TRIANGULAR
NUMBERS, Part 1: Preschool and Primary
Activities
by Fran Endicott Armstrong, Ph.D.
“Look, Mom. I did stickers!”

“Show me.”

“See.”

“Explain it to me.”

“Mom, look. I put one sticker here. I put two stickers here. I put three stickers here.”

“Are you sure you have three
stickers in that panel?”

“Yes. One, two, three.”

OK. It’s important to check. What about the next panel?”

“.4. See I put four stickers here. One, two, three, four.”

“Very good. You do have four pretty butterfly stickers
in that fourth panel.”

“And here I put five
stickers. One, two, three, four,
five. Two are butterfly stickers. Three are flowers.”

“Very nice counting. And lovely stickers. Did you write these numerals?

“Yes. The teacher had dots there. And I wrote my numbers over the dots.”

“That helped you write your
numerals correctly.”

“I wrote them nicely.”

“Yes, you did. How many stickers did you use on the whole
page?”

“I don’t know.”

“Can you count them?”

“Count all the stickers
on the whole page?

“Yes, please.”

OK. Onetwo, three—four, five, six—seven,
eight, nine, ten—eleven, twelve, thirteen, fourteen, fifteen.”

“Good job! You counted all the stickers on the
page. You used 15 stickers on this
page. Did you enjoy doing this sticker
page?”

“Yes. I’m going to do another one tomorrow!”


Thanks to my daughter Suzanne’s
preschool teacher, Laurie
Kleen, at that time Directress of A Growing Place Montessori School, I learned
an early introduction to Triangular Numbers.
Obviously, the purpose of this activity is not to learn about Triangular
Numbers. The objective of this activity is to put the number of stickers in
each horizontal panel as indicated by the numeral (1, 2, 3, 4, 5) at the left
side of the panel and then to also write each numeral correctly. When little preschool children first begin
doing this, given the limitations on their fine motor coordination, the
stickers may not line up in an orderly fashion.
See Figure 1. Sometimes in the
beginning the young child does not yet put the correct number of stickers in
each panel. Initially, there is no need
to make a big deal about this correctness.
As the child matures, s/he can be encouraged to check by counting
carefully and sometimes a sticker can be removed or added as needed. Also, as the child’s fine motor skills
develop further and s/he tends to place the stickers in a more orderly fashion,
it often becomes obvious that the array of stickers is triangular. When a child has the appropriate counting
ability, the teacher can ask how many stickers are on the whole page and then
the child can write that numeral as a sum below a total bar beneath the
numerals on the left. Then the child
can state that 15 is a triangular number.
See Figure 2.
I
queried my daughter’s Montessori teacher, Laurie, one day“Do you realize you
are doing Triangular Numbers? This
sticker worksheet shows that 15 is a triangular number since the stickers form
a triangular array.” Laurie admitted
that she hadn’t heard about triangular numbers until I brought this up. By coincidence, while my preschool daughter
was lining up 15 stickers in a triangular array, in my Business Mathematics
class at St. Louis Community College/Meramec I was teaching the Rule of 76
which involves Triangular Numbers (This will be dealt with in Applications of
Triangular Numbers in a subsequent issue.)
1,
2, 3, 4, 5, 6, 7, 8, 9, 10 stickers
One
can do the same activity with the numbers 1, 2, 3, . . . , 8, 9, 10 provided
one uses small enough stickers that a row of ten stickers will fit next to the
numeral “10” on a sheet of paper. The
little circle stickers of about a quarter inch diameter work well.
The
teacher can ask the youngster what is the total number of stickers on the
page. For rows 1 through 10 we have 1 +
2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55.
Therefore, 55 is the 10^{th} triangular number. See Figure 3.
Triangular Arrays
with Pegboards or Cuisenaire® Rods
Another
time that triangular numbers might come up is during free exploration with
pegboards or Cuisenaire® Rods or just rows of bears or pennies or whatever set
of uniform objects. A student might make
a triangular array in the form of
an
isosceles right triangle or in the form of a triangle
which
is not a right triangle.
·
·
· ·
· ·
· · ·
· · ·
· · · · · · · ·
This
provides an opportunity for the teacher to discuss triangular numbers. The total number of objects in the triangular
array represents a triangular number.
Note: Bowling pins are arranged
in a triangular array. 10 is the fourth
triangular number because 1 + 2 + 3 + 4 = 10, i.e., this is the sum of
the first four consecutive natural numbers.
When
a student constructs a triangular array on the pegboard, the teacher can ask
what shape the design suggests (“A triangle.”) and say “Yes, and since that
design suggests a triangle, we call the design a triangular array.” And the teacher can ask “How many pegs are in
the triangular array?” And when the
student tells what number of pegs, the teacher can say, “Because that number
forms a triangular array, we call that number a triangular
number.”
Similarly, if a student constructs a
triangular array with Cuisenaire® Rods, the teacher can ask “How many white
rods would it take to cover all the rods in the design?” and then explain that
such a number is called a triangular number.
Please
note that a triangular arrangement that has “holes” in it, such as below, is not
a triangular array and the number 9 is not a triangular number.
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· ·
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· · · ·
Making Observations
Encourage
students to use their triangular array of stickers or their arrangement of pegs
on the pegboard or of Cuisenaire® Rods to make observations. One way of provoking some observations is to
have the students place a large index card or a piece of cardboard horizontally
over their triangular array and simply cover all but the first row with one
object in it and then slide the card down one row at a time and notice what
number of objects is being added and also look at writing the sums as you
progress.
1

1

1
+ 2

3

1
+ 2 + 3

6

1
+ 2 + 3 + 4

10

1
+ 2 + 3 + 4 + 5

15

1
+ 2 + 3 + 4 + 5 + 6

21

1
+ 2 + 3 + 4 + 5 + 6 + 7

28

and so forth as far
as you want to go




More
will be said about this in a subsequent article in this journal featuring
making a Triangular Numbers Book.
Students
might make the observation that when progressing from the fourth triangular
number on to the fifth triangular number, one merely adds 5 to
the fourth triangular number, 10, thus yielding 15. This, of course, is because you are adding
another row of five objects to the triangular array to the four rows
which represent the fourth triangular number.
Likewise, to get the sixth triangular number, one adds a row of 6
objects and can simply add 6 to the fifth triangular number 15, thus arriving
at the fact that 21 is the 6^{th} triangular number, and so on. Please allow the students to make these
observations. Do not tell them these
observations and steal from them the opportunity to discover things for themselves
and to state their observations to others.
Students
might also notice that a convenient way of adding the series of numbers, let’s
say, for the bowling pin arrangement of the fourth triangular numbers is to add
the first and the last addends and then inward from there, i.e., 1 + 4 = 5 and
2 + 3 = 5 and 5 + 5 = 10. This corresponds
to physically moving the last row next to the first row and then placing the
second last row next to the second row.
Notice what happens with the fifth triangular number 1 + 2 + 3 + 4 + 5 =
15. Adding the top and bottom rows each
time or equivalently adding the first and last terms in the series each time, 1
+ 5 = 6, 2 + 4 = 6, and then there is a 3 left in the middle (because 5 is an
odd number. So 1 + 2 + 3 + 4 + 5 = 2(6)
+ 3 = 15. This is worth looking at for
subsequent triangular numbers. More will
be said about this in Part 2.
Conclusion
The above are some ways that
young children can be introduced to the mathematical idea of triangular
numbers. This also helps prepare
students for the idea of square numbers, another number concept related to the
idea of arranging a given number of objects in a particular geometric
shape. Triangular numbers are
particularly suitable for children in the primary grades since the concept, at
the beginning of its development, features addition. [Generating the formula for calculating the
nth triangular number for a given natural number n involves
multiplication.] Later students will
come to learn that a number of objects arranged in a rectangular
array represents a multiplication fact since the area of a rectangle is length
times width. [Please note: When discussing rectangular arrays
representing multiplication facts, it is important not to drop the word
“rectangular” from the phrase “rectangular array” and simply call them “arrays”
for the obvious reason that there are many different kinds of arrays—triangular
arrays, starshaped arrays, etc. The
word “array” simply means an arrangement without holes in it.] A square number is a particular instance of a
rectangular array since it represents multiplying a number by itself. Students could even go on to pentagonal
numbers and so forth.
In
subsequent issues of this journal will appear articles taking the concept of
triangular numbers further up through the mathematics curriculum. Please stay tuned to the MEGSL Confluence.
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Figure 1: A
young child’s early, but correct effort at placing the correct number of
stickers as
indicated by the numeral at the left
Figure 2: A
more mature child’s placement of the stickers and showing that the number of
stickers on the whole page is 15
Figure 3:
Stickers on the 1 through 10 page and the total number of stickers is 55
shown by Briana Heine, the author’s great niece.