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!

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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!

Let’s Build Some Bridges!

You’re either a genius, or the biggest idiot here.

– a colleague

Can’t I be both?

-me

It’s the first year of Keystone testing here in Pennsylvania, and everyone is adjusting to the fun changes.  And by fun, I mean time-consuming,  nightmarish organizational hoops to jump through provided by our wonderful state government.  This year, many of our 8th graders get zapped with the testing equivalent of Haley’s Comet: state grade-level testing, along with grade-level tests in writing and science….followed by the cherry on the sundae, this week’s Keystone Exam in Algebra 1.  It’s a shock they ever have time to actually, you know….learn stuff.  Hoping our kids don’t suffer too much bubbling withdrawal at the end of this week.

We have about 400 8th graders here, and many of them will take the Algebra 1 test this week.  Some, around 80, took algebra 1 as 7th graders and have already passed the Keystone.  Meanwhile, 40 or so are in a pre-algebra course and will take the Keystone next year as 9th graders.  So, what to do with 120 students, while their grade-level friends endure a state test.  For two days, and 4 hours, I have been given carte-blanhce, an emtpy slate, to keep 120 8th graders entertained.  And money!  Well, some money anyway.  What would you do?

My BridgeTomorrow morning, 120 8th graders will meet with me in the auditorium to learn their fate.  I have split the kids into 26 groups of 4 or 5 and gathered supplies for a popsicle-stick bridge-building contest.  The concept and many guidelines came from the site TryEngineering, which provides many neat and simple tasks for kids to encourage creativity and discovery.  I have worked with a colleague from our high school, who teaches an intro to engineering course, whose students found some great resources to share with the kids to get them excited about the project.  Two short and snappy videos they found from MIT feature simple bridge designs, with Lego-men being experimented upon:  Part 1 and Part 2.

SuppliesThe supplies are simple:  each group will receive 200 popsicle sticks, a glue gun, and glue sticks.  Teams will only receive the glue gun after they have drawn some sketches and discussed a plan for their design.  Most of today was spent organizing 26 boxes of sticks, and getting groups ready.  Groups will be graded on their design, how much load their bridge will hold, and how well they work together as a team.  And about those groups….all groups have a similar mix of “advanced” kids and “pre-algebra” kids, which I have assigned beforehand.  This mix led to the “genius or idiot” comment above from a colleague.  Yep, this could go badly.  But, it could go great!  Its too tempting to not try!

So, tomorrow we start building bridges.  My coach friend Gayle and I built a bridge on our own, which you see above, and we were quite proud of ourselves.  If time permits on Wednesday, we will test the strength of the bridges.  Our bridge snapped at 7000 grams.  But I am confident the kids will do a better job.

Looking forward to a fun, but chaotic, time the next 2 days!

Load Test 1

Load Test 2

A.P. Co-Teaching: Stats Meets Psych

For seven years before becoming an instructional coach, I taught Advanced Placement statistics.  I loved this course, as every day brought a new applied situation,  a new set of data, and a new, rich classroom discussion.  While many of my math colleagues have an aversion to teaching statistics (one friend from another school said to me “you’ll become the loneliest person in your department”), I think teaching the course gives an appreciateion for how we should be approaching data analysis in ALL math courses.  But that’s a post for another day.

This week’s stats twitter chat (#statschat, 9PM on Tuesdays) started with a discussion of the recently released AP Stats items, but later moved to post-exam activities.  As part of this discussion, cross-curricular options came up, and I mentioned a co-taught lesson I have developed with my AP Psychology colleague.  For a number of years, this teacher and I had discussed co-teaching a unit on experimental design, as the AP Psych course outline actually includes a nice chunk of material AP Stats students come to understand.  One section of the description, Research Methods, is right in the AP Stats wheelhouse:

  • Describe how research design drives the reasonable conclusions that can be drawn (e.g., experiments are useful for determining cause and effect; the use of experimental controls reduces alternative explanations).
  • Identify independent, dependent, confounding, and control variables in experimental designs.
  • Distinguish between random assignment of participants to conditions in experiments and random selection of participants, primarily in correlational studies and surveys.
  • Apply basic descriptive statistical concepts, including interpreting and constructing graphs and calculating simple descriptive statistics (e.g., measures of central tendency, standard deviation).

Yey for math featured in non-math subjects!  What a natural fit for a handsome, fun math teacher and a respected social studies teacher to join forces in a class lesson!  Some students comments that it seemed so out of context to have the two of us in the same class together.  Worlds colliding!  Dogs and cats shaking hands!

For two days, we led a discussion on correlation and causation, based on a curriculum module provided by the College Board.  In the “Teaching Statistics and Research Methodology” module, the section “A Lesson on Correlation” by Amy Fineburg was used as a framework for discussion.  Students were provided with an article to read beforehand and was used to generate discussion regarding student ideas of correlation, causation and experimental design.  Our pesentation to the students is given below, and was completed over 2 days.

Looking forward to sharing this experience at the AP Stats reading next month, along with a similar co-taught experience developed with my school’s AP Chemistry teacher.

Why is “Simplify” So Damn Complicated?

Making my classroom rounds this week, I came across a class reviewing concepts for the upcoming Pennsylvania Keystone Exams in Alegebra 1.  The PA Department of Education provides an eligible contect document with sample items on its website, and the class was working on the following question:

Item

Pretty standard problem.  Factor the numerator and denominator, cancel common factors, and you’re home.  But this class was struggling with the factoring review, so I stepped in with a different approach.  How about taking the given expression, and using a graphing calculator to evaluate it?  Sadly, the class was not familiar with the Table on their TI-84’s, but understood what it did right away:

Calc 1

Some nice discussions emerge here.  What’s with that “error”?  Is our calculator broken?  And some evidence over this function’s behavior emerges.  Note the slowly increasing values of y.

But how does this help us with the question at hand?  A number of students recognized that the correct answer would be the expression which had the same Y-values.  In essence, simplfying produces a different-looking expression with the same outputs as the original.  So, let’s try the answer choices.  Here’s A:

Calc 2

No dice.  Values are much different.  And a fantastic opportunity to discuss the difference between an output of zero, and an undefined output.  But eventually we get to D, and can check the tables:

Calc 3

Looks pretty good, butttt……..what’s with the errors?  And they seem different for some inputs.  But now we can review and discuss domain, and look at those pesky domain restrictions in a new light.

So, am I a bad person for bypassing the factoring review, and encouraging calculator use?  After the discussion, I reminded the class that factoring is a skill they need to have in their toolbox, but the alternate discussion of equivalent forms and assessing values was also worthwhile.  I feel good.


This classroom visit got me thinking about the nature of the word “simplify” in math class.  How often do we ask students to “simplify” in math class, and in what contexts?

Sometimes we want to simplify an expression:

Or maybe we want to simplify a rational expression:

Or perhaps we want so simplify a radical expression:

And make sure you simplify when there is a radical in the denominator (unless you are taking AP Calc, in which case we don’t care about such silliness)

For different situations, we have subtle differences in what it means to simplify, but is there a common goal of simplifying?  Is it just to make things look pretty?    And is a simplified expression always the most useful?  When is it not?

I’m curious if anyone has a short and snappy answer to “what does it mean to simplify an expression?”.  I invite you to participate and contribute your response on Todays Meet (click to participate).  If you have never used Today’s Meet, it is a nice, free way to gather responses.  Simply provide the link and start a conversation!  Feel free to share the link with your students a “bell ringer” activity.  If we get some responses, I’ll make a later blog post about them.

TodaysMeet

Last Night at Global Math….

Thanks to Megan Hayes-Golding for hosting last night’s session at the Global Math Department, where I shared some of my Tall Tales for Probability.  The recording is now available.  This was my second time presenting, and it is a unique experience.  I recognize that I talk WAY too fast at times, mostly because I am so excited to share my ideas.  But sitting alone on the couch talking to my laptop, and trying to assess reactions in the chat room make online speaking a wild ride.  I was the 2nd speaker of the evening, and enjoyed Chris Harrow‘s sharing the 4’s Game, and Chris Hunter‘s ideas for cooperative learning.  Always an uplifting experience to be around such excellent educators!

I had a few requests for the video of the hay bales.  I have put it on dropbox, but it is a BIG file.  Let me know if I need to zip it.