Category Archives: Class Openers

How I start my class each 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.