## Rise-Run Triangles

- Lesson

This lesson offers students a method for finding the slope of a line from its graph. The skills from this lesson can be applied as a tool to real-world examples of rate of change and slope.

To start the lesson, ask students what they already know about *slope*. They may know terms such as rate of change and rise over run. Often, students have recollection of these terms but don't remember or understand what they mean or how they relate to slope.

Ask students what it means to have positive or negative slope. Encourage a student to come to the front of the room and draw a line with positive slope. Ask classmates if they agree that the line has positive slope, and then ask how they can tell.

A line with positive slope is pointing upward as you look to the right. You always want to see if the line is pointing upward or downward on the right side of the graph, just as we read to the right.

Sketch these two lines with positive slope for students to see.

Ask students to tell you all they can about the two graphs. What's the same? What's different? Emphasize that although both lines have positive slope, there is something different about the direction in which they point. Explain that this description of how slanted a line is can be described by a number called its slope.

Now, draw a third line that has the same slope as the first line, but a different *y*-intercept. Ask students again for comparisons.

Students should eventually recognize that the third line has the same slope as the first line. Once they do, they are ready to think about the slope number as a description of how slanted a line is.

Use the Counting for Slope Activity Sheet for practice and enforcement.

Counting for Slope Activity Sheet

The activity sheet guides students through a process for finding the slope of a given line. Page 1 is meant to be completed as a class, so having an overhead slide of this page will be helpful.

Distribute the activity sheets and make sure each student has 1 or 2 colored pencils. Many students enjoy using a colored pencil to draw and shade the slope triangle, and doing so makes the lesson more memorable. You might ask students to use one color when they're drawing the triangle for a line with positive slope, and another color for triangles representing negative slope.

Shade in the slope triangles with students as shown below.

Encourage students to simplify their fractions on page 1 of the activity sheet. Point out that for each line, the simplified forms of the fractions are equivalent — no matter which two points on the line your student uses, or how large the triangle is, you get the correct slope.

On page 2, students are given the slope triangle in the first 3 examples (the top row). In the next 3 examples (middle row), they are given only the points to use to draw the triangle. In the last 3 examples (bottom row), students have to find the points themselves before drawing the triangle and determining the slope. The idea here is to gradually get students comfortable with finding the slope.

While students work on page 2, be sure that they:

- Simplify all fractions
- Determine which lines have negative slope and use a negative fraction to represent the slope of these lines.

This exercise provides students with the skill of finding the slope of a line from a graph. This skill can be applied to less abstract examples using real data from a table or a graph.

- Counting for Slope Activity Sheet
- Colored pencils (optional)

**Assessment Options**

- Provide students with data in table form and have them graph the points and find the slope of the line connecting the points. Ask students "What does the slope say about the trend of the data?"
- Ask students to explain how miles per hour can be seen as a slope? Explain how heart beats per minute can be seen as a slope.
- Provide a table of data for students to graph, such as the one offered below. Ask the students to graph the data and determine the speed of travel. Ask students, What are the units? How do the words miles per hour relate to what you see in the graph?
Distance From Philadelphia Time in Hours

(*x*)Distance in Miles

(*y*)1 55 2 85 4 145 8 265

**Extensions**

- Show students a given point on a coordinate grid and give them a value for the slope of a line. Ask them to draw a line that has that slope through the given point.
- How does the slope triangle method apply to horizontal and vertical lines? What does the slope-triangle look like for a vertical line? What does the slope-triangle look like for a horizontal line? What problems arise, and how do they affect the slope?
- How is slope formula consistent with the slope triangle method for finding the slope of a line? How does calculating
*y*_{1}–*y*_{2}relate to the height of the slope triangle? How does calculating*x*_{1}–*x*_{2}relate to the length of the slope triangle?

**Questions for Students**

1. Which, if any, of the fractions did you have to simplify when you found the slope of a line? How can you avoid the need to simplify a fraction?

[Answers will vary.]

2. Suppose you have identified 3 slope triangles for a line to help you find the correct slope. What can you say about the relationship between these triangles?

[The simplified forms of the fractions will be equivalent]

3. Explain the difference between a line with positive slope and a line with negative slope.

[A line with positive slope points upward as you look to the right. A line with negative slop points downward as you look to the right.]

4. Explain the difference between a line with slope 1/2 and a line with slope 2/1.

[The line with slope 2/1 will be steeper than the line with slope 1/2]

**Teacher Reflection**

- Describe what effect using slope triangles had on student understanding of slope.
- What difficulties did students have with this activity?
- How could you extend this activity to rate-of-change problems in the curriculum?
- Describe the effect using colored pencils had on student comprehension of slope.

### Learning Objectives

Students will be able to:

- Determine if the slope of a line is positive or negative.
- Express the slope of a line as a fraction.

### NCTM Standards and Expectations

- Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.

- Use graphs to analyze the nature of changes in quantities in linear relationships.

- Approximate and interpret rates of change from graphical and numerical data.

### Common Core State Standards – Mathematics

Grade 8, Expression/Equation

- CCSS.Math.Content.8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Grade 8, Functions

- CCSS.Math.Content.8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.