Maybe you've played around with putting marbles in various patterns on a Chinese checkers board. Each player starts with 10 marbles in a triangular pattern. The 10 marbles are in four rows inside
the triangle, and the triangle has 4 marbles on each side.
This same pattern is the position of the pins in 10-pin bowling. There's one pin in front, two behind it, three in the next row, and four in the last. On each side of the triangle there are 4 pins.
What if the triangle is bigger or smaller? Suppose the triangle has five marbles or pins or whateverlet's call them dotson each side. How many are there altogether? What if the
triangle only has 3 dots on each side? More generally, can you find out how many dots are there altogether in a triangle if you know how many dots it has on each side?
How can students in different grades approach this question?
Even the very youngest ones can answer it by drawing a figure and counting the dots. It takes a little dexterity to draw them well so that there aren't too many or too few in a row, so maybe arranging checkers (or other round figures all of the same size) into a triangle would work better.
The next step past counting could be addition. If you know that there are 10 dots altogether in a triangle with 4 dots on a side, then determining how many dots there are altogether in a triangle with 5 dots on a side can be done with addition instead of counting all the dots. There are other connections of this problem with addition, too. For instance, you can see 10 as 1 + 2 + 3 + 4, or as 4 + 3 + 2 + 1. Why is that? (Commutativity of addition: if you have 3 red apples and 5 green apples or if you have 5 red apples and 3 green apples, you've got 8 apples in all.)
Call a number triangular if it's the number of dots in one of these patterns. So, 10 is the fourth triangular number since it's the number of dots in a triangle of side length 4. What are the other triangular numbers? Here's the beginning of a table.
Can students find patterns? There are lots of them, but the most obvious one is that you get the next triangular number by adding the next number to the current triangular number. For instance, the 5th triangular number is 5 more than 10, the 4th triangular number.
Children a couple years older are learning multiplication. They might see another pattern. What is 10 the product of? 2 and 5. What is 15 the product of? 3 and 5. What is 21 the product of? 3 and 7? What is 28 the product of? 4 and 7. There's definitely a pattern there. But it's not apparent until you think of trying to see the triangular numbers as products.
Before getting on to the main question, which is how can you quickly compute large triangular numbers, like the 100th, there are some other interesting things about them.
If you add a triangular number to the next one, what do you get? Like 10 + 15 = 25, and 15 + 21 = 36. What's special about the numbers 25 and 36? And once you've found what's special, why does that happen? Is there a geometric explanation?
If you got that one, how about this one, a bit harder. Add 3 times one triangular number to the next one. For instance, 3 times 10 is 30, plus 15, gives 45. Or 3 times 6 is 18, plus 10, gives 28. What's special about 28, 45, etc? Why does that happen?
There are others as well. Try taking 1 plus 8 times a triangular number. Can you make up more?
Now it's time for the main question, at least for the upper grades. Suppose you want to know how many dots there are in a triangular configuration with a large number, say 100, dots on the side. How can you do it without without a lot of computations?
Look at the little ones to get an idea. What can you do to 5 to get the 5th triangular number, 15? What can you do to 6 to get the 6th triangular number, 21? What can you do to 7 to get the 7th triangular number, 28? Did you see the pattern? Does the pattern continue?
Why does that rule work? There are many answers to this question. Some are geometrical, some algebraic.
So, now you can compute the 100th triangular number.
How about a big one. Imagine a triangular arrangement of marbles, each one inch in diameter, to make a big triangle whose side is the length of a football field, 100 yards. How many marbles do you need?
Possible further directions. Try piling the marbles up into tetrahedrons. A tetrahedron with 2 marbles on each edge has 4 marbles (3 on the base, one on top). A tetrahedron with 3 marble on each edge has 10 marbles (6 on the bottom, 3 on the next level, one on top). Go from there. See if you can learn as much about the tetrahedral numbers as you know about triangular ones.
Are these numbers important?
They certainly are. Summing the numbers 1 + 2+...+ n generalizes to the sum of an arithmetic sequence. The particular triangular numbers are combinations of choosing among n things 2 at a time, also known as binomial coefficients. The tetrahedral numbers are combinations of n things taken three at a time. All of them are entries in Pascal's arithmetic triangle, central to mathematics. There is no branch of mathematics that does not use these numbers.
David E. Joyce
Department of Mathematics and Computer Science
Worcester, MA 01610