Find the rule from a table of values

Lesson

Tables are used everywhere in mathematics, usually to show data for two or more related quantities (represented by variables).

If we have a table of values relating two quantities, it is often useful to figure out if there is a relationship between them, and what the relationship is. If we can find such a relationship, it can be used to predict future values and patterns.

We will start by looking at simple relationships, where we can get from one variable to the other by a single operation ($+$+, $-$, $\times$× or $\div$÷​).

Example 1

Use the following table of values to write an equation for $y$y in terms of $x$x.

 $x$x $y$y $1$1 $2$2 $3$3 $4$4 $5$5 $5$5 $6$6 $7$7 $8$8 $9$9

Think: The $y$y values are larger than the $x$x values, so there must be addition or multiplication involved.

Do: To get from $x=1$x=1 to $y=5$y=5 we can either add $4$4 or multiply by $5$5. To get from $x=2$x=2 to $y=6$y=6 we can either add $4$4 or multiply by $3$3.

So the pattern is "add $4$4 to $x$x" (make sure to check that this works for the rest of the numbers). We can write this as an equation as $y=x+4$y=x+4.

Tips and Tricks
• Are the numbers in the second row smaller or larger than the numbers in the first row?
• If they are larger, then addition or multiplication is involved.
• If they are smaller, then subtraction or division is involved.
• Addition and subtraction are opposites, as are multiplication and division. If you figure out the pattern in the wrong order, you can reverse it by using the opposite operation.
• For example, if we did the previous question in the wrong order, we would have found that "$x=y-4$x=y4".
• Since addition is the opposite of subtraction, we could convert this to "$y=x+4$y=x+4" to get an equation for $y$y in terms of $x$x.

Example 2

Use the following table of values to write an equation for $q$q in terms of $p$p.

 $p$p $q$q $10$10 $15$15 $20$20 $25$25 $30$30 $2$2 $3$3 $4$4 $5$5 $6$6

Think: The $q$q values are smaller than the $p$p values, so there must be subtraction or division involved.

Do: To get from $p=10$p=10 to $q=2$q=2 we can either subtract $8$8 or divide by $5$5. To get from $p=15$p=15 to $q=3$q=3 we can either subtract $12$12 or divide by $5$5.

So the pattern is "divide $p$p by $5$5" (make sure to check that this works for the rest of the numbers). We can write this as an equation as $q=\frac{p}{5}$q=p5.

Notice!
It doesn't matter if the first row increases by more than $1$1 at a time, or if neither of the rows start from $1$1. We can still find a relation by looking at each pair of values.

Worked Examples

Question 1

Write an equation for $g$g in terms of $f$f

Question 2

Write an equation for $g$g in terms of $f$f.

Outcomes

NA4-9

Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns