6.01 Review: Patterns in number tables
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$5−3=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.
| 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 |
What is the difference in their ages?
Fill in the blanks.
- 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:
If the pattern continues on, the next point marked on the line will be
$\left(3,7\right)$(3,7)
A$\left(4,5\right)$(4,5)
B$\left(3,6\right)$(3,6)
C$\left(4,6\right)$(4,6)
DFill 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 $1$1 $\editable{}$ $\editable{}$ $5$5 $3$3 $\editable{}$ Choose the three statements that correctly describe this pattern:
The rule is $x+3=y$x+3=y
AAs $x$x increases $y$y increases
BThe rule is $y-3=x$y−3=x
CThe rule is $y+3=x$y+3=x
D