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.