I have recently visited the Montessori PSA conference in Vancouver, which was based in the Hammond Elementary school of Maple Ridge. In one of the workshops I got a chance to learn about the Montessori method for finding square roots without a calculator. The method is part of the Montessori elementary school curriculum.
Before I explain how square roots are found, I must discuss a prerequisite concept: a product of two binomials. Remember that a binomial is a polynomial of one degree. It has the form of (x+a).
Let’s consider the binomial product of (x+5) and (x+1), which expands to (x² + 5x + 1x + 5). Also, let’s assume that “x” is a 10. This product represents an area of a 15x11 rectangle.
We are going to look for patterns inside this rectangle. The 15x11 rectangle contains a 10x10 square, which corresponds to the x² term. Let’s isolate this square in the top-left corner, and consider how to identify the remaining sum 5x + 1x + 5 in the remaining part of the rectangle. Consider a layout of such a rectangle which uses beads, strings of beads, and a square of beads. (The beads are a basic math material introduced in Montessori pre-school).
The sum 5x + 1x represents the “tens” because we took “x” to be a 10. Since the square is a 10x10, the tens must be along its sides. In the diagram above there are 5 columns and 1 row adjacent to the square, each containing ten beads.
This leaves the bottom right corner, that touches the square at a corner. Because this position does not match the square on its sides, it is not forced to be a multiple of 10. This space is for the units, which are 5 in our case.
This seems to work with beads, but the lesson must be transferred to a pegboard. A pegboard is a board with holes, into which one must insert pegs. Its advantage is that it is easy to organize pegs into a certain pattern.
However, on a pegboard it would be tedious to lay out 100 or more pegs. To overcome this, a shortcut is used: a single red peg stands for a 100 units, and a single blue peg stands for 10 units. (A green peg stands for a single unit.) The previous 15x11 rectangle, when laid out on the pegboard becomes:
It looks so small — what happened? First, the 10x10 bead square was replaced by a single red peg. Second, the 10-bead columns and the 10-bead row have been replaced by the blue pegs. Third, the 5 green pegs represent the 5 single beads.
Although in the rest of this article I will assume x to be 10, notice that the peg pattern above would work for other values of x, as long as we modify the meaning of red peg and blue peg accordingly. For instance, if x was a 9 instead of a 10, then the red square should stand for a 9x9 square and a blue peg for nine pegs, arranged as a row or a column. The green pegs would still stand for units.
Armed with the method of representing (x+a)(x+b) on the pegboard, we can apply it to the problem of factoring a square number into its square roots. The first example was a rectangle because the two binomial factors were not the same: (x+5) is bigger than (x+1). But a square number is one which can be represented as a geometric square.
A square number can be represented as a product of two equal binomials: (x+a)(x+a) or (x+a)². Consider a representation of the number 225. This number is 15², or (x+5)² if x is equal to 10. The expression (x+5)² can be expanded a x²+5x+5x+25.
On the pegboard it would look like this:
The green square is 5x5, which has 25 units. Each of the blue sequences has 5 blue pegs. Finally, the single red square corresponds to the x² term, because it counts 10x10 or 100 units.
From the pegboard, if we look at the rightmost column, we can see that the square root of 225 is 1 blue peg and 5 green pegs. Because the blue peg stands for 10 units, we get 10+5 or 15 for the square root solution.
Also, let’s make this generalized observation: the two blue sequences are the same size. They are the “arms” of the red peg, wrapping the green square on two of its sides.
Let’s look at it algebraically in order to understand why the above pattern is happening. A square number is (x+a)(x+a) which expands to x²+xa+xa+a². This expression itself has two smaller squares x² and a². If we represent them with pegs as geometric squares, they must each have a spot within the big square (x+a)². We can place the x² in the top left corner, and the a² in the bottom right corner.
The square number a² is represented with green pegs. The square number x² is represented with a square of red pegs.
A single red peg corresponds to 10x10 units, which equals 100. A 2x2 red square corresponds to 20x20 units, which is 400. A 3x3 red square corresponds to 30x30 units, which is 900. And so on.
Also, notice that the term xa is both a multiple of x and a. This means that it can be placed adjacent to a side of either the red or the green squares.
Let’s try to work out how to place the number 225 on the pegboard without doing any algebra. We must have only one red peg, because a 2x2 square of red pegs already has a 400 value, which is more than 225.
Because we have only 1 red peg, there can only be a single row, and a single column of the blue pegs. The reason for that is that they must be on the side of the the red peg, like “arms”. The remaining value to place on the pegboard is 225 minus 100, or 125.
So, we must separate 125 into green and blue pegs, such that the greens form a square and the blues fit exactly along two of its sides. Does a 4x4 green square work? That would leave 125–16 or 109 units to represent with blue pegs. But, 109 is not divisible by 10. Therefore, a 4x4 green square doesn’t work.
We need to use enough green pegs, so that if we subtract their value from 125, the leftover should be representable by blue pegs. In other words, the leftover must be divisible by 10. The candidates for the number of green pegs are therefore: 5, 25, 35, 45, 55, 65, 75, 85, or 95. Of these, only 25 is a square number. (We need it to be a square number, because we have to represent it with a green square.) Therefore, we will have a 5x5 green square.
Voila, we now worked out all the information to lay out the pattern. The green square is 5x5, the red square is 1x1, and the remainder is filled by blue pegs. Here is the solution again:
In summary, finding the square root of a square number boils down to identifying the sizes of the two inner squares: the square of the hundreds, and the square of the units. Isn’t that interesting?
Also, the solution is the sum of their square roots, which are the sides of the squares on the pegboard. (On the diagram above, count the quantity in the right column: 1 blue and 5 greens give 10+5 or 15. The square root of 225 is 15.)
Let’s do this exercise again but with a larger number. What is the square root of 1764? Let’s approach this methodically, by starting with the red pegs. As we discussed, the red pegs should form an inner square. Notice that the most we can have is a 4x4 red square, because that’s already a value of 1600. (A 5x5 would be too large.) So, if we are using a 4x4 red square, this leaves 1764–1600 or 164 units remaining to arrange inside the overall square.
Again, we must subtract from 164 enough singles for the leftover to be divisible by 10, in order to represent that with blue pegs. The candidates are 4, 14, 24, 34, 44, 54, 64, 74, 84, 94, 104, 114, 124, 134, 144, 154. Of these only 4, 64, and 144 are square numbers, which we need if the greens are to be arranged as a square. Therefore, the choices for the green pegs are 2x2, 8x8, or 12x12. This gives us the possible leftovers of 160, 100, or 20 respectively to be filled with blue pegs. (I’m subtracting 4, 16, 144 from 164).
We must now choose between a 2x2 green square, 8x8 green square, or 12x12. In the first case there will be a 160 leftover, in the second a 100 leftover, and in the third 20 leftover. Which one is right?
To answer this question, remember that the blue pegs must be arranged symmetrically: half of them above the green square, and half to the side of the green square (bottom left corner of the overall square). If the leftover is 160, then 80 is half. If the leftover is 100, then 50 is half. Finally, if the leftover is 20, then 10 is half.
Because each half of the blue pegs is sandwiched between the red square and the green square, the value that they represent must be divisible by the size of a side of each square. We have a 4x4 red square, and if I put blue pegs along it, it would represent 40. So, half of blue pegs must be divisible by 40. Therefore, the half of blue pegs that works is 80, and not 50 or 10.
Note that we would also be able to filter out the wrong values, had we considered that each half of blue pegs must be adjacent to the green square that we are trying out. A 2x2 green square would require half of blue pegs (80 in that case) to be divisible by 20. That works. But, an 8x8 green square would require half of blue pegs (50 in that case) to be divisible by 80, which doesn’t work. Similarly, a 12x12 green square, would require half of blue pegs (10 in that case) to be divisible by 120, which doesn’t work too.
So, half of blue pegs must be 80, and we must have a 2x2 green square. That’s because, working backwards, if half of the blue pegs represent 80, then all the blue pegs represent 160, which leaves only 4 units to be represented by green pegs. Therefore, the green square is of size 2x2.
The solution representation of 1764 on the pegboard is:
The side with the blue-and-green pegs gives the answer of 4-blue and 2-green, or 4 tens and 2 units, which is 42. Check on your calculator that 42² is indeed 1764.
You may think that the process I have outlined is too hard for an elementary school student. It is not, because the thinking process I have outlined occurs interactively, while the Montessori student lays out the pegs on the board.
The steps that he would take are as follows. First, he takes just enough red, blue, and green pegs to represent the number 1764. He places them into separate red, blue, and green bowls. He knows that at any point he can exchange pegs of one colour for equivalent amount of another colour. For instance a red peg can be exchanged for 10 blue pegs, or vice versa. (Montessori program teaches this much earlier, between ages of 3 to 6, using beads, strings of beads, and 10x10 bead squares).
Second, the student tries to guess the sizes of green and red squares, by seeing if the blue pegs fit evenly in the space that is remaining. As he stumbles through this process, he begins to naturally notice the same kind of patterns I have outlined above.
As he gets familiar with the process, he may explore on his own. For instance, he can generate a new square number by changing the sizes of the inner squares, keeping the overall pattern.
For instance, given the layout for 1764 on the pegboard, he can reduce the 4x4 red square to a 3x3 red square, and increase the green square to a 3x3. He would fill the rest with blue pegs, and would notice that that they are also in a 3x3 pattern on each side. What is the square number represented by the overall new design? Three blue pegs and three green pegs give 33. The new square number is 33².
What does 33 squared equal to? He doesn’t even need a calculator. He can count the value of the pegs directly: 900 (red) + 9 (green) + 180 (blue) which equals 1089. (Did you know, dear reader, that 1089 is a square number?)
Another interesting realization is that there is a hierarchy of square numbers. Every square number is made up of two smaller square numbers plus a little extra. In other words, there is a recursive relationship of square numbers. Later, when he studies recursive relationship, this intuitive lesson will come to mind.
There are more general questions. For instance, is there only one unique arrangement of red, blue, and green pegs to represent a particular square number? Could there be a solution if I took a 3x3 red square instead of 4x4 red square, and then tried to fit the leftover? Can a square number be reworked into a rectangle?
If he explores that last question starting from a 15x15 square arrangement, he will see that it can be made into a 9x25 or 3x75 rectangle.
He can also consider patterns made solely from squares. A 2x2 red square, a 2x2 green square, and two 2x2 blue squares, all packed into a big square. What value does this design represent? It is 20 + 2 or 22 on a side, which is 22². Without a calculator, it is equal to 400 + 40 + 40 + 4 or 484.
Proceeding in this way, a student can learn first hand about patterns in the numbers, iterating back-and-forth between visual and numerical domains. Yet, another avenue of exploration is not to take the red peg to be a 10x10, and a blue peg to be 10. How about the red peg to be 3x3 or 9 and a blue peg to be 3? What would be the layout on a pegboard to represent 225 under such a system? A 2x2 red square would be 6x6 in value, or 36. A 3x3 red square would be 9x9 in value, or 81. A 4x4 red square would be 12x12 in value, or 144 . A 5x5 red square would be 15x15 in value, which is 225. Therefore, in this system no green or blue pegs are necessary.
Also, because each red peg has value 9, it means that 225 is divisible by 9. Furthermore, because each red peg is 3x3, the number 225 is divisible by 3.
What we have essentially done in the preceding exploration, is switch from base 10 to base 3 arithmetic. Having done this exercise, a student will much better understand the significance of positional notation of numbers and the meaning of units, tens and hundreds.
This is the kind of thinking that makes future mathematicians. It is understanding what is really going on. It is applied mathematics, grounded in physical reality.
It is important for a student to understand how the knowledge that he learns can be applied. Although mathematicians in the last 400 years exclaimed that their math has no application whatsoever, not only it is a wrong philosophy, it is anti-pedagogical.
Maria Montessori said that infants and young children must learn everything through “the hand”, by literally touching the things that they are supposed to understand by merely seeing. For instance, to learn the shape of a circle, a child must trace the edge with his finger.
The same principle of touching applies even for older children who are learning math. By manipulating the pegs, children make numbers real. It is more vivid and memorable to see that a square is made of two smaller squares, than to see algebraically that (a+b)² is made of a² +2ab + b², so that the two smaller squares are a² and b².
Algebra is a great invention, but we must make sure that students are ready to learn it. After all, neither ancient greeks nor Egyptians had a fully fledged algebra.
Maria Montessori wrote that a child must trace the development of civilization in his own process of development. He must interact with raw nature, forests, take care of plants and watch them yield fruit. Only after such experience he can appreciate more abstract knowledge that he will receive in elementary school.
Similarly, a child must learn mathematics through tracing its historical development. Educators must show him a manual way to solve problems, before introducing shortcuts like algebra.
Consistent with this, the overall Montessori math program is a gradual introduction to abstract concepts. At infancy, an infant is taught through sensation a variety of geometric shapes. He handles them and puts them into his mouth. Then, he is taught shapes by matching them onto a template. At 2 years old he is taught to compare shapes by dimensions: big or small, tall or low, long or short. Once he can compare two items, he is prompted to sort several of them according to a dimension — for instance, from shortest to longest.
Have you heard the expression “God made the natural numbers, but everything else is the creation of man?” Actually, the natural numbers are the creation of man too. The only thing that is metaphysical is the attribute of “length”. We measure length with other lengths, by putting copies of a smaller length along a bigger length. (Thus, the Montessori exercise of comparing and sorting lengths is the foundation of all of mathematics.)
Once he understands relative relationships of length, a Montessori student child is introduced to working with beads. He is taught to associate different amounts of beads with numbers from 1 to 10. When he gets beyond 10, he is taught to exchange ten individual beads for a single string of ten beads. When he gets beyond ten such strings, he is taught to exchange them for a 100 bead square. When he has a good grasp of working with beads, he is introduced to the pegboard.
The Montessori math curriculum is of a literally hands-on nature. Although the student learns ever higher and higher levels of abstraction, each level is motivated by a physical need of increased organization. For instance, it is more organized to keep sticks of increasing length in a pattern than to have them lying around haphazardly. And, it is more organized to keep a single 10-bead string than ten individual beads. (Compare with how adults organize their wallets: they prefer to carry large bills more than small bills, and small bills more than pocket change.)
Maria Montessori noticed, with surprise, that children as young as infants like organization. They enjoy putting learning materials back into their proper place. If you ever visit a Montessori daycare, you will see how calm and organized everything else. It is not the teacher who keeps the place neat, but the children themselves.
The pegboard is a new step in organization of ideas that the Montessori student learns. He will eventually learn algebra, but with this foundation he will be able to appreciate how invaluable algebra is. In particular, he will understand how the abstract symbols of “x”, “a” and “b” relate to physical reality, because he can go from algebra, to pegs, from pegs to beads, from beads to lengths.
There are more things that children learn with the pegboard, than products of binomials. For instance, they also learn how to find divisors, by representing the same number as differently sized rectangles.
Would you like to teach math to your child using the Montessori method? If your child is not in a Montessori math program, you can still apply these ideas at home. Montessori learning materials can be made by hand, or can be bought on Amazon, or bought second-hand in a local Montessori Facebook group. A classic wooden 30x30 pegboard costs $65 on Amazon. If your child is too young for the pegboard, you can train him using earlier Montessori materials that I have outlined.