Category Archives: Uncategorized

Area Models and Completing the Square

I’m nearing the end of my time with my 9th graders, with this week dedicated to moving beyond factoring as the sole method for solving quadratic equations and towards more general methods like completing the square.

Late in May, David Wees shared materials which challenge students to investigate the relationships between “standard form” and “completing the square” form (aside – does anyone agree on proper terms for these?) using area models to build representations.  Given that I use area models often to introduce polynomial multiplication, I was eager to maintain consistency in the student understanding.


But before we dove into David’s lesson, I wanted students to revisit their understanding of area models.  In this Desmos Activity Builder lesson I created, students shared their interpretations of area models and worked in pairs to investigate non-square models.  In one of the final screens, students argued for the “correct” interpretation of a model.


Using the Desmos teacher dashboard, we could see clear visual arguments for both representations.  This was valuable as we ended the lesson for the day, and tucked that nugget away for Monday, when we would begin to formalize these equivalencies.


After the weekend, students worked independently through David’s Completing the Square lesson. Not only did students quickly move through the area models and the dual representations, the debates between students to explain how to move from one representation to the other were loud and pervasive.  I’m also loving how many of my students have started to use color as an effective tool in our OneNote-taking (below).


At the end of the sheet, all students completed problems which translate standard form to vertex form with no support from me (“no fuss…no muss”).  It dawned on me that something amazing had happened….my students had figured out completing the square without my ever talking about completing the square.

Tomorrow we’ll tackle those pesky odd-number “b” terms, but my students own this already!

Another Pascal’s Triangle Gem

At a recent local conference, Jim Rubillo, Annie Fetter and I were saying our good-byes at the end of a fun evening, when Jim’s puzzly side emerged…

What proportion of the numbers in Pascal’s Triangle are even?

Every time I talk to Jim, he’s bound to have a neat problem for me to chew on.  The last time, he shared a fun task involving the harmonic series. Take a few minutes and think about this Pascal’s Triangle scenario…I’ll even leave you some spoiler space.

Jim RubilloAt the ATMOPAV Spring Conference last month, Jim shared an entertaining talk titled “Gambling, Risk, Alcohol, Poisons and Manure – an Unfinished Life Story”.  The talk led the attendees on a journey through the history of statistics, starting with games of chance and the meeting of Chevelier du Mere and Blaise Pascal, through the introduction of formal inference procedures developed at the Guinness brewery, and to identifying statistical abuses in the present day.

Jim is a life-ling educator and former Executive Director of NCTM who happens to live quite close to me.  It was a thrill having him share his ideas with the group.

Had enough time to think about this Pascal question?  Spoiler time is up!

So, which rows are in Pascal’s Triangle are we talking about here?

In theory, we are talking about “all” of the rows in the infinite Pascal’s Triangle, which makes this a bit tricky to think about for kids (and adults as well!).  But Jim shared with me slides which show the proportion of evens in increasing numbers of rows of the triangle.  You will notice that as the number of rows grows, the proportion of even entries also increases, and approaches 1.  What a neat result!  Below is an animated gif I made using a Pascal’s coloring applet which shows the increase in the proportion of even (white space) numbers in increasing rows.

Pascal Zoom

For your class, this is a fun opportunity to talk about the parallels between Pascal’s Triangle, Sierpinski’s Gasket, and fractal area.

Already looking forward to my next encounter with Jim!

Drinking the Statistical Power Kool-Aid

For my colleagues who teach AP Stats, there are few phrases more terrifying:

Today I am teaching Power.

Power: a deep statistical concept, but one which often gets moved towards the back of the AP Stats junk drawer.  The only mention of power in the AP Stats course description comes under Tests of Significance:

Logic of significance testing, null and alternative hypotheses; p-values; one- and two-sided tests; concepts of Type I and Type II errors; concept of power

So, students need to understand the concept of power, but not actually compute it (which is itself not an easy task).  Floyd Bullard’s article “On Power” from the AP Central website provides solid starting points for teachers struggling with this concept; specifically, I appreciate his many ways of considering power:

  • Power is the probability of rejecting the null hypothesis when in fact it is false.
  • Power is the probability of making a correct decision (to reject the null hypothesis) when the null hypothesis is false.
  • Power is the probability that a test of significance will pick up on an effect that is present.
  • Power is the probability that a test of significance will detect a deviation from the null hypothesis, should such a deviation exist.
  • Power is the probability of avoiding a Type II error.

This year, I tried an activity which used the third bullet above, picking up on effects, as a basis for making decisions.


Photo Mar 06, 12 11 56 PMArriving at school early, I got to work making 3 batches of Kool Aid.  During class, all students would receive samples of the 3 juices to try.  Students were not told about the task beforehand, or where this was headed. Up to now, we had discussed type I and type II error, so this served as a transition to the next idea.


All students received cups and as they worked on a practice problem I circulated, serving tasty Kool Aid – don’t forget to tip your server!  I told students to savor the juice, but to pay attention: I promised them that this first batch was made using strict Kool Aid instructions.  Think about the taste of the juice.


Next, students received a drink from “Sample A”.  Their job – to assess if this new sample was made using LESS drink mix than the baseline batch.  Also, I varied the amounts of juice students received: while some students were poured full cups, some received just a few dribbles.  To collect responses, all students approached the board to contribute a point to a Sample A scatterplot, using the following criteria:

Photo Mar 06, 8 50 05 AM - CopySample size: how much juice you were given

Evidence: how much evidence do you feel you have to support our alternate hypothesis – that Sample A was made with LESS mix than the baseline?

As you can see, the responses were all over the place – a mixture of “we’re not quite sure” to “these are strange directions” to “I just don’t trust Lochel – something’s up”.  But the table has been set for the next sample.

Sample A: it was made with just a smidge less mix than the baseline.  So I wasn’t totally surprised to see dots all over.


Photo Mar 06, 8 50 05 AMI poured drinks again from this new sample, and again varied the sample sizes.  I asked all students to think about their evidence in favor of the alternate, and wait until everyone tasted their juice before submitting a dot.

And check out those results!  Except for a few kids (who admitted they stink at telling apart tastes), we have universal support in favor of the alternate hypothesis.

Sample B: this was made with 1/2 the suggested amount of drink mix.  Much weaker!


This activity made the discussion of power much more natural.  In particular, what could occur during a study which would make it more likely to reject the null hypothesis, if it deserves rejecting?

Larger sample size: smaller samples make it tough to detect differences

Effect size: how far away from the null is the “truth”.  If the “truth” as just a bit less than the null, it could be difficult to detect this effect.

In terms of AP Stats “concepts of power”, this covers much of what we need.  Next, I used an applet to walk students through examples and show power as a probability.  And like most years, this was met with googly eyes by many, but the foundation of conditions which would be ripe for rejecting the null was built, and I was happy with this day!

Suggested reading: Statistics Done Wrong by Alex Reinhart contains compelling, clear examples for teachers who look to lead discussions regarding P-value and Power.  I recommend it highly!