Category Archives: High School

EdCamp and My Amazing Principals

alg-ad-snickers-betty-white-jpgThe first week of district PD – lots of meetings, scads of “sit and get” messages, and every administrator making sure their voice is heard.  I suspect what I am describing is not unique to my area. A great opportunity to energize is lost, and the “grind” begins. And I haven’t been too shy about expressing my displeasure through tweets when I am frustrated – I can be kind of a jerk (think of those Snickers ads…better?…better…).

This week my school admin team got it right. And I feel fortunate to work with them. 

For about a year, my school’s admin team had kicked around the concept of a school-wide EdCamp. To be honest, I never thought it would see the light of day…there are just too many other things loaded into the calendar.  So an invitation to work with a team of teachers to organize a high school-wide EdCamp was a true surprise…then the work began.

We planned 3 morning sessions, followed by lunch and prizes.  But beyond the structure of the day, we had a lot of talking-up to do.  Would our teachers, many of whom had never been to an EdCamp, understand the concept?  Would people propose sessions? Could we engage the curmudgeons in our teaching ranks? At our opening faculty meeting, we showed a brief video to help teachers understand the EdCamp concept, then talked it up over the next 2 days.


The morning of the conference many teachers suggested ideas, asked questions and thought about what they’s like to learn. In the end, we had a nice variety of topics and it felt like there was something for everyone!


As I walked around during the sessions, I was thrilled to see rooms filled with discussions, and teachers from different departments with an opportunity to engage.  I know there is no possible way every to reach everyone, but I hope it was a day of professional learning for most.

On my end, I offered a session “Activity Builder for the Non-Math Crowd” which seemed to be of use to those assembled. Then later, a session where we just did math – problems from Open Middle, Nrich, Visual Patterns and others let math teacher talk, learn, and think about engaging problems for their classrooms. You can download the problems I shared with this link.

Thanks to the fantastic people I work with for letting me be part of this: Baker, Dennis, Kristina and Melissa.

And a big thanks to the HH admin team: Dennis, Ralph, Tracey and JZ.  I appreciate the opportunity, and promise not to complain again….until the next time…..



Last Week I Refused to Teach Factoring

The students in my Freshman Honors class have certain expectations for how a math class works – a teacher lectures, there’s lots of drill practice, and then a test. Breaking this mold, and causing them to think of themselves as reflective learners, is one of my many missions. So this past week, when confronted with factoring, I simply refused to lecture.

My 9th graders have seen factoring before, but it was back in 7th grade, and it was only a surface treatment. So after a brief opener where we discussed what a “factor” means (both numerically and algebraically), I dropped the bomb –

  • I’ve posted your learning targets online
  • I’ve posted videos, resources and practice problems if you need them
  • I’ve set up online practice if you need it
  • You have a timed quiz on Friday (we started on Tuesday)


Panic….apprehension….incredulous looks….

So, you’re not going to teach us?

Nope.  Now get to work.

Here are some details of what I posted:


  • F1: I can identify and factor expressions which involve greatest common factors.
  • F2: I can efficiently factor trinomials of the form ax2+bx+c, where a = 1.
  • F3: I can factor trinomials of the form ax2+bx+c, where a does not equal 1 (or zero).
  • F4: I can identify and factor perfect square trinomials.
  • F5: I can identify and factor “difference of squares” expressions.
  • F6: I can factor expressions which may represent a combination of F1 to F5.
  • F7: I can factor expressions “by parts” (or “by grouping”) when necessary.
  • F8: I can factor expressions which are the sum (or difference) of two cubes.


Each learning target featured a video – some from Khan Academy, and some from other sources I searched for – but I attempted to provide a variety of methods. Some featured grouping as a primary means, others demonstrated the box method or the diamond.  This was the most important aspect of this learning experience: I wanted students to experience a variety of approaches, evaluate them, and make a personal decision about what worked best for them.  The students did not disappoint.

I also posted other online resources, such as worked examples and flowcharts.  One of my favorite resources – Finding Factors from nrich, was also included. Finally, I created an assignment on DeltaMath for each learning target, and a final jumbled assignment. The end of each day featured an exit ticket quiz and recap, to assess progress and provide “next steps” during the week.


Some students latched onto factoring by grouping for every quadratic, and explained their reasoning to their peers.  Many of these same students later in the week found more confidence in their number sense and chose to group only for “tricky” problems. One student was particularly insistent that the box method was the best was to go for all problems. Others found the diamond method helpful – which led to deep conversations about number sense and how to make searches more efficient. And in one fascinating conversation, a student discovered a “trick” he had found online. The group debated the merits of the method, tried some practice…but as nobody in the group could figure out why the method worked, they quickly dismissed it.  Good boys!!!

In the end, the quiz scores were great.  But beyond the scores, I feel confident that the students have made choices about their learning, assessed and revised their thinking, and can move forward using their new tools.


Today I asked students to reflect upon their learning experience, and provide me feedback.

What was your overall feeling about last week’s learning method?  (1 = “Please never do that again”, 5 = “I loved it – do it more”.)


Describe something you LIKED about last week’s classes, and why you liked it.

  • I liked being able to choose what i wanted to do. I could focus on my weaknesses and do less problems on what i was good at. I also appreciated the practice problems.
  • I liked that if you knew a topic you could move on and didn’t have to wait for someone else or the next day of class.
  • I liked that I could learn and do problems at my own speed.

Describe something you DIDN’T LIKE about last week’s classes, and why you didn’t like it.

  • I did not like that you did not explain how to factor
  • I didn’t have as much instruction from the master of factoring. {note – I suppose this is me?}
  • the teacher wasn’t involved

This last comment intrigues me…and I’m not sure if I should be bothered by it…I don’t think I should be.  In many respects, I feel I worked harder during the classes, as students were all over the place.  But I also realize students don’t see all of this going on around them.  I’ve become intrigued by how I can be less of a teacher and more of a facilitator in my classes, and this was a solid step forward I feel.

Now, off to plan to not lecture tomorrow….

How I Stumbled Into Math Modeling Without Even Realizing It.

We started a unit on counting principles this week in my 9th grade honors class – permutations, combinations – eventually leading to the binomial theorem.  Since my  classes had used Desmos Activity Builder a few times and were familiar with the need to enter a 5-character code to start an activity, I planned to ask the following question as a class opener:

How many different 5-character DesmosActivity Builder codes exist?


This problem would have likely met my intended goal of having kids think about the fundamental counting principle in a real-world context.  It also would have taken about 10 minutes of class time, and have been forgotten about by the next day.  It felt like I was missing an opportunity to develop a deeper discussion.  A slight tweak to the question added just the right layer:

Activity codes for Desmos Activity Builder currently have 5 characters, as shown here.  When will Activity Codes need to expand to 6 characters?

And now we have a problem which requires a bit more than a quick calculation.  To start, I asked students to work in their teams to make a list of information they would need to help solve this problem.  This was not easy or comfortable for them – but a preliminary list of questions emerged from group discussions:

  • How many 5-character codes are there?
  • Are codes used less on weeekends and summers?
  • Can letters repeat in codes?
  • How many codes a day are used?

This was a good start to set kids in motion to think about how to solve the problem.  I’m hoping they will think about new questions or revise their questions as we go along…the class did not disappoint!


As kids worked, clarifying questions came up – some of which I just didn’t know the answer to, and hadn’t really thought about:

Mr. L, are there any zeroes in codes? Kids might confuse them with the letter O.

Mr. L, I don’t see any L’s in the codes?

Excellent observations, and restrictions we need to think about in our calculation. A tweet to the Desmos crew lent some clarity, and added more restrictions!

Thank for the intel, Eli!


This was tricky for my class. To help, I reminded students that when we started the semester, codes were 4 characters.  When did the Desmos 5-character era begin?  A quick scroll through my history (shown here) provides some info. After further interrogation from my class, I shared that Activity Builder started around July of last year with 4-character codes.  Add this to our bucket of helpful info.



Writing a draft solution was the next task for students.  But instead of turning it in to me immediately, I formed class teams of 3 where students shared their drafts and ideas.  I used this opportunity to build teams of students who I observe don’t often interact or chat.  From here, I gave students another day to think about their explanation – keeping in mind that there are no right answers to this question, only answers we can defend. But it still feels like we are missing a key piece in this problem……


The next morning as students were mingling before the bell, I looked across the room at the laptop of Jacob – one of my more insightful, but also introverted, students:


It’s the mother lode!

The google trends graph for student.desmos.  Yes! Yes! Yes!  Stop everything kids, we need to talk!  Jacob – tell us all about this graph. How does this new info factor into our estimates?  What should we do with it?  Is this going to continue?  And with this, I gave the class an extra day to think about their responses, share, and dig deeper.  And while many students simply estimated a growth rate by doubling or tripling their computed rate (this is fine with me), I am getting some responses which far exceed my expectations – like Jacob, who developed a growth function and evaluated integrals (did I mention this is a 9th grade class????)


Yep, this was definitely better than my originally intended problem!



Compute Expected Value, Pass GO, Collect $200

Photo Oct 11, 7 08 49 AM.jpgExpected Value – such a great time to talk about games, probability, and decision making!  Today’s lesson started with a Monopoly board in the center of the room. I had populated the “high end” and brown properties with houses and hotels.  Here’s the challenge:

When I play Monopoly, my strategy is often to buy and build on the cheaper properties.  This leaves me somewhat scared when I head towards the “high rent” area if my opponents built there.  It is now my turn to roll the dice.  Taking a look at the board, and assuming that my opponents own all of the houses and hotels you see, what would be the WORST square for me to be on right now?  What would be the BEST square?

For this question, we assumed that my current location is between the B&O and the Short Line Railroads.  The conversation quickly went into overdrive – students debating their ideas, talking about strategy, and also helping explain the scenario to students not as familiar with the game (thankfully, it seems our tech-savvy kids still play Monopoly!).  Many students noted not only the awfulness of landing on Park Place or Boardwalk, but also how some common sums with two dice would make landing on undesirable squares more likely.


After our initial debates, I led students through an analysis, which eventually led to the introduction of Expected Value as a useful statistic to summarize the game.  Students could start on any square they wanted, and I challenged groups to each select a different square to analyze.  Here are the steps we followed.


First, we listed all the possible sums with 2 dice, from 2 to 12.

Next, we listed the Monopoly Board space each die roll would causes us to land on (abbreviated to make it easier).

Next, we looked at the dollar “value” of each space.  For example, landing on Boardwalk with a hotel has a value of -$2,000.  For convenience, we made squares like Chance worth $0.  Luxury Tax is worth -$100.  We agreed to make Railroads worth -$100 as an average.  Landing on Go was our only profitable outcome, worth +$200. Finally, “Go to Jail” was deemed worth $0, mostly out of convenience.

Finally, we listed the probability of each roll from 2 to 12.

Now for the tricky computations.  I moved away from Monopoly for a moment to introduce a basic example to support the computation of expected value.

I roll a die – if it comes out “6” you get 10 Jolly Ranchers, otherwise, you get 1.  What’s the average number of candies I give out each roll?

This was sufficient to develop need for multiplying in our Monopoly table – multiply each value by its probability, find the sum of these and we’ll have something called Expected Value.  For each initial square, students verified their solutions and we shared them on a class Monopoly board.


The meaning of these numbers then held importance in the context of the problem – “I may land on Park Place, I may roll and hit nothing, but on average I will lose $588 from this position”.

HOMEWORK CHALLENGE: since this went so well as a lesson today, I held to the theme in providing an additional assignment:

Imagine my opponent starts on Free Parking.  I own all 3 yellow properties, but can only afford to purchase 8 houses total.  How should I arrange the houses in order to inflict the highest potential damage to my opponent?


I’m looking forward to interesting work when we get back to school!

Note: I discussed my ideas about this topic in a previous post.  Enjoy!

Pulling In To the Station

My school isn’t 1-1 with technology yet, though there are rumblings we will get there next year….or the year after….or 2031…anyway, it’s time to get techy!  My new classroom features 4 computer stations in the back – nice to have, but not super-helpful with classes of about 24 each. Station-model classroom structure has been super-helpful in my pre-calculus class in the first month. Besides the chance for all students to participate in rich technology-based activities, I’ve had the opportunity to carve out valuable small-group time with students.  Here’s an example:

In our first pre-calc unit, we review functions and their shirts, folding in new ideas like the step function, piecewise and even/odd functions.  My objective for the class was for students to consider functions in varied forms.  As students entered class, playing cards were drawn to establish their groupings, so there were 3 groups of 7 or 8.  With 15 minutes on the classroom clock, students started on their first station:

  1. Group 1 gathered in a small group with me in a circle of desks, where we worked through proving functions even or odd, and sketching their graphs.
  2. marbleslideGroup 2 worked at the computer stations on a Desmos Marbleslides featuring quadratic functions, with many students pairing up to work together. If you have never tried a Marbleslides, run and play now – we’ll wait for you to come back…
  3. Group 3 worked out in the courtyard (hey, my new classroom leads outside – which is nice) on a group task involving a piecewise function.

After groups had rotated through all 3 activities, we had time to recap / share and assess our learning over the hour.  Here’s why I need to do this more:

  • The small group station let me touch base with every student, assess strengths, find out what we need to work on, and provide feedback to everyone.
  • Marbleslides is sneaky awesome! When students begin to obsess over function shifts and how to restrict domains and don’t want to peel away from their computer, you know something is going right.
  • Class went fast! It felt like the mixed practice from Let It Stick was now becoming part of my classroom culture.
  • My pre-calc is mostly 11th and 12th graders, who have had a pretty traditional classroom experience in their math lives.  I can sense they appreciate that something difference is happening.
  • All students are responsible for their learning.  Even the least-active task, the piecewise function, was used the next class for sharing out and a jumping-off point.


Activity Builder – Classroom Design Considerations

This past summer, our forward-thinking math-teacher-centric friends at Desmos released Activity Builder into the wild, and the collective creativity of the math world has been evident as teachers work to find exciting classroom uses for the new interface. Many of these activities are now searchable at – you’re welcome to leave now and check them out – but come back…please?

Its easy to get sucked in to a new, shiny tech tool and want to jump in headfirst with a class. I’ve now created a few lessons and tried them with classes which range from the “top” in achievement, to my freshmen Algebra 1 students. In both cases, I’ve settled upon a set of guiding principles which drive how I build a lesson.

  • What do I want students to know?
  • What path do I want them to take to get there?
  • How will my lesson encourage proper usage of math vocabulary?
  • What will I do with the data I collect?
  • How does this improve upon my usual delivery?

It’s the last question which I often come back to. If making a lesson using Activity Builder (or incorporating any technology, for that matter) doesn’t improve my existing lesson, then why am I doing it?

One recent lesson I built for my algebra 1 class asked students to make discoveries regarding slopes and equations of parallel and perpendicular lines. Before I used it with my class, a quick tweet 2 days before the lesson provided a valuable peer-review from my online PLC.  It’s easy to miss the small things, and some valuable advice regarding order of slides came through, along with some mis-types. The link is provided here in the tweet if you want to play along:

The class I tried this with is not always the most persistent when it comes to math tasks, but I was mostly pleased with their effort. Certainly, the active nature of the activity trumped my usual “here are bunch of lines to draw – I sure hope they find some parallel ones” lesson.

As the class finished, I called them into a small huddle to recap what we did. This is the second lesson using Activity Builder we have done together.  In the first, the students didn’t know I can see their responses, nor understand why it might be valuable.  In this second go-round, the conversation was much deeper, and with more participation than usual. In one slide, the overlay feature allowed us to view all of our equations for lines parallel to the red line:


We could clearly see not only our class successes, but examine deeper some misunderstandings.  What’s happening with some of those non-parallel lines?  Let’s take a closer look at Kim’s work:

Parallel 2

What’s going on here? A mis-type of the slope? The students were quite helpful towards each other, and if nothing else I’m thrilled the small group conversation yielded productive ideas in a non-threatening manner – it’s OK to make errors, we just strive to move on and be great next time.  The mantra “parallel lines have the same slope” quickly became embedded.

The second half of the lesson was a little bumpier, but that’s OK.  Before questions regarding slope presented themselves in the lesson, storm clouds were evident when the activity asked students to drag a slider to build a sequence of lines perpendicular to the blue line.  Observe the collective responses:


So, before we even talk about opposite reciprocal slopes, we seem to have a conceptual misunderstanding of perpendicular lines.  I’m glad this came up during the activity and not later after much disconnected practice had taken place.  In retrospect, I wish I had put this discussion away for the day and come up with a good activity for the next day to make sure were all on board with what perpendicular lines even look like, but I pressed ahead.  We did find one student who could successfully generate a pair of perpendicular lines, and I know Alexys enjoyed her moment in the sun.


What guiding principles guide you as you build activities using technology? How do they shape what you do?  I’m eager to hear your ideas!



Estimation and Anchoring

A recent post by my Stats-teacher friend Anthony, “Wisdom of the Crowd“, reminded me of an estimation activity I have used many times in my 9th grade Stats class.  The activity is based on a chapter from John Allen Paulos’ book A Mathematician Reads the Newspaper.

You’ll need two groups of students; 2 different classes will do.  Each student uses an index card or a scrap of paper to write responses to 2 survey questions. I warn the students beforehand that the questions may seem strange: just do your best to answer as best you can.

  • Question 1: Do you believe the population of Argentina is MORE or LESS than 10 million people?
  • Question 2: Estimate the population of Argentina.

Allow a few moments between the questions for the inevitable blank stares and mumbling.  Then collect the responses.

For the second group, you will ask the same two questions, except that the first question will replace 10 million with 50 million.  After you have data from both groups, write it on the board or print it and hand it out. It’s time to analyze and compare. Challenge students to communicate thoughts about center and spread. Also, which group’s data do they feel does a better job of estimating question 2?  It’s a neat activity, and while you will receive some strange responses as estimates, and students will generally guess higher on question 2 if they have been anchored to the 50 million number.  Some guidelines for this activity are avilable.  Have fun!

According to Google, the actual population of Argentina is around 41 million.