Linear patterns worksheet.pdf - Learning on the Loop - Free Printable

You’ve uploaded a worksheet titled “Linear Relationships – Patterns, Rules and Graphs” with 5 problems. Since I can’t see the image (as you mentioned), I’ll solve each problem based on the text you provided — which appears to be the full content of the worksheet.

Let’s go problem by problem.

---

## Problem 1: Bridge Construction

> A construction firm builds bridges using spans, as shown in the diagram:
>
> - 1-span → 4 struts
> - 2-span → 7 struts
> - 3-span → 10 struts

a) Complete the table:

| Number of spans (n) | 1 | 2 | 3 | 4 | 5 |
|---------------------|---|---|---|---|---|
| Number of struts (S) | 4 | 7 | 10 | ? | ? |

We observe:
- From n=1 to n=2: +3 struts
- From n=2 to n=3: +3 struts

→ This is an arithmetic sequence with common difference = 3.

So:
- n=4 → 10 + 3 = 13
- n=5 → 13 + 3 = 16

✔ Completed Table:

| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| S | 4 | 7 | 10 | 13 | 16 |

---

b) Write a rule to calculate S for a given n.

We have:
- When n=1, S=4
- Each additional span adds 3 struts.

This is linear:
S = 3n + 1

Check:
- n=1 → 3(1)+1 = 4 ✔
- n=2 → 3(2)+1 = 7 ✔
- n=3 → 3(3)+1 = 10 ✔

✔ Rule: S = 3n + 1

---

c) Is this pattern discrete or continuous?

Since we can only have whole spans (you can’t build half a span), and struts are counted as whole numbers, this is a discrete pattern.

✔ Answer: Discrete

---

d) Graph the pattern for n=1 to n=5.

You’d plot points:
(1,4), (2,7), (3,10), (4,13), (5,16)

It’s a straight line with slope 3, starting at (1,4). Since it’s discrete, use points (dots), not a connected line (unless instructed otherwise).

---

e) A bridge requires 50 spans. How many struts?

Use the rule: S = 3n + 1

Plug in n = 50:
S = 3(50) + 1 = 150 + 1 = 151 struts

✔ Answer: 151 struts

---

## Problem 2: Painter A’s Charges

> Table:
>
> | Hours (h) | 1 | 2 | 3 | 4 | 5 | 6 |
> |-----------|---|---|---|---|---|---|
> | Total Cost ($) | $70 | $90 | $110 | ? | ? | ? |

a) Complete the table.

Look at the pattern:
- h=1 → $70
- h=2 → $90 → +$20
- h=3 → $110 → +$20

So, hourly rate = $20/hour.

Then:
- h=4 → 110 + 20 = $130
- h=5 → 130 + 20 = $150
- h=6 → 150 + 20 = $170

✔ Completed Table:

| h | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Cost | 70 | 90 | 110 | 130 | 150 | 170 |

---

b) Give the rule for total cost (T) for the first hour.

Wait — “for the first hour”? That seems odd. Probably means: rule for total cost T based on hours h.

From above:
- T = 20h + 50? Let’s check.

At h=1: 20(1) + 50 = 70 ✔
h=2: 20(2)+50 = 90 ✔
h=3: 20(3)+50 = 110 ✔

So, T = 20h + 50

Why +50? Because even for 0 hours, there’s a base fee of $50 (maybe setup fee).

✔ Rule: T = 20h + 50

---

c) Is this pattern discrete or continuous?

In real life, painters might charge by the hour or partial hour. But since the table uses whole hours, and the context implies billing per hour, it’s likely discrete unless specified otherwise.

But technically, if they can charge for fractions of an hour, it could be continuous. However, based on the table and typical school problems, we say:

✔ Discrete

---

d) Use the graph to determine how much for 2.5 hours.

If we graphed the points (1,70), (2,90), etc., and drew a line through them (even though discrete), we could extend to h=2.5.

Using the formula:
T = 20*(2.5) + 50 = 50 + 50 = $100

✔ Answer: $100

---

e) What is Painter A’s hourly rate?

From the increase per hour: $20 per hour.

✔ Answer: $20 per hour

---

f) How much for 7.5 hours?

T = 20*(7.5) + 50 = 150 + 50 = $200

✔ Answer: $200

---

## Problem 3: Seedling Growth

> Seedling grows 1 cm per day after the first day. Initial height = 0.5 cm.

a) Fill in the table for days 1–7.

Day 1: 0.5 cm
Day 2: 0.5 + 1 = 1.5 cm
Day 3: 1.5 + 1 = 2.5 cm
... and so on.

So, height H = 0.5 + 1*(d - 1) = d - 0.5

Check:
- d=1 → 1 - 0.5 = 0.5 ✔
- d=2 → 2 - 0.5 = 1.5 ✔
- d=3 → 3 - 0.5 = 2.5 ✔

✔ Table:

| Day (d) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---------|---|---|---|---|---|---|---|
| Height (H) | 0.5 | 1.5 | 2.5 | 3.5 | 4.5 | 5.5 | 6.5 |

---

b) Graph growth over 7 days.

Plot points: (1,0.5), (2,1.5), ..., (7,6.5)

Straight line with slope 1, starting at (1,0.5). Again, since days are whole, use discrete points.

---

c) Rule for relationship between d and H.

As above:
H = d - 0.5

Or: H = 1*d - 0.5

✔ Rule: H = d - 0.5

---

## Problem 4: Matchstick Pattern

> Rule: M = 2n + 1, where M = number of matchsticks for nth pattern.

a) Create a table for first 5 patterns.

n | M = 2n+1
--|---------
1 | 2(1)+1 = 3
2 | 2(2)+1 = 5
3 | 2(3)+1 = 7
4 | 2(4)+1 = 9
5 | 2(5)+1 = 11

✔ Table:

| Pattern (n) | 1 | 2 | 3 | 4 | 5 |
|-------------|---|---|---|---|---|
| Matchsticks (M) | 3 | 5 | 7 | 9 | 11 |

---

b) How many matchsticks in the 50th pattern?

M = 2(50) + 1 = 100 + 1 = 101

✔ Answer: 101 matchsticks

---

c) Show this relationship on a graph.

Plot points: (1,3), (2,5), (3,7), (4,9), (5,11)

Straight line with slope 2, y-intercept 1. Since n is discrete, use dots.

---

## Problem 5: Another Matchstick Pattern

> Rule: M = 8n + 4

Create a table for first 5 patterns.

n | M = 8n+4
--|---------
1 | 8(1)+4 = 12
2 | 8(2)+4 = 20
3 | 8(3)+4 = 28
4 | 8(4)+4 = 36
5 | 8(5)+4 = 44

✔ Table:

| Pattern (n) | 1 | 2 | 3 | 4 | 5 |
|-------------|---|---|---|---|---|
| Matchsticks (M) | 12 | 20 | 28 | 36 | 44 |

---

Draw the graph.

Plot points: (1,12), (2,20), (3,28), (4,36), (5,44)

Line with slope 8, y-intercept 4. Discrete points.

---

## ✔ Final Summary of All Answers:

Problem 1:

a) S: 4,7,10,13,16
b) S = 3n + 1
c) Discrete
d) Graph points (1,4) to (5,16)
e) 151 struts

Problem 2:

a) Cost: 70,90,110,130,150,170
b) T = 20h + 50
c) Discrete
d) $100 for 2.5 hours
e) $20/hour
f) $200 for 7.5 hours

Problem 3:

a) H: 0.5,1.5,2.5,3.5,4.5,5.5,6.5
b) Graph points (1,0.5) to (7,6.5)
c) H = d - 0.5

Problem 4:

a) M: 3,5,7,9,11
b) 101 matchsticks for n=50
c) Graph (1,3) to (5,11)

Problem 5:

a) M: 12,20,28,36,44
b) Graph (1,12) to (5,44)

---

Let me know if you want printable tables, graphs drawn as ASCII art, or explanations for any part!