 Build algebraic expressions I

Lesson

As we have seen before, we can represent patterns using sequences. Each of these sequences follows some rule that describes the pattern. We will now see how to use algebra to describe patterns by a rule, even when we don't have a sequence.

Coming from patterns

Let's look at a particular pattern of flowers growing in pots.   There are $3$3 flowers in one pot, $6$6 flowers in two pots, $9$9 flowers in three pots, and so on.

We can see that the rule for the number of flowers will be "three times the number of pots." If we let the variable $n$n represent "the number of pots" then the rule is simply "three times $n$n." So we can write this rule algebraically as $3n$3n.

What if we want to know how many flowers will be in $12$12 pots?            We know that the rule is "three times the number of pots," or more simply $3n$3n, so the number of flowers will be $3\times12=36$3×12=36.

Where there's a rule, there's a way

We can use algebra to describe rules even without having a pattern or a sequence to follow. Suppose that there are $6$6 people on a bus to start with, and $1$1 additional person gets on at each stop.

We can describe the number of people on the bus by the rule "$6$6 plus the number of stops". If we let $x$x represent "the number of stops," then the rule will become "$6$6 plus $x$x", which we can write as $6+x$6+x.

Using this rule, we can find how many people are on the bus after any number of stops. For instance, we know that after $8$8 stops there will be $6+8=14$6+8=14 people on the bus.

Examples

Question 1

Let $x$x represent "a number of strawberries."

a) Write an expression in terms of $x$x for "four more than a number of strawberries."

Think: $x$x represents a number of strawberries, so "four more than a number of strawberries" is the same as "$4$4 more than $x$x".

Do: We can write "$4$4 more than $x$x" as $4+x$4+x.

b) There are a certain number of strawberries in the picture below. Use the picture to find the value of $4+x$4+x.       Think: $x$x represents a number of strawberries. How many strawberries are in the picture?

Do: There are $7$7 strawberries in the picture. So the value of $4+x$4+x will be $4+7=11$4+7=11.

Question 2

Let $k$k represent "a number of pumpkins."

a) Write an expression in terms of $k$k for "five times a number of pumpkins, plus another one."

Think: $k$k represents a number of pumpkins, so "fives times a number of pumpkins, plus another one" is the same as "$5$5 times $k$k, plus another $1$1".

Do: We can write "$5$5 times $k$k, plus another $1$1" as $5k+1$5k+1.

b) The picture below shows a number of pumpkins. Use the picture to find the value of $5k+1$5k+1.        Think: $k$k represents a number of pumpkins. How many pumpkins are in the picture?

Do: There are $8$8 pumpkins in the picture. So the value of $5k+1$5k+1 will be $5\times8+1=41$5×8+1=41.

Worked Examples

Question 1

Let $p$p represent "the number of tennis balls in a pack".

1. John has five tennis balls left when he opens a new pack. Write an expression in terms of $p$p to represent "the number of tennis balls in a pack plus five".

2. The picture below shows how many tennis balls are in the pack that John opened. Use this to find the value of $p+5$p+5. Question 2

Let $x$x represent "the side length of an equilateral triangle".

1. The perimeter of an equilateral triangle is equal to three times its side length. Write an expression in terms of $x$x for "three times the side length".

2. The picture below shows an equilateral triangle with a side length of $8$8 cm. Use this to find the value of $3x$3x (the perimeter of the shape). Question 3

Let $m$m represent 'a number of apples'.

1. Write an expression in terms of $m$m to represent "five times a number of apples, plus another five".

2. If we now know how many apples there are, shown in the picture below, find the value of $5m+5$5m+5. Outcomes

NA4-9

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