Here are two examples of pre-algebra Montessori “work”
The Peg Board
The Peg Board has many uses, here is the example we worked on this week. The purpose is to take a square and make it into a smaller square that is equivalent to the larger square.
For this example, we took the number 625.
First, my son "D" takes the units pegs (units are always identified as the green color, tens are blue and hundreds are red.) At this age, he is already well prepared to have the amounts represented by just a color – so there is some abstraction in this work that has come with years of being in the Montessori classroom using these representations.
Here is the peg board with the units pegs in a square that is equal to 625.
Second, he now counts over 10 pegs from the side, and replaces each row of 10 with a blue peg (which represents 10.)
Third, he continues with this process of replacing rows of 10 pegs, with a blue peg.
This process continues until the smallest square possible is built – equivalent to the original square of 625! From here we can see that 625 is equal to 25²
He learns to document this process as well as he goes along. Which you'll see in the next example.
Working with Squared Numbers
For this one, I’ll use the same numbers for sake of ease.
Materials needed include:
- Montessori Decimal Bead set (up to the ten bars)
- Montessori number tiles (I take them from the checkerboard multiplication set we have)
- Pipe cleaners – used for the “()” in the problem
- Cuts of paper with “+” and “X” on them.
- Montessori “hundreds” squares (each represent 100 beads)
Working to find the answer to 25²
First, we agree on a few things:
We agree that 25² = (20 + 5)² = (20 + 5) X (20 + 5)
Now he can work on the problem visually to come to his answer.
Step 1: He lays out the problem using the actual amounts shown by the beads. (See part of the problem built out visually below)
Step 2: Once he has the two rows of the problem both written and shown visually with the beads and tiles, he begins to use the hundreds squares and beads to find and show the answer to the problem.
As he finds the answers, he finds he will need 4 hundreds boards (represented by the squares) and then he finds how many 10 bars he will need - and places them in the binomial square shape (which he is already familiar with from previous works.
So he finds that he needs 400 + 100 + 100 + 25
400 - represented by the 4 100s squares
100 + 100 represented by the 10s bars
25 represented by the 5 bars (light blue bars have five beads on them)
His written notation looks like this:
From here he can work through the answers to multiple problems, practicing the steps over and over as well as practicing writing out the problems in his math notation workbook.
I didn’t know how to do this until Jr. High school, (and even then it was pretty shaky) but perhaps if someone had shown me how all squared numbers actually make a square – I might have found it interesting and concrete enough to try a bit harder. I’m looking forward to practicing this myself so I can help my daughter prepare for algebra the same way!
If you'd like more examples of this with more detailed instructions or pictures, feel free to contact me and I'll be happy to share examples to help you along!