slope and lines - wp-media.tellibrary.org · we read a line from left to right just like we read...

Slope and Lines

Inquire: Riding the Slopes

Overview Ramps, ladders, and stairs are all built using the concept of slope. You want to travel a certain vertical distance with respect to a certain horizontal distance. This can be accomplished with appropriate slope, but how do we measure slope, and how much is too much slope? Slope can even have a direction depending on whether it is positive or negative. Slope isn't limited to building objects. It can be used to observe a rate of change in two quantities. An example would be a bank account starting at $5000 losing $250 every month. We can even represent this change graphically if we use slope and points. By the end of this lesson, the student will be able to find slope and graph lines using slope.

Big Question: How do I find slope and use it to graph lines?

Watch: Applying Slope To have a line, you need two points. With these points, we can create a slope. This can be understood as “rise over run” or “change in y over the change of x," but it is important to understand the meaning of how quantities can change with respect to other quantities. This is typically called a rate of change. A scientist wants to observe how fast a burner can heat a tube of liquid. The tube starts at 50 degrees Fahrenheit. After 3 seconds, the tube is at 56 degrees. After 10 seconds, it is 70 degrees. The scientist has recorded two points worth of information: the temperature at 3 and 10 seconds after he has started heating the tube. These can be written as coordinates, but let’s talk about the change between them. The difference in temperature is 70 minus 56, which is 14 degrees. The difference in time is 10 minus 3, which is 7 seconds. This means that the tube has increased 14 degrees over 7 seconds. This can be simplified to 2 degrees every second. This helps us understand the effectiveness of the burner but also helps us predict the temperature of the tube. Let’s say the tube will bust beyond 100 degrees. Will the tube still be safe after being heated for 20 seconds? Since we know that the tube heats up 2 degrees every second, we can calculate that the tube will heat up 40 degrees over 20 seconds. If we add this to the starting temperature of 50, this puts the tube at a safe temperature of 90 degrees. Quantitative Literacy Module 3 - Slope and Lines Copyright TEL Library 2018

In places with colder temperatures, an understanding of slope is important to keep the house in order. The pitch of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. If the roof is too steep, it minimizes the amount of space in the house. The steepness of the roof can be recorded by how tall the roof is by how long it is. What rates can you think of that can be written as slopes?

Read: Rise Up to Run Forward In mathematics, the ‘tilt’ of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples where you see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill. In this section, we will explore the concept of slope.

Defining Slope To find the slope of a line, we need to measure the distance along the vertical and horizontal of a line. The vertical distance is called the rise and the horizontal distance is called the run. See the figure below.

The slope of the line is m = rise/run. If the rise goes up, it is positive, and if it goes down, it is negative. The run will go from left to right and be positive. We read a line from left to right just like we read words in English. If the line is going up, it has a positive slope. If the line is going down, it has a negative slope. See the figure below.

Finding Slope Using a Graph To find the slope, we must count out the rise and the run.

Find the Slope of a Line from its Graph Using m = rise/run:

Quantitative Literacy Module 3 - Slope and Lines Copyright TEL Library 2018

1) Locate two points on the line whose coordinates are integers. 2) Starting with the point on the left, sketch a right triangle, going from the first point to the second point. 3) Count the rise and the run on the legs of the triangle. 4) Take the ratio of rise to run to find the slope, m = rise/run. Example 1: Find slope in the figure shown

A figure of steps one, two, and three will be shown after step three. Step 1: We will plot two points on the line whose coordinates are integers. In this example, we will use (0, -3) and (5, 1); however, these are not the only coordinates we could use. Step 2: From (0, -3), we will sketch a vertical line up and a horizontal line over until we create a right triangle with (5, 1). Step 3: Count the rise (4 squares up) and run (5 squares right). Since we use the directions "up" and "right," the rise and run are positive integers.

Step 4: Substitute the values of rise and run in the slope formula: m = rise/run = 4/5. The slope is 4/5. This means every time the line goes up 4, it goes right 5.


A Different Point of View Example 2: Find slope of the figure shown

Step 1: Locate two points with integer coordinates. We will use (0,5) and (3,3). Step 2: We will sketch a right triangle starting at (0,5) and ending at (3,3). Step 3: Our rise is down 2 and run is right 3. Down is negative, so our rise is -2 and our run is +3. See the figure below for our progress so far.

Step 4: Using the slope formula, m = -2/3. This means the slope of the line is -2/3. This can mean "as y decreases by 2, then x increases by 3" or "as y increases by 2, then x decreases by 2." What if we had the same line, but used different points? Example 3: Find the slope in Example 2 using the points (-3,7) and (6,1)


The rise would be −6 and the run would be 9. Then m = −6/9 which simplifies to −2/3. Remember, it does not matter which points you use — the slope of the line is always the same.

Slope of Vertical and Horizontal Lines Horizontal and vertical lines have special slopes. We can find their slopes like we did before. Let's use y = 4 as an example of horizontal lines.

With points (0,4) and (3,4), the rise is 0 and the run is 3. Using the slope formula, m = rise/run = 0/3 = 0. The slope of the horizontal line y = 4 is 0. In fact, all horizontal lines have a slope of 0. This makes sense because horizontal lines do not have any tilt. Now, let’s use x = 3 for our vertical line.


With points (3,0) and (3,2), the rise is 2 and the run is 0. Using the slope formula, m = rise/run = 2/0 = undefined. Since we can not divide by 0, the slope of the vertical line x = 3 is undefined. Moreover, the slope of any vertical line is undefined. Anything going down a line like this would fall off! Below is a quick reference for slopes:

Slope Formula (Two Points) We can find slope without graphing. We still need two points to find slope, but we can label them using subscripts like so: (x1, y1) and (x2, y2). These are the coordinates for points 1 and 2. We can find rise by finding the difference between y's, and we can find run by finding the difference between x's. This gives us a different version of the slope formula:

m = rise/run = (y2 - y1)/(x2 - x1) = (y1 - y2)/(x1 - x2)

*Be sure your set up matches the formula you use exactly* Example 4: Use the slope formula to find the slope of the line between the points (1,2) and (4,5). Let’s call (1,2) point #1 and (4,5) point #2. This means x1 = 1, y1 = 2, x2 = 4, and y2 = 5. We will use the formula m = (y2 - y1)/(x2 - x1). m = (5 - 2)/(4 - 1) m = 3/3 = 1 The slope of the line is 1.


We can check our solution with a graph.

In the graph, our rise and run are 3. This gives us the same fraction that showed up in our work before: 3/3 = 1. Example 5: Use the slope formula to find the slope of the line through the points (−2,−3) and (−7,4). Let’s call (-2,-3) point #1 and (-7,4) point #2. This means x1 = -2, y1 = -3, x2 = -7, and y2 = 4. We will use the other formula m = (y1 - y2)/(x1 - x2). m = (-3 - 4)/(-2 - (-7)) m = (-7)/(5) = -7/5 The slope of the line is -7/5. Note that it doesn’t matter which point you call point #1 and which one you call point #2. The slope will be the same. Try the calculation yourself.

Reflect Poll: Step-By-Step Slope How helpful have these steps been in helping you find slope between two points?

● They are not helpful at all ● They help a little ● They help sometimes ● They are very helpful

Expand: Help Me, Equation. You’re My Only Slope. A key application for slope is using it to graph a line. A typical situation for this is when you have a point and a slope, but we can also use equations to help us with this. Graph a Line with Point and Slope


We can graph a line by plotting a coordinate as a starting point and using the definition of slope (rise over run) to create the rest of our line. Graphing lines in this way is sometimes called the point-slope method.

Point-Slope Method: 1) Plot the point. 2) Starting at the point, use the rise and run from the slope to mark a second point. 3) Connect the points with a line. Example 1: Graph the line passing through the point (1,−1) whose slope is m = 3/4. First, let’s plot the point (1,-1). Then, we can interpret the slope 3/4 as “up 3 and right 4." Finally, we can use that to plot a second point and connect the points together with a line that goes through them both. This is shown in the image below.

Example 2: Graph the line with y-intercept of 2 whose slope is m = −2/3. First, y-intercept of 2 is just the point (0,2). We can plot that point. Because of the negative, the slope of -2/3 can be interpreted in two ways: “down 2 and right 3” or “up 2 and left 3." We will use the first interpretation to plot the second point. As shown in the figure below, we will finish the problem by drawing a line through both points.


Slope-Intercept Form When given the equation in a specific form, we can immediately find the slope and a point (the y-intercept). The slope-intercept form of an equation is:

y = mx + b; where m = slope and y-intercept is (0,b) Example 3: Find the slope and y-intercept of the equation y = 2x + 3. Since the equation is in slope-intercept form, the slope is 2 and the y-intercept is (0,3). Example 4: Identify the slope and y-intercept of the line with equation y = −2/3x - 5. Since the equation is in slope-intercept form, the slope is -2/3 and the y-intercept is (0,-5). Example 5: Identify the slope and y-intercept of the line with equation x + 2y = 6. This equation is not in slope-intercept form, so we can not find the slope and y-intercept as it is. We could solve for y and rewrite the equation, but that will be beyond the scope of this lesson.

Graph Using Slope-Intercept Form Equations in slope-intercept form are created by finding and using a starting point and a rate of change. A rate of change is another interpretation of slope that shows how much one quantity changes with respect to another quantity. An example of rate of change is getting paid $10 for every hour or drinking 100 mL of water every 5 minutes. The starting point serves as the y-intercept for the graph. We are going to graph using slope-intercept form. You will find it is similar to when we had a slope and a point. Example 6: Joseph is tracking the distance between himself and a scooter. The scooter starts 2 miles away from Joseph and is traveling at 4 mph. The scooter passes Joseph and continues to travel at 4 mph away from him. Graph the distance between Joseph and the scooter as a line using the equation y = 4x − 2. First, we want to find and plot the y-intercept. Since the equation is in slope-intercept form, the y-intercept is (0,-2). Next, we will find and use the slope to plot a second point. Because the slope is 4, we will interpret the fraction 4/1. From our first point, the second point is up 4 and right 1. As shown in the figure below, we will connect both of our points with a line going through them.


Lesson Toolbox

Additional Resources and Readings A short game that helps you practice finding slope using graphs

● Link to resource: https://www.quia.com/rr/79713.html?AP_rand=1411752438 A game allowing you to practice finding slope using two points

● Link to resource: https://www.mathgames.com/skill/8.32-find-slope-from-two-points

An application video about slope ● Link to resource: https://www.youtube.com/watch?v=vQvlFx3-hrA

Lesson Glossary rate of change: another interpretation of slope that shows how much one quantity changes with respect to another quantity

Check Your Knowledge 1. Find slope in the graph shown below.


a. 2/5 b. 5/2 c. -5/2 d. -2/5

2. What is the slope of the equation y = 3?

3. What is the slope for the two points (-3,3) and (4,-5)?

a. 7/8 b. 8/7 c. -7/8 d. -8/7

Answer Key: 1. A 2. 0 3. D

Citations

Lesson Content: Authored and curated by Kashuan Hopkins for The TEL Library. CC BY NC SA 4.0

Adapted Content: Title: 4.4 Understand Slope of a Line. Openstax. License: CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line Title: 4.5 Use the Slope–Intercept Form of an Equation of a Line. Openstax. License: CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:tegV4wXG@4/Use-the-SlopeIntercept-Form-of

Attributions “Figure 3” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line Quantitative Literacy Module 3 - Slope and Lines Copyright TEL Library 2018

“POSITIVE AND NEGATIVE SLOPES” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “m=rise/run” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “Solution. m=rise/run” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “Slope” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “Solution. (0,5), (3,3).” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “Solution. (−3,7) and (6,1)” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “ y=4” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “Slope 0” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “QUICK GUIDE TO THE SLOPES OF LINES” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “Solution. (1,2) and (4,5)” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “Solution. (1,−1) whose slope is m=¾” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “Solution. y-intercept 2 slope is m=−2/3” By OpenStax is licensed under CC BY 4.0 Link to resource: https://cnx.org/contents/[email protected]:ukK0bN45@2/Understand-Slope-of-a-Line “Solution. y=4x−2” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:tegV4wXG@4/Use-the-Slope-Intercept-Form-of-an-Equation-of-a-Line


slope and lines - wp-media.tellibrary.org · we read a line from left to right just like we read...

Documents