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### "TRIANGULAR NUMBERS, Part 1: Preschool and Primary Activities"

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.  One--two, 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 10th 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.

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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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### 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 6th 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.

# 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, star-shaped 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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