Let's analyze the image and solve the problems step by step.

a. Count the total number of circles (within the five rows):

There are 5 rows of circles. Let's count the total:

• Row 1 has 1 circle.
• Row 2 has 2 circles.
• Row 3 has 3 circles.
• Row 4 has 4 circles.
• Row 5 has 5 circles.

Total number of circles = $1 + 2 + 3 + 4 + 5 = 15$

b. Count (from top to bottom) the number of circles in the first four rows:

• Row 1: 1 circle
• Row 2: 2 circles
• Row 3: 3 circles
• Row 4: 4 circles

Total number of circles in the first four rows = $1 + 2 + 3 + 4 = 10$

c. Count (from top to bottom) the number of circles in the first three rows:

• Row 1: 1 circle
• Row 2: 2 circles
• Row 3: 3 circles

Total number of circles in the first three rows = $1 + 2 + 3 = 6$

d. Find the difference between the numbers found in items b and c:

Difference = $10 - 6 = 4$

e. Count the number of circles shaded dark:

From the image, it appears that 3 circles are shaded in red (dark circles).

f. What is special about the numbers found in items a, b, c, and e?

The numbers found in items a, b, c, and e are 15, 10, 6, and 3, respectively. These numbers form the triangular numbers:

• 15 = $T_5$ (the 5th triangular number)
• 10 = $T_4$ (the 4th triangular number)
• 6 = $T_3$ (the 3rd triangular number)
• 3 = $T_2$ (the 2nd triangular number)

h. Show on the diagram that $15 = 3 \times (10 - 6) + 3$:

Let's break this equation down:

• $10 - 6 = 4$
• $3 \times 4 = 12$
• $12 + 3 = 15$

So, $15 = 3 \times (10 - 6) + 3$ is mathematically true.

Would you like further details on any part of this? Here are 5 related questions to deepen understanding:

1. How are triangular numbers formed, and why are they relevant in this image?
2. Can you derive a formula for the $n$-th triangular number?
3. What patterns emerge from subtracting triangular numbers?
4. Can you generalize the expression $15 = 3 \times (10 - 6) + 3$ for other triangular numbers?
5. How can this arrangement of circles represent real-world scenarios or geometric shapes?

Tip: Triangular numbers represent the sum of natural numbers up to a certain point, and they often appear in geometry, combinatorics, and algebra.