Using the Progression of Roots to Introduce Radical Numbers

We’ve been working on our Grade 7 Waldorf Geometry block. When planning for the year, I tend to leave the best for last. I consistently over plan and so that means we end up doing the best for the first block of the year. That’s what happened this year. This lesson is the lesson on radicals using the Spiral of Theodorus also called the progression of roots.

In this post you’ll find:

A very natural way for us to learn numbers is to first start counting, sorting, and sequencing by those numbers. We do this naturally when we are learning the natural numbers. It’s also helpful to do this with numbers beyond the natural numbers, such as multiples or fractions. In this lesson, it served as a use of the Pythagorean theorem that we had done previously, and as a formal introduction of the radicals.

The Pythagorean theorem is one of the most used geometry theorem. As such, I wanted to spend quite a bit of time on that theorem before we moved into this exercise. The previous two days before we were exploring, what exactly the Pythagorean theorem was saying, making observations of the different visual proofs of that theorem, and then placing those observations in the theory itself into our main lesson books.

This works at the stage for the progression of lesson that we eventually use Pythagorean theorem to find the diagonals of all the rectangles were constructing.

The Lesson

Constructing the Rectangles

We begin the lesson with a series of rectangles that we constructed on kite paper. The first rectangle is a square with the side unit of one. Our square actually measured four cm, but we need that unit to easily see the progression of roots for the square to be one. It doesn’t matter if it’s one inch, or 4 cm segment. 

After the square was constructed, we found the diagonal of the square by connecting to the opposite vertices of the square. At this point, we cut our square and half along the diagonal. The key of the construction of the remaining rectangles is that the link of one side stays that one unit, for us, which is 4 cm, and the length of the other side is the previous rectangles diagonal. We made a series of roughly 11 rectangles and corresponding pair of triangles in this way. One of the triangles served to construct the spiral later, and the other set of triangles served to show the approximate link of the diagonal.

Constructing the Spiral

After we cut out all of our triangles, which we kept in order as we kept them out, we first assembled our spiral. The key to simulating the spiral is to make sure that the diagonal of one rectangle, lines up with the link of the next triangle. You’ll remember from the construction of the rectangles; we were using the previous rectangles diagonal for the length of our next rectangle.

Calculating the Lengths of the Diagonals

At this point, we went through and calculated the length of each of the diagonals in radical form. Using the Pythagorean theorem that we had learned in the previous two days, we started with the first square. Because our first square sides are both one and one squared is still one, then the diagonal squared is equal to two. This means that the diagonals equal to the square root of two.

When we move to the next right triangle in our spiral, one of the sides remains one which squared is also one, and the other side is the square root of two squared gives us two. When we sum those we have 1+2, which is three. Three is equal to the hypotenuse squared, and the link of the diagonal would be the square root of three. It continues along this way with each diagonal measuring the next radical. We are essentially counting in radicals: square root of two, square root three, square root of four, etc.

Sequencing the Radicals

At this point, we made a table to the side of our spiral with a list of radicals in order on the right. Spent a little bit of time discussion numbers, and where we might see whole numbers as a square root. Then we went through with each of our triangles, starting with the largest because we were overlapping them, and line them up on a line at the bottom of our page. For each triangle, we laid the hypotenuse along the number line so that we could get an estimation of what that radical number would be, and where it would fit in our line. With the radical of square numbers, we were able to take the square root of that number and no exactly where on the line it should be. We went along, we glued each of the triangles hypotenuse along the number line and used scientific calculator to find the square root of each of the other diagonals. We filled out the information into our table as we went along.

Notice and Wonder

Various points during the activity, we were pause and do ‘notice and wonder‘ and talked about what we noticed. One of the things that we noticed was that the distance between the triangles when the hypotenuse was lined up, became less and less radicals. We also noticed in our chart that, there were more and more radicals in a predictable pattern between our numbers as we increased in the radical of natural numbers.

Extrapolation

We only had 10 or 11 triangles, but we had 17 radicals listed in our chart. We use the understanding that we had made from our observations to guess what the amount would be for those radicals. Then we went back with the scientific calculator and filled in exactly what those radicals were to see how close we were in our estimations.

Video

Sources

I have seen the progression of roots and several different places, but the biggest source for this was Polypad.