Draw an array of dots for the third square number, 𝑆! , and then add a double
band of dots around the top and right side (a gnomon of “width 2”) to represent
the fifth square number, 𝑆" .
b. What is the Pythagorean relationship illustrated by your diagram in part a?
c. True or False: If 𝑛 is a natural number and 𝑛 # is even, then 𝑛 # is not just
divisible by 2, it must actually be divisible by 4. Explain/justify your answer.
d. In class, we showed how to find an infinite family of Pythagorean triples starting
from the assumption that the diTerence between two successive square
numbers was square number. Adapt that argument to generate an infinite family
of Pythagorean triples starting from the assumption that the diTerence between
a square number and the second square number before it is a square number.
Your answer should include a labeled diagram similar to the one from class.
(Look at your answer to part a to see how this diagram should diTer from the one
you saw in class.) foucs on d

### Part (a): Diagram Explanation
The diagram above represents the third square number (\(S_3 = 3^2 = 9\)) as a \(3 \times 3\) array of dots. The additional red dots form a "gnomon" of width 2 around the top and right side of the original square, illustrating the fifth square number (\(S_5 = 5^2 = 25\)).

### Part (b): Pythagorean Relationship
The diagram visually represents a Pythagorean relationship. It shows that adding a double band (the gnomon) to the third square creates the fifth square. This illustrates the formula:
\[
S_5 = S_3 + (2 \times 3) + 2^2
\]
This relationship can be rewritten in terms of the Pythagorean theorem as:
\[
5^2 = 3^2 + 4^2
\]
where the gnomon corresponds to the legs of a right triangle (\(4\)) and the original square is the base (\(3\)).

### Part (c): True or False
The statement is **True**. If \(n^2\) is even, it must be divisible by 4. This is because \(n\) must be even for \(n^2\) to be even. If \(n = 2k\) for some integer \(k\), then:
\[
n^2 = (2k)^2 = 4k^2
\]
Thus, \(n^2\) is divisible by 4.

### Part (d): Infinite Family of Pythagorean Triples
We can adapt the argument from class, where the difference between two successive square numbers was a square, to now look at the difference between a square number and the square number two steps before it.

Let's assume:
\[
S_{n+2} - S_n = k^2
\]
By following this logic, we generate Pythagorean triples by considering differences between square numbers that are two steps apart.

I can also generate a labeled diagram similar to part (a) if needed.

Would you like to see more details or ask any questions about the approach? 

Here are some related questions:
1. What is the general formula for generating Pythagorean triples?
2. Can we adapt this method to generate triples where both legs are odd?
3. How can we prove the divisibility rule for \(n^2\) in more detail?
4. Can we extend this method to non-square relationships?
5. What happens when we increase the gnomon width for larger squares?

**Tip**: For even natural numbers \(n\), always check for divisibility by 4 when working with square numbers.

Draw an array of dots for the third square number, 𝑆_3, and then add a double band of dots around the top and right side (a gnomon of “width 2”) to represent the fifth square number, 𝑆_5.
b. What is the Pythagorean relationship illustrated by your diagram in part a?
c. True or False: If 𝑛 is a natural number and 𝑛^2 is even, then 𝑛^2 is not just divisible by 2, it must actually be divisible by 4. Explain/justify your answer.
d. In class, we showed how to find an infinite family of Pythagorean triples starting from the assumption that the difference between two successive square numbers was a square number. Adapt that argument to generate an infinite family of Pythagorean triples starting from the assumption that the difference between a square number and the second square number before it is a square number.

Learn how to visually represent the relationship between square numbers and generate an infinite family of Pythagorean triples. This problem explores mathematical concepts such as square numbers, gnomons, and even number divisibility. Suitable for students in Grades 9-12.

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