Algebra

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 $\div$÷).

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

$x$x |
$1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
---|---|---|---|---|---|

$y$y |
$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`=`y`−4". - 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`.

- For example, if we did the previous question in the wrong order, we would have found that "$x=y-4$

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

$p$p |
$10$10 | $15$15 | $20$20 | $25$25 | $30$30 |
---|---|---|---|---|---|

$q$q |
$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`=`p`5.

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.

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

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

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