Exploring Pascal's Triangle worksheet with patterns and discussion questions.

Let’s go step by step to solve each question about Pascal’s Triangle.

---

Question 1: What is one pattern that you found in the triangle? Describe it.

Looking at the triangle:

- The first and last number in every row is always 1.
- Each number inside the triangle (not on the edge) is the sum of the two numbers directly above it.

For example:
- Row 3: 1, 2, 1 → the “2” comes from 1 + 1 above it.
- Row 4: 1, 3, 3, 1 → the first “3” is 1 + 2, the second “3” is 2 + 1.
- Row 5: 1, 4, 6, 4, 1 → 4 = 1+3, 6 = 3+3, next 4 = 3+1.

So one clear pattern: Each interior number is the sum of the two numbers above it.

Also, the triangle is symmetric — left side mirrors the right side.

✔ So we can say: *One pattern is that each number (except the 1s on the edges) is made by adding the two numbers just above it.*

---

Question 2: Complete the next two rows of the triangle.

We are given up to row 6 (if we count the top as row 0):

Row 0: 1
Row 1: 1, 1
Row 2: 1, 2, 1
Row 3: 1, 3, 3, 1
Row 4: 1, 4, 6, 4, 1
Row 5: 1, 5, 10, 10, 5, 1

Now let’s do Row 6:

Start with 1.

Then add pairs from Row 5:

- 1 + 5 = 6
- 5 + 10 = 15
- 10 + 10 = 20
- 10 + 5 = 15
- 5 + 1 = 6

End with 1.

→ Row 6: 1, 6, 15, 20, 15, 6, 1

Now Row 7:

Start with 1.

Add pairs from Row 6:

- 1 + 6 = 7
- 6 + 15 = 21
- 15 + 20 = 35
- 20 + 15 = 35
- 15 + 6 = 21
- 6 + 1 = 7

End with 1.

→ Row 7: 1, 7, 21, 35, 35, 21, 7, 1

✔ Double-check: All additions correct? Yes. Symmetric? Yes.

---

Question 3: Can you explain (in words) how the triangle is made? In other words, the recursive process for completing each row?

Yes! Here’s how:

- Start with a single 1 at the top (that’s row 0).
- Every new row starts and ends with 1.
- For any number in between, look at the two numbers directly above it (from the previous row), and add them together to get the new number.
- Repeat this for each row.

It’s like building a pyramid where each brick (number) is supported by the two bricks above it — you add those two to make the new one.

Example: To get the middle number in row 4 (which is 6), you take the two numbers above it in row 3: 3 and 3 → 3 + 3 = 6.

✔ Simple rule: *Start and end with 1; everything else is the sum of the two numbers above it.*

---

Question 4: Is there a pattern in the diagonals?

Yes! Let’s look at the diagonals going from top-left to bottom-right (or vice versa — they’re symmetric).

First diagonal (all 1s):
1, 1, 1, 1, 1, 1, ... → obvious pattern.

Second diagonal:
1, 2, 3, 4, 5, 6, 7, ... → counting numbers!

Third diagonal:
1, 3, 6, 10, 15, 21, ... → these are called triangular numbers. They come from adding consecutive integers:
1 = 1
3 = 1+2
6 = 1+2+3
10 = 1+2+3+4
etc.

Fourth diagonal:
1, 4, 10, 20, 35, ... → tetrahedral numbers (sums of triangular numbers).

So yes — each diagonal has its own special number pattern, starting with all 1s, then natural numbers, then triangular numbers, etc.

✔ Answer: *Yes! The diagonals show patterns: first diagonal is all 1s, second is counting numbers (1,2,3,4...), third is triangular numbers (1,3,6,10...), and so on.*

---

Final Answer:

1. One pattern: Each number (except the 1s on the sides) is the sum of the two numbers directly above it. Also, each row starts and ends with 1, and the triangle is symmetric.

2. Next two rows:
- Row 6: 1, 6, 15, 20, 15, 6, 1
- Row 7: 1, 7, 21, 35, 35, 21, 7, 1

3. How the triangle is made: Start with 1 at the top. Each new row begins and ends with 1. Every other number is found by adding the two numbers directly above it from the previous row.

4. Yes, there are patterns in the diagonals:
- First diagonal: all 1s
- Second diagonal: 1, 2, 3, 4, 5, 6, 7… (counting numbers)
- Third diagonal: 1, 3, 6, 10, 15, 21… (triangular numbers)
- Fourth diagonal: 1, 4, 10, 20, 35… (tetrahedral numbers)