The weather is still reaching almost into the 100s each day, and the sun is coming up later each morning. I am hoping that we'll be able to play tennis mid-day and practice lacrosse and soccer in the evenings as the days cool down a bit. I am also hopeful that I'll be able to tweak my morning schedule so that I will be able to see the sun and will not have to go jogging with a head-lamp on. And I am hopeful that our family schedule will become routine again very soon.
Along these lines, I have a ton of school stuff to share, but I just haven't had the time to sit down at the computer to modify all my pics and to write about what we've been doing. A full posting routine will come in time, but for now, here is just a tiny bit of what T's been doing with our cubing materials.
T has finished the Passing from One Square to Another lesson sequence and can pass to a non-successive square and express this process algebraically. For example: we can pass from the square of 3 to the square of 7. (I realize now that I wanted to get a shot of this before I posted about it, but sorry, no shot.) So, numerically he gets: 7^2 = 3^2 + 2(3*4) + 4^2; and algebraically he gets: (a+b)^2 = a^2+2(a+b) + b^2.
T also finished the next lessons in the squaring sequence, Squaring a Sum, where we do somewhat the same process using bead-bars. T had a big ah-ha moment when he figured out that the rectangles were equivalent to the 2(ab) he's been seeing over and over again. Sorry, no pictures of this either.
The next lessons in the sequence would be Squaring with Hierarchical Value. In this sequence we revisit the peg board and a bit of geometric multiplication. I decided to move on to Passing from One Cube to Another and do the Squaring with Hierarchical Value lessons concurrently.
This work uses our cubing materials. They are from Alisons I think. This material is incredibly heavy. Only I can actually move it. I need to keep it on our top shelf since our weird shelves are organized from bottom to top so that the littlest guy can reach everything he needs. This box is a really wonky weird shape which I hate. The entire material was super expensive and it came four months later since it was on back-order for ever and ever. Anyway, the quality is nice and the kids are very intrigued.
The material consists of one huge box with dividers to fit each cube. There are 27 cubes of 1. All cubes and squares are color coded according to the Montessori beads. There are then 27 squares of 2, and one cube of 2. The squares are all one unit thick. There are also 27 squares of each size (3-9) and one cube of each size (3-9) as well. The albums note that the cubes should be a shade darker than their corresponding squares...but this doesn't really seem to be apparent in our set.The first lesson is passing from one Cube to Another: Passing from One Cube to its Successive Cube. Above we have the cube of 5 (lt. blue) and the cube of six (purple.) We then worked to turn the cube of five into the cube of six. You can see that T added a square of five on three sides and a length of 1*5 to three edges and the cube of 1 to one corner.
These are the pieces taken apart he used to create the cube of 6.
Then we added writing to describe what we were doing. Here he wrote that he used the cube of 5, three squares of 5, three lengths of 5, and the cube of 1. He verified numerically that all of that does indeed equal the cube of 6, or 216.
The album suggests in the first lesson to use tickets and name individually each piece we used to build the cube of 6 and then move to writing down the entire equation on a single sheet of paper. T didn't need to do this step and just wrote down his equivalent equation on paper in one step. I think that it may have been because he had been doing something similar with the passing from one square to another and that this looked similar to the binomial cube he used in primary. Later on we will derive the algebraic description of what T is constructing here.
T decided to carry on and pass from one cube to a non-successive cube. Here he is going from the cube of 4 (the yellow) to the cube of 6 (the purple.)
Here he is verifying that these cubes are indeed equivalent or equal on all sides.
And these are the pieces he used to construct his cube of 6. The cube of 4, three pieces consisting of two squares of 4, three prisms of 2x2x4, and the cube of 2.
This was T's equation describing all of the pieces he used to build his cube of 6.
T here made the cube of 9, from the cube 6. I don't remember if he wrote this one down.
The next lesson will be cubing a binomial using beads. Later we will include algebra and lastly link our mathematical description to the sensorial material T used in the primary class, the binomial cube.
D decided to get in on the construction action too.
We'll be back with more school stuff soon!