GM Lesson 055 Tables of Values from Formulae

Learning Intentions

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

• Construct a table of values from a formula.
• Use technology or a spreadsheet to generate repeated calculations.
• Interpret patterns shown in a table of values.

Prerequisites

Students should already be able to:

• Substitute numerical values into formulae.
• Evaluate formulae using correct order of operations.
• Identify the input and output quantities in a formula.
• Use simple spreadsheet formulas such as multiplication, powers and cell references.

Key Idea Summary

A table of values shows how the value of a formula changes as one input changes.

For a formula such as

we choose values of , substitute each one into the formula, and calculate the corresponding value of .

The table shows a pattern: every time increases by , increases by .

A spreadsheet can be used to complete repeated calculations efficiently. For example, if the value of is in cell A2, then the formula could be entered as:

=3*A2+2

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 table of values usually has:

• an input column, such as , , or ;
• an output column, which is the value calculated from the formula;
• a clear heading showing the formula being used.

To construct a table of values:

1. Identify the input variable.
2. Choose or use the given input values.
3. Substitute each input value into the formula.
4. Calculate the output value.
5. Look for a pattern in the table.

Worked Example 1: Constructing a Table from a Linear Formula

Construct a table of values for

using .

The value of increases by each time increases by .

This is a linear pattern because the first differences are constant.

Worked Example 2: Constructing a Table from a Non-linear Formula

Construct a table of values for

using .

Round answers to one decimal place.

The area does not increase by the same amount each time.

This is a non-linear pattern because the first differences are not constant.

Worked Example 3: A Financial Formula Table

A simple interest investment earns interest according to

where is the principal, is the annual interest rate as a decimal, and is the number of years.

An investment of $ earns simple interest at per year.

Construct a table showing the interest earned for years.

First write the formula with the known values:

	$
	$
	$
	$
	$

The interest increases by $ each year.

This table shows a linear pattern because the amount of extra interest is constant each year.

Worked Example 4: Spreadsheet Method

A phone plan costs $ per month plus $ for each extra gigabyte of data.

The formula is

where is the monthly cost in dollars and is the number of extra gigabytes.

Construct a table for .

	$
	$
	$
	$
	$
	$

In a spreadsheet:

Cell	Entry
A1	g
B1	C
A2	0
A3	1
A4	2
B2	=25+8*A2

The formula in B2 can be filled down to calculate the remaining values.

The table shows that every extra gigabyte increases the monthly cost by $ .

Understanding Checks

Check 1

For the formula

complete the missing values.

Check 2

For the formula

explain what happens to when increases.

Check 3

A spreadsheet has values of in column A. The formula is

What spreadsheet formula should be typed into cell B2 if is in cell A2?

Check 4

A table made from a formula has output values:

What pattern is shown in the output values?

Exercises

Simple Familiar Exercises

Exercise 1

Construct a table of values for

using .

Exercise 2

Construct a table of values for

using .

Exercise 3

Construct a table of values for

using .

Exercise 4

Construct a table of values for the circumference formula

using .

Round answers to one decimal place.

Exercise 5

Construct a table of values for

using .

State the pattern shown in the values of .

Complex Familiar Exercises

Exercise 6

The cost of hiring a trailer is $ plus $ per hour.

The formula is

where is the total cost in dollars and is the number of hours.

Construct a table for .

Then describe the pattern in the total cost.

Exercise 7

The area of a circle is given by

Construct a table for .

Round answers to one decimal place.

Explain whether the pattern is linear or non-linear.

Exercise 8

A rectangular prism has volume

where the base area is square centimetres.

Construct a table showing for .

State the relationship between height and volume.

Exercise 9

A simple interest investment uses

A principal of $ is invested at simple interest per year.

Construct a table showing the interest earned after years.

Exercise 10

A swimming pool is filled at a rate of litres per hour. There are already litres in the pool.

The formula is

where is the number of litres in the pool and is the time in hours.

Construct a table for .

Use the table to determine when the pool first contains more than litres.

Homework Problems

Problem 1

Construct a table of values for

using .

Problem 2

Construct a table of values for

using .

State whether the pattern is linear or non-linear.

Problem 3

The cost of printing posters is $ setup plus $ per poster.

The formula is

where is the number of posters.

Construct a table for .

Problem 4

The surface area of a sphere is given by

Construct a table for .

Round answers to one decimal place.

Problem 5

A student uses a spreadsheet to calculate values from

The values of are placed in cells A2 to A7.

Write the spreadsheet formula that should be typed into cell B2.

Then explain how the rest of the table can be completed efficiently.

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