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?

Don’t talk about 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.

nrich factors

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….

BookOne 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?

STUDENT: Do you have any advice for us? Please?

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.


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.


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.

speed dating


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.

squaresI 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!

Class Opener – Day 67 – Verifying Inverse Functions

We’re finishing up our unit on function operations.  Yesterday we developed the definition of inverse functions (using only linear functions) and graphed to demonstrate the symmetry. Time to see what we have learned:


Many students’ instincts let them down on the first pair, believing them to be inverses. But after my prodding that they somehow verify the inverse relationship, we soon verified that f(g(x)) did not produce the result we desired.   The second example was then complete easily.

But what about that third problem?  They sure seem like inverses.  One student offered his proof for the pair:

They are inverses just because I know.

Sometimes ideas in math are just that obvious, and maybe we don’t need to prove them specifically.

On the board, we “proved” that both f(g(x)) and g(f(x)) both seem to simplify as x.  And a few numerical examples help show this:

  • f(g(5)) = 5
  • g(f(10)) = 10
  • g(f( -6 )) = 6……. ruh roh……

Students in my class have not been exposed to a formal definition of the square root function, and this led to a nice discussion of absolute value, and the need to restrict the domain in order to consider inverses. Planting seeds for algebra 2, which many of my students will take next semester, is always a bonus.

Class Opener – Day 66 – Surprising Coin Patterns

A short post today, as I am out the door for a meeting with our NCTM local group, ATMOPAV. Please check out our website, where we have information on local awards, and house our award-winning newsletter!

I enjoy giving problems with solutions which go against our instinct. In statistics, there are many opporunities for this, and today’s opener in my AP class seemed innocent enough:

Which will more likely occur first in a string of coin tosses: HTH or HTT?

After a few moments of debate, there was universal agreement that the two patterns are equally likely, and therefore we should have an equal expectation of seeing them occur first in a string.  But the correct answer goes against this intuitive notion.

Peter Donnelly’s TED Talk – “How Stats Fool Juries” is easily digestible for the high school crowd. I show it over 2 days, first to present the coin-tossing problem. Then in our next class meeting I will show the second half, where conditional probabilities and the multiplication rule make appearances in courtroom trials.  In the video below, fast-forward to about the 5:30 mark if you want to learn about the coin-tossing problem, or watch from the beginning for some statistics humor.