Monthly Archives: May 2013

More Great Conics Project News!

UPDATE – a newer post concerning this project, with rubric can be found with this post.

I can tell it’s the end of the school year, and math teachers are looking for fun math projects to do with their classes, as the search terms which get people to my blog contain lots of references to “conics projects, “math art projects” and the like.  The searches have led to many hits to my conic sections art project blog post from last June.

At my home high school, this is the second year we have done our long-standing conic sections art project using the Desmos calculaor, and this year’s submissions have raised the bar considerably.  The most improvement has come from working with students to restirct domains, which has made more complex drawings easier to manage.  Here are a few to share, but look for an announcement from Desmos, with whom I will present a webinar on June 6 and give you some ideas for getting started with your class.

First off, a tiger, which took over 100 individual equations to create.  Stunning!

Tiger

Next up, an ambitious student who took 87 different picture “slides” to create this animated gif.  I wish I was half this creative when I was 15!

Falling Man

Check out the recording from the Global Math Department for more information, and be on the lookout for webinar information on the 6th!  Meanwhile, let me check some of our other Algebra 2 classes for some promising projects!

The Binomial Theorem Jigsaw

Is there a rule in math which encapsulates more great stuff than the binomial theorem?  Increasing powers of x, decreasing powers of y, a bunch of terms to look at…and hey, what’s this….combinations, you say?  I’m in.  But please don’t remove all the fun discovery moments there are to be had here.  Here’s a jigsaw activity which was handed down by some of my “senior” and now-retired colleagues, and is now infused with my need to get kids moving around and discovering things.  I hope you enjoy it.

The students will need some background on a few things here.  In particular, students should already be fluent with multiplying binomials, and have some familiarity with combinations.  Copy the first 4 pages of the binomial theorem Jigsaw Activity and have them ready to go.  Now it’s time for some movement.

Arrange classroom desks into groups of 4.  If your class roster number is not a multiple of 4, then you can have some 5’s, and we will deal with them soon.

binomial

In each group, have the students number themselves from 1 to 4.  It is important that each group have one of each number, and that students know their number. If you have any groups of 5, then allow for two “1’s” in a group.  This will be helpful, as #1’s actually have the trickiest job in this task (but don’t tell them that yet!).

binomial1

Next, hand out the packet of tasks.  The packet looks intimidating, but each student is only responsible for completing one page.  All students who are “1’s” are responsible for completing Task 1.  2’s will complete Task 2.  3’s = Task 3.  And 4’s have Task 4.  On your command, students will break from the group, and move to a new location and meet with all oftheir  similarly-numbered classmates to complete the task together.  After all students have completed their task, they will re-join their group and share their findings.  It is important that students understand the need to complete their task thoroughly and thoughtfully, as their group success depends upon it!

binomial2

Now, send all the numbered students to meet with their group and complete their task:

SUMMARY OF THE TASKS:

  • In Task 1, students are asked to expand (x+y)^4, starting with (x+y)^2 and working their way up.  This task usually takes the longest for groups to complete, and requires attention to detail.  In the end, groups are asked to list the co-efficients of the terms they get.
  • In Task 2, students are asked to list the sample space for 4 tossed coins.  The task is not difficult, but requires attention to detail in making sure all possibilities are provided.  Groups are then asked to list number of ways to get 0 heads, 1 head, 2 heads, etc., in the 4 coins.  Note, groups often give probabilities here….we want the COUNTS!
  • In Task 3, students complete a number of rows in Pascal’s Triangle.  The task in not tricky, but requires attention to detail in filling out the small boxes.  The group is then asked to list the numbers in row 4 of the triangle.
  • In Task 4, students are asked to compute a number of combinations by formula.  I usually try to have 4-function calculators at the ready for this group, and want them to really focus on the formulas.  In the end, the group is asked to list the combinations of 4 items taken 0, 1, 2 3 and 4 at a time.

The tasks often take about 20-30 minutes for groups to complete and check.  Bringing groups back together and having them share their findings with their teammmates often takes another 20-30 minutes, so this may need to be done over 2 days.

binomial3The big reveal occurs as groups begin to realize that all 4 tasks have the same “answers”:

1-4-6-4-1

We’re now ready to start exploring the amazing connections between 4 ideas (binomials, coins, Pascal’s Triangle, combinations) which seemed quite different math-wise, but have some strong connections.  Now the ideas come fast and furious as we explore the connections.  Page 5 of the packet provides some guiding questions, but don’t feel tied to the linearity of my questions.

Let’s think about (x+y)^5…..

  • How many terms will it have? 6
  • How will the powers of x and y behave?
  • How can we find the coefficients?  Pascal’s Traingle!
  • How else can we find the coefficients?  Combinations!

How about coin flipping:

  • If we flipped 6 coins, how many items would be in the sample space?  2^6
  • How many of these possibilities will have all heads?  1
  • How can we find out the number of ways to get exactly 2 heads?  Pascal’s Triangle

Eventually, this will become the basis for “assembling” the Binomial Theorem.  You’ll be surprised at how much of it your students will be able to piece together after this activity.  Your job is to just help out with some symbols and some common language.

Math Review: Secret Phrase Scavenger Hunt

Another great night of learning at the Global Math Department last night, where Matt Vaudrey and Megan Hayes-Golding shared their ideas for Exam Reivew That Doesn’t Suck.  Enjoy the playback and admire Matt’s enthusiasm for teaching, and great ideas for keeping kids engaged.  And thanks to Megan for her continued willingness to facilitate and share.

One of my favorite test reivew activities is a Secret Phrase Scavenger Hunt.  The problems here are from a review of inequalities, but can be easily adapted for many grades and courses.  Full disclosure: while looking through files for this example, I was shocked to discover that this Word document is one of the oldest files in my network drive, from September 2000!  Maybe I need to edit stuff more, or maybe this activity is just plain perfect.

Here’s the idea: take a sheet of probems, and assign each problem a “secret letter”, so that problems completed in order will spell out the “secret phrase”.

Inequal sheets

Make an index card for each answer, and tape them around the room or the hallway.  I usually place the letters on the back of the cards, but for this activity they appear on the front.

Cards on board

After the sheets are handed out, teams complete problems and may get up at any time to hunt for answers.  I usually assign students to teams for this activity, and it is interesting to observe different approaches.  Some teams will complete all problems together, then hunt for all solutions.  Other teams will complete some problems, hunt for solutions, then go back to work.  Another approach is to split up the problems – “divide and conquer”.  This apporach often leads to Civil War as one or two students in a group will invariably make enough errors to bog down the process.

The winner is the first team to come to me with the “Mystery Phrase”.  To keep the ball rolling, I will often give award to the first 2 or 3 teams to find the phrase. Here are a few tips for setting up your hunt:

  • Don’t be afraid to make your phrase something goofy.  After students fill in a few letters, they may try a “Wheel of Fortune” approach.  Unpredicatable phrases avoid this some.  For my inequality sheet, I chose the phrase “TWO HAWAIIAN UKELELES” – not easy to guess.
  • Make harder problems the “key” letters in your phrase.
  • Adding a few “distractor” cards – cards that are not solutions to any problems – also works nicely.
  • If students come to me with an incorrect phrase, I do not tell them where they went wrong.  It is up to the group to re-visit their problems and troubleshoot mistakes.

Hope you and your class enjoy the phrase hunt!