Lesson 10 Point the Way Solidify Understanding
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
Sione said: I also noticed that the pattern was growing by two each time. I saw that Pattern 1 just has
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
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.
Sione said, “I used the pattern I saw in the table to write another equation. This time, it’s
Zac said, “I don’t even need the table. I think
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
Use Sione’s method to find an equation for each of the tables below.
What information do you need to write an equation using this method?
Write the equation of the line shown in the graph using this method.
Use this method to write the equation of the line with slope
Zac has a challenge: “I’ll bet you can’t use the pattern to write the equation of the line through the points
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:
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.
Point-slope form comes from the slope formula:
With a little rearranging:
- point-slope form of a line
- Bold terms are new in this lesson.
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.
Create a graph and explicit equation for the given information.
The first term is
The first term is
Rewrite each expression by combining like terms.