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.

Saturday, March 22, 2014

Part 2 Wk 9, March 20, 2014

Happy Spring to You!

I am still sniffling and attached to the tissue box, but my energy seems to be coming back slowly. Lots of tea is helping.

Somehow with the illness I am not taking as many photos. Maybe the sniffles keep me from shooting clear stills in low light. 

Anyhow, this is what we've been up to otherwise.
Math: multiples using the algebraic peg board
Language: command cards, grammar box work, vocabulary development
Geography: time zones chart
Walking on the line and other toddler antics
Okay, I figured out why T wasn't getting the prime factors stuff. It was because we had skipped a step. The instructions for creating Chart C, from the KotU math album, indicate that you should underline each prime factor with a red pencil. So we did that. I explained to T that a factor is only divisible by it itself and one. T defines a factor as a number that has no other multiplication problems after it; as in, nothing times nothing can equal that number. (For example we recorded 6 = 3*2 on Chart C.  6 is not a prime number because something times something does equal 6 (something other than one). On Chart C after the number 3, there are no such equations, thus indicating to T that 3 is a prime number.)

So after that was taken care of I realized that we had skipped the first multiples lesson using the algebraic peg board and had done the factors lesson first. So we skipped back and did the multiples lesson.
In this lesson we are finding the least common multiple (LCM) for three numbers. This lesson is a continuation of the work of multiples and common multiples, or in our case, the work we did with Charts A, B and C. The lesson in our album (KotU) suggest we find the LCM for 2, 3, and 4 first. In these photos, T is finding the LCM for 6, 7, and 8. It is 168. 
In the initial stage, (sorry, really fell off the horse with the photos, these are really not illustrative of the lesson) we began by inserting 6 green pegs in the first column, 7 green pegs in the second column, and 8 green pegs in the third column. (This description is not a first lesson description. The LCM for {2,3,4} is much simpler to construct than the LCM for {6,7,8}.) Then we look and see that the three columns are not equal in length or quantity. So we add pegs in sets of 6, 7, or 8 to the column that is most lacking until the columns are equal in length and quantity. 

In T's problem he got very excited about the exchanging. In his problem he had the opportunity to use all three hierarchies of pegs. You'll notice around the time you get to 5 sets of 6 green pegs you run out of room to build your column on the board. (The board contains 30 holes in a vertical column pattern and there are 30 columns on the board. Somehow this makes the board just a tad bit to large to be a comfortable work when you have seven-year-old arms.) This means you need to exchange at some point to note quantities larger then 30. One blue peg equals ten green pegs. So you would exchange each set of 10 green pegs for one blue peg which is what T did in each of these photos you see.
Here in the right most column he is counting by 6's. He is in the middle of an exchange perhaps, because he has only 20 represented in that column, or two blue pegs. In his middle column, he is counting by 7s. He has represented 35 pegs here and had to exchange 30 green pegs for 3 blue pegs. 

The little black strips are a short lived material in this lesson. Typically they are used to divide the sets. You would put a little black strip between sets of 6 green pegs. After while, you don't need them any longer and the child will remember what is a full set. 
Here T is up to 60+ and still adding sets of 6, 7, and 8 to try to make the columns equal in length and quantity. 
Finally T got to a point where he had 10 blue pegs. He immediately lit up and realized that he needed to exchange them for a red 100 peg. Then we kept going until I used my iPhone to figure out how high we needed to count...LCM {6,7,8} = 168. 

T liked this lesson much better than the factors lesson. I wonder if he'll like the factors lesson better the second time he sees it. Oh, and I think that we ordered this material from Montessori Outlet and the quality of everything is satisfactory.
 Independent geometric solid exploration.
This is little D's favorite little buddy.
And here are little D's vocabulary objects. We play little games after he knows all of the names of the objects. I'll ask him which ones are edible, which ones are alive, which ones live in the water, which ones fly in the air, which ones are smooth, or which ones are hard. We might do a preposition game like, "please put the octopus in the basket." We aren't focusing on key sounds yet, just vocabulary development. I rotate these objects each week and play different games with him almost daily. Sometimes T or S will play these games with him as well.

I collected these objects from various places over the last year. We ordered some from Montessori Services and Safari Toobs, and I collected many from the kid's toy sets. The tin is from a local thrift store.
This also made it into our day somehow. D picked this getup out all by himself. He is looking into a lamp that we were using to shine on the sandpaper globe. (The light was turned off.)

This was the other part of that lamp fiasco. S and I were working on the Time Zone Chart from the KotU geography. She has been present at all our elementary geography lessons this year, but for some reason this one didn't stick. I will not go into the lesson too much here, since I am going to have to present this one again, but generally, we were talking about simplified time zones and who is experiencing breakfast, lunch and dinner while we are in our schoolroom. 

Those little disks at the top of the map are wooden circles from the craft store and I just punched out printed clocks showing the time on the hour, color coded them with a sharpie, and Mod-Podged them on. The map I downloaded from the KotU discussion boards, pasted it to poster board and laminated it with contact paper. The black strips are to denote nighttime hours and I cut these out of black poster board. The album indicates that we should place white strips over the regions experiencing daylight, but I wasn't sure why we wouldn't want to see the actual regions experiencing daylight. Why cover them with white poster board strips if the focus here is the relationship between the time of day and the map zones?
Anyway because the concept wasn't 'sticking', we moved on to a free form project; "lets illustrate our day." After drawing illustrations of the activities we do during the day and night I figured we'd assign times to them, and then look to see what other people around the world in other time zones are doing at the same time we are doing our activities. I think this might get us back to this lesson and then we can revisit the map and the clock faces.
 It took this...
 and this....
 and this...
and this...
to get to this. T is still finding errors and omissions.
 I finally replaced the blue line on the floor of the classroom this week, and look who was first to get on.

Here he is running a little bit which is not so wonderful in the classroom.
We have been working on command cards this week from the Dwyer booklet sequence. After the child has learned all the symbols that correspond to our key 40+ English sounds, and has had ample time to write, we move to reading. S is a beginning reader for sure, but I think it may be more a matter of confidence than remembering the sounds. These command cards are most certainly boosting her confidence. 

In this lesson, I am writing the words right in front of her so the element of communication is still present. I say something like, "do you know what I am thinking? I am going to let you know what I am thinking without even speaking a word." Then I write in my silly cursive what I am thinking (it is difficult to write in a straight line and with even strokes while sitting on the floor with three children crowding around in anticipation) taking care that to include only phonetic words, words with the key phonogram sounds and puzzle words we've learned. S read the message above and then with her brothers promptly engaged in the following...
Yes, language can be fun and silly.

The other things we did this week, which for some reason I didn't get to photograph:
T has been working hard with the bells and can grade the C major scale bells proficiently. (He will shuffle the order of the 8 bells in the C major scale and then put them in order from lowest to highest by comparing them to each other. In T's case, typically only tapping each bell once.) He has good tone recognition and even can identify the middle C and high C with a single tap intermingled among other tones. There is a matching exercise he does where he separates five brown bells, taps one, remembers the tone, walks around the classroom, talks with someone, and then finds the matching white bell which is standing on the bell table. He hasn't needed to tap that first brown bell for a reminder at all. I guess all that humming I endure for HOURS EVERY DAY may be paying off, even though it still drives me nuts.

All three children are doing more metal inset work. T and S are really coming along with their shading abilities. The points of interest I stress as I give them a lesson on shading are: it is a comfortable hand motion, there is no stretching, use correct pencil grip at all times, the pressure should be comfortable as well, not too hard and not too soft, and aim to make caterpillars. If the area to be shaded is wide, take a caterpillar swath and shade in that portion from left to right. Return to the left side of the page and shade in another caterpillar swath as if the second caterpillar is resting on top of the first one. Before, their work was choppy, contained white spaces, and frequently extended beyond the lines. Now their work is much smoother and they are beginning to explore more complex patterns with the insets.

Some of S's bulbs out back are beginning to grow and we can see their green leaves up above the brown leaf mulch. She is VERY excited. I wonder how many will come up. I remember she really stuffed that bed full of bulbs in our 90+ degree heat wave last October!

And that is it for the week!

Thursday, March 20, 2014

Part 1, Wk 9 March 17, 2014

I was sick last week, and I was still sick the beginning of this week. A head cold. I didn't get a lot done on any front, except a lot of dragging myself here and there. We homeschooling moms don't get sick days.

Anyway, I am hoping that after a snowy Monday start to a busy week that my energy will... well, re-energize. 
D started off with the geometric cabinet demonstration tray. (The tray pictured above isn't the demonstration tray and it isn't complete, there should be a sixth circle, not a blank square.) I showed little D how to trace the perimeter of the inset with his left pointer and middle finger before tracing the perimeter of the yellow cut out and replacing the figure inside the cut out. He did this very easily with each figure before replacing the tray back in the cabinet.

This sensorial work prepares the child for future geometry work and for future handwriting (nomenclature will be introduced later).

S is still working on those lefty cutting skills. Here she is cutting out paper dolls. Her tripod pencil grip has come a long way since we started using lefty scissors. And yes, it could have been practice and time that helped her improve as well.
Still working on these pin maps! VERY close to done. More on this subject in a later.
In an effort to get to the algebraic peg board lessons, T sat down and did the Calculation of Multiples tables a and b, and the Table of Factors c, both pages, in a couple of hours. My KotU albums say that this is work should take the child a few days to complete. Not for T. When there is a goal, like the peg board, there is a will and a way. I hope that these multiplication tables are a bit more fixed up there in his brain. I pulled out the bead bars for this this, but he didn't want to use them. He prefers his fingers when he is skip counting.

These tables may look like worksheets, but I believe their point is a little different. There is no drill. The child can use whatever manipulative material, or fingers, to complete the problems. And these tables will be used as a reference and control for later lessons.
Here S is tracing her sandpaper letters with her eyes closed. This makes a HUGE difference in her ability to hand write them on paper with a pencil. After a few sandpaper letter tracings, she can write the letter proficiently. (We use very small pieces of paper. Like 1/8 of a piece of copy paper, on which you can only fit three letters.) You can also see her hand trace the sandpaper letters with more confidence, and refined movement. She was never able to write these letter forms on paper after tracing them with her eyes open. I wrote more about this in this post.
This is a brown stair extension lesson. Before this lesson, the child will already be able to remove the brown stair from the shelf and build it independently from random order on a single work rug.

In this lesson, we moved the brown stair from the shelf to the the first work rug and set it down in random order. We set up another rug on the other side of our shelving island, out of sight from the first rug. Then we built the brown stair on the second rug one piece at a time.

First we assessed which piece of all the pieces on the first rug was the "largest." We took that piece and began building the stair on the second rug on the other side of the shelving island. Then we looked at the remaining pieces on the first rug and selected the second "largest" piece to continue our build. Several times D selected an incorrect piece and each time he noticed that it was incorrect when he compared his selection with the rest of the completed stair. This material is control of error is the child, so he/she must be able to see visual discord and that the shapes do not graduate correctly.

There is an awful lot of walking to and fro during this lesson. I can see why this the perfect activity for the little ones with lots of energy. This little guy worked with lots of focus and concentration.

For reference, D started his Montessori training informally at home 6 months ago in September 2013. He has never had any traditional classroom training. He is now 2 years and 9 months old, and his two older siblings have had formal Montessori training.


 Carrying the brown stair prisms on your head is not a requirement.

 He likes to "walk down" the stairs to compare their dimensions.
And this always happens too. I am very glad we don't have to share these stairs with anyone else. I wonder if I hadn't called them stairs if he would have thought that they were something to walk down, though he does prefer to feel things with his feet.
He also constructed the pink tower in the same way as the traditional first lesson.


D did the metal insets and just astounded me. I didn't even see him do these. But I asked him and he said that he had done them. I wrote more about when to introduce metal insets in this post. I had thought that he was too young to do the metal insets, but this example may have proved me wrong.

I guess all those stairs, walking around the shelving island, building the pink tower, and handwriting really tired him out.

S invented these new characters the "Super Puppies." She writes it "Soopr Pupesz." Everyone has a mask and a cape. I am eager to read more about their adventures. 
This is S's new amaryllis, with the crazy Spanish moss "hair" around the top of it.
T is working on factors now. I haven't gotten the feeling that he understands quite what a factor IS yet. He is comfortable with the geometric representation, but gives me this blank look when I ask, so what are the factors for 12? There is time yet to understand everything.

He as able to follow the lesson and break down 30 and find all of its prime factors using the peg board for assistance.
At first glance this may not really seem like the Dwyer reading scheme, nor that this activity has anything to do with reading at all. But it does. S, the girl who doesn't like to read, read upwards of 30 words. All were verbs, written in cursive, on little slips of paper and she had to "crack the code" and "perform the action." "Cracking the code" made her puff up with pride. "Performing the action" made her laugh like a crazy person and fall down.

I can see now how these kinds of command card activities really catch the attention of the child, and how this kind of exercise really demonstrates the communication element of writing and reading. We are communicating each time we write something down and hoping someone will "hear" our thoughts without us ever speaking them aloud. I believe the fact that language is for communication is one of the intrinsic notions that is lost in the traditional educational frenzy. In the traditional education setting so much of the focus is on the mechanics and not the reason why children might learn how to read and write in the first place.
I hope that in between falling down these children understood that I was trying to communicate with them.