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3.01 Review: Patterns in number tables

Lesson

Patterns all around us

We see patterns all around us in the world. From the growth of money in a savings account to the decay of radioactive materials. It can be extremely helpful (and fun) to figure out how to get from one number in a pattern to the next. Knowing how a pattern works can help us make important predictions and plan for the future. A simple pattern (or sequence) is formed when the same number is added or subtracted at each step. Let's take a look at the examples below:

This is an increasing pattern, where $2$2 is added at every step

This is a decreasing sequence, where $3$3 is subtracted at every step

To find the next number that follows in a pattern, it's as simple as figuring out what the pattern is and applying it to the last number. For example, the next number in the decreasing pattern above would be $5-3=2$53=2. We could continue this pattern forever if we wanted to!

 

Representing patterns in tables

A table of values can be a nice way to organize a pattern. Below is a drawing of a pattern of flowers.

A table can be generated to count the number of petals visible at a given time, based on how many flowers are present.
 
Number of flowers    $1$1 $2$2 $3$3 $4$4
Number of petals $5$5 $10$10 $15$15 $20$20

Notice that the number of petals is increasing by $5$5 each time - in particular, the value in the table for Number of petals is always equal to $5$5 times the value for Number of flowers. Therefore, we could generate a rule for this table to say:

$\text{Number of petals}=5\times\text{Number of flowers}$Number of petals=5×Number of flowers

Or to write it more mathematically:

$y=5x$y=5x

where $y$y represents the number of flowers and $x$x represents the number of petals.

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 $x=10$x=10 into the rule to find $y=5\times10=50$y=5×10=50 petals. So even though there were only $1,2,3$1,2,3 and $4$4 flowers present in the picture above, the rule has determined that there would be $50$50 petals visible when there are $10$10 flowers present.

Let's explore some different patterns in the practice questions below!

Practice questions

Question 1

Nadia knows that she is younger than her father, Glen. The following table shows her dad's age compared to hers.

Nadia's age Glen's age
$1$1 $24$24
$5$5 $28$28
$10$10 $33$33
$20$20 $43$43
$30$30 $53$53
    • When Nadia was $1$1 year old, Glen was $\editable{}$ years old
    • When Nadia is $49$49 years old, Glen will be $\editable{}$ years old.

Question 2

A catering company uses the following table to work out how many sandwiches are required to feed a certain number of people.

Fill in the blanks:

Number of People Sandwiches
$1$1 $5$5
$2$2 $10$10
$3$3 $15$15
$4$4 $20$20
$5$5 $25$25
    • For each person, the caterer needs to make $\editable{}$ sandwiches.
    • For $6$6 people, the caterer would need to make $\editable{}$ sandwiches.

Question 3

Consider the pattern shown on this line graph:

Loading Graph...

  1. If the pattern continues on, the next point marked on the line will be $\left(\editable{},\editable{}\right)$(,).

  2. Fill in the table with the values from the graph (the first one is filled in for you):

    $x$x-value $y$y-value
    $0$0 $3$3
    $\editable{}$ $\editable{}$
    $\editable{}$ $\editable{}$
    $\editable{}$ $\editable{}$
  3. Choose all statements that correctly describe this pattern:

    The rule is $x+3=y$x+3=y

    A

    As $x$x increases $y$y increases.

    B

    The rule is $y-3=x$y3=x

    C

    The rule is $y+3=x$y+3=x

    D

    The rule is $x+3=y$x+3=y

    A

    As $x$x increases $y$y increases.

    B

    The rule is $y-3=x$y3=x

    C

    The rule is $y+3=x$y+3=x

    D

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