Making sense of patterns Linear relations Making sense of patterns

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.

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

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!)

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

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 4 minutes? 10 minutes? Explain why.

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

Let’s begin with visual patterns Check out this series of pictures: Figure 1 Figure 2 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?

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)

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

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

Homework: Page 217 #4, 5, 6

From words to equations Written patterns From words to equations

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.

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

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

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

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