Lesson 9

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.

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.

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.)

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.

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.

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)

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.

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.

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?

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.