1. Making sense of patterns
Linear relations
Making sense of patterns

2. What are linear relations?
A linear relation is one where at least two things are related to each other; they have a pattern that we can quantify with numbers and operations, and show on a graph.
The linear relation can be written as an equation. There are always two variables:
A y variable and an x variable
A linear relation always produces a graph with a straight line.

3. Say what?? Let’s start with an easy example: What is the relation???
There are four baskets of kiwis.
Basket 1 contains 5 kiwis
Basket 2 contains 10 kiwis
Basket 3 contains 15 kiwis
Basket 4 contains 20 kiwis
What is the relation???
First, let’s describe it:
there are 5 extra apples in each basket
Second, let’s create a table of values:
Third, let’s develop an equation:
If we take the basket number, and multiply by five, we get the number of kiwis
This can be represented by:
k = 5b
How many kiwis are in basket #22?
basket # (b)
Kiwis (k) 1 5 2 10 3 15 4 20

4. Ready for more?? Here is a trickier example:
Let’s say you have a huge oil super tanker that has to make an emergency stop.
Now, when the super tanker is full, it travels at a maximum speed of 30 km/h (honest!). That’s pretty hard to stop quickly.
It takes 15 minutes to actually bring the tanker to a complete stop. (hmmm – if you were about to hit an iceberg, I guess you wouldn’t be able to avoid it – - bad environmental disaster!)

5. What’s the relation? So – what two things are related here?
Aha – time and speed
The table below shows the speed of the super tanker during the emergency stop:
time (min)
Speed (km/h) 30 3 24 6 18 9 12 15

6. Thinking about it…
time (min)
Speed (km/h) 30 3 24 6 18 9 12 15
Describe the pattern you can see in the table. What do you notice about how the numbers are changing?
What do you think the speed will be at minutes? 10 minutes? Explain why.

7. Creating an equation Look carefully at the table of values.
What equation can I set up to figure out missing speeds?
Speed = -2t + 30
time (min)
Speed (km/h) 30 3 24 6 18 9 12 15

8. Let’s begin with visual patterns
Check out this series of pictures:
Figure Figure Figure 3 Figure 4
Describe the pattern
Create a table of values to represent the linear relationship between the number of squares and the figure number
Write a linear equation to represent this pattern
How many squares will be in figure 12?

9. Let’s begin with visual patterns
The pattern is increasing in each figure
A table of values would be:
Expand the table to figure out the pattern:
The equation is: s = 3n – 2
figure number (n)
number of squares (s) 1 2 4 3 7 10
figure number (n)
number of squares (s)
pattern
what do we do to the figure number?
what do we do to the result? 1
multiply by 3 (=3)
subtract 2 (3 -2 = 1) 2 4
multiply by 3 (=6)
subtract 2 (6 -2 = 4) 3 7
multiply by 3 (=9)
subtract 2 (9 -2 = 7) 10
multiply by 3 (=12)
subtract 2 (12 -2 = 10)

10. Try it on your own:
Write an equation to represent the number of circles in relation to the figure number:
figure 1 figure 2 figure 3 figure 4
Remember:
Describe the pattern
create a table
expand it to figure out the pattern
Create the equation

11. what did we then do to get c?
Response:
Describe the pattern
In each figure, the number of circles increases by 2
create a table
expand it to figure out the pattern
Create the equation
c = 2f -1
figure (f)
circles ( c ) 1 2 3 5 4 7
figure (f)
circles ( c )
pattern
what did we do to f?
what did we then do to get c? 1
multiply by 2
subtract 1 2 3 5 4 7

12. Homework:
Page 217
#4, 5, 6

13. From words to equations
Written patterns
From words to equations

14. Let’s try it with written patterns
Patterns can be found in the written words. Try this one:
A bead design has an arc shape
Row 1 has seven beads. All the beads are red.
Row 2 has five additional beads. All the beads are green.
Row 3 has five additional beads. All the beads are blue.
The pattern repeats, with five beads added to each successive row.

15. Written patterns First, create a table of values:
Try to create the equation that shows the pattern. You can do that by extending the table to help determine the pattern.
The equation is b = 5n + 2
row number (n)
number of beads (b) 1 7 2 12 3 17 4 22
row number (n)
number of beads (b)
pattern
multiply n by 5
add 2 to the result 1 7 5 2 12 10 3 17 15 4 22 20

16. Written patterns A single rectangular banquet table seats six people.
The tables can be connected end to end.
For each additional table added to the row, four additional people can be seated.
Create a linear equation.
First, create a table of values
Second, extend the table to try determine the pattern
Third, create the equation

17. Written patterns
Here is the table of values for the information provided:
Try to create the equation that shows the pattern. You can do that by extending the table to help determine the pattern.
The equation is s = 4t + 2
tables (t)
seats (s) 1 6 2 10 3 14 4 18
tables (t)
seats (s)
pattern
multiply t by 4
add 2 to the result 1 6 4 2 10 8 3 14 12 18 16

18. Homework:
Page 217 and 218
#7, 8, 9