In this section we’ll focus on exactly how Pathematics™ works and touch on the some of the mathematical information that’s coded into the Pathematics™ Runway.
The Pathematics™ Runway is essentially a large-scale arrangement of the numbers from zero to 100. Alongside the number line is an array of color-coded shapes that allow students to quickly visualize natural mathematical patterns.
Let’s look at the lower end of the Runway to see how the shapes and colors work.
Imagine yourself standing on the red number 1 in the zero row, or “Z-row” as we call it. If you take one step forward, you’ll be on the red dot on Row 1. If you take four more steps,you’ll be on the red dot on Row 5. The ones column is very simple: you’re just counting by ones as you make your way to 100. Every row has a red dot in this column, since the number one is a factor for every number.
Now imagine standing on the green number 2 in the Z-row. If you’re only allowed to step on the green dots, you simply can’t step on Row 1 (no green dot there!). You have to step on Row 2, then Row 4, Row 6, and so on. As you step through the numbers, you’re counting by twos, or stepping only on the multiples of two (which are called even numbers). If you count your steps as you go, you’ll form the “equation” that carried you to your destination. For instance, if Number 2 steps 4 times, he’ll end up on Row 8, peforming the equation 2 x 4 = 8. Note how the word “times”, like we say when we read the equation, takes on a new meaning here. Eight is just two, four times!
Of course, someone playing the role of Number 4 could solve the same equation. If you’re yellow Number 4, standing on the Z-row, you’d take just two steps (Row 4 and Row 8 ) to reach Row 8. Again, 2 x 4 = 8.
Each of the ten factors on the Z-row work this way: with every step onto one of your colored dots, you’re stepping to the next multiple of your number. Number 10 takes 4 steps (BIG steps!), and she finds herself on the gold dot on Row 40, showing that 10 x 4 = 40. After a little practice, you learn the colors associated with each of the number. You know that when you see a purple dot in a row, then that number is a multiple of six.
In addition to all those circles, you’ll notice that there are occasional squares on the Runway. They represent the perfect squares of each of the ten factors. For instance, when someone playing the role of Number 5 takes 5 steps, she ends up on the light blue square at Row 25. She just performed the equation 5 x 5 = 25, or five squared. In the figure above, we see the red square for 1 x 1 = 1, the green square for 2 x 2 = 4, and the pink square for 3 x 3 = 9. The next one would be a yellow square at Row 16.
If you focus on the individual rows on the Runway, you’ll see that you can tell all the factors for a number at a glance. For instance, Row eight has a colored dot in columns 1, 2, 4, and 8. These are the only four factors of eight. Take a look at the section of the Runway and make sure you understand how the factors 1 through 10 are coded into pattern.
Now we’re going to look at some of the more advanced features of the Runway.
Not only are the factors 1 through 10 coded into the Pathematics™ Runway; all the factors up to 100 are represented. Looking closely at a typical number row, like 90, we can see how the larger factors are represented.
The little dots in the borders of the circles (or squares) are the key to the larger factors. Looking back at the first figure (with Rows zero through 14), see how factor 11 is coded into Row 11. In Column 1, where we find all the red dots, we see a big red dot in the circle (just like all the other dots in that column, since every number is divisible by 1). There is also a little red dot in the border to represent the factor 11. If you moved up to Row 22, the second multiple of 11, you’ll find exactly the same little red dot in this position, and you’d find the same thing at 33, 44, 55, and so on. You can jump through the numbers in steps of 11 in exactly the same way you can jump by twos, fives, and nines!
Note that a larger factor does not have to have a big dot in the middle; it’s often an empty circle that has the little dot on the border. In Row 13, for instance, the factor 13 is indicated by the little red dot on the border of the empty circle in Column 3. Again, a red dot tells us that the tens place is 1 (since red is always the color for 1), and we know the ones place is three since our circle is in the third position, or column. You’ll find a repeat of this exact element on Row 26, the next time that factor 13 shows up. On Row 39, the third multiple of 13, you’ll find the little red dot, but it’s on a circle with a big pink center dot, since 39 is also a multiple of 3.
The little dots may be a bit confusing at first, but once you get the hang of it, it’s wonderfully convenient to be able to find all the factors of a number (like 90 in the illustration above) at a glance. Most of us would be able to quickly figure out that 90 has the factors 1, 2, 3, 5, 9, 10 and 90, but it’s not so easy to come up with some of those other factors. Once you learn the secrets of the color coding, they’ll practically jump off the Runway. Most students enjoy the “hunt” for these factors; it’s a puzzle that’s fun to solve.