Here is a bit more of our work. We've been spending more time in the classroom lately and I have a corresponding increase in the volume of digital photography I need to edit. (Yes, the crazy person in me edits all my photos.)
I cut the last post in half because I thought, "no one is going to sit down to read about 90 photos of work." And so the concept of two installments was born. Here is the second installment.
This week has been the week of new works. I attribute this to my more effective, perhaps, planning and studying. And perhaps the increase in number of classroom hours has helped us do more new lessons and continue pushing along the threads we've started. I do hope that our momentum will continue in this manner for a bit.
S is back at the fraction insets. I don't even remember the last time S worked with this material. I think it was a long, long, long time ago. Anyway, I believe last time we worked on nomenclature and naming each fraction family. The inset with one piece in the family is called a "whole." The inset with 9 pieces in the family is called "ninths." She actually remembered these names. In this lesson, we worked on writing the names of the fractions.
First, I introduced the denominator. I told her that the "family name," or the number of pieces in that family, goes below the black line. She proceeded to put all the denominators written on paper tickets in place near their corresponding inset.Then we wrote some numerators, which represent the number of family members we have. Here you can see that S wrote numerators other than one in several instances.
Then she traced and colored in her complete fractions in her working notebook.After writing the names of fractions, S placed prepared name "tags" on each fraction (like T did here.) Next we will move on to adding fractions with like denominators with sums less than 1 whole.
Now that I am looking at these photos, it seems like a round ring is really a tricky item to polish. No wonder children start out with flat items like plates and the like. Even so, D did just fine. Incidentally, this is my grandfather's college class of '38 ring.
This is the 4-cubed-bead chain. S has counted the 100 chain, the 1,000 chain (and here) and all the squared-short-chains. Now she is starting on the long-cubed-chains. Since her progress in this area has been a little slow so far, I decided to try and ignite some interest in this corner with what else but little trinkets. (S really needs the counting practice, and the practice with seeing multiples. But, I am suspecting, she is a little past the sensitive period for this work, so we are trying to inspire a bit more intensity.)
S loves, small shiny objects. Here she chose to place a shiny trinket at each point in the chain that equals another square.
And now that I am looking at her pictures, her number tags aren't really where they are supposed to be which means she wasn't counting accurately. Humn. We need to figure out what is going on in this area.
UPDATE: So, I think I understand the REAL root reason S has been ignoring the bead chains. She doesn't remember her numbers. She doesn't remember which hierarchies go where. She doesn't remember her teens, or her tens, or any other number for that matter. She doesn't see the patterns in the base-ten system. Okay, now I need to re-think, re-work, and review with S. I'll keep you posted on what happens next in this department.
I last posted about suffixes here. In this extension activity the child will place this Suffix Chart 2 in an opposite corner, note a root word he would like to write down and then return to a remote table to use colored pencils to write down the word's root and suffix.
Here we have all sorts of interesting penmanship and capitals in all places not needed, and zero regard for the notepaper lines. Maybe I should get the kids blank sketch books next time with no lines. Anyway, T is writing down his root words in regular pencil and his suffixes in colored blue pencil.
Leaving the chart across the room helps the child work on short term memory.S also got to the bow tying frame last week. S finally got some sneakers that were tie-sneakers so we decided it was time to learn how to tie our shoes. (Up until now, she's avoided tie sneakers by wearing, crocs, cowboy boots, Uggs, ballet-flats, velcro sandals, wellies, slippers, you get the idea that she has a lot of footwear alternatives that do not involve tying any bows.)
This lesson is part of the Care of Self part of the Practical Life primary album. There are a million steps to this particular lesson so I will not go into all of them here. I remember being in awe of the number of steps there are when I watched the CGMS training video for this lesson. Each step is simple and pretty intuitive.
In the shot above, S is pulling the tails of the bow to pull apart the loops.
Here S is loosening the half knots. And of course now that I look at our lesson maybe we should have tied it from top to bottom. But I think that the child, especially an older one, may tie, or untie, the knot closest to them. D even has a bit of a hard time reaching the top of the frames.
Here S has smoothed the ties to either side making everything neat.
And here, S opened the clothes showing the open frame. Young children might trace the rectangle frame with two fingers before closing the clothes.
Then, S made a loop in the orange ribbon and held it low, close to the half knot. Then we wrap the green ribbon around to the front, counter clockwise, and then tuck it under itself to create the beginnings of the second loop. After that tuck-under is secure, we switch hands, and the left hand will hold the orange loop and the right hand will whole the green loop and we pull the knot tight adjusting by pulling the tails and again the loops if necessary.
S was very proud of herself.
T's lesson here comes from the commutative and distributive laws of multiplication sequence. He has already seen, twice, the commutative and distributive law in both beads and simply cards. (In fact, he went a lot further in this numeration sequence last year, but with our drawn out summer-time relocation-half-way-across-the-country, a lot of this knowledge was forgotten.) Anyway, the shot above was a quick review with our arithmetic signs.
The problem above is a sum multiplied by a sum. The multiplicand is displayed in beads and the multiplier is represented by the grey number cards. Our black card stock parens indicate where we need to add and where we need to multiply.
The child would then create small groups of each bead color under the multiplicand on the left. The first group would include 5 purple 6-bars, and 5 brown 8-bars. The second group would include 9 purple 6-bars and 9 brown 8-bars. The child would then add up all the beads to find their final answer. I think what is happening above is that T isn't using beads to make partial products anymore and is doing the math in his head. So, 6 taken 5 times equals the 30 he wrote down on his paper.
Then we progressed to this, still on paper without bead partial products.
And then we progressed to this, which is all on paper without any beads or cards.
He found this "on-paper" way a lot more efficient.
The goal of this particular lesson is abstraction on paper, but our next lesson will take us back to golden beads again. This time we'll be using beads to really examine what happens when you multiply particular hierarchies by other particular hierarchies which is work that prepares the child for algebra, squaring and square roots.
This is a Practical Life, Care of the Environment lesson I did with D. (I secretly made this mess all by myself. D was actually pretty horrified when he saw this mess.)
This is the "how to use a dustpan and brush lesson." We got this child-sized brush and dustpan from Montessori Services, though I suspect that you could find a mini one at the dollar store or any big-box store locally.
First, I invited D to come and help me clean something. We went downstairs with the dustpan and brush and that is when he saw the messy floor. I disconnected the dustpan and brush, since they nest, and squatted on the clean part of the floor near the mess. I used the brush to sweep the crumbs into the dustpan, which I held to the side, using small, slow, sweeping strokes. I showed him how to sweep the crumbs into the dustpan continually until the line of debris at the lip of the dustpan disappears. After carefully tipping the dustpan back to level, I carried it and brush to the trash can, stepped on the trash can pedal to lift the lid and carefully tipped the dustpan to let the crumbs slide out into the trash can. I used the brush to brush the dustpan to rid it of any remaining debris.
Pretty much the entire time I was presenting, D spun around on his butt on the tile, sulking, and telling me HE wanted a turn.
I'd say he did a so-so job. It was kind of like he had foggy goggles on and couldn't really see all the crumbs. I guess that is what a mommy with a vacuum is for.
After our congruent and similar work here, we moved on to equivalency. Equivalent figures are not the same shape or size, but share the same value. In the shot above, both the triangle and the rectangle represent 1/8 of the same square area.
After exploring the metal insets, T and S both picked a couple of equivalent pairs to trace onto paper, cut out, and glue into their working notebooks.
Here, D felt compelled to grade the equivalent sets. I guess we have been doing a lot of grading lately.
And then I taught them the mathematical sign for equivalent. These blue and black figures are T's work.
These figures are S's work.
T and I began the Squares and Cubes of Numbers sequence in the KotU Math album. This is the Notation of Squares lesson. I prepared small blank laminated tickets for this lesson and let me tell you, they worked like a charm. The dry-erase marker just rubs right off, and you can use the little ticket again for the next notation of cubes lesson, or the next child in line.
Anyway, we started the short bead chain of 5. Here it is light blue, and there are (5) 5-bars linked together with jump rings. I snaked up the linear chain so it looks like the "square of 5" which is that light blue square of beads all wired together. T counted one side of the square and noted that there were 5 beads. And then we counted another side of the square and noted that there were 5 beads which indicated that the square was 5 by 5, or 5x5 beads. I also showed him the traditional square notation which is 5^2. (I didn't have time to Google the HTML for superscripts. Maybe I will by the time we get into this work a little deeper and I am typing up squares and cubes all the time.)
Then we took out my prepared tickets and the beads from the bead cabinet the quantities that each ticket represented. Here you can see that we can write 6^2, 6x6, and 36, which all mean 6-squared.
T illustrated all of this on graph paper as well. The circles on the left represent the folded chains and the squares on the right represent the bead squares.
The next day we moved on to cubes. This is the cubed 4-bead chain. I snaked the long bead chain around to demonstrate that it is equal to 4 4-squares.
Then we stacked the (4) 4-squares on top of each other to show that together they equal the cube of 4.
Then we wrote the different ways to represent 4^3.
T used prepared tickets to label each of our bead cubes. From left to right we have, the 10-cube, the 8-cube, the 5-cube, the 7-cube (white), the 2-cube, the 6-cube, the 3-cube, and the 9-cube. I don't know where the 1-cube is.
As you can see, we might need a little more practice remembering our cubed values.
These notation lessons are preparation for squaring and cubing work to come.
I like to adhere to the album lessons pretty strictly, but this work was sort of out of way. D and I did a little bit of story sequencing here. I read the booklet and D looked at the picture cards and placed the in order. (These, if you haven't guessed, are our biome readers.) Or, he looked at the back of the card at the card code number (all are 1-6) and put them in order that way. Somehow he picked up those number forms.
This was S's compound word lesson. The row of books at the top demonstrate that compound words are really two words joined together to make a new word. Our demonstration included a workbook, a cookbook, a checkbook and a notebook.
Then I took these small objects T used last year, and remade the lesson for S. At first we explored the items in the box and then I encouraged her to figure out what to do with them. After a tiny bit of prodding, but no give-aways, she figured out that if you combine two of the items, they "make" the name of the third item. She was very proud of her discovery.
And finally, we cracked the first lessons in the decimal fraction sequence. Here we are introducing smaller than 1. Before starting decimal fractions the child should have experience with fraction work (specifically the concept of a part of a whole), long multiplication (especially multiplying by tens and dealing with zeros) and the decimal system (especially with the Montessori color coding and card and bead work.) T has solid experience in all of these areas, but hasn't completed any of these threads.
T and I pulled the green unit bead from our racks and tube set up and the whole fraction inset and 1/10 fraction inset from our fraction drawers. T and reviewed that the unit bead is the same as the whole fraction inset. Each material represents 1-whole. Then, we pulled out the light blue, decimal fraction cube from the decimal fraction materials and the 1/10 fraction inset and laid them side-by-side. I told T that when we divide up 1 whole into 10 equal parts, each part is 1/10 of that whole. I also told him we can represent 1/10, or 1 of ten equal parts of the green unit bead, with the light blue cube, and we call the cube one-tenth. At this point he was just super excited to be using a new material.
After this we placed our linear lay-out with just beads and cubes. (No cards yet.) To the right of the unit we placed, 1 tenth, 1 hundredth, 1 thousandth, 1 ten-thousandth, 1 hundred-thousandth and 1 millionth. Just as above, if you cut up a 1 tenth slice into ten parts, each part is equal to 1 hundredth and so on. As you can see above, each of these "part of a whole" quantities were represented by a light green, light blue, or orange cube.
To the left of the green unit bead, T placed the hierarchies greater than one: ten, hundred, thousand, ten-thousand, hundred-thousand and million. To make 10, we multiply the unit by ten. To make 100, we multiply ten by ten, and so on. These quantities were represented by the regular green, blue and red beads.
Initially we were only discussing this concept. In the next step we introduced the number symbols that represent each of these quantities.
In the photo above, we did an intermediary step the album says not all children need to complete. I made the printed chart on the computer and pasted it to some card stock. Since it isn't a material that will be handled much I didn't bother to laminate it. But this strip does further reinforce the dividing up by 10 by naming the decimal quantities in fraction form.
Then we added decimal cards. We borrowed some from the bank-game for this first presentation. First we laid out above the beads all quantities larger than 1: 10, 100, 1,000, 10,000, 100,000 and 1,000,000. Then, we "flipped" the cards upside down, over the unit. So, the blue 10 card became 01 (upside down) and then we placed a black paper punch dot for the decimal point to create 1 tenth. T thought that turning the cards upside down was pretty cool,
Then we replaced the upside down bank game cards with the cards from the decimal fraction material.
And finally, we played a game. I took all the notation cards and shuffled them up. Then T had to put them over the bead/cube quantities they represented and say their name. He got the name wrong a bunch, but then always smiled and corrected himself.
Next up, we will break out the decimal board and start forming and reading decimal quantities to prepare for adding, subtracting, multiplying and dividing decimal fractions.
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Recently, a friend alerted me to a blog post out there that now troubles me. I don't really know why it makes me feel troubled, so I am working on sorting this out, if it turns out to be worth sorting out at all. In the meantime, reading that post did make me feel compelled to give notice to the two readers out there who read this blog, who might not know, that what I write here is only a compilation of my own interpretations of, and opinions about, the thoughts about the Montessori philosophy, and life in general, I feel are worthy of forming opinions about.
I also want to thank, deeply, all of the people who help me along this homeschooling journey. (You know who you are.) Thank you for all you do.