Introduction
One area of mathematics that we use algebra for is writing equations to describe patterns in numbers. We use pronumerals to express how the numbers vary as the pattern continues, capturing the entire pattern in a single equation..
Write rules for relationships
Patterns can be represented in lots of different ways: a number pattern, a rule, an equation, or a table of values. We can see how these different representations interact using the example of the two times table.
The number pattern for the two times table is: 2,\,4,\,6,\,8,\,10,\,...
We say that the number 2 has a place number of 1 (since it is in the first position), that 4 has a place number of 2 (since it is in the second position), and so on. This lets us express the pattern with a rule: "The value is double the place number".
To convert this rule into an equation we start by rewording it using numbers and pronumerals. Since "the value" changes as the pattern continues, we give it a pronumeral name (let's choose x), and since "the place number" also changes as the pattern continues, we also give it a pronumeral name (let's choose y).
This lets us rewrite the rule more mathematically:
| The value | is equal to | the place number | multiplied by | two |
|---|---|---|---|---|
| y | = | x | \times | 2 |
We have made an equation. Since we are multiplying a pronumeral by a number, we can simplify it even further:y=2x
Now that we have this equation, we can find any number in this pattern by substitution. If we choose a value for the place number x, we can find the corresponding value y. For example, when we replace x with y the equation becomes:y=2\times 10=20
which tells us that 20 is the 10\text{th} number in the pattern. We can fill out a whole table of values using substitution to check:
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| y | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 |
The equation produces the pattern in the bottom row.
Each of these descriptions (pattern, rule, equation, table) of this relationship can be used to construct another. But for now we will focus on finding the equation from the other three kinds.
For any pattern represented by a table of values, the change in the y-values should always be compared to the change in the x-values.
This table has a change of 4 in the y-values for every 2 in the x-values:
As an equation, we can say that 2y=4x, which is the same equation as y=2x. In other words, the y-values change by 2 when the x-values change by 1.
Importantly, the rule is not y=4x.
Examples
Example 1
Which of the following equations matches the rule "the value of y is six less than the value of x"?
Example 2
Vanessa opens a bank account and deposits \$ 300. At the end of each week she adds \$ 10 to her account.
Complete the following table to show the balance of Vanessa's account over the first four weeks.
| \text{Week }(W) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Account total }(A) |
Which of the following equations represents the relationship between Vanessa's account total (A) and the number of weeks (W) for which she has been adding to her account?
Patterns can be represented in lots of different ways: a number pattern, a rule, an equation, or a table of values.
Each of these descriptions (pattern, rule, equation, table) of a relationship can be used to construct another.
For any pattern represented by a table of values, the change in the y-values should always be compared to the change in the x-values.