Mathematics Standard • Year 11 • Module 1 • Lesson 5

Building Formulas from Patterns and Tables

Build fluency identifying the starting value and the repeated change in tables and patterns, writing a linear formula, and testing it against known values.

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete the general linear pattern.

output = ____________ value + ____________ × input. (Two blanks.)

Q1.2 In a table, the input column goes 0, 1, 2, 3 and the output column goes 12, 17, 22, 27. State the starting value and the repeated change.

Starting value = ____________ Repeated change per input step = ____________

Q1.3 A table has outputs 2, 4, 8, 16 for inputs 1, 2, 3, 4. Is a simple linear formula appropriate? Circle: Yes / No. Briefly say why: ____________________

Stuck? Revisit lesson § Check the Differences — if the change is NOT constant (here +2, +4, +8 doubles), the relationship is not linear.

2. Worked example — build a formula from a delivery table

Follow each step. The starting value comes from input = 0 (or is found by working back from input = 1).

Problem. A delivery company's quote table:
k (km): 0, 1, 2, 3, 4 | C ($): 10, 13, 16, 19, 22. Build a formula for C in terms of k, then test at k = 3.

Step 1 — Identify the starting value (k = 0).

When k = 0, C = 10. Starting value = 10.

Reason: input = 0 is the cleanest reading of the "before anything has been added" value.

Step 2 — Identify the repeated change per 1-step increase in k.

10 → 13 (+3), 13 → 16 (+3), 16 → 19 (+3), 19 → 22 (+3). Change = +$3 per km.

Reason: every constant input step adds the same amount, so the relationship is linear.

Step 3 — Write the formula.

C = 10 + 3k (C = cost in dollars, k = kilometres)

Step 4 — Test against a known row (k = 3).

C = 10 + 3(3) = 19 ✓ matches table.

Conclusion. The cost formula is C = 10 + 3k.

3. Faded example — fill in the missing steps

A printing service quote table is shown below. Build a formula and test it.
p (pages): 0, 1, 2, 3 | C ($): 25, 27, 29, 31. 4 marks

Step 1 — Starting value (p = 0):

C(0) = ____________ → starting value = $ ____________

Step 2 — Repeated change per extra page:

Change = +$____________ per page (constant? ____ Y / N)

Step 3 — Formula (define your variables):

C = ____________ + ____________ × p

Step 4 — Test at p = 3:

C = ____________ + ____________ (3) = ____________ ✓

Conclusion. Cost formula: C = ____________ + ____________ p.

Stuck? Revisit lesson § Worked Example 1 — Build a formula from a delivery table.

4. Graduated practice — build, test, interpret

For each, identify the starting value and the repeated change, then write the formula. Test it against the LAST row before reporting.

Foundation — direct read-off from input 0 (4 questions)

Q	Table (input, output pairs)	Formula
4.1 1	(0,5), (1,7), (2,9), (3,11)	y = ________ + ________ x
4.2 1	(0,12), (1,16), (2,20), (3,24)	y = ________ + ________ x
4.3 1	(0,100), (1,90), (2,80), (3,70)	y = ________ + ________ x
4.4 1	(0,0), (1,6), (2,12), (3,18)	y = ________ + ________ x

Standard — practical situations + test (6 questions)

Define both variables. Show the test calculation at one row of the table.

4.5 A hire company charges $30 plus $8 per hour. Write a formula for total cost C after h hours, then test at h = 5. 2 marks

4.6 Mia starts with $40 and saves $15 each week. Write a formula for her savings S after w weeks, then test at w = 6. 2 marks

4.7 A printing company charges $25 plus $2 per page. Write a formula for total cost C for p pages, then find the cost for 18 pages. 2 marks

4.8 A pattern has terms 7, 11, 15, 19 for term numbers 1, 2, 3, 4. Write and test a formula for term T at term number n. 2 marks

4.9 A water tank starts with 500 L and is being drained at 12 L per minute. Write a formula for volume V (L) after t minutes, then test at t = 10. 2 marks

4.10 A pattern has terms 9, 14, 19, 24 for term numbers 1, 2, 3, 4. Write a formula and test it using term 4. 2 marks

Extension — first listed output is NOT the starting value (2 questions)

4.11 A table has these (n, T) pairs: (1, 8), (2, 13), (3, 18), (4, 23). The input column starts at 1, not 0. (a) Find the starting value (T at n = 0) by working backwards. (b) Write the formula T = a + bn. (c) Test at n = 3. 3 marks

4.12 A table has outputs 5, 10, 20, 40 for inputs 1, 2, 3, 4. (a) Calculate the changes between rows. (b) Explain in one sentence why a simple linear formula (output = a + bx) cannot fit. 3 marks

Stuck on 4.11(a)? The change per step is +5, so going BACK one step from (1, 8) means subtracting 5: T at n = 0 is 8 − 5 = 3.

5. Self-check the easy 3

Tick the first three once you've verified your method works.

For 4.1 I checked the change is +2 each time, so the formula y = 5 + 2x correctly gives 11 at x = 3.

For 4.3 I recognised the change is −10 each step, so the formula is y = 100 − 10x (not y = 100 + 10x).

For 4.4 I noticed (0, 0) means starting value = 0, so the formula simplifies to y = 6x with no constant term.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — General linear pattern

output = starting value + rate × input.

Q1.2 — Reading the table

Starting value = 12 (output when input = 0). Repeated change = +5 per input step. Formula: y = 12 + 5x.

Q1.3 — Doubling pattern

No, a linear formula is not appropriate. The changes (+2, +4, +8) are not constant — the output doubles each step (geometric, not linear).

Q3 — Faded printing example

Step 1: C(0) = 25. Starting value = $25.
Step 2: Change = +$2 per page. Constant? Y.
Step 3: C = 25 + 2 × p.
Step 4: C = 25 + 2(3) = 31 ✓.
Conclusion: C = 25 + 2p.

Q4.1-4.4 — Foundation formulas

4.1: y = 5 + 2x. 4.2: y = 12 + 4x. 4.3: y = 100 − 10x. 4.4: y = 0 + 6x, i.e. y = 6x.

Q4.5 — Hire company

Let C = total cost ($), h = hours. C = 30 + 8h. Test h = 5: C = 30 + 40 = $70.

Q4.6 — Mia's savings

Let S = savings ($), w = weeks. S = 40 + 15w. Test w = 6: S = 40 + 90 = $130.

Q4.7 — Printing

Let C = cost ($), p = pages. C = 25 + 2p. For 18 pages: C = 25 + 36 = $61.

Q4.8 — Pattern 7, 11, 15, 19

Change = +4 per term. Starting value (n = 0) = 7 − 4 = 3. T = 3 + 4n. Test n = 4: T = 3 + 16 = 19 ✓.

Q4.9 — Water tank

Let V = volume (L), t = minutes. Change = −12 L/min. V = 500 − 12t. Test t = 10: V = 500 − 120 = 380 L.

Q4.10 — Pattern 9, 14, 19, 24

Change = +5. Starting value (n = 0) = 9 − 5 = 4. T = 4 + 5n. Test n = 4: T = 4 + 20 = 24 ✓.

Q4.11 — Input starts at 1

(a) Change = +5 per step. Working back from (1, 8): T(0) = 8 − 5 = 3.
(b) T = 3 + 5n.
(c) Test n = 3: T = 3 + 15 = 18 ✓.

Q4.12 — Doubling table

(a) Changes: 5 → 10 (+5), 10 → 20 (+10), 20 → 40 (+20). The change DOUBLES each step.
(b) A simple linear formula y = a + bx requires a constant change per step. Since the change here grows (+5, +10, +20), the pattern is exponential (the output doubles), not linear, and cannot be modelled with this form.