## Question:

You will be asked to find out the difference between triangular and square numbers in order to obtain positive answers.

Diagrams, photos or pictures of concrete materials can be included to support your hypotheses, explanations and reasoning.

Discuss with your colleagues how you might extend the investigation.

Charles Lovitt, in his paper Investigations as the Central Focus for Mathematics Curriculum, discusses the failures and recommendations of the problem solving movement. He suggests that children in classrooms are given the chance to be mathematicians through mathematical investigations.

## Answer:

Investigating the Difference Between Square and Triangular Numbers

Triangular numbers can be formed by forming triangular counters.

Triangular numbers are 1, 3, 6, 10, 15, 21 28, 36, 45 e.t.c.

These numbers form in a basic sequence that follows the product between the nth number (Posamentier and Lehmann, 2009).

For example, in order to obtain the fourth triangular number we will need to stack of 4 times 5 divided by 2. This gives us 10.

This gives us the fourth triangular numbers, 10.

Figure 1: Formation triangular numbers

You can find square numbers in 1, 4, 9, 16, 25, 26, 39, 49, 64 and 81. 100 is another example.

Square numbers are based on the concept of consecutive odd numbers.

Square numbers are the sum of all consecutive number up to nth factors. (Bardos & Carbaugh 2010).

For example, the square number 3 is the sum the first three consecutive odd numbers. This is 1+3+5 =9.

Figure 2: Forming Square numbers

This experiment can use multilink cubes to show clear differences between the square and triangular numbers.

You can also use the consecutive numbers to create square numbers.

Figure 3: Square numbers are formed from triangular numbers

The above sequence shows how adding two triangular number after each other creates a square number.

As an example, 1+3 = 4.

The difference is that 1 and 3 are both triangular number.

Similar rules apply to other numbers.

The sequence shows us that to get an nth square number we must add the sum of nth and n-1 triangle numbers (Koshy (2014)).

For example, the 5th Square Number will be the sum 5th Triangular Number and 4th Triangular Number which is 15+10 = 25.

References

Amazing Math Projects That You Can Make Yourself: Numbers and Geometry.

Chicago: Nomad Press.

Pell-Lucas and Pell-Lucas numbers, with applications.

Mathematical surprises and amazements: Fascinating numbers and notable numbers.