One area of mathematics that we use algebra for is writing equations to describe relationships in numbers. We use variables to express how the numbers vary as the pattern continues, capturing the entire relationship in a single equation.
Exploration
Relationships between variables 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 relationship 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 variables. Since "the value" changes as the pattern continues, we give it a variable name (let's choose $x$x), and since "the place number" also changes as the pattern continues, we also give it a variable 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 variable by a number, we can simplify it even further:
$y=2x$y=2x
Now that we have this equation, we can find any number 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 values 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 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:
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The differences between the $y$y-values is always $8$8, because the $y$y-value increases by $8$8 each time.
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 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.
For any relationship represented by a table of values, the change in the $y$y-values should always be compared to the change in the $x$x-values.
This table has a change of $4$4 in the $y$y-values for every $2$2 in the $x$x-values:
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As an equation, we can say that $2y=4x$2y=4x, which is the same equation as $y=2x$y=2x. In other words, the $y$y-values change by $2$2 when the $x$x-values change by $1$1.
Importantly, the rule is not $y=4x$y=4x.
Example 2
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 variables 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:
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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:
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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
Which of the following equations matches the rule "the value of $y$y is six less than the value of $x$x"?
$x=y-6$x=y−6
A$y=x-6$y=x−6
B$y=6-x$y=6−x
C$y=x+6$y=x+6
D
Question 2
Vanessa opens a bank account and deposits $\$300$$300. At the end of each week she adds $\$10$$10 to her account.
Which of the following tables shows the balance of Vanessa's account over the first four weeks?
Week ($W$W) $0$0 $1$1 $2$2 $3$3 $4$4 Account total ($A$A) $\$300$$300 $\$310$$310 $\$320$$320 $\$330$$330 $\$340$$340 AWeek ($W$W) $0$0 $1$1 $2$2 $3$3 $4$4 Account total ($A$A) $\$0$$0 $\$10$$10 $\$20$$20 $\$30$$30 $\$40$$40 BWeek ($W$W) $0$0 $1$1 $2$2 $3$3 $4$4 Account total ($A$A) $\$10$$10 $\$310$$310 $\$610$$610 $\$910$$910 $\$1210$$1210 CWeek ($W$W) $0$0 $1$1 $2$2 $3$3 $4$4 Account total ($A$A) $\$300$$300 $\$300$$300 $\$300$$300 $\$300$$300 $\$300$$300 DWhich of the following equations represents the relationship between Vanessa's account total ($A$A) and the number of weeks ($W$W) for which she has been adding to her account?
$A=300$A=300
A$A=300+10W$A=300+10W
B$A=300W+10$A=300W+10
C$A=10W$A=10W
D
Question 3
Use the table of values below to write an equation for $y$y in terms of $x$x.
| $x$x | $5$5 | $6$6 | $7$7 | $8$8 | $9$9 |
|---|---|---|---|---|---|
| $y$y | $2$2 | $4$4 | $6$6 | $8$8 | $10$10 |
Question 4
Which of these options show an increasing pattern?
$7,6,11,3,9$7,6,11,3,9
A$6.5,6.6,6.7,6.8,6.9$6.5,6.6,6.7,6.8,6.9
B$25,22,19,16,13$25,22,19,16,13
C
Question 5
Use the rule to complete the table of values:
"The starting number is doubled, then $4$4 is subtracted."
Starting Number ($N$N) $12$12 $13$13 $14$14 $15$15 Answer ($A$A) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Which of these equations describes the rule above?
$A=2\times\left(N-4\right)$A=2×(N−4)
A$A=N\times2-4$A=N×2−4
B$A=N^2-4$A=N2−4
C$A=2+N-4$A=2+N−4
D
Consider the following pattern:
This is a decreasing pattern, where $3$3 is subtracted at every step
To find the next number that follows in a pattern, it's as simple as finding the pattern and applying to the last number. For example, the next number in the decreasing pattern above would be $5-3=2$5−3=2.
Practice questions
QUESTION 6
Find the next number in the sequence:
$7$7, $9$9, $11$11, $13$13, $\editable{}$
Using tables to find the next term
Below is a drawing of a simple pattern:
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| Flowers $n$n | $1$1 | $2$2 | $3$3 | $4$4 |
| Petals $p$p | $5$5 | $10$10 | $15$15 | $20$20 |
In this pattern, $n$n represents the step number in the pattern (that is, the number of flowers) and $p$p represents the total number of petals at that step.
Notice that the number of petals are increasing by $5$5 each time - in particular, the value of $p$p is always equal to $5$5 times the value of $n$n. Therefore, the generating rule for this pattern must be $p=5n$p=5n.
This rule can now be used to predict future results. For example, to calculate the total number of petals when there are $10$10 flowers present, substitute $n=10$n=10 into the rule to find $p=5\times10=50$p=5×10=50 petals. So even though there were only $1,2,3$1,2,3 and $4$4 flowers present in the pattern above, the rule has determined that there would be $50$50 petals visible when there are $10$10 flowers present.
Practice questions
Question 7
Matches were used to make the pattern attached:
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Complete the table:
Number of triangles ($t$t) $1$1 $2$2 $3$3 $5$5 $10$10 $20$20 Number of matches ($m$m) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Write a formula that describes the relationship between the number of matches, $m$m, and the number of triangles, $t$t.
How many matches are required to make $77$77 triangles using this pattern?