Monday, March 24, 2014

Algebraic Peg Board - first lesson to find the LCM

So, someone!! asked for an Algebraic Peg Board lesson. So here is the first lesson. (I'll post more about this material, if people are interested, as we progress through the lesson sequence.)

I am using the Keys of the Universe elementary math albums.

The child should already be familiar with the concept of multiples, common multiples which was started soon after Primary.
First, T filled out these tables from the albums before we even started the algebraic peg board work on multiples and factors. These tables may look like drill, but they aren't really. The child may use any material he/she wishes to complete these tables if they don't already know their multiplication facts. In addition, the child constructs these tables for future reference as a control of error. (I apologize for weird dark pictures...I was trying to take these shots at night.)

Table A is a calculation of multiples of 2 through 10 up to a product of 50.

Table B is a calculation of multiples up to 100.
After completing table a and b, T then used them to complete table c, which indicates factors. Then we circle each prime number in red.

After completing the tables, we set out our algebraic peg board materials: pegs in green, blue and red, with little cups that are color coded as well, small black strips of card stock, and a peg board which is made of wood and has 30 rows and 30 columns of punched holes, and, not pictured, number tiles.

The first lesson in the album suggests finding the lowest common multiple for 2, 3, and 4. The lowest common multiple is the smallest number that fits even groups of 2, 3, and 4. 
We are going to build multiples of 2, 3, and 4 and we set out our number tiles accordingly at the top of the columns. Under each number tile, we build even groups of 2, 3, and 4. (We use green pegs because we are dealing with units.) Under each group we place a black strip of card stock. We will continue building groups of pegs until all three columns are even in length.

At this point the 2s column is the shortest so we will continue adding to this column first. Here I added an even group of 2.
Here we see that the columns are again not all even and the 3s column is the shortest. Therefore, we need to continue adding even groups of 3s to the 3s column.
We proceed in this manner, adding even groups to the shortest column, until all three columns are the same length.

We see that the lowest common multiple, (LCM) for 2, 3, and 4 is 12, or LCM {2,3,4}= 12

At this point the child may continue to find LCMs for other groups of numbers. Keep in mind that larger numbers can get unwieldy and the child shouldn't loose track of the process.
In this new scenario I am finding the LCM for 4, 5, and 6. You'll see that I added even groups until I ran out of space in the column. This means that I need to exchange unit green pegs for blue pegs which represent 10s.
I ran out of room here at hole number 30.
I exchanged 20 green pegs for two blue pegs in the 4s column and then 30 green pegs for three blue pegs in the 5s and 6s column. (Since I had 28 green pegs in the 4s column to start and then traded out for 2 blue pegs, I had 8 green pegs left over I kept at the bottom of the column.) 

Now I can continue adding unit pegs and exchanging until I find that all columns are equal in value and length.

It is important to look at the value of the columns when deciding which one to add to next. Sometimes the shortest column may contain a larger quantity than a longer column. In the photo above, the 5s column is the shortest, but this column contains the largest quantity, 40, represented by 4 blue pegs. The 4s column is the longest but represents the smallest quantity, 28. We must add even groups of 4 to the 4s column.

Additionally, as soon as we add 10 green pegs to a single column we can immediately exchange them for a single blue peg.
Finally, we find that the lowest common multiple for 4, 5, and 6 is 60, or LCM {4,5,6} = 60.

Hope this helps a tiny bit. 

This is the only the first algebraic peg board lesson. There are several more that lead to abstraction, or finding LCM's with paper and pencil only and I'd be glad to go over them if any one is interested.

4 comments:

  1. Fascinating - thank you so much for writing this up!

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  2. Doing this right now. In the writeup it says when the numbers exceed 10 use the pegs hierarchically, and that's where I'm getting ready confused at. Do you exchange whenever you exceed 10 or do you only do it when you run out of pegboard space? When the numbers are 2 digits, what do you do? I would do it in two columns. But if that's the case, when you're exchanging for single digits, do you put it in one or two columns?

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  3. Guavarama - you can do the exchanging for 10 either way. Inititally we start exchanging when we run out of space, but quickly the children realize they can exchange their FIRST 10 for a 10-peg as they get to it.

    Once you have reach 10, you have the blue peg to represent 10, then in the column below you have the unit pegs for 11, 12, 13, etc. until you get to 19, then add one more and those 10 unit-pegs can be exchanged for another 10-peg.

    Each muliple (no matter how many digits the multiple is) only gets one column. If you end up with 10 10-pegs, exchange for a 100-peg; thus even very high numbers will "fit" in the one column.

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