Building Formulas from Patterns and Tables

Use starting values and repeated changes to construct formulas from tables, patterns and practical cost situations.

45 min Algebra Formulas and equations Lesson 5 of 13

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Printable worksheet

Open this lesson's worksheet

Use the printable version for identifying starting values, repeated changes and testing formulas against tables.

Think First

A delivery company charges $10 before travel begins, then adds $3 for each kilometre. How could you write a formula for any number of kilometres?

Type your first formula and define the variables.

Write your first formula and define the variables in your book.

Write your formula in your book

Saved

Know

A starting value is the amount present before the repeated change begins.
A repeated increase or decrease becomes the variable term.
A formula should be tested against known table values.

Understand

Not every increasing table has the same repeated change.
The first output is not always the starting value.
Context determines what each variable means.

Can Do

Identify the repeated change in a table.
Write a formula from a pattern or practical context.
Test a formula using known values.

f

Starting Value Plus Repeated Change

output = starting value + rate x input

Use when each equal step in the input changes the output by the same amount.

$C = 10 + 3k$

$C$ = cost in dollars, $k$ = kilometres

Core Content

1. Look for Equal Input Steps and Equal Output Changes

A table supports a simple linear formula when equal increases in the input produce equal increases or decreases in the output.

Cost, C

$12

$17

$22

$27

Change from previous row

-

+5

The starting value is $12 when $b = 0$. The repeated change is $5 per box, so $C = 12 + 5b$.

Common error: The first listed output is only the starting value if the input is 0.

Worked Example 1

Build a formula from a delivery table

Total cost, C

Column B

The starting value is $10 because the cost is $10 when $k = 0$.

The repeated increase is $3 for each extra kilometre.

Formula: $C = 10 + 3k$.

Test: if $k = 3$, then $C = 10 + 3(3) = 19$, which matches the table.

Worked Example 2

Build a formula from a savings pattern

Mia starts with $40 and saves $15 each week. Write a formula for her savings after $w$ weeks.

Let $S$ be the savings in dollars after $w$ weeks.

Starting value: $40.

Repeated change: +$15 each week.

Formula: $S = 40 + 15w$.

Communication habit: Define both variables: $S$ is the savings amount, and $w$ is the number of weeks.

Worked Example 3

Check whether a formula fits a table

A student claims the formula for this table is $P = 6n + 2$.

P

Column B

Test $n = 1$: $P = 6(1) + 2 = 8$.

Test $n = 2$: $P = 6(2) + 2 = 14$.

Test $n = 3$: $P = 6(3) + 2 = 20$.

The formula fits all listed values.

2. Check the Differences Before Assuming a Linear Formula

If the output changes by different amounts, the simple "starting value plus rate times input" model may not fit.

Output

2

4

8

16

Change

-

+2

+4

+8

Reasoning check: This table is increasing, but the repeated change is not constant. Do not force a linear formula.

Activity

Build and Test Formulas

A hire company charges $30 plus $8 per hour. Write a formula for total cost $C$ after $h$ hours.
A pattern has values 7, 11, 15, 19 for term numbers 1, 2, 3, 4. Write and test a formula.
A table has outputs 5, 10, 20, 40 for inputs 1, 2, 3, 4. Explain why a simple linear formula is not suitable.

Complete the formula-building practice in your book.

Revisit the Delivery Formula

The delivery company formula is $C = 10 + 3k$. The $10 is the starting cost and the $3 is the repeated cost for each kilometre.

Explain the changed formula in your book.

Assessment

MC

Multiple Choice

Random questions from the lesson bank - feedback appears immediately.

SA

Short Answer

Build formulas and test them against known values.

1. 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. 4 MARKS

Answer in your book.

2. The values 9, 14, 19, 24 match term numbers 1, 2, 3, 4. Write a formula and test it using term 4. 4 MARKS

Answer in your book.

3. Explain why checking differences is important before writing a formula from a table. 2 MARKS

Answer in your book.

Pattern Builder

For each table, say the starting value, the repeated change and one test value before choosing a formula.

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