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Day 101: Logs in Life Peer Review

My precalculus students completed a project on logs and logistics function in real world.  Before break, they signed up for a topic where logs or logistic functions are used. Students signed up for two possible topics and referenced them with an online site, which they submitted through a Google Form. Since Google Forms are time stamped, I able to use a first come, first served approach to topic assignment. I allowed up to 3 people to select the same topic, but after that, they got their second choice. Out of two classes, I only had 4 students that needed to determine a new topic.


Here is a brief synopsis of what was required:

THE TASK: The end product of your webquest will be a BROCHURE describing what you have learned about your chosen application of logarithms. • Brochure style (tri-folded) • Typed in your own words • Description and history of the application (e.g. “what is a decibel?” and “who invented the Richter scale?”) • Description of mathematics used in calculations – specifically logarithms with an example • Description of at least one career in which this application is used • At least one relevant graphic Citation of other websites used beyond the ones given (just list the url)

Students submitted both a physical copy and an electronic one.  I created groups of brochures with different topics and none created by a particular group member.

Today they did the peer review of the brochures, again using a Google Form.  I used radial buttons for the rubric part of the assessment. There was also a required comment section. I asked students to give both a positive comment about an intellectual aspect of the topic presentation and a constructive criticism. Part of their grade is their review of their peers and their peers’ review of their pamphlet.  You could have heard a pin drop.

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We had 10 minutes left in class, so I tried something on the fly…never know how that will go. I asked the groups to discuss and determine the most interesting brochure. They then passed that one on to the next group.  Maybe this will be part of the whole process next year.

Day 84: Hands on Polar Equations

Many years ago (I don’t want to say how many but it was before we had access to graphing calculators), I had my students hand draw various polar equations.  Having technology so readily available now, this project had been shelved and collecting dust. I decided today that I wanted my precalculus students do experience a tactile activity today.  Although I love polar equations, it just didn’t fit the timeline this semester (maybe we’ll revisit next semester).  But why not give them an initial exposure to Polar Equations and their graphs as a way to review polar coordinates just before finals…and besides, the graphs are fun and unexpected.

I wrote up the activity and made some decisions about how to have students make posters.  A few years ago our department invested in large boxes of poster paper (you can see the poster paper in many of my posts) which was a pretty cheap investment for the amount of paper…and the box seems to last for a couple of years at least.  I wanted the paper to be a square in order to find the center and create the polar graph paper.  Once I folded to create the square, I thought, why not use the flap for added info.  Here is my prototype that I had posted on the front white board.


I then wrote up the directions, including a chart for my students to record the r-values for their equation. I did not tell them the names of the graphs, just handed out various equations; each student got a unique one. I was ambitious with what I wanted them to do afterwards (and we didn’t get that far), but I would keep the “interesting facts” part of the activity.

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Then I handed out the directions and away they went.  I had hoped they would finish in the class period, but some of my students are very meticulous so it took a while for them to create the polar graph paper on the poster.  I am still pondering how to improve that aspect of the activity.  Anyway, here are some photos of the activity in progress:


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Hopefully they will finish on Monday.  Happy Week-end

Day 80: Folder Trig

One activity I started last year to get students thinking about constraints and possibilities of an application problem is what I call the “Folder Activity.”  This time around, I wanted my students to be thinking about what makes a problem sinusoidal and what information from the situation helps to create a mathematical model (equation).  In addition, I want them to be thinking about what kinds of questions could be asked about the situation that can be answered with the graph and/or the function equation.

To set up this activity, I needed lots of sinusoidal application situations that are different from the typical situations of ferris wheels, merry-go-rounds and oscillating springs.  The internet is a great source for these and I found many!  I then “pull apart” the problem situation from the questions asked.  The students only get the problem situation…here is one example of the eight different problems I had available:

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First of all, students open the task card on their iPad.  This includes group roles, materials needed, the task and the product.  Here is a copy of the tasks and folder layout.

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The groups then get a folder, a quarter sheet headed with “Known Information,” a second quarter sheet headed with “Mathematical Relationships,” and a half sheet of graph paper.  At this point they read and discuss the problem information, determine known information and identify potential useful mathematical relationships…they know not all of the possible mathematical relationships will be used, but it is helpful to think about them.  The discussions are rich, the misconceptions get “fixed” most of the time, and once the questions are actually asked, they already have a plan to answer them.

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Once the information is organized and glued to the folder, then the group can pick up the actual questions and begin working on them.

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The final product is the folder with the solution written out neatly…each person’s handwriting needs to be apparent in the solution write-up.  All is glued in the proper places and finally the group evaluates to what extent they used the 8 Math Practices along with a 1-2 sentence summary for each practice.

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Day 72: Reviewing for the Big Trig Assessment

We are finally at the end of the introduction to trigonometry in precalculus…and to have an assessment that covers the breadth of the topics, my teaching partner and I decided we would create a two-day assessment, one day being non-calculator part and the other day being a calculator part; so each day is written as a class period assessment.  In order to get ready, my students were given a review sheet, but in class, having me drone on about the topics and going over problems can be deadly.  So….we created 8 Trig Stations (see the post from last year to get some more details) that cover the 8 big topics.  Students are in groups of 3-4 and have 10 minutes to work at each station…even if they don’t finish, they move on.  This year, students requested to take photos of the questions if they didn’t finish along with the answers (found in the orange envelopes).

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Most students found the experience helpful for identifying the areas they needed to focus on for studying.  This year I even taped some of the conversations (unbeknownst to the group) so I could hear them use some of the mathematical practices including Sense-make, Reason, Argument, Model, Tools, Precision of language use, and Structure.  They were very enlightening and encouraging!


What do you do to check on student use of the math practices?

Day 67: Pirates of the Caribbean Visit an AP Stats Classroom

Today, as we are finishing up our study of Combining Random Variables, I decided to use an activity I found on Chuck Bakers website about Pirate Liar Dice.  We first watched a clip from the Pirates of the Caribbean where the pirates are playing Pirate’s Dice, a game in which each player has 5 dice and a cup.  Check out the clip:

Then my students talked about what happened during the game (some said they didn’t have a clue) but most were able to recall the progression of the bids: 4 fours, 4 fives, 5 fives, 8 fives, 12 fives.  They still weren’t sure what was happening, so we then watch this “Liar’s Dice in 60 Second” to clarify the rules:

Finally, they played.  I don’t have enough dice for each student to get 5, so we used our Nspires and the randint( command to simulate rolling 5 dice.  Then the bidding became.  They had fun for sure and there were some subtle probability ideas percolating.


Our debrief after the game included the following questions:

  1. What were some things you thought of as you made bids?
  2. How did you know how many dice were left in play?
  3. How did you keep track of what dice the other players may have had?
  4. How improbable did a bid have to be for you to challenge it?
  5. When was it easier to make a safe bid?
  6. When was it harder to make a safe bid?

One of our vice-principals (who taught AP Stats in a former life) happened to be walking by so he popped in and joined in the discussions.  Fun times for all, and hopefully a memorable experience for my students.

Day 60: Trig Graphs in Under 15 Minutes!

I want my precalculus students to be able to quickly AND accurately graph sinusoidal functions, regardless of the mode, the form or the apparent complexity of the equation.  After all, there are really only up to 5 transformations that can happen to a sinusoidal function: reflection, vertical and/or horizontal dilation, and vertical and/or horizontal translation.  And I want them to be able to clearly and accurately identify the transformed quadrantal points by their coordinates.  Here one of my students chose to color-code the parts of the graph.  The other shows how to determine the horizontal coordinates.  Both of these are done using Notability on the iPad and submitted to Schoology, a learning management system (LMS) our district uses.

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I continue to use Foerster’s approach to graphing sinusoidal functions (sometimes referred to as the window method) as I find students have the most success AND understanding.  As part of the development, I want students to create a point-rule and a verbal description of the transformations on the parent function that result in the final graph.  But the practice can be boring and potentially fraught with errors in thinking if students begin graphing in isolation.  So I use complex instruction via a group poster to solidify the process before my students practice on their own.  I have a set of 6 colored, various sided dice in a green-topped container.  Student roll the dice and it determines the equation.  Then they have lots of other things to do once they get the equation.


This year’s students were even more ready to complete these graphs.  I think we did a little more sense-making prior to actually starting to graph and this made it easier for them.  I also used the Online Stopwatch to help them monitor their time.

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I even had one of my students, who proclaimed early in the year that I had students do way TOO MANY posters, say that he now likes doing posters because they help him understand the ideas better.  Glory be, he made my day!!

What mini-triumphs have you experienced lately?

Day 59: Gettin’ Triggy with Hedbanz

I needed a fun but meaningful way to practice writing equations for sinusoidal functions. I found this idea last year for practicing quadraticspracticing linears and practicing rationals.  I figured I could adapt it for trig functions but I just didn’t have the energy to put it together.  This year I was determined to get it done!


So what is this fabulous activity, you say?  Have you heard of the game, Hedbanz: The Quick Question Game of What Am I? The basic premise for kids try to guess the object that is in their headband based on clues from other players. When I announced that we were going to play Hedbanz, my students were utterly goofy in their excitement.

Initially, I thought I could use Infinite Algebra 2 by Kuta Software to quickly generate graphs.  Unfortunately, I couldn’t control the attributes nor the way graphs were drawn (all horizontal scales were in π forms and max/mins were not easy to determine).  They just didn’t fit what I was looking for; that is, they just weren’t “friendly” enough for this high-paced activity.  I then perused various precalculus books looking for good graphs that students could somewhat readily determine the critical attributes.  I found 32 different graphs, some in radians, some in integer form with clearly determinable extrema.

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To put together the game, I made the graphs approximately the same size, printed them, cut them out and glued them to different colored 3 x 5 index cards, in groups of 4, although I didn’t end up using the cards as a group activity.  I numbered the cards 1 – 32 on the back of the graphs.  I also found sets of 8 neon colored girls elastic headbands for $1 at our local dollar store, so I bought 5 sets for a total of 40.  I also made a set of answers that I posted at the front of the room for students to self-check.


The kids just loved the activity and the face-to-face discussion was awesome.  I heard things like: “what’s my amplitude/” “what the period?” “what’s the phase shift if I want to write a cosine function? “Hey, the amplitude is NOT the maximum value…you made me get this wrong!”

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Sam Shah has the most detailed approach to using this type of activity with your students.

Day 56: Time to Review Probability!

We are getting ready to test the probability chapter in AP Statistics and it’s time to review.  It’s time to use Review Stations again (used to review the summer work this fall). For each review station, I used my acrylic stand-up frames (I purchased last year at the Dollar Store), a laminated set of problems, an envelop with answers (for group self-checking) and an extra calculator.  At some stations, I also included a laminated first semester formula sheet.  I found problems that reflect common student mistakes or misconceptions as well as a smattering of AP exam questions.


I used to put the stations at each group, but with my bigger science room I am using the counters and space to get my kiddos up and moving.  So super quick to set up and take down between classes!


What kinds of activities to you use for reviewing that is student-centered and self checking?  I’d love to hear about other ideas!

Update 12/4: These review stations (along with some of the other activities we did during the chapter) sure resulted in one of the best performances on the probability chapter ever!  Bravo!!

Day 52: Unit Circle Game

Getting students to know the special Unit circle values can be such a tricky thing.  It is some rote memorization, but hopefully rooted in sound conceptual foundation.  But how to have students practice in a somewhat meaningful way.  Well, the Unit Circle Game is one way.

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Last year, I thought the game didn’t have much impact.  But a student shared via information I ask for when I write letters of recommendation, that she found the game particularly helpful for her to visualize the values on the Unit circle.  So I decided to do it again…but with some modifications in the play to make the game run more smoothly.  And it did!  Great Friday activity.


Day 51: Human Venn Diagrams

Decided today that I wanted to get the kids up and moving a little in AP Stats.  We are working on probability right now, and wanted to help make the connection between Venn diagrams and two-way tables in a personal way.  I have two big balls of thick colorful yarn, magenta and baby blue.  Cut out two long pieces and tied into circles.  Also used painter’s tape to make a gianormous two-way table on my classroom floor.  As my students walked into the class, they were already curious about what we were doing today…yay, instant engagement.


I had two sentence cards made up with MALE and PETS.  Simple, but hoped there would be some overlap and some students that didn’t fit either.  I put the two cards on the floor and they created the Venn diagram.  I then asked what each group represented.  Pretty simple.  Then I asked them to rearrange themselves onto the two-way table…using the same cards as the table labels.  Again asked what each region meant as well the complement of various descriptors.  Then we did the whole activity again, but they determined the two descriptors.  Very interesting and organic discussion about what works for descriptors and what needs to be thought through; for instance, ACT vs SAT for one descriptor or for both, or number of siblings, etc. We then did notes and they instantly were able to make the connections.

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One thing I would change is perhaps ask a few probability questions based on the Venn diagram or two-way table, including “and” “or” and “given that” verbiage. Overall a very successful and minimal prep activity.  It’s a keeper for me!

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