just gibberish? Is it even true? (note – this is exhibit A as to why I was a bad person today – of course I know where it came from, but it’s time to dangle the bait in front of the kiddies)

Many students dove into their calculators to verify the equation, and there were quite different results:

In my morning class, a few students quickly “verified” that the equation was true, and the rest trusted them. Bless them…..I can now dangle more bait….

In my afternoon class, students were a bit more careful. You’ll find that the two sides of the equation share many leading digits, but the equal-ness falls apart in the later digits. One cunning student dicovered the Nspire will give a conclusive “false” when presented with the equation. This is shown below, along with the full calculations:

So now, exhibit B of “I am a bad person”. I then gave both classes the following challenge: I will give you a candy prize if you can name any positive integers which satsify the following:

, where n is a positive integer greater than 2.

I really admire the students who tried here, even those who pretty much ignored anything else we were trying to get done. The agony when they came oh so close to a solution, only to see it break apart. I really can’t let this go on, can I?

ME: Yes…give up.

STUDENT: Come on…give me a hint here.

ME: I am…give up, it’s the best advice I can give you.

Letting students off the hook, we shared a brief discussion of Fermat’s Last Theorem, and why the Simpsons’ folks were so clever with this “near miss” in their Halloween episode.  Many stayed behind after class to hear more about some of math’s long-standing mysteries, and how exactly Fermat’s Theorem was eventually proven. After my bout of evilness, it was a pretty cool day.

## Class Opener – Day 69 – There’s No Opener :-(

All 3 of my classes have a quiz today. And with a half-day of school today due to parent conferences at the elementary level, there’s just no time for a class opener in any of my classes today.

Sorry….

But wait…there must be something I can share with you all! This week’s freshman class offered a number of great activities which provided much-needed practice, along with team building and class-vs-class trash talk.  This week’s unit focused on functions: specifically, operations with functions , inverses and domain/range.

SPEED DATING WITH FUNCTIONS

To set up this activity, students ripped apart my classroom, moving desks into two long rows, with tables facing each other.  Then, each student took a card, which offered an expression, such as “2x+5”, “x^2+2”, “-3x+1”.  Students on one side of the table were assigned the role of f(x), while the other side of the table was g(x).  This was written on the board at the end of the long table so students could remember.

Next, I held up a dry-erase board with a function operation.  With their speed-dating partner, the pair completed the operation.  Here are some sample tasks:

• Find (f+g)(x)
• Find f(g(x))
• Find (f/g)(x) and state any domain restrictions
• Find g(f(n+1))
• Find g(f(g( -2 )))

Partnerships reached an agreement on their answer, and I encouraged pairs to write down any problems which caused trouble so we could review them later.  After each question, all student rose from their chairs, and moved one chair over in a clock-wise direction, which ensured that students worked with a host of different partners and functions.  The entire activity took about 30 minutes, and was a fun review for our quiz today.

FUNCTION INVERSE MATCHING

A dash of creativity and preparation can turn a mundane worksheet into a classroom filled with action and sharing.  In this activity, I created a worksheet (using Kuta software) of 25 linear functions, where students are asked to find the inverse.  Giving all of these for students to do would not be the most thrilling task of their high school careers, yet we need some practice with identifying inverses.  So, here’s what I did:

• Take the questions and their answers, cut them all apart, seprating question from answers.  Place them in a baggie to store. Shake up the baggie.
• My students sit in groups, so each group was given a handful of the slips of paper.
• Project a stopwatch on the screen.
• Have the class select a class captain.
• When I say go (and start the stopwatch), the goal for the class is to match up all 25 functions with their inverses.  I give no rules for how they need to organize this, other than that the problems need to be in order, as I have left the question numbers with the original problems.
• The class which completes the task quickest wins class snack.

This week, my morning class completed the task in over 12 minutes. My afternoon class was a bit more organized from the jump and took only 10.  The teamwork and whole-class participation was exactly what I was looking for!  The questions I used this week are given below.  Enjoy!

## Class Opener – Day 68 – Some Special Squares?

A semester goes by so quickly under block scheduling, and after tommorow I only have 3 full weeks left with my freshmen. Many of them will move to Algebra 2 next semester, and I’m hoping to put them in a good place to remain successful in their high school math careers by thinking about the “why” – how everything we learn in math is connected, and form new connections when new ideas are encountered.

These freshmen are one of our first groups to have had Algebra 1 in 7th grade, followed by Geometry in 8th grade. One difference I have observed with this structure is that my current 9th graders seem less confident with their algebra skills than past classes.  I have to be careful with this generalization, as it’s easy to fill into a “the sky is falling” trap – maybe this year is really no different than previous years, but it sure feels that way.  In particular, I sensed a good bit of uneasiness this week when multiplying binomials or factoring trinomials made an appearance during this past week of work on function operations.  Next week, we’ll take a deeper look at polynomials as a segue into algebra 2.  Today’s opener is taken from the awesome Nrich site, titled “Plus Minus”, and I hope to build some connections from their existing knowledge of “difference of squares” patterns. You can find many class resources for this problem there.

I asked students to think about the equations on the board and tell me what they noticed. Does the information on the board help them generate any additional entries? The class was divided into two camps: those who dove into their calculators to “guess and check” for more, and those who observes some patterns in both the numbers and the equations as a whole.

The class did develop some additons to the list, but I didn’t notice many students making a connection to any previous knowledge….until….

I notice that 55 and 45 add up to 100, and 105 and 95 add up to 200

Oh yes, yes, yes….tell me more…..please…….

But the second half, where we look at the differences, was missing.  I’m not going to force the issue yet, as I want them to find it.  Later in class, we did a “read and recognition” activity to get unlock some of their trapped algebra knowledge.  10 questions, 30 seconds each, all relating to a factoring pattern.  Many of their “difference of squares” cobwebs were dusted off, but we still have some work to do.

We’re going to keep these inetresting number patterns on the board for the next few days, maybe we’ll add some to the list. It will be interesting to see how we grow in comfort over the next week!