Ed Tech 4 Math

Yesterday, as I was eating breakfast, I watched a few segments of the Today Show.  And I happened to catch a segment about keeping your kids engaged while they are stuck inside during a snowstorm.  Friday’s weather forecasted the east coast to be hit with a snowstorm again this weekend. 

I wasn’t really paying that much attention yet, but I think plug for the upcoming segment said something about technology being a part of 5 ways you can keep your children busy when they can’t go outside this weekend.  Goodhousekeeping magazine was sponsoring the segment, as the full story is in this month’s issue.

So, I thought, okay, this might be interesting.  Let’s see what they have to say.

·         Make a digital photo album at Shutterfly, CVS Photo Center, or OurHubBub

·         Get a high-quality iPod dock station

·         Make popcorn by putting the kernels in a specially made bowl that allows you to pop them in the microwave

·         Join a DVD movie exchange service

Yes, I know that’s only 4.  But that’s all we were shown.

When I heard about the idea of using Shutterfly, I thought, sure, that is fun and engaging.   It also allows parents to bond with their kids and tell stories about the memories preserved in the digital pictures.  And you can tie in a bunch of math without even knowing it – cropping, changing the resolution, placement of the photos.   I myself am a fan of Shutterfly, and thought this a worthy idea.

I was surprised when the next idea and recommendation was to buy a high-quality iPod dock station.  And I think Meredith Viera was a little surprised to.  She said something on the lines of “Oh, to listen to music while you read a book perhaps.”   I love music, and usually you can hear something playing in the background no matter where I am at.  But, go out and buy a high-quality iPod dock station right before a snowstorm so that your kids are engaged?  Meredith even questioned the price and practicality of having a two-foot tall dock station.    

The next two ideas obviously went together.  Make yummy, delicious, and healthier popcorn in a bowl that allows you to pop the kernel instead of using the pre-made popcorn bags.   Why not?  Personally, I like air-popped popcorn better.  I might even go find one of these new popping bowls.    And now that you have your healthy popcorn, put your kids in front a television to watch a movie that you just happened to receive in the mail before the snowstorm hit.  (Kung Fu Panda was the DVD most visible among the stack of 20 or so DVDS.)

You’ve got to be kidding me.  These are the best ways to engage your kids if they can’t go outside?

I was quite dismayed, and bothered, that this was the advice that the Today Show and Goodhousekeeping were giving.

There didn’t seem to be much thought in the quality of how to use these gadgets as activities to stimulate the brain and open the door to learning (except for the digital photo album idea) while stuck inside because of a snowstorm.

So, let’s see what we can do with these 4 (not 5) recommendations.

·         While listening to music through a high-quality iPod dock station, spend 60 minutes reading your favorite childhood stories to your children.  When you are done, choose your favorite short story, or selection, and insert the text to create a Wordle (see my previous post Math Vocabulary Becomes Art). 

·         Choose a topic such as the ABCs of math or science, and have your children search the house (and even outdoors if the snow stops falling) to take pictures of those items.  Then, use a digital photo album service to create a mini-photo book of their ABCs.   Oh, and don’t forget to have some music playing in the background!

·         Plan in advance to have a copy of movies that will inspire your child to be the best they can and embrace learning and the world around them.  I recommend 3 documentaries: Spellbound (2002), Paperclips (2004), and Mad Hot Ballroom (2005).  Pop some popcorn and sit together and discuss the issues and stories while doing everything you can to not dance while watching young kids from Brooklyn learn how to ballroom dance, or spell as any words as you can during the 1999 National Spelling Bee.

For those of you on the east coast, I hope the storm wasn’t too bad.  And I hope you found ways to keep your kids busy!

P.S.  I love living in Arizona!

The New Year’s Eve ball has dropped in Times Square and we are now 4 days into 2009.

With the start of another year, many of us take time to reminisce about times past and reflect upon the achievements of those who came before us.

Here’s a little something for us technology enthusiasts:

2009 marks the 40^th birthday of the public debut of Douglas Engelbart’s computer mouse.

40 Years of the Mouse is currenlty being featured on MSN chronicling the debut and evolution of the computer’s most revered sidekick.

The mouse has come a long way since December 1968, and I have used many of them: the Microsoft Mouse 2.0 to the Apple iMac Mouse to a Logitech wireless mouse.

Although I mainly work off of a laptop, I can’t imagine not having my little wireless mouse with me!  To be honest, I feel like am without a limb if I don’t have my mouse with me; I just can’t work as efficiently without it!

For more on the mouse:

How Stuff Works:  How Computer Mice Work

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 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!

There’s only so much you can do with a cardboard box.  But, technology can ease and enhance the delivery of meaningful math lessons such as the one my colleague, Mrs. A, is planning for her 8th grade students on finding the surface area of rectangular prisms. 

 

Last week Mrs. A. shared with me her idea of using realia such as cereal boxes or soda can cases to unwrap, or break apart, in order to show students what would be considered the net of the prism shaped object.  A worthy idea and one I’d recommend using.

 

We then talked about how these unwrapped prisms could be used to derive the formula for finding the surface area. 

 

• Take the unwrapped box and place it on large graph paper
• Trace the unwrapped box, including the creases, onto the large graph paper (this gives you the net of the box)
• With each square measuring one unit, count the area for each section (there should be six) of the unwrapped box
•  Add the areas together to determine what would be considered the surface area

 

Our discussion continued with a walk-through of what to consider and where things could go wrong (I have taught a similar lesson and know first hand what to avoid doing!).  Basically, it came down to this: unwrapping the boxes is great; however, when unwrapped, the faces of the prisms do not all have straight edges!  There are little flaps that are used in order to glue all the sides together.  And this can cause a few problems in the overall lesson design when it is used as one of the first lessons in studying the formula.

 

Like I said, there’s only so much you can do with a cardboard box!

 

Consider what an extension of this lesson would be. What if Mrs. A wanted to show the students what would happen to the surface area if she doubled the height of her cereal box?  Or what if she wanted the same cereal box to have a base area half that of the original?  Can there be two cereal boxes with the same surface area but different base areas?

 

Virtual manipulatives are available for students (and teachers) to quickly make changes to the variables (height, length, incline, base area, etc.) of geometric objects and observe the results.  Shodor Interactive and Explore Learning’s Gizmos both offer stand-alone virtual manipulatives that provide an opportunity for changing the dimensions of 3-D objects and showing how those changes affect the surface area and/or volume.

 

Adaptive Curriculum uses what can be called dynamic modeling in a series of flashed-based Activity Objects for surface area and volume of prisms, pyramids, cylinders, and cones. 

 

Each of these Activity Objects provides excellent visuals, explanations, and exploration of dynamic modeling as it relates to surface area and volume.  By working with these Activity Objects, students can stay engaged and focused on the math and the relationships that are formed as the variables change for each 3-D object.  These Activity Objects are a perfect complement to any lesson and are worth the time to be used in the classroom!

 

 

 

 

Go to www.adaptivecurriculum.com for a 30-day free trial and to learn more about the following Activity Objects:

• Observing Changes in the Surface Area of Cylinders
• Observing Changes in the Surface Area of Prisms
• Observing Changes in the Surface Area of Pyramids
• Observing Changes in the Surface Area of Cones
• Observing Changes in Volume of Square Prisms
• Observing Changes in Volume of Cylinders
• Observing Changes in Volume of Pyramids

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.

“It was much different in our times…” is a cliché that we often hear from our elders. But now, it is my turn to say it. When I was in high school, which corresponds to the late 1980’s, we did not have the luxury of exploiting technology in our math classes. In order to sketch the graph of a function, we had to perform a series of tedious steps which would yield a graph that we were never a hundred percent sure of, and unfortunately, we would have to count on the sketch presented in the text book. I can also remember going over a number of books and tens of graphing exercises prior to a really scary precalculus exam.

Once when I was in high school, I came upon a question similar to this one:

How many real solutions does the equation given by e^x = x^3 + 4 have?

At first sight, it really did not ring a bell and I desperately looked up the solution: Sketch the graphs of y = e^x and y = x^3 + 4 on the same set of coordinate axes and observe the number of intersection points. This was a complete awakening for me (and would later constitute a major corner stone in my math teaching career: I wrote a book, based upon how to exploit the capabilities of the graphing calculator, which was recognized by many as well as the US Department of Education).

Now, if I make the following claim, what would you think?

It is possible to solve any type of equation or inequality with a graphing calculator, whether it is algebraic, trigonometric, exponential, logarithmic, polynomial, or transcendental, in a similar way without having to perform tedious steps.

With a graphing calculator, yes, it is possible. However, it would be such a waste not to exploit other useful functions of these brilliant handheld computers that are given the modest name of “graphing calculators”. With our mathematical knowledge and what is already there within those handheld giants, the sky is the limit to what can be performed, mathematically speaking!

For instance, let us consider the following question:

What is the equation of the parabola that passes through the points (2, 0), (4, 6), (-3, 20)?

Here is one approach to the solution: Let the parabola be y = ax^{^}2+ bx + c. Plug in the points and you get the three equations given by 4a + 2b + c = 0, 16a + 4b + c = 6 and 9a – 3b + c = 20. Now you have a linear system of three unknowns and three equations that you can use to solve for a, b and c to find the correct solution: a = 1, b = – 3 and c = 2.

Now, how can you use technology to help you? Can you use it to solve the linear system for a, b and c? Yes, indeed you can. But let me propose a rather “radical” approach. Does your graphing calculator perform quadratic regression? I bet it does. I suggest that you use quadratic regression to find the equation of the parabola; since you have three points and no more, quadratic regression will exactly give you the parabola that passes through all three of these points and in one easy step. I believe this approach is much better than what I used to be doing when I was in high school since it saves lots of time avoiding the danger of making a mistake while solving the system of three equations for the three unknowns.

Consequently, in the level of civilization attained today, we do have the luxury of using technology. All it takes is to use this option smartly by combining the capabilities of technology with our knowledge of mathematics. Thus, we can not only increase the quality of our teaching, but also speed up the teaching and learning process. Here is a humble opinion of mine: There will be times when the difference between a fair math teacher and a good one will depend on how competently that teacher employs technology in the classroom. Believe me, my dear friends, those times have already arrived!