6.03 Writing rules for relationships

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.

 

Exploration

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$2, $4$4, $6$6, $8$8, $10$10, $\dots$…

We say that the number $2$2 has a place number of $1$1 (since it is in the first position), that $4$4 has a place number of $2$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 $x$x), and since "the place number" also changes as the pattern continues, we also give it a pronumeral name (let's choose $y$y).

This lets us rewrite the rule more mathematically:

The value	is equal to	the place number	multiplied by	two
$y$y	$=$=	$x$x	$\times$×	$2$2

We have made an equation. Since we are multiplying a pronumeral by a number, we can simplify it even further:

$y=2x$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 $x$x, we can find the corresponding value $y$y. For example, when we replace $x$x with $10$10 the equation becomes:

$y=2\times10=20$y=2×10=20,

which tells us that $20$20 is the $10$10th number in the pattern. We can fill out a whole table of values using substitution to check:

$x$x	$1$1	$2$2	$3$3	$4$4	$5$5	$6$6	$7$7	$8$8	$9$9	$10$10
$y$y	$2$2	$4$4	$6$6	$8$8	$10$10	$12$12	$14$14	$16$16	$18$18	$20$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.

 

Worked examples

Example 1

Consider the pattern of triangles made out of matchsticks.

Construct an equation describing the relationship between the number of matchsticks and the number of triangles.

Think: We will need to choose some pronumerals to represent the number of matchsticks and triangles, then we want to find some rule relating the two values.

Do: We can start by letting $M$M represent the number of matchsticks and $T$T represent the number of triangles.

We can see that each triangle uses exactly three matchsticks and all of the matchsticks are only used for one of the triangles.

So we can deduce that there are:

"three matchsticks per triangle"

Rewording this rule so that it's easier to translate into an equation gives us:

The number of matchsticks	is equal to	three	times	the number of triangles
$M$M	$=$=	$3$3	$\times$×	$T$T

Simplifying the algebraic expression gives us the equation:

$M=3T$M=3T

Reflect: The visual presentation of this pattern made it easy to deduce the rule just by looking at it. We could have chosen $x$x and $y$y for our pronumerals, but we can choose anything we like, and choosing $M$M and $T$T made it easier to keep track of which was which.

 

Example 2

Consider the table of values:

$x$x	$1$1	$2$2	$3$3	$4$4	$5$5
$y$y	$8$8	$16$16	$24$24	$32$32	$40$40

Construct an equation describing the relationship between $x$x and $y$y.

Think: In order to deduce the rule for the pattern we want to see how the value for $y$y changes as $x$x changes.

Do: We can see how the value for $y$y changes as $x$x changes by looking at the differences between the $y$y-values in the table. We represent this with an arrow linking one to the next:

$x$x $1$1 $2$2 $3$3 $4$4 $5$5 $y$y $8$8 $16$16 $24$24 $32$32 $40$40

 

The differences between the $y$y-values is always $8$8.
This means that the $y$y-value increases eight times faster than the $x$x-value, so maybe the rule is:

"the $y$y-value is equal to eight times the $x$x-value"

We can test this by adding another row to our table:

$x$x	$1$1	$2$2	$3$3	$4$4	$5$5
$y$y	$8$8	$16$16	$24$24	$32$32	$40$40
$8x$8x	$8$8	$16$16	$24$24	$32$32	$40$40

The values in the row for $y$y match the values in the row for $8x$8x. We can express this using an equals sign:

$y=8x$y=8x

Reflect: Since the pattern was represented as a table of values it was easier to find the pattern rule by noticing a repeated difference in the $y$y-values. We then tested whether the repeated difference worked as the rule by adding $8x$8x as a row to our table. Since this row matched our y-values we could confirm that this was the correct rule.

 

 

Example 3

Consider the pattern of pentagons made out of matchsticks.

Construct an equation describing the relationship between the number of matchsticks and the the number of pentagons.

Think: This is similar to Example 1. We want to first choose some pronumerals to represent the variable values in our pattern. After that we can look at the pattern to see if there is some easy rule to follow. If we can't find one then we can try constructing a table of values.

Do: We can start by letting $M$M represent the number of matchsticks and $P$P representing the number of pentagons.

The first way we could try doing this question is to notice that we can divide the matchsticks in the pattern into groups, like so:

Looking at the pattern like this, we can see that there are four matchsticks per pentagon and one matchstick leftover.
This means that the number of matchsticks will always be equal to four times the number of pentagons, plus one more.

This gives us the equation:

$M=4P+1$M=4P+1

But what if we don't see this pattern?

Another way to answer this question is to construct a table of values from the pattern. This will give us:

$P$P $1$1 $2$2 $3$3 $4$4 $M$M $5$5 $9$9 $13$13 $17$17

 

Looking at the differences between the $M$M-values we can see that for every change of $1$1 in the $P$P-value there is a change of $4$4 in the $M$M-value.

So we have reason to believe that the rule could be:

"the number of matchsticks is four times the number of pentagons" 

We add a row for $4P$4P into our table to check:

$P$P $1$1 $2$2 $3$3 $4$4 $M$M $5$5 $9$9 $13$13 $17$17 $4P$4P $4$4 $8$8 $12$12 $16$16

 

The values for $4P$4P don't match the values for $M$M so this is not the correct rule. But we are close - notice that the values in the row for $M$M are always one more than the values in the row for $4P$4P.

This tells us that the rule for the pattern is:

"the number of matchsticks is equal to one more than four times the number of pentagons"

Converting this into an equation gives us:

The number of matchsticks	is equal to	one	more than	four	times	the number of pentagons
$M$M	$=$=	$1$1	$+$+	$4$4	$\times$×	$P$P

Simplifying the algebraic expression gives us the same equation we had before:

$M=4P+1$M=4P+1

 

Reflect: In this example we could either deduce the rule visually or construct a table of values and proceed from there.

When looking for the rule visually we noticed a common difference in the shapes.

When looking for the rule in the table of values we calculated the common difference between values, checked if that was the rule by adding a new row to the table and then used those new values to find the correct equation.

 

As demonstrated by the different approaches taken in the worked examples, the method by which we find the rule and equation for a pattern changes depending on what kind of pattern is presented.

Now that we have had some practice, try applying these different methods to the questions below.

Practice questions

Question 1

Question 2

Question 3