In about 6 hours Eastern Standard Time, the magnificently decorated New Year’s Eve Ball will once again descend 77 feet (23 meters) over the course of a minute₁ to symbolize the passage of time while millions of people gather together to ring in the year 2009.

There haven’t been too many New Year’s Eves that I haven’t witnessed this televised traditional event.  And as a kid, was always awed by how the ball would drop as time ticked away on the television screen.  Since the dawn of the new millennium, the New Year’s Eve ball has truly been a magnificent piece of artwork from the designers at Waterford Crystal, and I am ever more still eager to watch its brilliance descend from the heights of the New York skyline.

2009 gets a make-over

This year, the ball is a 12 foot (in diameter) geodesic sphere based on a truncated icosahedron, weighing in at 11,875 pounds.  It is covered by 2,668 Waterford crystals and illuminated with 32,256 LED lights (about 3 times more than last year). And even will all this, the ball is more energy efficient than ever, consuming only the same amount of energy per hour as it would take to operate two traditional home ovens₂.

I can’t wait to see the show that will be put on this year as the ball descends, and with the enhancement of technology is capable of producing 16 million colors and billons of kaleidoscope patterns.  It’s sure to be a spectacular show!

In the classroom

The design and new facts of the New Year’s Eve ball lends itself to a variety of math problems to be solved.  One in particular that comes to mind is in regards to the overall surface area of the ball.

The New Year’s Eve ball is not quite a sphere, it’s a geodesic sphere (think of EPCOT at Disneyworld).  Its surface is made up of many triangles, with these triangles having different dimensions because of the curvature of the sphere₃.  So, we would need to know the dimensions of every kind of triangle and how many of those different triangles to calculate the surface area.

But, if we simply considered the New Year’s Eve ball as being a sphere, we can use the formula .  After the calculations, we find the surface area is approximately 452.39 square feet.

S = 4(Pi)r².  With what we know about the 2009 ball, our radius is 6 feet.  Therefore, S = 4(Pi)6²

What if we were to fill the New Year’s Eve ball with a bunch of confetti?  How much confetti can the ball hold?

As with surface area we have a formula that we can use to fine the volume: V = 4/3(Pi)³

I’d much rather explore the volume of sphere with a 3-D interactive animation that helps me to build conceptualization of the concept.

Adaptive Curriculum offers such a resource in “Volume of a Sphere.”  In this Activity Object dynamic modeling is used to derive the formula for the volume of a sphere from the formula for the volume of a pyramid.  As the user changes the number of pyramids in the sphere they observe the relationship between the sum of the volume of the pyramids and the volume of the sphere.  The visualization of deriving this formula assists students in understanding where the formula came from and also strengthens reasoning abilities.  The final visualization reminds me a little of the New Year’s Eve ball!

For more on the New Year’s Eve ball:

• Time’s Square Alliance
• Modern Marvels: New Year’s Eve Ball Drop
• New Year’s Eve Ball Grows Bigger and Brighter

References:

1. Times Square Ball.  http://en.wikipedia.org/wiki/Times_Square_Ball  Retrieved December 31, 2008.
2. Time’s  Square Alliance.  http://www.timessquarenyc.org/nye/nye_ball.html   Retrieved December 31, 2008.
3. Geodesic dome. http://en.wikipedia.org/wiki/Geodesic_dome   Retrieved December 31, 2008

So, I’m sitting in the conference room with two fellow instructional coaches, and I just showed them Wordle.  And all three of us are intrigued with how we can use this Internet based classroom tool.

Wordle is a tool for displaying words as a graphic image.  The size of the words is a relative indicator of their frequency of use. You can read more about Wordle on Ed-tech-4-Science.

In Wordle:  Seeing Science Images as Art, Dr. Rillero, I was immediately intrigued by what Wordle creates with text. Here’re my Wordle art for this blog:

So, why would a classroom teacher want to spend time on Wordle? 

The word analysis and count feature gives teachers important information as to how often academic content vocabulary terms are used in a text.  In regards to assessments, knowing how many times content vocabulary is used can help the teacher make sure he/she is using the same terminology in their instructional delivery.

This week is benchmark assessment week in the Avondale School District.  So, I immediately thought of the math assessments hundreds of students will be taking on Monday.

I thought it would be interesting to “see” what words were most often used in the 6^th grade assessment.  Knowing that this quarter’s focus was on fractions, I was expecting to see words related to fractions as the largest ones in the art creation.  

I should note that I included only the text for the questions stems, not the answers.  This is what I got:

I was really surprised that “ate” was one of the largest words.  Using the word count feature, I found out it was used 13 times.  “Multiply” was used 10.  This tells me that many of the fraction problems are given in the context of eating some type of food!  Maybe a redesign of some of these problems is necessary.  Fractions can be used in many other contexts! I wonder what kind of Wordle art the AIMS (Arizona Instrument to Measure Standards) practice tests would create?

In the Math Classroom

I can easily see a math teacher using the information provided by Wordle in a data analysis unit.  The word count provided with each Wordle art is a good source of data to be graphed and analyzed.  Students can use current event articles posted online from sources such as CNN, FOXNews or BearEssential News as the text, and do an analysis of the word usage.  The data can be interpreted (and displayed) in many different ways:  content vs. everyday language, frequency of word usage, or even determining the frequency of how many words with 3, 4, or 5 letters.   

The possibilities are vast with Wordle.  Share your thoughts and Wordle art with us!

So, I bet you were wondering what happened with the 8^th grade students from Mr. K’s class? 

Well, plans changed.  As they do so often in our daily lesson plans!

I ended up spending Friday afternoon with Mr. K and two of his 7^th grade classes.

The focus of the week’s lessons was on probability, and Mr. K was still determined to use Adaptive Curriculum as part of his instructional delivery with the SmartBoard.

Adaptive Curriculum has a few Activity Objects for probability:

• Probability with Tree Diagrams
• Find the Given Probability
• Playing with Probability
• Analyze Experimental Probability Using Graphs
• Experimental and Theoretical Probability

After briefly previewing these Activity Objects with Mr. K during his morning prep period, he decided to use “Find the Given Probability.”

Mr. K had success in both his classes in using this Activity Object.  One of my favorite moments was when we first started the Activity Object and all the students were dead silent and watching the SmartBoard as the introduction was given.  After some time went by and students were being given a chance to come up to the SmartBoard, I heard comments of “Oh, you got it!” and “That’s awesome!”  It got even better when students were working together with each other to solve the problems and could barely stay in their seats for want of getting to the SmartBoard and solve the problem!

At the end of the lesson, Mr. K asked the students to reflect on their learning.  Here’s what a few students had to say:

“I learned an easier way to do probability.  The good thing about the activity is that you’re basically making your own problems.  It was really fun.  I loved the project.”  W. B

“I learned to use probability in a better way.  It gives a good challenge.  I really liked it.”  P. G.

“I learned that you have to multiply the smaller probabilities to get a final one.  I liked the animations and interactive learning.  I would recommend this program to any math teacher.”  J. G.

Sometimes you just need a new way to “see” the math.

On Monday, I observed what was to be an 8^th grade math lesson on solving for angles of triangles. 

I watched Mr. K’s 50-minute class period go by with homework being corrected and recorded, a few problems from the homework reviewed, and a start at classifying triangles.

In the middle of explaining the relevant terms (scalene, isosceles, acute, obtuse, etc.) Mr. K stopped, as there appeared to be some confusion about the relationship between the interior angles of a triangle.  So, he had the students cut out a triangle and complete the following:

• Label each angle as 1, 2, and 3.
• Cut off the corners of the triangle, making sure you can still read the numbers.
• Arrange the cut corners by matching angles 2 and 3; and then angle 1 to 2.

After this, students were asked to observe the arrangement.  The conclusion was that the sum of the angle measurements in the triangle totals 180 degrees, and that was true for all triangles.  This can be observed because the straight edges of the triangle all match up and form one edge, or a straight angle.

Here’s a clip from TeacherTube on this same activity:

 Triangle Angle Sum

The triangle activity Mr. K had the students complete was a good way to review previous learning.  It was hands-on and focused on conceptualization.  In fact, it was already used in the direct instruction of the lesson the previous week.

But, the lesson just didn’t seem to go the way Mr. K wanted.

Maybe it was because Monday was the first day back after the Thanksgiving break or maybe it was that these 8^th grade students just weren’t interested in math on a Monday morning.  Or maybe they just needed to “see” the math in a different way.

I talked to Mr. K after the lesson about the overall engagement of the students and the activity they worked on, and I asked him to stop by my office after school as I had a resource to show him that I thought would help him in his next lessons.

We looked at Adaptive Curriculum’s “Type of Triangles” in which dynamic modeling is used to create different triangles so that students can observe the changes in angle and side measurements as it relates to classification. 

I chose this Activity Object not just because it focuses on the content being addressed in Mr. K’s lesson, but it allows for excellent use of Mr. K’s Smart Board, which would allow the students to get more engaged and involved in the lesson about the relevant vocabulary.

The plan was that we would use this Tuesday with his two classes. 

This morning, we played around with “Types of Triangles” a little bit more and also looked at “Interior and Exterior Angles of Triangles.”  Mr. K was excited about both of these Activity Objects and we played and discussed them for about 40 minutes.  Mr. K decided that he wanted to spend some more time with these Activity Objects before using them with the class and we made a new plan to use them on Friday.

I’m excited that Mr. K is excited! And I’m looking forward to spending more time in his classroom on Friday. 

I’ll let you know how it goes with the students on Friday and how Adaptive Curriculum’s Activity Objects allowed students to “see” math in a new way. 

Check back this weekend for an update on the lesson!

Tired of playing the same old board games or just sitting around watching movies?

Try Blokus!

This multi-award winning (2004 Teacher’s Choice Award and Mensa Best Mind Game Award of 2003, just to name a few) is one of the best board games that I have come across!

Blokus, a game of strategy, was created in 2000 by Bernard Tavitian, a French mathematician.  The game stems from the Four Color Theorem and the use of polyominoes as the pieces, and is a bit reminiscent of Tetris.  Not only is this game highly engaging and competitive, it builds on your spatial reasoning and logic skills.

Winning the game may seem simple:  be the first to place all your pieces (polyominoes) on the game board.

The only restriction is that each new piece you place on the board must touch another piece of the same color only at the corners (hence the connection to the Four Color Theorem)! 

If you don’t have time to run out to the store before Thanksgiving Day arrives, you can play online and compete against players around the world.

Or, you can download Blokus World Tour (for a small fee) and play on your computer.  The graphics and competitions are much better at Blokus World Tour and you just can’t help but to keep playing until you win each of the different tournaments.  I’m currently at playing Tournament 6 – The Berlin Showdown – where best out of five of the Classic 2 Player and Duo games wins you the tournament.

Once you begin playing, I guarantee that you’ll be hooked!  And hopefully you’ll get to claim the title of being the first player in your family to place all their pieces on the board!  A title I proudly hold! 

Oh, did I mention that you get 50 bonus points if the very last piece you place is the single square piece?

Read more about what people are saying about Blokus at Blogus. 

This morning, when I went online (my browser opens to MSN), and had scanned the homepage any faster, I would have missed two unique articles!

At first, the article “20 Thing You Didn’t Know About … Pencils” flashed before my eyes.   So, naturally, being the math teacher that I am, I opened the link and read about 20 things I didn’t know about pencils.  Dean Christopher (Discovery Magazine) lists some very interesting factoids about our beloved pencil.

Did you know …

• The average pencil has enough graphite to draw a line about 35 miles long
• The first American pencil factory opened in 1861 in New York City
• The word pencil derives from the Latin “penicilus,” meaning “little tail”

You can read the rest of Christopher’s pencil facts at “20 Things You Didn’t Know About … Pencils“.

And if pencils weren’t interesting enough, the next headline to catch my eyes was “ ‘Smoot’ reaches new heights in MIT.”

I can’t remember why I opened this link, but when I did, I read about the 50^th anniversary of using a “Smoot” as a unit of measurement.  In 1958, Oliver Smoot and his fraternity brothers at MIT measured the Harvard Bridge using Oliver as the unit of measurement!    They found that the bridge was approximately 364.4 Smoots long (Oliver measured 5 feet and 7 inches).  Smoot later became the chairman of the American National Standards Institute.

So, how could I not look for more information on the Smoot?

I didn’t really come across anything more than what I had already read.  But, I did find a nice article from Cross & Crescent, a publication from Smoot’s fraternity.  And I found on Google calculator that 35 feet = 6.26865672 Smoots.

Ferris Bueller said it best:  “Life moves pretty fast. If you don’t stop and look around once in a while, you could miss it.”

At the beginning of the school year, one of my fourth grade teachers, asked me if I would come to her class and introduce the Product Game to her students as a way of having some fun while practicing multiplication facts (we’ve done this for the past 3 years). 

Each time I visited or walked through Miss Rogowski’s classroom this past quarter, we did a little update with the students on the progress they were making with working on their multiplication facts and built up the excitement of me coming to class to “play a fun game” with them and the ice-cream party they would have at the end of the year as a celebration for their mastery of multiplication facts.   It doesn’t take much to excite fourth graders!  It got to the point that when I walked in, some students would mention right away where they were…one student in particular continually reminded me that he had already memorized all his 12s! 

Yesterday marked the end of the first grading quarter.  So, Miss Rogowski and I decided that it would be a perfect day to play the Product Game.

The object of the game

The Product Game is fairly easy to play.  Get four squares in a row (horizontally, vertically, or diagonally) by using the factors 1 – 9 to create products located on the game board.

NCTM’s Illuminations website [1] features the online applet that the Connected Mathematics Project created to use the Product Game online.  There are no bells and whistles to this applet, but it’s worth a mention and recommendation to all math teachers because the game is fun – I can attest to that! 

I first came across the Product Game when I taught sixth grade math using the Connected Mathematics Project [2] textbook series.  The game was part of the Investigations in Prime Time: Factors and Multiplies (1996), the book focusing on Number Sense (GCF, LCM, prime numbers, etc.).  I always had success with students in playing the game and couldn’t see just using it in sixth grade.  I’d recommend the game to any group of students who need a way to practice their multiplication facts!

Adaptations to the game

The Connected Mathematics Project revisits the Product Game in the seventh grade series in Accentuate the Negative (1996).  This time, positive and negative factors are used to play the game.

Teachers can easily enough change the format of this game to fit the needs of their students. You can recreate the game to fit the needs of any students:

• Change the factors to be used to 4 – 12 (don’t forget to change the products in the game board!)
• Make the game board smaller and decrease the number of factors for students who are struggling.
• Go back to the good old days of playing bingo and use some of the variations of that game to win:  four corners or blackout.
• Play in trios

Please feel free to post any other ideas on how to adapt the Product Game to fit the needs of students!

References:

[1] National Council of Teachers of Mathematics.  Illuminations. The Product Game.

[2] Prime Time: Factors and Multiples. Connected Mathematics Project. G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips. Dale Seymour Publications (1996), pp. 17‑25.

Last week, I had the pleasure of observing an 8^th grade classroom.  The planned activity was the next part of an ongoing study of data collection and representation.

On this particular day of the lesson, the students were given the task of conducting a survey in order to collect and display data in a circle graph.    Students were provided with a handout and were asked to do the following:

• Write a question
• Determine 6 choices
• Survey the class and complete a frequency table
• Convert data to fractions, decimals, and percents
• Divide the circle graph into quarters, then fifths
• Divide up circle graph into percents
• Create another type of graph to represent the data

After a clear explanation of the task, the class of 15 students stood up and began to collect data.  For about 20 minutes, the students worked together and discussed the data they needed to collect.  Then, the students sat down and began work on the construction of their circle graphs.

Overall, the students were engaged and were having a good success rate at completing the task.  The teacher made the task meaningful to the students in the sense that they had ownership in the creation of the survey topic and who they asked to collect the data.

Enhanced with technology

Adaptive Curriculum, the award winning, online learning environment, offers a similar Activity Object called “Circle Graphs.”  In this multi-part activity, students are first asked to plan their 24-hour day by choosing from a variety of events (typical of a middle school student) and determine the amount of time to be spent on each event.  Then, students divide a 24-hour clock according to the hours selected for the events.

The 24-hour clock is then recreated as a circle graph. To do this, students need to determine the angle measurements needed for each section of the circle graph by finding the fraction of the 24-hour day each event needs and then multiplying by 360⁰ to get the actual angle measurement.  With using sliders on the circle, students can easily draw the correct angle measurements needed for each section of the graph.

In Section 2 of this Activity Object, students design their own data set and categories and then practice creating a circle graph with the new set of data. 

“Circle Graphs” would be a great complementary activity for the 8^th grade students to work on; in fact, I would recommend that it take the place of the handout and paper-and-pencil task that the teacher provided.  The Activity Object provides meaning to learning and applying math to the middle school student; it is more engaging as it allows for ease of the creation of the circle graph with intelligent explanations and feedback on the construction as it is needed.  After completing “Circle Graphs”, students will have gained practice and knowledge on how to construct circle graphs without unnecessary time spent on attempting to draw a circle graph with accuracy.

You can view more of Adaptive Curriculum’s Activity Objects at www.adaptivecurriculum.com. 

As with many mathematical operations, most of us have been taught the rules.  And most probably wouldn’t be able to explain what we did to get the answer.

Multiplying integers is an example.

An explanation for 2 x -4 = -8

One way to explain this rule is to look at the pattern that results from a series of multiplying.

2 x 3 = 6

2 x 2 = 4

2 x 1 = 2

2 x 0 = 0

What do you notice?  The first factor remains constant.  The second factor decreases by 1.  And the product decreases by 2. 

If we continued the patterns,

2 x -1 = -2

2 x -2 = -4

2 x -3 = -6

As we analyze the patterns in the multiplication series, one can conclude that when you multiply a postivie integers by a negative integer, the product is a negative integer. Therefore, 2 x -4 = -8.

An alternate explanation enhanced by technology

Making an array is often a means of explaining multiplication.  But it is mostly only used with positive whole number and in the elementary grades.  The National Library of Virtual Manipulatives offers Rectangle Multiplication of Integers [1] an online applet for creating arrays with negative integer factors. 

Take a look at 2 x -4 = -8 

With the use of a coordinate grid and the ease of moving the slider along the axes, students can create arrays with both positive and negative integers.

In this explanation, the first factor represents a value on the y-axis and the second factor represents a value on the x-axis.  The array is created in Quadrant II and is colored red.

Likewise, here’s -2 x -4 = +8  

This array is created in Quadrant III and is colored blue.

With continued exploration of integer multiplication using arrays, students will more likely be able to make meaning of the operation.  Taking notice of where the array is created on the coordinate grid and the positive or negative factors used to create the array, students will be able to understand why 2 x -4 = -8, but  -2 x -4 = +8.

Representation to conceptualization

Often times, the rules of mathematical operations don’t mean much to our students.  It’s not until they can physically manipulate an object or create a visual representation that students make meaning of the operation.

NCTM  states:

“Representations should be treated as essential elements in supporting students’ understanding of mathematical concepts and relationships … representation associated with electronic technology create a need for even greater instructional attention to representation.”[2]

References:

[1] Rectangle Multiplication of Integers.  National Library of Virtual Manipulatives.

[2]  The National Council of Teachers of Mathematics.  Principals and Standards for School Mathematics.  2000.  Page 66.