Lesson 6 Point the Way Solidify Understanding

Learning Focus

Find patterns that are useful in writing equations for linear functions.

Are there different ways to write the equation of a line?

What does each part of an equation tell us?

Open Up the Math: Launch, Explore, Discuss

Grandma Billings is making a quilt and piecing together a number of blocks. Zac and Sione are watching and thinking about predicting the number of quilt blocks in this pattern:

Zac said: I noticed that two blocks are being added each time. I thought it would be easier if I had the $y$ -intercept, so I imagined what “Pattern 0” would look like by subtracting $2$ . That left $1$ block at $x = 0$ , so I wrote $B = 2 n + 1$ .

Sione said: I also noticed that the pattern was growing by two each time. I saw that Pattern 1 just has $3$ blocks. Pattern 2 has $3$ blocks plus $2$ groups of $1$ block on either side. Pattern 3 has $3$ blocks plus $2$ groups of $2$ blocks. So, I wrote $B = 2 (n - 1) + 3$ .

1.

What does $2 (n - 1)$ mean in Sione’s equation?

2.

Are the equations that Zac and Sione wrote equivalent? How do you know?

Marcus joined the conversation and noticed this: I was focusing on the strip of blocks in the middle and looking at Patterns 3 and 4. I think that you could also write $B = 2 (n - 2) + 5$ , and $B = 2 (n - 3) + 7$ .

3.

Do you agree with Marcus? Are these equations all equivalent to Zac’s equation?

Sione said, “What is going on here? I’m going to look at a table to see if I can see why this is working.” Sione made this table and noticed a pattern in the table and the equations they had written.

$n$	$0$	$1$	$2$	$3$	$4$
$f (n)$	$1$	$1 + 2$	$1 + 2 + 2 = 5$	$1 + 2 + 2 + 2 = 7$	$1 + 2 + 2 + 2 + 2 = 9$

Sione said, “I used the pattern I saw in the table to write another equation. This time, it’s $f (n) = 2 (n - 4) + 9$ .”

Zac said, “I don’t even need the table. I think $f (n) = 2 (n - 5) + 11$ is another equivalent equation.”

4.

Examine the table and compare it to the equations that have been written. Explain how Sione might have been using the table when he found the equation $y = 2 (n - 4) + 9$ .

5.

Use Sione’s method to find an equation for each of the tables below.

a.

$x$	$y$
$32$	$50$
$33$	$55$
$34$	$60$

b.

$x$	$y$
$26$	$78$
$28$	$70$
$30$	$62$

c.

$x$	$y$
$- 15$	$32$
$- 14$	$39$
$- 13$	$46$

6.

What information do you need to write an equation using this method?

7.

Write the equation of the line shown in the graph using this method.

8.

Use this method to write the equation of the line with slope $- 3$ that contains the point $(- 1, - 5)$ .

9.

Zac has a challenge: “I’ll bet you can’t use the pattern to write the equation of the line through the points $(1, - 3)$ and $(3, - 5)$ . Try it!”

10.

Sione is still thinking about the pattern: “I wonder if we could use this pattern to graph lines, thinking of the starting point and using the slope.” Try it with the equation: $f (x) = - 2 (x + 1) - 3$ .

Starting point:

Slope:

11.

Zac wonders, “What is it about lines that makes all this work?” How would you answer Zac?

Ready for More?

Grandma Billings has started piecing her quilt together and has created the following growth pattern:

See if you can find at least two equivalent equations to model the number of squares in the pattern.

Takeaways

Point-slope form comes from the slope formula:

With a little rearranging:

Vocabulary

point-slope form of a line
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned a new and efficient pattern for writing the equation of a line. The method can be used with a table, a graph, or any two points on the line.

Retrieval

Create a graph and explicit equation for the given information.

1.

The first term is $2$ , and each term increases by $3$ .

Equation:

2.

The first term is $2$ , and each term increases by $3$ times the previous term.

Equation:

Rewrite each expression by combining like terms.

3.

$24 - 10 (2 x + 8) + 62 + 7 x$

4.

$7 (2 x + 1) - 5 x - 3 (3 x + 1)$