Lesson 9
Equations of Lines
9.1: Remembering Slope (10 minutes)
Warmup
This task reviews the concept of slope. This work will lead to the development of the pointslope 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 wholeclass discussion.
Student Facing
The slope of the line in the image is \(\frac{8}{15}\). Explain how you know this is true.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
The goal of the discussion is to highlight the expression \(\frac{2(\text6)}{5(\text10)}\). 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: \(\text{vertical distance} \div \text{horizontal distance}\). 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 \(2  (\text6)\) and \(5  (\text10)\).
Now write out the slope as \(\frac{2(\text6)}{5(\text10)}\). It’s important that students see this expression in preparation for their work in the next activity. 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.)
9.2: Building an Equation for a Line (15 minutes)
Activity
In previous activities, students developed equations for circles and parabolas. In this task, they use similar methods to develop the pointslope form of the equation of a line.
Student Facing
 The image shows a line.
 Write an equation that says the slope between the points \((1,3)\) and \((x,y)\) is 2.
 Look at this equation: \(y3=2(x1)\)
How does it relate to the equation you wrote?
 Here is an equation for another line: \(y7=\frac12 (x5)\)
 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 a line passes through a particular point \((h,k)\).
 Write an equation that says the slope between point \((x,y)\) and \((h,k)\) is \(m\).
 Look at this equation: \(yk=m(xh)\). How does it relate to the equation you wrote?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
If students struggle with the first question, suggest they label the lengths of the legs of the triangle in the diagram.
If students write the first equation as \(\frac{3y}{1x}=2\), explain that \(y3\) and \(x1\) are equivalent to \(\text1(3y)\) and \(\text1(1x)\), respectively.
Activity Synthesis
The goal of the discussion is to ensure students understand the pointslope form of the equation of a line. Here are some questions for discussion:
 “What set of points does the equation \(y3=2(x1)\) represent?” (It is the set of points that has a slope of 2 with the point \((1, 3)\). That is, it’s the line with slope 2 that goes through \((1,3)\).)
 “The equation \(yk=m(xh)\) is called the pointslope form for the equation of a line. What do \((x,y), (h, k),\) and \(m\) represent?” (\((x,y)\) represents any point on the line. We substitute in a particular point \((h,k)\). The letter \(m\) represents the slope of the line.)
 “Why do we subtract the \(k\) from the \(y\) and the \(h\) from the \(x\)?” (This gives the lengths of the legs of the slope triangle.)
Ask students to add this definition to their reference charts as you add it to the class reference chart:
The pointslope form of the equation of a line is \(yk=m(xh)\) where \((h,k)\) is a particular point on the line and \(m\) is the slope of the line. (Definition)
Tell students that in previous courses, they learned multiple ways to write the equation of a line, including slopeintercept form, \(y=mx+b\), and standard form, \(Ax+By=C\). For the rest of this unit, students should feel free to use whatever form is easiest for the given problem.
Design Principle(s): Support sensemaking; Optimize output (for explanation)
Supports accessibility for: Conceptual processing; Language
9.3: Using PointSlope Form (10 minutes)
Activity
This task allows students to practice writing and reading equations in pointslope form. Monitor for a variety of answers for the last part of the first question to highlight during the synthesis.
Launch
Design Principle(s): Cultivate conversation
Supports accessibility for: Organization; Attention; Socialemotional skills
Student Facing
 Write an equation that describes each line.
 the line passing through point \((\text2, 8)\) with slope \(\frac45\)
 the line passing through point \((0,7)\) with slope \(\text\frac73\)
 the line passing through point \((\frac12, 0)\) with slope 1

the line in the image
 Using the structure of the equation, what point do you know each line passes through? What’s the line’s slope?
 \(y5=\frac32 (x+4)\)
 \(y+2=5x\)
 \(y=\text2(x\frac58)\)
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
Another way to describe a line, or other graphs, is to think about the coordinates as changing over time. This is especially helpful if we’re thinking tracing an object’s movement. This example describes the \(x\) and \(y\)coordinates separately, each in terms of time, \(t\).
 On the first grid, create a graph of \(x=2+5t\) for \(\text2\leq t\leq 7\) with \(x\) on the vertical axis and \(t\) on the horizontal axis.
 On the second grid, create a graph of \(y=34t\) for \(\text2\leq t\leq 7\) with \(y\) on the vertical axis and \(t\) on the horizontal axis.
 On the third grid, create a graph of the set of points \((2+5t,34t)\) for \(\text2\leq t\leq 7\) on the \(xy\)plane.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
If students struggle with identifying the point that the line passes through in the last 2 parts of the second question, suggest that they look back to the first question. For lines with points that included the number of 0, how can those be rewritten so that the 0 doesn’t appear? Do any of those forms look similar to the equations in the second question?
Activity Synthesis
The goal of the discussion is to demonstrate that there are many equivalent ways to write an equation for any given line. Focus the synthesis on the last part of the first question. Invite previously identified students to share their answers. Record and display these answers for all to see. Possible examples include:
 \(y3=\frac12 (x2)\)
 \(y5=\frac12 (x6)\)
 \(y=\frac12 x+2\)
 \(y0=\frac12(x(\text4))\)
If time allows, graph each answer using Desmos or other graphing technology that allows graphing of implicit equations. Point out that these are all different ways to describe the same line. Any point on the line can be substituted for \((h,k)\), and the equation can be put into slopeintercept form by rearranging. Challenge students to choose 2 answers and use rewriting to show they are equivalent.
Lesson Synthesis
Lesson Synthesis
Display this image for all to see:
Ask students:
 “What do you notice?” (All the lines intersect at \((4,2)\). 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.”
 \(y=2\)
 \(x=4\)
 \(y2=\frac13(x4)\)
 \(y2=\text\frac{1}{3}(x4)\)
 \(y2=\text3(x4)\)
 \(y2=3(x4)\)
Invite students to share the equations 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: Slopeintercept is best, because it’s easy to graph a line in this form. Pointslope is best, because you can use any point in it, not just the \(y\)intercept.)
9.4: Cooldown  Same Slope, Different Point (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
The line in the image can be defined as the set of points that have a slope of 2 with the point \((3,4)\). An equation that says point \((x,y)\) has slope 2 with \((3,4)\) is \(\frac{y4}{x3}=2\). This equation can be rearranged to look like \(y4=2(x3)\).
The equation is now in pointslope form, or \(yk=m(xh)\), where:
 \((x,y)\) is any point on the line
 \((h,k)\) is a particular point on the line that we choose to substitute into the equation
 \(m\) is the slope of the line
Other ways to write the equation of a line include slopeintercept form, \(y=mx+b\), and standard form, \(Ax+By=C\).
To write the equation of a line passing through \((3, 1)\) and \((0,5)\), start by finding the slope of the line. The slope is \(\text\frac{4}{3}\) because \(\frac{51}{03}=\text\frac43\). Substitute this value for \(m\) to get \(yk=\text\frac{4}{3}(xh)\). Now we can choose any point on the line to substitute for \((h,k)\). If we choose \((3, 1)\), we can write the equation of the line as \(y1=\text\frac{4}{3}(x3)\).
We could also use \((0,5)\) as the point, giving \(y5=\text\frac{4}{3}(x0)\). We can rearrange the equation to see how pointslope and slopeintercept forms relate, getting \(y=\text\frac{4}{3}x+5\). Notice \((0,5)\) is the \(y\)intercept of the line. The graphs of all 3 of these equations look the same.