Blog Archives

Day 94: Class Discourse Sparked by Navigator Quick Poll

One of the things I’d like to use more often in my Precalculus classes are multiple choice questions to promote class discussion.  Today as part of the opener I gave my first MC question.  Fairly straight forward question because I wanted to get the kiddos logged into the Nspire Navigator, which we haven’t done for a while.  The students wrote down on their Opener-Exit slip the answer along with any calculations they needed to do to determine the best choice. Then the Navigator, through a Quick Poll, collects their answers. Once I stop the poll,  I can project the responses in a bar chart.


As you can see, there were two popular answers.  I asked my students in their groups to discuss their choices and try to determine which one was actually the correct answer along with supplying a reasoned argument for their group’s choice.  Some great conversations!  I then resent the Quick Poll to see if they could arrive at the correct answer without my confirming the answer.

Day 39: Confused about Fractions


Today in precalculus, I was beat over the head about how important it is for students to truly understand the conceptual underpinnings of learned procedures.  I asked students to evaluate a difference quotient expression where the function was the inverse function:

Simplify the expression  when  :   Then evaluate when x = 3 and h = 0.

Did I learn a lot about what students had memorized poorly as well as misconceptions.  What a great opportunity to explore via asking the right questions their misunderstandings as well as give them tools for figuring things out when they are not sure how the mechanics work.


This problem also gave us an opportunity to revisit the idea of an undefined value when it is indeterminant in nature; i.e. 0/0 and how that is related to the point discontinuity.  Good stuff today!

Day 28: What Type of Discontinuity?!

This was a red-letter day in precalculus…at least for the teacher.  As an opening question, I asked my class this simple question:

Is x = 3 the location of an infinite discontinuity or a removable discontinuity?  Explain how you know.

As I was walking into class to set up, I decided to change it up.  I asked the students to answer the question silently and individually.  Then, once everyone had an answer, I asked them to move to one side of the room or the other depending on their answer of vertical asymptote versus removable discontinuity (hole).  Then as a group, they needed to come up with one convincing mathematical reason their choice was correct.  You can see the two huddles below.

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Then I was the scribe as each group gave a reason.  After each group commented, the other group needed to give a convincing statement to either support their stance OR refute the other group’s statement…thus the different colors.  After each group gave a statement, they had time to re-huddle and/or change sides.  And then we repeated the process.  I was after student voice and academic discourse targeted at each other and not me as the conduit.  They desperately wanted the answer, but I didn’t give it.  So they had to resort to the definitions and theorems to add sophistication and depth to their arguments.


Although this took longer that I had originally planned for the opener (and I had to adjust the subsequent lesson) the end result was that the students, through disciplined discourse, were able to convince themselves of the answer AND had a much deeper understanding of the underlying concepts.  It was so exhilarating to see the metamorphosis of their graph of the ideas behind vertical asymptotes and point discontinuities.  They were so proud, and somewhat surprised that they could actually reason through to the correct answer.  I never did tell them 😛

Day 18: Normal Calculations with a Twist


We are working on the concept of z-scores and the associated skill development in AP Statistics.  I found this awesome and easily implemented activity at Teaching Statistics blog.  This particular self-checking activity was shared at the Made4Math blog.


There are problems on the front and answers on the back, but the twist is that the answer on the back goes with a different problem.  So students pick any first card, work out the problem, find the answer on a different card, then turn it over for their next problem to solve.  The set has a nice mix of given a data value, find the probability, or given the probability, find the data value.  We actually spent two days on this practice activity (approximately 15 minutes each day): the first day they did the problems using the Standard Normal Table without a calculator and the second day they used the calculator commands: normcdf( and invnorm(.


Day 158: The Eccentric Ellipse

Today was a red-letter, AWESOME day as a teacher…in fact, I was pinching myself continually because I thought I was in a dream.  As I mentioned last week in post Day 155: Coneheads, my colleague used taped string on the whiteboards to get his students to develop the geometric definition of an ellipse…and I wanted to do that too!!  But I had already committed to the patty paper activity.  So what to do?  I just wanted to have fun too!

Since we had not done any of the formal notes I had planned to do yesterday, I saw a wonderful opportunity!  I put up 6 stations of taped string around the room; I had the foci at different distances along an imaginary horizontal major axis, one with a vertical major axis and one with a slanted major axis.  All of the strings measured 100cm (and were they surprised when they all measured their major axis!).  I also put out 5 different color markers, a yardstick and an extra piece of tape. At this point, I wasn’t sure what I would actually do.

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I asked my students to divvy themselves up between the 6 stations.  Of course, they immediately picked up a marker and drew in the ellipses.

Then, with a time restriction of 5 minutes, I asked them to identify these 5 things on their ellipses and color-code them.  They were not to use their iPad to go to the internet at this time.  They had to discuss what they thought these words meant and label their ellipse.  A little whining, but very little.  They talked and looked around at what the other groups were doing and saying, and revised their thinking and/or stood firm to their original thoughts.  Love seeing the tilted heads and moving arms!

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After this initial time, I had them go back to their seats so we could generate some generalizations.  Some things they said were:

  1. Distance from one focus to a point on the ellipse added to the distance from that same point to the other focus is always the same.
  2. The closer together the foci are, the more the ellipse looks more like a circle.
  3. String length = major axis length
  4. eccentricity is the ratio between the major axis and the minor axis.


From yesterday’s exploration, they knew the first conjecture was true, although it was nice to have them check this around their personal ellipse.  We looked at the various ellipses around the room and determined that the second conjecture seemed to make sense (and a nice lead in to the idea of eccentricity).  They weren’t so sure about the third conjecture.  So I sent them back to a different ellipse (not their own) with the tasks to verify the three conjectures AND to see if they could find a relationship between the semi-major axis, the semi-minor axis, and the focal radius.

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More great discussions and most did the typical “string forming an isosceles triangle” approach to find the relationship.  Again, the deep discussion with probing questions and sense-making almost brought tears to my eyes!

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We came back together as a class and debriefed once again (we had a block class today which means a 90 minute period).  Last of all, I had originally put this cartoon up for the exit slip today, but actually used it to explore eccentricity.

eccentricity comic

I simply asked if they could determine the eccentricity of their conic using the cartoon as their guide.  I reminded them that they could only use measurements we had already discussed: major-axis, semi-major axis, minor axis, semi-minor axis, and the focal radius. Absolutely wonderful discussions.  Most groups came up with the ratio of the focal radius to either the semi-major axis or the semi-minor axis.  They argued/reasoned through why they thought the denominator was the semi-major axis.  YA-HOO!

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Back in March, I attended and blogged about a one-day workshop with some of my school colleagues  given by Dan Meyer on Intellectual Need in the Classroom.  Looking back, I realize now that I used Dan Meyer‘s three essential components of an engaging and relevant lesson: Start a fight, turn up the math dial slowly, and create a headache – to provide the “aspirin.” And maybe that’s why I got the comment from one student, “That was the best lesson I’ve ever had!”

What is one spectacular lesson of your career?  What made it so?  Have you been able to replicate it in subsequent years?  I’d love to hear about your Red-Letter day in teaching.

Day 128: Revisiting Linear Motion

How do you connect “old” ideas (that is, previous learning) with “new” or unfamiliar situations?  I’m always trying to build new dendrite connections between the old and the new.  Linear motion via parametrics is just one example.  As our opening activity, I asked students this question:  Write the vector equation and the parametric equations of the line through (-7, 4) and perpendicular to 2x + 3y = 12.  Draw a diagram of the situation.  

I purposefully left out the t-interval to force a discussion of why it is important and how do we write parametric equations when there is no clear one-unit increment of time.  And what a rich discussion we had!  I found out what “stuck” from yesterday’s Geogebra exploration, drew in vectors (yay for review!) and really looked deeply at the variety of parametric equations that are correct even if they look different.


I was soooo thrilled when two students shared different x(t) equations (green vs. red).  Immediately, students took sides (Creating a fight, as Dan Meyer would say).


As a class we compared and contrasted what was written: same starting x-coordinate, different rates of change.  This led to a discussion of how one was 2 and the other was 7….it came down to the arbitrarily chosen “end” point.  One student used the slope and found the next point along the line; the other student used the y-intercept.  Cool, huh?!  This  drove us to explore whether both could be correct and why.  Can we use both to determine a specific location, and how are they related.  You can see the results of our queries.


How have you facilitated a deep discussion based on student responses?  And what turned the discussion into a deep one?

Day 120: Candy Store for Math Teachers

I just love this conference.  There is so much to see, do and learn.

Opening speaker was Jo Boaler talking about Fixed and Growth Mindsets in Mathematics

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Then the rest of the day I attended a variety of sessions using and not using technology, using and not using the math practices, using and not using engagement techniques.  It is really interesting to see how certain strategies in presenting make such a difference

Once strategy I found intriguing is “My Favorite NO-Learning from Mistakes,” presented by Sharon Bruce.  Have to think about how to make this happen in the secondary upper math classroom.  Of course, with a good idea, there is always a way!


Day 112: Ungraded Check-In

I’ve talked earlier about check-ins here and here, but today I decided to try something new.  I had the students complete a check-in in the allotted time.  Since they are color-coded by highlighter markers, I had them switch papers with their table partner.  Here are the two versions:

CI 5.2a  CI 5.2b

You can see the two versions are similar, but results are different.

Then the “yellow” papers regrouped together and the “orange” papers did the same.  Then I asked them to check the work of their partner’s paper with the new group.  Two things happened: they worked a new set of problems AND kid conversation  happened.  It was a wonderful way to have students experience all problems on both versions.  More importantly, they asked for clarification about certain ideas while also deepening their understanding of the properties.  Writing out suggestions for correcting mistakes also helped them solidify the main ideas.

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What are some different ways you have kids “talk about math ideas?”


Day111: Time to Check-In with My Students

My students did the check-in today.  These are a mini-formative assessment on a half-sheet of paper.  Usually the problems are from in-class work and/or homework problems and are used as a gauge of my students’ skill acquisition.  I give 10 minutes for the students to complete.  They are quick and relatively easy to grade so turn around time is short.  Great little device for immediate feedback.


My students are serious about these check-ins, but they barely impact their grades.  They are worth between 5-8 points in the 20% category, so I see them as a win-win activity: they get immediate feedback without waiting until a quiz or a test and I get timely information about how the kiddos are doing in a semi-assessment situation.

Day 103: Share the Rainbow

The activity we did today in AP Stats was an Introduction to the Logic of Hypothesis Testing using Skittles.  I wrote this activity after being inspired by Adam Pethan’s video Hypothesis Tests: Introduction.  He had a wonderfully simple way of using a real life scenario (that used food) and gave me an awesome activity that connected sampling distributions to this new idea.


Because I wanted (needed) the sample size to be controlled and the sample proportion to be the same for all students, I used Adam’s random sample proportion of yellow skittles as the basis for building the logic of the hypothesis test.  They needed to draw the population distribution (labeled correctly) and write both hypotheses correctly with correct symbols (this is the FIRST time they have ever seen a Null or Alternative hypothesis).  They had to show me their answers on these first questions before they could get Skittles.  It gave me a chance to check every single hypothesis along with symbols and notation…great formative assessment.

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Once the student wrote the two hypotheses correctly along with the hypothesized population distribution, they could get a mini-cup of Skittles to munch on while they continued with the activity.20150211_105647 20150211_105839

During our study of sampling distributions, I emphasized ad nauseam what the probability meant and had the kids write an interpretation of the probability they calculated in their own words based on the mean of the sampling distribution AND the sample statistic comparison.  In particular, the focus was on the idea of the sample being “unusual” in our sampling distribution as reflected by the probability we calculated dovetailed easily into today’s lesson.  They determined what their level of tolerance for an unusual sample value would be based on the probability (area).  This will lead in nicely to alpha levels later in the unit.

skittle activity

Then they calculated the probability using the sample value and the constructed sampling distribution (of course they checked the conditions to build the distribution!!)  But looking over their submissions, we still have to work on testing the Normal condition…but we have months to do this, right?  Formative assessment is so great for highlighting misconceptions and missing details, isn’t it?  I also gave a silent yelp of joy as my students talked, discussed, argued, clarified and focused on understanding the big ideas.

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The last part of the activity reviewed confidence intervals again since the kiddos are having their test tomorrow on this topic.  Very few questions to me, but lots of intense discussion about how/why to approach these questions.  I would say that the 7 of the 8 math practices were in evidence today: sense-making, reasoning, argument, modeling, using tools, precision of language and calculations, and attending to the inherent structure of the problem.

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Finally, they submit their results electronically in Schoology so I can look at the results and determine the next steps.  All in all, I was really pleased with the success of this first-time activity.  I did work out the problems ahead of time, but using with students is always eye-opening.  Some tweaking is needed, but not as much as some of my first-time activities need.  How do you vet  your activities (make a careful and critical examination of them) before you use them for the first time?


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