Lesson 9

Equations of Lines

9.1: Remembering Slope (10 minutes)

Warm-up

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.

Student Facing

Line. Points plotted at -10 comma -6 and 5 comma 2.

The slope of the line in the image is \(\frac{8}{15}\). Explain how you know this is true.

Student Response

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Activity Synthesis

The goal of the discussion is to highlight the expression \(\frac{2-(\text-6)}{5-(\text-10)}\). 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 \(2 - (\text-6)\) and \(5 - (\text-10)\).

Now write out the slope as \(\frac{2-(\text-6)}{5-(\text-10)}\). 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 point-slope form of the equation of a line.

Student Facing

  1. The image shows a line.
    Line, y intercept = 1, slope = 2. Points plotted at 1 comma 3 and x comma y.
    1. Write an equation that says the slope between the points \((1,3)\) and \((x,y)\) is 2.
    2. Look at this equation: \(y-3=2(x-1)\)
      How does it relate to the equation you wrote?
  2. Here is an equation for another line: \(y-7=\frac12 (x-5)\)
    1. What point do you know this line passes through?
    2. What is the slope of this line?
  3. Next, let’s write a general equation that we can use for any line. Suppose we know a line passes through a particular point \((h,k)\).
    1. Write an equation that says the slope between point \((x,y)\) and \((h,k)\) is \(m\).
    2. Look at this equation: \(y-k=m(x-h)\). How does it relate to the equation you wrote?

Student Response

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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 \(\frac{3-y}{1-x}=2\), explain that \(y-3\) and \(x-1\) are equivalent to \(\text-1(3-y)\) and \(\text-1(1-x)\), respectively.

Activity Synthesis

The goal of the discussion is to ensure students understand the point-slope form of the equation of a line. Here are some questions for discussion:

  • “What set of points does the equation \(y-3=2(x-1)\) 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 \(y-k=m(x-h)\) is called the point-slope 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 point-slope form of the equation of a line is \(y-k=m(x-h)\) where \((h,k)\) is a particular point on the line and \(m\) is the slope of the line. (Definition)

A line with point h comma k on an x y axis.

Tell students that in previous courses, they learned multiple ways to write the equation of a line, including slope-intercept 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.

Speaking: MLR8 Discussion Supports. Use this routine to help students produce statements about the terms in the point-slope form equation of a line. Provide sentence frames for students to use when they interpret \((x,y)\), \((h,k)\), and \(m\), such as: “_____ represents _____.” and “It looks like _____ represents….”
Design Principle(s): Support sense-making; Optimize output (for explanation)
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: slope, \(y\)-intercept, and point-slope form.
Supports accessibility for: Conceptual processing; Language

9.3: Using Point-Slope Form (10 minutes)

Activity

This task allows students to practice writing and reading equations in point-slope form. Monitor for a variety of answers for the last part of the first question to highlight during the synthesis.

Launch

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion about the many equivalent ways to write an equation of a line. After students find the equation of the graph of the line, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their equations. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the equations of the line.
Design Principle(s): Cultivate conversation
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 2 parts to complete for each question. Chunking this task into more manageable parts may also help students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

  1. Write an equation that describes each line.
    1. the line passing through point \((\text-2, 8)\) with slope \(\frac45\)
    2. the line passing through point \((0,7)\) with slope \(\text-\frac73\)
    3. the line passing through point \((\frac12, 0)\) with slope -1
    4. the line in the image

      Line, y intercept = 2, slope = 1 over 2. 
  2. Using the structure of the equation, what point do you know each line passes through? What’s the line’s slope?
    1. \(y-5=\frac32 (x+4)\)
    2. \(y+2=5x\)
    3. \(y=\text-2(x-\frac58)\)

Student Response

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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, \(t\).

Coordinate plane 
Coordinate plane 
Coordinate plane 
  1. On the first grid, create a graph of \(x=2+5t\) for \(\text-2\leq t\leq 7\) with \(x\) on the vertical axis and \(t\) on the horizontal axis.
  2. On the second grid, create a graph of \(y=3-4t\) for \(\text-2\leq t\leq 7\) with \(y\) on the vertical axis and \(t\) on the horizontal axis.
  3. On the third grid, create a graph of the set of points \((2+5t,3-4t)\) for \(\text-2\leq t\leq 7\) on the \(xy\)-plane.

Student Response

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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:

  • \(y-3=\frac12 (x-2)\)
  • \(y-5=\frac12 (x-6)\)
  • \(y=\frac12 x+2\)
  • \(y-0=\frac12(x-(\text-4))\)

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 slope-intercept 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:

5 lines on a grid that intersect at a point.

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.”
    • \(y=2\)
    • \(x=3\)
    • \(y-2=\frac13(x-3)\)
    • \(y-2=\text-\frac{1}{3}(x-3)\)
    • \(y-2=\text-3(x-3)\)
    • \(y-2=3(x-3)\)

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: 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 \(y\)-intercept.)

9.4: Cool-down - Same Slope, Different Point (5 minutes)

Cool-Down

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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 \(\frac{y-4}{x-3}=2\). This equation can be rearranged to look like \(y-4=2(x-3)\).

Line, y intercept = -2., slope of 2. Points plotted at 3 comma 4 and x comma y.

The equation is now in point-slope form, or \(y-k=m(x-h)\), 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 slope-intercept 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 \(\frac{5-1}{0-3}=\text-\frac43\). Substitute this value for \(m\) to get \(y-k=\text-\frac{4}{3}(x-h)\). 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 \(y-1=\text-\frac{4}{3}(x-3)\).

We could also use \((0,5)\) as the point, giving \(y-5=\text-\frac{4}{3}(x-0)\). We can rearrange the equation to see how point-slope and slope-intercept 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.