Varying inputs in a digital formula table | Year 7

01 / LEARN

Understand the idea.

A table records repeated substitutions

An input-output table records repeated substitutions into the same formula. For $C = 3 n + 5$ , substitute each chosen $n$ afresh and place its cost in the matching row or column. Label the output in dollars so the numbers retain their meaning.

Increasing $n$ by one adds one extra three-dollar item, so $C$ increases by three dollars; the fixed five-dollar fee is unchanged. This constant increase comes from the stated formula. Other formulas, such as an area rule with two changing dimensions, need not behave this way.

Cost rule C = 3n + 5 dollars
Number n	1	2	3	4
Cost C (dollars)	8	11	14	17

A spreadsheet uses the same operations

A spreadsheet performs the same operations using cell references. If A2 holds $n$ , the formula =3*A2+5 multiplies that input by three and adds five. The equals sign starts the formula, and the asterisk means multiplication.

Check one output by hand before copying a formula. When copied down, a relative reference normally changes from A2 to A3 and onward. Keep required brackets, as in =2*(A2+4). A finite table can suggest a pattern, but the supplied rule determines the intended calculation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Three different inputs

Use $C = 4 n + 7$ dollars at $n = 0$ , 2 and 5.

Worked solution

1
Use zero as the input.
$C (0) = 4 \times 0 + 7 = 7 dollars$
No items leaves only the fixed seven-dollar cost.
2
Use two as the input.
$C (2) = 4 \times 2 + 7 = 8 + 7 = 15 dollars$
Two items contribute eight dollars, and the fixed cost is added once.
3
Use five as the input.
$C (5) = 4 \times 5 + 7 = 20 + 7 = 27 dollars$
The same formula is applied independently to the third input.

Answer

7 dollars, 15 dollars and 27 dollars.

EXAMPLE 02

A changing increase

A rectangle is $w$ cm wide and $(w + 2)$ cm long. Find $A = w (w + 2)$ for $w = 1$ , 2 and 3. Are the increases equal?

Worked solution

1
Calculate the area for width one.
$A = 1 (1 + 2) = 1 \times 3 = 3 {cm}^{2}$
The corresponding length is three centimetres, so area is width times length.
2
Calculate the area for width two.
$A = 2 (2 + 2) = 2 \times 4 = 8 {cm}^{2}$
Both dimensions use the new width input.
3
Calculate the area for width three.
$A = 3 (3 + 2) = 3 \times 5 = 15 {cm}^{2}$
Substitute three into both occurrences of $w$ .
4
Compare consecutive increases.
$8 - 3 = 5 {cm}^{2}, 15 - 8 = 7 {cm}^{2}$
The increases are unequal because both the width and length change, rather than only a fixed-rate item count.

Answer

3, 8 and 15 cm²; the increases are unequal.

03 / YOUR TURN

Try it yourself.

Check your understanding

Cell A2 contains 4. What does =2*A2+3 calculate?

Show answer & reasoning

A2 supplies the input four, so =2*A2+3 calculates $2 \times 4 + 3 = 8 + 3 = 11$ . Multiplication is completed before addition.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Use $y = 2 x + 1$ at $x = 0$ , 1, 2 and 3. List the outputs.
2 marks
Use $P = 4 s$ for square side lengths 2, 5 and 8 cm.
2 marks
Cell A2 holds $n$ . Write a spreadsheet formula for $5 n + 6$ .
1 mark
For $C = 3 n + 5$ , explain the cost change when $n$ rises from 4 to 5.
2 marks
Use $A = n^{2}$ at $n = 1$ , 2, 3 and 4. Are consecutive outputs equally spaced?
2 marks
A student uses =2*A2+4 for $2 (n + 4)$ . At input 3, find the student's answer and the correct answer, then correct the formula.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Use $y = 2 x + 1$ at $x = 0$ , 1, 2 and 3. List the outputs.
2 marks
Substitution gives $1$ , $3$ , $5$ and $7$ respectively.
1, 3, 5, 7
Use $P = 4 s$ for square side lengths 2, 5 and 8 cm.
2 marks
Multiply each side length by 4: $8$ , $20$ , $32$ .
8 cm, 20 cm, 32 cm
Cell A2 holds $n$ . Write a spreadsheet formula for $5 n + 6$ .
1 mark
Use * for multiplication and A2 for the input.
=5*A2+6
For $C = 3 n + 5$ , explain the cost change when $n$ rises from 4 to 5.
2 marks
$C (4) = 17$ and $C (5) = 20$ .
The $3 n$ term increases by 3; the fixed fee does not change.
An increase of 3 dollars.
Use $A = n^{2}$ at $n = 1$ , 2, 3 and 4. Are consecutive outputs equally spaced?
2 marks
The differences are 3, 5 and 7.
They are not constant.
1, 4, 9, 16; no.
A student uses =2*A2+4 for $2 (n + 4)$ . At input 3, find the student's answer and the correct answer, then correct the formula.
3 marks
The student's calculation is $2 \times 3 + 4 = 10$ .
The intended calculation is $2 (3 + 4) = 14$ .
Include brackets around the sum.
Student: 10. Correct: 14. Formula: =2*(A2+4).

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Curriculum reference & lesson scope

Australian Curriculum v9.0 / SA Curriculum Mathematics

Year 7 variables, expression notation, substitution, simple equivalence and formula tables. Digital methods are explained with supplied data; the topic test works by hand.

Official curriculum source ↗

In this lesson

Understand the idea.

A table records repeated substitutions

A spreadsheet uses the same operations

See the method in action.

Three different inputs

Use zero as the input.

Use two as the input.

Use five as the input.

A changing increase

Calculate the area for width one.

Calculate the area for width two.

Calculate the area for width three.

Compare consecutive increases.

Try it yourself.

Check your understanding

Keep practising