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Day 70: Trig Jeopardy

Aren’t Google Searches great! I wanted a quick but engaging way to welcome back my precalc students after our long two week break.  We have a “Big-A Trig Assessment” on Thursday and Friday this week and I wanted a fun way to ease back into the school and trig routine.  I did this search this morning: trigonometry jeopardy powerpoint and found that Ms. McClellen of Charlotteville City Schools had a great powerpoint for reviewing the basics of our Chapter 4 Trigonometry topics; unfortunately, her website wasn’t available so I could give her more and full credit.

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I did some tweaking to add in topics that I wanted to review and it was so easy to do!

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We had lots of fun reviewing the first day back.

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Day 63: It’s Time for Clock Buddies!!

I love having students interact with each other, chatting about mathematics and sense-making about new concepts while connecting skills to those concepts.  But “turn to your partner” or “discuss in your groups” can get stale after a while.  I needed some new strategies for getting my kiddos face-to-face.  I found some good ones, but a keeper is Clock Buddies (Partners).

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I shared about the process of setting up clock buddies last year, but forgot how easy it is to have “extra kids” after they all sign up. This year with my AP Statistics class, I followed the same process with that same result…isn’t that the definition of insanity?

Well, I reflected afterwards about how I could make the process smoother.  So with my precalculus students, instead of the “free for all” of signing up, I had them find a 1 o’clock buddy that wasn’t in their group.  Then they sat back down.  Next they found a 2 o’clock buddy that was in their group. For 3 o’clock, they found someone sitting in an “odd” group if they were sitting in an “even” group; then they sat back down. Etcetera, etcetera.  This worked much better and had fewer individuals left out early in the process.

Day 57: Non-Threatening Forced Participation

Once in a while I employ what I call “forced participation” strategies for getting random students involved in the conversations in a somewhat non-threatening way.  Some ways I’ve used in the past:

  1. Use randint( on my TI Nspire to select a random number associated with a student.  Sometimes I have that student hit the button to select the next participant.
  2. Drop a coin on the seating chart.
  3. Select a tongue depressor with students names on them
  4. Dave Bunker’s Random Name Selector (uses Macromedia Flash Player 8)

I saw a new teacher to our building use cards on a ring with the student’s photo on one side; the cards are color-coded for each class.  Then I grab the ring from my front board, and randomly flip to a card and that student is selected.

I also had students write down activities they are involved in at school on the back.  I also could have them share other interesting things about themselves to get to know them better.  The cards are hung in a convenient and accessible spot at the front of the room.  Just another tool in my toolbox.

What are some “tried and true” strategies you have used to encourage students to participate?

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.

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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(.

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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.

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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.

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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 157: It’s an Eclipse – NO, an Ellipse!

The patty paper activity we did today is so relaxing, yet kinestheticly engaging and intellectually satisfying for most students.  Art Mabbott initially shared these activities informally during one of my TI User Groups, followed by using the TI Nspire documents to make sense of what had happened during the paper folding.  I just loved the chance to have some hands-on activity within a higher level math that I seized the opportunity.  And the students get a chance to talk to each other both about math and other things…we don’t get a chance to do that often.

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After the experience, I spent some time rethinking/redoing the activity while writing guided explorations for the parabola, ellipse and hyperbolas.  Here’s a snippet of the ellipse exploration:

  1. You will need a sheet of patty paper with a circle filling most of the paper. Does not have to be centered on the patty paper.
  2. Locate the center of the circle. Call it point C.  Describe your method:
  1. Choose a point on the circle and call it point P.
  2. Mark a point inside the circle that is somewhere between the center of the circle and the circle itself (it doesn’t necessarily have to be in line with P and C)– this will be a focus. Call this point F.
  3. Fold point P (the point on the circle) onto point F (the focus point). Make a good crease.  With your pencil, draw in the line on the fold.  Also draw the line FP.
  • What is that fold line you drew?  Think back to geometry.  There are two correct answers.
  • Write on the fold line that this line is the …. (ask your teacher).
  1. Pick another point on the circle. Repeat the process from step 5 and make another good crease. You do not need to draw in the line on the fold, though.

As the kiddos were folding away, they were making conjectures about what they were seeing: is it another parabola?  my lines aren’t making anything! Cool, it might be a circle.  No wait, it’s too flat.

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What I really like about using the patty paper is that the fold lines are easy to see and the kids can write on it too.  Once the class finished their folding, we then explored some patterns and relationships with some guiding questions that eventually lead to the geometric definition of an ellipse.

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Looking at the guiding questions, I need to think more about how “leading” they are or how “open-ended” they are.  My sense in listening to the discussions was they weren’t sure what they were trying to do.  I need to rethink these questions for next year:

  1. Now, go back to your original point P on the circle. Draw in the radius of the circle using point P and point C.  The radius will intersect the original crease line at a point.  Call this point E.  Describe some properties of this point E:
  1. Using a ruler, measure segment CE, FE and CP.
  • What relationship do you notice between these three segments?
  • Is it true for other points that have the same properties as point E?  Explain
  • What has this to do with the Focal Radii definition of an ellipse?

Do any of you have some thoughts about these questions or what could be asked differently to get students sense-making, organizing, analyzing and reasoning to impose structure to this situation?

 

 

Day 135: Rock, Paper, Scissors Anyone?

Do you remember playing Rock, Paper, Scissors as a kid? As an adult? Against a computer?  I’ve seen various lesson plans around RPS and probability of winning, but none seemed to fit the level and sophistication I was looking to use with my AP Stats class.  Well the New York Times science section had a great article Rock Paper Scissors: You vs. the Computer.  If you haven’t played the computer, its a kick and I highly recommend it!

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Then I remembered reading Bob Lochel’s post Rock-Paper-Scissors and Two-Way Tables.  And found his second post, Chi-Square Tests: Rock Paper Scissors.   I liked the idea, but the gold came in the response sections of both posts.  Doug Page  shared a worksheet he has developed for using the Rock, Paper, Scissors applet and also a Google Form for having students submit results.  I stole blatantly! RPS 1RPS 2
Since my students use the iPad, they can’t access Flash animations, so I had to assign the play as homework.  I hate to do this because so many kids just don’t follow-up and complete the assignment – and we don’t get the data we could if we collected in the classroom.  And the same happened with this class, but at least we had enough to continue with the introduction to Chi-Square test of Homogeneity.  I think next year, they will use their own data rather than the class data…or compare their results to the class results.  Will have to think more about this next year.

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I required the students to calculate the components by hand – they need to know where the components come from and how they are related to the final χ² statistic.  Once we finished this problem, we tackled another problem using the calculator.  Because of the hand-calculating of the components earlier, they then understood what the expected values matrix meant and the components matrix.

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How do you incorporate electronic experiences to develop engagement?

 

Day 130: Going In Circles

Short post today as I hope to join the #Statschat on twitter…every Thursday night at 9pm EST hosted by Bob Lochel.

Today in Precalculus, we ventured into parametizing circular motion.  Good ol’ Geogebra to the rescue again!  I just have to send out a big THANKS to all of you who create and post such fabulous documents!  With little (to no) intervention from their teacher (why was I even there?!), my precalc scholars developed the parametric equations almost seamlessly based on their “playing” with the figures and sliders.  They were so quick that I had to ask them “what would happen if we wanted to go around the circle three times as fast?”  And they jumped on it like a bunch of math-jackals!  You can see in the fourth photo their almost immediate result.

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Once they had the conceptual understanding, we moved into the practice/application phase of the lesson.  They worked on the following problems:

Merry-Go-Round Application

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A rider jumps on a merry-go-round of radius 20 feet at the pictured location. The ride rotates at the constant angular speed of ω = -p/7 radians/second. The center of the platform is located 50 feet East and 50 feet North of the ticket booth for the ride.

  1. What are the parametric equations describing the location of the rider?
  2. Where is the rider after 18 seconds have elapsed?
  3. How far from the ticket booth is the rider after 18 seconds have elapsed?

Applying Circular and Linear Motion

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A six foot long rod is attached at one end A to a point on a wheel of radius 2 feet, centered at the origin. The other end B is free to move back and forth along the x-axis.  The point A is at (2, 0) at time t = 0, and the wheel rotates counterclockwise at 3 rev/sec.

  1. As the point A makes one complete revolution, indicate in the picture the direction and range of motion of the point B.
  2. Find the coordinates of the point A as a function of time t.
  3. Find the coordinates of the point B as a function of time t.
  4. What is the x-coordinate of the point B when t = 1? You should be able to find this two ways: with your function from part (c), and using some common sense (where is point A after one second?).
  5. Find the first two times when the x-coordinate of the point B is 5.
  6. Write and solve one question about this situation.

As a side note, I have our tech trainer setting up a PD on Geogebra because lots of my mathies in my building are interested because of what we’ve done with our kids so far.  I am SO thrilled!

Day 125: Intellectual Need in the Math Classroom

I can’t believe I’m out of the classroom AGAIN, but I just couldn’t miss the opportunity to hear and interact with Dan Meyer in his workshop: Intellectual Need in the Math Classroom offered by our local educational service district.  So many good things today, but I did come away with one nugget that I’ll be ruminating on for the rest of the year (teaching life?!).  Dan started the workshop by posing the question: How do we engage students in difficult mathematics?  And he suggested the three biggest responses (by teachers and textbook companies) are:

  1. Make math real world
  2. Make math job related
  3. Make math relevant

We then took a humorous look at these “suggestions” in action: interesting covers (because we want the kids to like what we have to offer, so we’re desperate), have career interviews or real-world connections dropped into the middle of a unit (make the work seem related to job acquisition), try to have real-world problems (do something to a real-world picture to link it to the math we’re studying – but almost lying to the kids?) or try to take an uninteresting problem and re-work it to try to connect to what kids might relate to (but does an image of Starbuck’s coffee make the problem about exponential growth any more engaging?).

In Dan Meyer fashion, he offered a Dandy Candy Video and activity (which can be found at 101Questions) to look at strategic moves to engage students at the beginning of the class.  And through the debrief of our experiences, we revisited the question:  How do we engage students in difficult mathematics?  And Dan’s answer is:

  1. Start a fight
    • instigate an intellectual (or emotional) fight
    • get them arguing with each other and you
    • use student answers to get their opinions out
  2. Turn the math dial up slowly
    • start with their everyday experience
    • make the problem vague and bring in the math as needed
    • have students guess high and low answers, best and worst, etc. along the way
    • slowly add vocabulary and layers of “math” to the experience
    • you can always add to their experience.  You can’t subtract what has been done. So think before you give the “math” component
  3. Create a headache – to provide the “aspirin.”
    • ask questions to get the students to think more deeply
    • ask student to describe how to do something without the math tools – they’ll “beg” you for it eventually, if you don’ violate #2
      • for example, ask students to describe a precise location without a precise tool (grid system.
      • in the Dandy Candy example, ask to determine “the best.”
    • challenge their thinking to the next level: more precise, more efficient, best way

It was an invigorating day, with lots of conversation and pushing on our everyday practice.  I really appreciated Dan’s disclaimer that you don’t do this every day – you’ll burn out quicker than a flame in a hurricane (my analogy).  He suggests only one of these kinds of activities per unit.  Lure the students to the mathematical water through an engaging and meaningful activity, but then its okay to provide direct instruction as needed to develop skills and sense-making activities to develop concepts.  I have always believed balance between activity-based instruction and direct-instruction makes for the most productive and growing classroom.

I’m always thrilled to talk with other educators to learn new things and today provided that opportunity for me.  It was a delightful and fruitful day.

 

 

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.

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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.

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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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