Triangular Numbers
In all explorations, students worked in groups of two or three. The exploration began with the reading of a passage on triangular numbers from Thomas Heath's A History of Greek Mathematics: From Thales to Euclid (1921):
The particular triangle which has 4 for its side is mentioned in a story of Pythagoras by Lucian. Pythagoras told someone to count. He said 1, 2, 3, 4, whereon Pythagoras interrupted, “Do you see? What you take for 4 is 10, a perfect triangle and our oath." (p. 77)
After reading this passage and viewing the accompanying diagram (see Figure 3a), students modeled the first four triangular numbers using the Cuisenaire rods in accordance with the passage in slightly different ways (see Figures 3bd). They then proved the triangular number formula \[1+2+3+\cdots+n={\frac{1}{2}}n(n+1)\] via mathematical induction.
Figure 3: Triangular numbers \(1, 3, 6, 10\) using Cuisenaire rods
Odd Numbers (Symmetric LShaped Gnomons)
Prior to the construction of square numbers, class time was dedicated to the modeling of the consecutive odd numbers \(1,3,5,\dots\) as symmetric Lshaped gnomons (see Figure 4a):
All this was known to Pythagoras. The odd numbers successively added were called gnomons; this is clear from Aristotle's allusion to gnomons placed round 1 which now produce different figures every time (oblong figures, each dissimilar to the preceding one), now preserve one and the same figure (squares). (1921, p. 77)
Students noted that the only Cuisenaire rod that is already in the desired gnomon format is the white rod representing 1. Students in all groups indicated that they must combine various Cuisenaire rods to form gnomons representing odd numbers larger than 1. This activity led to a diversity of representations (see Figures 4be). Through a series of debates and considerations, students in all groups eventually agreed on the model in Figure 4e as the one in agreement with the passage they discussed.
Figure 4: Odd integers \(1, 3, 5, 7, 9, 11\) with Cuisenaire rods
Gnomons and Square Numbers
Having already modeled the gnomons of square numbers (see Figure 4e), it was time to combine them to form a square number at each step while "preserv[ing] one and the same figure (squares)" (1921, p. 77).
Consider the square with \(n\) dots in its side in relation to the next smaller square \({(n1)}^2\) and the next larger \({(n+1)}^2.\) Then \(n^2\) exceeds \({(n1)}^2\) by the gnomon \(2n1,\) but falls short of \({(n+1)}^2\) by the gnomon \(2n+1.\) Therefore the square \({(n+1)}^2\) exceeds the square \({(n1)}^2\) by the sum of the two gnomons \(2n1\) and \(2n+1,\) which is \(4n.\) (1921, p. 81)
Using the template given in the text (see Figure 5a), students developed a gnomonic representation of the first six square numbers (see Figure 5b). For \(n\ge2,\) we can see the desired representation showing that the square \({(n+1)}^2\) minus the square \({(n1)}^2\) equals the sum of the successive gnomons \(2n1\) and \(2n+1\) at each step (see Figure 5c).
Figure 5: Square numbers with Cuisenaire rods
Gnomons and Oblong Numbers
The next task was to model oblong numbers; that is, numbers of the sequence
\[2,\,\,2+4, \,\,2+4+6,\dots, \,\,2+4+6+\cdots+2n.\]
For this purpose, students first focused on representation of the even integers with the Cuisenaire rods. As was the case for the odd numbers, students offered different visualizations of the even numbers \(2, 4, 6,\dots\) (see Figure 6).

a 


b 

Figure 6: Even integers \(2, 4, 6, 8, 10, 12\) with Cuisenaire rods
Students then studied the text along with the accompanying template (see Figure 7a) for the purpose of generating the oblong number sequence:
While the adding of the successive odd numbers as gnomons round 1 gives only one form, the square, the addition of the successive even numbers to 2 gives a succession of 'oblong' numbers all dissimilar in form, that is to say, an infinity of forms (1921, p. 83).
When it came to the modeling of the oblong numbers, the majority of the students selected the model proposed in Figure 6b as the most suitable representation in agreement with the description in the historical resource:
It is to be noted that the word έτερομήκης (‘oblong’) is in Theon of Smyrna and Nicomachus limited to numbers which are the product of two factors differing by unity, while they apply the term προμήκης ('prolate', as it were) to numbers which are the product of factors differing by two or more (1921, p. 83).
Figure 7b depicts the oblong numbers generated based on the gnomonic structure given in Figure 6b. Other groups preferred the simplified \(n\times(n+1)\) rectangular representations in which they used the minimum number of Cuisenaire rods corresponding to each oblong number (see Figure 7c).
Figure 7: Oblong numbers \(2, 6, 12, 20, 30, 42\) with Cuisenaire rods.
Students in all groups recognized the relation between the \(n\times n\) square number pattern (see Figure 5b) and the \(n\times(n+1)\) rectangular oblong number pattern (see Figure 7), which they easily verified by the distributive property as \(n\times(n+1)=n^2+n\). That is, the \(n\)th oblong number is \(n\) more than the \(n\)th square number (see Figure 8).
Figure 8: Squareoblong relation: the \(n\)th oblong number = the \(n\)th square number \(+\,n.\)