Unit 5, Lesson 4
Equations of Lines
Integrated Math 1
4.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
This task reviews the concept of slope. This work will lead to the development of the point-slope form of a linear equation, in the next activity.
While students work, monitor for students who draw a slope triangle and for those who use a slope formula.
Launch
Arrange students in groups of 2. Tell students that there are many possible answers for the question. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Activity
NoneThe slope of the line in the image is . Explain how you know this is true.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of the discussion is to highlight the expression . Seeing the coordinates subtracted in this way will help students as they work through upcoming activities.
Ask students what slope means. Students may describe slope as the steepness of the line, the rate of change in a linear relationship, or “rise over run.” Tell students that one way to think about slope is that it is the quotient of the lengths of the legs of a slope triangle: . A right triangle drawn between any two points on a line will produce the same slope result.
Invite a student who drew a slope triangle to share their work. Display the student’s slope triangle for all to see, or draw one of your own. Ask the students how they can calculate the lengths of the legs of the triangle. As students describe how to do so, label the legs "" and "."
Now write out the slope as . It’s important that students see this expression in preparation for their work in an activity that follows. If any students used a slope formula, ask them how the formula relates to this expression. Ask students if the order of the numbers matters. (The order must be consistent. Because we started with the 2 in the numerator, we have to start with the 5 in the denominator.)
4.2
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
In this activity students develop the point-slope form of the equation of a line.
Launch
NoneActivity
None- The image shows a line.
- Write an equation that says the slope between the points and is 2.
- Look at this equation:
How does it relate to the equation you wrote?
- Here is an equation for another line:
- What point do you know this line passes through?
- What is the slope of this line?
- Next, let’s write a general equation that we can use for any line. Suppose we know that a line passes through a particular point, .
- Write an equation that says the slope between points and is .
- Look at this equation: . How does it relate to the equation you wrote?
Lesson Synthesis
Display this image for all to see:
Ask students:
- “What do you notice?” (All the lines intersect at . One line is horizontal and one is vertical. The other 4 lines are either slanted upward or slanted downward.)
- “Write the equations of at least 3 different lines shown.”
Invite students to share the equations that they wrote. Record and display their responses for all to see. Use graphing technology to show that the equations they wrote match the image.
Ask which form of the equation of a line students prefer. (Any preference students state is valid. Sample responses: Slope-intercept is best, because it’s easy to graph a line in this form. Point-slope is best, because you can use any point in it, not just the -intercept.)
Student Lesson Summary
The line in the image can be defined as the set of points that have a slope of 2 with the point .
An equation that says point has slope 2 with is . This equation can be rearranged to look like .
The equation is now in point-slope form, or , where:
- is any point on the line.
- is a particular point on the line that we choose to substitute into the equation.
- is the slope of the line.
Other ways to write the equation of a line include slope-intercept form, , and standard form, .
To write the equation of a line passing through and , start by finding the slope of the line. The slope is because . Substitute this value for to get . Now we can choose any point on the line to substitute for . If we choose , we can write the equation of the line as .
We could also use as the point, giving . We can rearrange the equation to see how point-slope and slope-intercept forms relate, getting . Notice that is the -intercept of the line. The graphs of all three of these equations look the same.