Category Archives: Middle School

4 Engaging Ideas From Twitter Math Camp

This past week, over 100 math teachers descended upon the Drexel University campus for Twitter Math Camp 2013.  It was a fantastic opportunity to meet people I had communicated with via Twitter for some time, make new friends, and share math ideas.  It’s a real rush to hang out with colleagues who share similar ideals on math instruction, and a commitment to improve our practices.  Check out the hashtag #tmc13 on Twitter to look back on some of the action and reactions, and find new math folks to follow.

While there’s so much to share from TMC13, I know there are many math friends who couldn’t attend who are looking forward to hearing about the goings-on, so in this post I share 4 ideas from this year’s Twitter Math Camp I am eager to try in my classroom right away.

EliELI’S BALLOONS – Followers of the blog know that I am a big fan of the Desmos online graphing calculator.  The highlight of the week for me, and I suspect for many, was having Desmos founder Eli Luberoff model a lesson using his creation.  Eli’s enthusiasm for sharing Desmos, and his sincere desire to work with teachers to improve the interface, are infectuous.  There were many “oooh” and “aah” moments from the assembled group, and a loud cheer for the “nthroot” command…yes, it’s a pretty geeky group!  (thanks to @jreulbach for tweeeting out the great picture of Eli showing off Desmos’ position when you Google “graphing calculator”)

Eli’s lesson idea has a simple and engaging premise:

  • Hand out balloons
  • Blow up the balloon.  For each breath, have a partner record the girth of the balloon
  • Consider the data set

Balloons

That’s it.  No worksheet.  No convoluted instructions.  Eli walked us through an exploration of the data set using Desmos, using the table to record the data, and considered various function models: is a square root model?  Is it logarithmic?  The group eventually settled upon a cube root as the proper model – and how often in class do we encounter data best modeled by a cube root?!  Since the explanatory (air entering the balloon) is volume, and the response variable (girth) is linear, the cubic model makes perfect sense. Fun stuff.  But wait…there’s more!  Eli then analyzed the fit of curve by looking at the squares of the residuals.  Click the graph below to check out my best-shot recreation of Eli’s presentation, and play around with the fit of the curve by toying with the “a” slider.

More great new additions to Desmos are coming.  Thanks to Eli for letting us preview some of them!  Was a pleasure meeting you and hearing about your fascinating story.

GLENN’S PROBLEM POSING – Glenn Waddell is a colleague I feel I have a lot in common with, in that we have both experienced the frustrations of trying to “spread the word” to colleagues of the great new ideas, and strong need, for inquiry-based mathematics.  In this session, Glenn presented a framework for problem posing in mathemtics which can be employed equally-well with real-life problems (see the “meatball” example in Glenn’s Powerpoint, which was adapted from a Dan Meyer “math makeover” problem) or with a garden-variety drill problem.

The framework asks that teachers lead students in a discussion that goes beyond just the problem in front of us.  Think about the many attributes of a problem, list them, consider changes to them and their consequences, and generalize results.  Glenn suggests the book “The Art of Problem Posing” by Brown and Walter as a resource for getting started, which employs the problem posing framework.

Framework

Glenn led the group through an exploration of a quadratic equation, where we started by listing its many attributes.

Glenn Problem

Now we consider changes to attributes:

  • What would happen if there were a “less than” sign, rather than equals?
  • What would happen if the last sign were minus?
  • What is it were an x-cubed, rather than x-squared?

There’s no limit to the depth or number of adaptations, and that’s why I like this method of problem posing for all levels of courses.

Download Glen’s presentation on the TMC wiki, and explore the wiki to get the flavor of many of the sessions.

A PAIR OF LESSONS FROM MATHALICIOUS

Logo

If you have never visited Mathalicious, go now….take a look at some of the free preview lessons, and you will become lost in the great ideas for hours.  THEN, make sure you sign up and get access to all of the engaging lessons.  Here is a company that is doing it right: lessons come with a video or visual hook, data which naturally lead to a discussion of tghe underlying mathematics, and just the right amount of structure to encourage students to contribute their thoughts and ideas.  At TMC, Mathalicious founder Karim Kai Ani led the group through two lessons.  A brief summary is given here, but I encourage you to check out the site and subscribe….you’ll be glad you did.

The “Romance Cone” – What is the appropriate age difference between two romantic partners?  Is there a general rule?  A fun lesson, “Datelines” on Mathalicious, where students explore a function and its inverse, without using those scary-looking terms.  I have been looking for an opening activity for our Algebra 2 course, which brings back ideas of function, inverses and relationships, and looking forward to trying this as a my first-day hook.  Also a great activity for Algebra 1.

PRISM = PRISN? – I have led my probability students through an exploration of false positives in medical testing for many years, and I like how this activity puts a new twist, and some great new conceptual ideas, on the theme.  “Ripped from the headlines”, this lesson challenges students to consider government snooping, and the flagging of perhaps innocent citizens.  If a citizen is flagged, what is the probability they are dangerous?  How often are we missing potentially dangerous folks in our snooping?  What I really liked here was the inclusion of Venn Diagrams, with sets representing “Flagged” and “Dangerous” people, where the group was asked to describe and compare the diagrams.  Fascinating discussions, and a good segue into Type I and Type II error for AP Stats if you want to take it that far.

This lesson does not appear to be available on the Mathalicious site yet, (update from Mathalicious – will be released in the Fall) but will be using it when it is completed!  Later that day, the TMC teachers broke into smaller groups to gain behind-the-scenes access to the Mathalicious writing formula.  Thanks to Kate and Chris for sharing, listening, and giving us all the opportunity to contribute ideas.

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