7.3 - More Linear and Non-Linear Patterning | MFM1P Gr 9 Applied Math - Free Printable

Let's solve the "Feeding Frenzy" problem step by step.

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Part A: Feeding Frenzy

We are given a pattern of tables and chairs. Each term number represents a configuration with increasing numbers of tables placed side-by-side, and chairs arranged around them.

We'll analyze the pattern to determine:

1. The number of chairs for each term.
2. The first differences.
3. The type of relationship.
4. An equation relating chairs $ C $ to term number $ n $.
5. Graph the relationship.

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## Step 1: Complete the Number of Chairs Column

Let’s examine the pattern.

Term 1:

- 1 table
- Chairs: 6

> Visual: One rectangle (table) with 2 chairs on each of the longer sides (top and bottom), and 1 chair on each end.
>
> So: 2 (top) + 2 (bottom) + 1 (left) + 1 (right) = 6 chairs

Term 2:

- 2 tables joined side-by-side
- Each table has two long sides and two ends.
- But when tables are joined, the inner edges don't have chairs.
- So, only outer edges get chairs.

Let’s count:
- Top: 2 chairs per table → 2 × 2 = 4
- Bottom: same → 4
- Left end: 1 chair
- Right end: 1 chair
- Middle join: no chairs

Total: 4 + 4 + 1 + 1 = 10 chairs

> Wait — let’s double-check from image:
> - Top: 4 yellow circles
> - Bottom: 4 yellow circles
> - Left: 1
> - Right: 1
> → 4+4+1+1 = 10

✔ Yes.

Term 3:

- 3 tables in a row
- Top: 3 tables × 2 chairs = 6
- Bottom: 6
- Left end: 1
- Right end: 1
- Total: 6 + 6 + 1 + 1 = 14

✔ Matches visual.

So far:
| Term | Number of Chairs |
|------|------------------|
| 1 | 6 |
| 2 | 10 |
| 3 | 14 |

Now, let's find the pattern.

Each time we add a table:
- We add 2 chairs to the top
- We add 2 chairs to the bottom
- The ends remain the same (only leftmost and rightmost have end chairs)
- So adding a new table adds 4 chairs?

Check:
- From 1 → 2: 6 → 10 → increase of 4
- From 2 → 3: 10 → 14 → increase of 4

Yes! So this is an arithmetic sequence with common difference 4.

So:
- Term 4: 14 + 4 = 18
- Term 5: 18 + 4 = 22
- Term 6: 22 + 4 = 26

Now fill in the table:

| Term Number | Picture (description) | Number of Chairs |
|-------------|------------------------|------------------|
| 1 | 1 table | 6 |
| 2 | 2 tables | 10 |
| 3 | 3 tables | 14 |
| 4 | 4 tables | 18 |
| 5 | 5 tables | 22 |
| 6 | 6 tables | 26 |

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## Step 2: Create the First Difference Column

First differences = difference between consecutive terms.

| Term | Chairs | First Difference |
|------|--------|------------------|
| 1 | 6 | |
| 2 | 10 | 10 - 6 = 4 |
| 3 | 14 | 14 - 10 = 4 |
| 4 | 18 | 18 - 14 = 4 |
| 5 | 22 | 22 - 18 = 4 |
| 6 | 26 | 26 - 22 = 4 |

✔ All first differences are 4, constant.

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## Step 3: What type of relationship exists? Explain.

This is a linear relationship.

Why?

- The first differences are constant (always 4).
- This means the number of chairs increases at a constant rate as the number of tables increases.
- So, the relationship between term number $ n $ and number of chairs $ C $ is linear.

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## Step 4: Find an equation relating $ C $ and $ n $

We know:
- It's linear: $ C = mn + b $
- Common difference = slope $ m = 4 $
- When $ n = 1 $, $ C = 6 $

Use point-slope or substitution:

$$
C = 4n + b
$$

Plug in $ n = 1 $, $ C = 6 $:

$$
6 = 4(1) + b \Rightarrow b = 2
$$

So the equation is:

$$
\boxed{C = 4n + 2}
$$

Let's verify:

- $ n = 1 $: $ 4(1) + 2 = 6 $ ✔
- $ n = 2 $: $ 4(2) + 2 = 10 $ ✔
- $ n = 3 $: $ 4(3) + 2 = 14 $ ✔
- $ n = 4 $: $ 4(4) + 2 = 18 $ ✔

Perfect!

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## Step 5: Graph the relationship

We will plot points $ (n, C) $:

| $ n $ | $ C $ |
|--------|--------|
| 1 | 6 |
| 2 | 10 |
| 3 | 14 |
| 4 | 18 |
| 5 | 22 |
| 6 | 26 |

On the grid:
- Label x-axis: Term Number ($ n $)
- Label y-axis: Number of Chairs ($ C $)
- Plot the points and draw a straight line through them.

The graph will be a straight line with:
- Slope = 4
- Y-intercept = 2 (but since $ n $ starts at 1, it doesn’t start at origin)

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✔ Final Answers:

#### 1. Completed Table:

| Term Number | Picture | Number of Chairs |
|-------------|------------------|------------------|
| 1 | 1 table | 6 |
| 2 | 2 tables | 10 |
| 3 | 3 tables | 14 |
| 4 | 4 tables | 18 |
| 5 | 5 tables | 22 |
| 6 | 6 tables | 26 |

#### 2. First Differences:

| Term | Chairs | First Difference |
|------|--------|------------------|
| 1 | 6 | |
| 2 | 10 | 4 |
| 3 | 14 | 4 |
| 4 | 18 | 4 |
| 5 | 22 | 4 |
| 6 | 26 | 4 |

#### 3. Type of Relationship:
- Linear relationship because the first differences are constant (4), indicating a constant rate of change.

#### 4. Equation:
$$
\boxed{C = 4n + 2}
$$

#### 5. Graph:
- Plot the points: (1,6), (2,10), (3,14), (4,18), (5,22), (6,26)
- Draw a straight line through them.

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Let me know if you'd like a visual sketch of the graph or help drawing it on paper!