Fractals, Triangles, and Kites Oh My!

Mandel_zoom_08_satellite_antennaAhhh, the STAAR test is complete for another year.  Sadly, some topics such as these are put on the back burner…   But now that burner is on!  The last two months of school we’ll be digging deeper into some concepts that we skimmed over while covering the curriculum, while also covering topics to prepare them for 9th grade.

Next week, we’ll be designing and constructing Tetrahedral Kites.  We’ve been building these kites since the mid 90’s in the classroom.  Students  (ok not all but I’d say most) enjoy the time spent working together, creating the kites and then going to a nearby park to fly them later in May. And every year, I’ve got a few students who will fly kites for the first time!

 

To introduce the unit, we usually start with the Sierpinski Triangle.   However, this year180px-Britain-fractal-coastline-50km, I jumped into the concept of Fractals.  These are figures that are considered self-similar when there can be found a point in the figure that contains a copy of the entire figure.  There are many websites out there.  Of course, there is no need at an 8th grade level to share too much of the equations behind the design of the fractals.  But we can discuss the fractional relationships and the correlation with measurements (Mandelbrot fractal), the history of fractals (Cantor’s Dust) and the associations between fractals and the mathematics and sciences of the world, such as with the Coastline dilemma of Great Brittain.

 

While viewing a PowerPoint and parts of the video – The Mandelbrot Set ( a video including a segment from @profkeithdevlin ) of different examples, the students use whiteboards and markers to create Cantor’s Ternary Set.  Then we design Koch’s snowflake.  The whiteboards make it easy to erase and add to the line segments.   Another day we discover how easy to make Heighway Dragons out of strips of paper.

IMG_1046

 

Then, onto Sierpinski’s triangle.

IMG_1048

The YouTube videos can be eerie, but fascinating, and so those played on the screen as we designed our triangles.   And as we finish, we build our own Sierpinski triangle along the hallway

.IMG_0553

And now, this week, the building of 3-d tetrahedron kites will commence this week. Looking forward to it!  We’ll discuss volume, surface area, fractions, the relation between the kites and Sierpinski’s Triangle…  and of course the history and science of kites and flying.

 

I’ll add photos here later in the week.  😉

 

 

Week 8 Poly-Gnomials Speed Dating?

poly-gnomialsSo, it’s been a few weeks since I last wrote in this blog.  I wish there was a good excuse!   We’ve been going through Topic 5 and 7 of the Algebra 1 Pearson textbook…. journaling and practicing all the joyous methods and types of problems pertaining to Exponents, and multiplying polynomials.

Tomorrow, is the day!  I will try out Speed Dating with the Algebra kids.  I’ve  read a few blogs about this, and have tried to wrap my head around it.

Here’s my method to start with:

Students sit in desks that are facing opposite each other in two lines (which may curve around the room).  I have made cards with the questions on one side, and the answers on the other.  There are two sets of cards.  “Table #” and “Seat #”.   So Jim North and Bob South sit across from each other. Next to them sit Sarah North and Susie South and then Lina North and Linus South.  The “North” students receive the “Table Cards” and the “South”  receive the “Seat Cards.”   The students at each table also have a cup that is turned upside down.

The students work the 3 problems from the card on paper.  When they are finished, they turn the cards over and check their answers. They both have the same problems at the first of this “Speed Dating.”  The teacher goes around and verifies that every student can solve the problems that they received.  (Between their partner and themselves, this shouldn’t take long).  When they are good, they flip upright the cup, which signals to the teacher that they are ready.  These students will be considered the Master of their own card, and should be able to explain how to get to the answers.

Once all the cups are upright, the the teacher rings a bell, or asks the “Seat” students to move.  So in our example above,  Bob South, Susie South, and Linus South move to the next table.  (Make sure to remove all backpacks from the area so that it’s easier to rotate).  The North “Table” students will remain in at their seat.  So now, Jim will work with who ever the seat “south” student sat prior to his “table”. Sarah will work with Bob, and Lina will work with Susie. Linus will work with whomever “Table 4” student is. They turn their cups upside down, switch cards, do the problems, and check their work.  If they have questions, they can ask the opposite student who is the Master of his own card. When both students have the correct answers, the cups are turned upright, and the teacher sees that students are ready to rotate (again, the “Seat” students will always move…  the “Table” students remain)….

Below are pictures of the problems with answers and the cards with answers are written on the back

I CAN’T WAIT TO TRY THIS!

So, thanks to SumMathMadness and Kate Nowak for this idea!

Week 3 Taking Note of Exponents

beethoven-5Week three was busy with tests and exponents for both Math 8 and Algebra 1.  Explanations of exponents just can’t be done without students seeking out the patterns of numbers, esp how powers of ten grow and decrease and also powers of 2.  In our interactive notebooks, we took notes listing our patterns.  We watched the Powers of 10 by Charles and Ray Eames…an oldie but a goody.  Stopping half way to discuss positive exponents and the next day discussing negative exponents.  Then we jumped into scientific notation as well.

So yesterday, the students walked into Beethoven’s 5th playing in the class.  We reviewed our patterns of exponents, and then went into discussing powers of 2.  Remember we had discussed Raja Rice a while back listing powers of 2 (positive). So we went the other direction discussing the negative exponents and their values.  And then I mention how much I wish the orchestra and band teachers could rewrite the musical notes using exponents.  (with a wink of course).   We look at the first couple measures of Beethoven’s 5th and discuss how the exponents apply to the beats of the notes  (changed it to 4/4 time to make more sense).  (insert picture here)   At the end of the class, I continued the masterpiece as they worked on their assignment….  no one mentioned the music…  I may try a little Vivaldi next week.  🙂

Can’t forget that we also discovered “Anything to the power of zero is 1” again by using patterns.  I keep asking the cheerleaders to cheer CMS and holding up their hand as a zero..  “CMS to the zero power!  We’re number 1!”….   And I expect those football players to raise their arm and make a 0 with their hand after scoring a touchdown!…  no one’s taken me up on that yet…  (wink wink).    cms-0

If there’s anything I’ve worked at this so far is wait time, and asking the students to talk to their “shoulder partner” prior to giving out an answer.  I LOVE it!  Let those proverbial crickets chirp while their thinking… there’s more going on than just silence after a question.