Prime Number Manipulatives

tcm_blog_button2For those of you who are curious what we were actually doing in class in yesterday’s post (Record with a Document Camera and a Flip), we were “fingerprinting” the factors of composite numbers with their prime compositions. The wooden “prime number tiles” are created using the backs of Scrabble tiles and the white tiles you see on the board are the student version of the prime number tiles (each of them cut out a sheet of their own prime number tile manipulatives at the beginning of class).  Just for the record, I would like to confess to destroying more than 5 Scrabble games in the last 6 months.

I tell students that the prime numbers are like the building blocks of the whole numbers, in the same way that the nucleotide bases (adenine, guanine, cytosine, and thymine) are the building blocks of DNA. It was ironic, then, that at the end of our prime factorization fingerprinting, the board looked (to me) the way an agarose gel electrophoresis looks.

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You can also use the prime number tiles to line up the prime factorizations of two numbers (leaving spaces when the prime is not in both numbers) and read the GCF as the intersection of the two lines of tiles and the LCM as the union of the two lines of tiles.

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I’ve tried explaining these concepts in writing before (circling common factors and using colors or highlighting to show the steps), but the ability to easily rearrange the prime numbers after they’ve been found makes all the difference.  Students find the prime factorization using a factor tree or short division, then place out all the tiles for each number.  At this point, they can rearrange each row into least-to-greatest order.  Then line up the primes, bumping the whole row when there is no overlap in primes between the composite numbers.

We also were able to look at more interesting problems now that students could “see” the inner workings of the factorizations better.  For example, find a pair of numbers with a GCF of 42 and an LCM of 4620 (you may not use the “trivial” answer of 42 and 4620).  By constructing a “board” of prime number tiles, starting with the factorization of the GCF and the LCM, students begin to see which primes have to be in the two numbers and which can only be in one of the two numbers.  Note that there will be more than one answer here … which is another interesting discussion to have!

I’ve never seen “prime number tiles” in any math manipulative kit … they were just something I created last semester.  I think this method of using Prime number tiles would also be helpful with explaining how to find the GCF in the factoring unit of algebra, and with explaining how to find the LCD in fractions.

Feel free to print the prime number tiles and use them in your own classes if you’d like.

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Interactive Simulations from PhET

PhET (Physics Education Technology) consists of a group of scientists, software engineers, and science educators from the University of Colorado at Boulder, who are striving to create effective, interactive learning tools.  Their work spans the fields of physics, biology, chemistry, earth science, and mathematics.  Much care has been taken in the design of each simulation.  The developers use a research-based  strategy to implement the most effective visual cues for learning.  User-interviews are held routinely.   Members of PhET have learned that animated responses are effective, as well as,  the use of a “click and drag interface”.  There’s more:  each simulation comes with lesson plans that have been submitted by instructors.  The simulations are free and require flash and java to run.

Here is an applet on estimation:

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Here is Calculus Grapher, which shows the relationship between a function and its derivative.

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And a Plinko Probability applet that shows a histogram that approaches the Binomial Distribution.

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Algebra Balance Scales

There are lots of “games” out there about solving equations, but I haven’t found a single one that is more than algebra homework dressed up with pretty packaging.  The “games” are all of the same format.  We’ll give you problems, you give us answers and we’ll reward you (or your character) if you get them right.  These are not teaching games, these are just more of the same kind of practice that you would find in an algebra text.

There is one applet that is worthy of mention, though.  The Algebra Balance Scales from the National Library of Virtual Manipulatives is quite good.  It isn’t billed as a game, but when you’re using it, you feel like you are playing a game because you’re interacting with the algebra on the screen.

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I recorded an example to show my students how it works.

An interesting assignment for an online or hybrid class would be to have THEM record an example explaining the process (you could, for example, use Jing like I did) and turn in the link.

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Fibonacci Sequence in Siftables

“We’re on the cusp of this new generation of tools for interacting with digital media that are going to bring information into our world on our terms.” – David Merrill

Jump to 2:20 to see the math example of a Fibonacci sequence in this TED Talk called Siftables.

 

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