GM Lesson 056 Two-Way Tables for Formulae with Two Variables

Learning Intentions

By the end of this lesson, students will be able to:

• Construct two-by-two tables for formulae with two variable quantities.
• Use a spreadsheet to organise repeated calculations.
• Interpret how changing two variables affects the value of a formula.

Prerequisites

Students should already be able to:

• Substitute numerical values into formulae.
• Evaluate formulae using the correct order of operations.
• Construct a table of values from a formula with one changing input.
• Use basic spreadsheet formulas such as multiplication, addition and cell references.

Key Idea Summary

A formula with two variable quantities has an output that depends on two changing inputs.

For example, the formula

can represent the total cost, , of hiring equipment for hours at a rate of $ per hour.

A two-way table organises repeated calculations by placing:

• one variable across the top row
• the other variable down the first column
• the calculated outputs inside the table

This allows us to compare how the output changes when either variable changes.

For example:

hours

In a spreadsheet, the same structure can be created by entering the formula once, then copying it across and down.

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Direct Instruction

A two-way table is useful when a formula contains two quantities that may change.

Suppose a formula has the structure

where and are both variable quantities.

Instead of calculating one value at a time, we can organise several calculations in a table:

Each inside entry is found by multiplying the row value by the column value.

For example, when and ,

A spreadsheet can do this efficiently if the row and column headings are used as inputs.

Worked Example 1: Constructing a Two-Way Table by Hand

A delivery company charges according to the formula

where:

• is the total cost in dollars
• is the number of deliveries
• is the cost per delivery in dollars

The company is considering two delivery rates: $ and $ per delivery.

Construct a two-way table for and .

Because the rate can change, rewrite the formula as

where:

• is the rate per delivery
• is the number of deliveries

Number of deliveries,

Calculate each inside value:

Completed table:

Number of deliveries,

Interpretation:

• Increasing the number of deliveries increases the total cost.
• Increasing the rate per delivery also increases the total cost.
• The greatest cost occurs when both variables are largest.

Worked Example 2: Using a Spreadsheet Structure

A casual worker earns

where:

• is total earnings
• is the hourly rate in dollars
• is the number of hours worked

Construct a two-way table for hourly rates of $ and $ , and hours worked of and .

Set up the table:

Hours worked,

Calculate the values:

Completed table:

Hours worked,

Spreadsheet method:

Cell	Entry
A1	Hours
B1	24
C1	28
A2	5
A3	9
B2	=B1*A2

Then copy the formula from B2 across and down, adjusting the cell references as needed.

Interpretation:

If the hourly rate increases while the number of hours stays fixed, the earnings increase.

If the number of hours increases while the hourly rate stays fixed, the earnings also increase.

Worked Example 3: A Formula with Addition and Multiplication

A taxi fare is modelled by

where:

• is the fare in dollars
• is the flag fall in dollars
• is the distance travelled in kilometres
• is the cost per kilometre in dollars

Construct a two-way table for distances of km and km, and kilometre rates of $ and $ .

Distance,

Calculate:

Completed table:

Distance,

Interpretation:

The fare increases when the distance increases.

The fare also increases when the charge per kilometre increases.

Because of the fixed flag fall, the fare is not found by multiplying the two variables only.

Understanding Checks

Check 1

A formula is given by

A student constructs this table:

Explain what calculation was used to obtain the value .

Check 2

For the formula

which quantity is fixed and which quantity could be treated as variable?

Check 3

A formula is

where is length and is width.

Complete the missing value:

Length,
		?

Check 4

In a two-way table, one variable is placed down the first column and the other variable is placed across the top row.

Explain why the inside entries are the outputs, not additional input values.

Exercises

Simple Familiar Exercises

Exercise 1

Use the formula

to complete the table.

Length,

Exercise 2

Use the formula

where is total cost, is the rate in dollars and is the number of items.

Complete the table.

Number of items,

Exercise 3

A worker earns

where is earnings, is hourly rate and is hours worked.

Complete the table.

Hours worked,

Exercise 4

Use the formula

where is volume, is base area and is height.

Complete the table.

Height,

Exercise 5

For the formula

complete the table.

Distance,	Fixed charge $ and rate $

Then explain why this is not yet a two-way table.

Complex Familiar Exercises

Exercise 6

A taxi fare is modelled by

where is the fare in dollars, is the distance in kilometres and is the cost per kilometre.

Complete the table.

Distance,

Exercise 7

A rectangular garden has area

Complete the table.

Length,

Give each answer in square metres.

Exercise 8

A simple pricing rule is

where:

• is the total cost in dollars
• is a fixed setup fee
• is the price per item
• is the number of items

Complete the table.

Number of items,

Explain which change has a greater effect in this table: increasing from to , or increasing from to .

Exercise 9

A cylinder has volume

Construct a two-way table for:

• and
• and

Round answers to one decimal place.

Height,

Explain why changing has a larger effect than changing .

Exercise 10

A spreadsheet is being used for the formula

The values of are placed in column A, and the values of are placed in row 1.

A student enters the formula =A2*B1 in cell B2.

Explain what the spreadsheet should calculate in B2.

Then describe what needs to happen to the formula when it is copied across and down.

Homework Problems

Homework 1

Use the formula

to complete the table.

Length,

Homework 2

A worker’s weekly earnings are modelled by

Complete the table.

Hours worked,

Write one sentence comparing the effect of increasing hours and increasing the hourly rate.

Homework 3

A hire company charges according to

where:

• is the total cost in dollars
• is a fixed booking fee
• is the number of days
• is the hire cost per day

Complete the table.

Number of days,

Homework 4

A rectangle has area

A student says:

“If the length and width both double, the area also doubles.”

Use a two-way table with and , and and , to test whether the statement is correct.

Homework 5

A formula is

Construct a two-way table using:

• and
• and

Then identify the smallest and largest output values.

Next: GM Lesson 057 Technology for Linear and Non-linear Relationships