Hitting the Slopes An adventure on the bunny hill of linear equations! Start

Hitting the Slopes

An adventure on the bunny hill

of linear equations!

Start

What would you like to learn about?

1Calculating the slope of the line,given two points

2Solving for y

(slope-intercept form)and graphing

the line

3Determining the

equation of the linefrom the graph

The definition of slope

Given two points, (x1,y1) and (x2,y2),

the slope of a line is

determined by this equation:

where m = slope.

We think of slope as the change in y

divided by the change in x.

12

12

xx

yym

−−

=

Look at anexample

Example of finding slope.

Find the slope of the line between the points (0,6) and (5,10).

1. Remember the formula.

2. Identify which numbers represent the given variables.

3. Substitute into the formula and simplify.

The slope of the line extending through

the points (0,6) and (5,10) is .

12

12

xx

yym

−−

=

61 =y 52 =x 102 =y

05

610

−−

=m5

4=m

5

4

Take a Quiz

01 =x

Definition of Slope Quiz

Click on the correct answer below:

Slope is a number that represents the sum of y divided by the sum of x

Slope is a number that represents the change in x divided by the change in y

Slope is a number that represents the change in y divided by the change in x

Hooray!You are correct, congratulations!

Slope is a number that represents the change in y divided by the change in x.

We can also think of it as:

Since the y-axis is the vertical axis on the coordinate grid, we think of the change in y as “rising” and since the x-axis is the horizontal axis, we think of the change in x as “running”.

run

rise

Try calculating

slope

Oops!

Remember, slope is defined as the change in y divided by the change in x. It is important to understand that change means difference. In math, the term difference tells us to subtract. The answer you chose contained the word “sum”, which means to add.

I got it!Take the quiz

Again.

Still a bit confused, take me back

to the definition.

Oops!

You are close, but let’s recall the definition: Slope is the change in y divided by the change in x. We think of slope as . Since the y-axis is the vertical axis on the coordinate grid, we think of the change in y as “rising”. Since the x-axis is the horizontal axis on the coordinate grid, we think of the change in x as “running”.

So, it’s important to rise first and then run!

run

rise

Still a bit confused, take me back

to the definition.

I got it!Take the quiz

Again.

Calculating Slope

What is the slope of the line

between (5,2) and (10,1)?

5

1−

2

518

*Be sure to click on the blue part of the button, not in the white box*

Congratulations!

Yes, the correct slope is .

In this case, the slope was in fraction form, but sometimes this won’t always be the case.

If, for example, the slope of a line was 4. We should think of it as a fraction:

5

1−

1

4

Uh Oh!

Let’s recall the steps to finding slope:

1. Remember the formula.

2. Identify which numbers represent the given variables.

3. Finally, substitute the numbers into the formula and simplify!

12

12

xx

yym

−−

=

21 =y 102 =x 12 =y51 =x

Still confused, take me back

to the beginning.

I got it!Return to the quiz.

The Graphs of Slopes

Let’s graph the

previous examples.

Example 1:

The slope between (0,6) and

(5,10) is .

Example 2:

The slope between (5,2) and (10,1) is .

Example 3:

The slope between (-1,2) and

(-2,-2) is 4.

5

4

5

1−

Plotting Points & Graphing Lines1. In order to graph the

lines, we must first plot each of the points.

2. Then draw a line through the points, adding arrows at each end to represent a line (rather than a line segment).

3. Now, start at the bottom of the two points. If you use the slope to “rise and run”, you should end at the second point.

rise

4

run 5

rise

4

run 1

rise

1

run -5

Take a quiz

Graph Quiz

Given the slope, click on the correct graph:

7

3=m

*Be sure to click on the blue button, not the graph when choosing an answer*

You’re Right!

Yes, if you start at the bottom point, then rise 3 and run 7 you will end at the other point!

Think about this…

You could also start at the top point, run -3 and rise -7 (which means go down three and left 7). This works because . Both fractions simplify to .

7

3

7

3

−−

=

7

3

Not Quite!

Since all of the points were already plotted and the lines were graphed, we just need to focus on how to get from one point to another. Remember to rise from the bottom point and then run to the other point. The number of places you “rise” is the numerator of the slope. The number of places you “run” is the denominator. In the quiz, remember to run 3 and rise 7.

I’m still a bit confused,return to the example.

I got it!Take the quiz again.

Where you see Slope outside of the Classroom

Engineers & carpenters consider slope when determining the pitch of a roof (or how steep it is). A pitch of 8/12 means that the roof rises 8” for every 12” (or 1’) it runs.

See more examples

rise

8’’

run 12’’

Where you see Slope outside of the Classroom

Ski hills have a variety of slopes. The incline of slopes vary from gentle, like a bunny hill, to steep, like a black diamond.

Congratulations, you have successfully completed Part 1! What is your next step?

Stop (I will complete the next section(s) another time)

Continue to next section(Part 2-solving for y and graphing)

Equations in Slope-Intercept Form

42 += xy24

3−= xy33 +−= xy

The three equations above are all in slope-intercept form.

The slope is the number in front of the x-variable,

and the y-intercept is the number after the x-variable.

slopeslope

slopey-intercept y-intercept y-intercept

Slope-Intercept FormAn equation is written in slope-intecept

form if it is of the form:

y=mx+b• The equation must be solved for y.• m is the slope of the line, we think of slope as a

fraction, .• The y-intercept is represented by the variable, b.

◦ b can be positive (y=mx+b), negative (y=mx-b or y=mx+-b), or zero (y=mx)

• The y-intercept is the point (0,b), where the graph of the line crosses the y-axis.

run

risem =

Solving for y.If an equation is not in y-intercept form, follow these steps to solve for y:

1. Is y positive?2. Is y by itself?3. Is y on the left side of the equal sign?4. Is the equation exactly in y=mx+b form?

If you answered no to any of the questions, manipulate the equation so the answer becomes yes, and then move on to the next question.

Take a Quiz

Slope-Intercept Form Quiz

Which of the following equations are in slope-intercept form?

a) b)

c) d)

24

5+= xy

27 −−= xy

42 =− xy

03

52 =+− yx

a a and b a and c a and d All of the above

Absolutely!!

Correct, in both equations y is:• positive• by itself• on the left of the equal sign• and, in the form of y=mx+b

27 −−= xy24

5+= xy and

You are correct that is in slope-intercept form. However, this is not the only equation that is. Ask yourself the 4 key questions again:

What other equation(s) can you answer yes to all four questions?

You’re So Close!2

4

5+= xy

1. Is y positive?2. Is y by itself?3. Is y on the left side of the equal sign?4. Is the equation exactly in y=mx+b

form?

Still confused, return to examples & explanation.

I got it!Take the quiz again.

Not Quite

Let’s examine the four key questions again:1. Is y positive?

Yes, y is positive in each equation.

2. Is y by itself?Take a look at each equation again and . Is y all alone

on one side of the equal sign? NO! In which equation is this false?

3. Is y on the left side of the equal sign?Yes, in both equations y is on the left side of the equal sign.

4. Is the equation exactly in y=mx+b form?No, both equations are not in slope-intercept form. Which one isn’t?

24

5+= xy

Still confused, return to examples & explanation.

I got it!Take the quiz again.

42 =− xy

Not Quite

Let’s examine the four key questions again:1. Is y positive?

Yes, y is positive in each equation.

2. Is y by itself?Take a look at each equation again and . Is y all alone

on one side of the equal sign? NO! In which equation is this false?

3. Is y on the left side of the equal sign?Yes, in both equations y is on the left side of the equal sign.

4. Is the equation exactly in y=mx+b form?No, both equations are not in slope-intercept form. Which one isn’t?

24

5+= xy

Still confused, return to examples & explanation.

I got it!Take the quiz again.

03

52 =+− yx

Uh Oh!Although more than one equation is in slope-intercept form, not all of the equations are. Ask yourself the four key questions of determining if an equation is in slope-intercept form:

1. Is y positive?2. Is y by itself?3. Is y on the left side of the equal sign?4. Is the equation exactly in y=mx+b

form?

Still confused, return to examples & explanation.

I got it!Take the quiz again.

Slope-Intercept FormRemember, an equation is written in slope-intecept

form if it is of the form:

y=mx+b• The equation must be solved for y.• m is the slope of the line, we think of slope as a

fraction, .• The y-intercept is represented by the variable, b.

◦ b can be positive (y=mx+b), negative (y=mx-b or y=mx+-b), or zero (y=mx)

• The y-intercept is the point (0,b), where the graph of the line crosses the y-axis.

run

risem =

Graphing Lines from equations in Slope-Intercept Form

Once an equation is in slope-intercept form, graphing the line is a breeze!

y=mx+b

First, use the y-intercept to plot the point (0,b). This shows where the line crosses the y-axis (where the x-value is zero).

Next, use the slope to “rise and run” to another point on the graph.

See anexample

Example of Graphing

1. Identify the y-intercept.

The number after the x is +3, so the y-intercept is a positive 3. The graph intersects the y-axis at (0,3).

2. Identify the slope and think of it as a fraction.

The number before the x is -3, so the slope of the line is -3. We think of it as:

1

3−

33 +−= xy

3. Plot the y-intercept, then use the slope to find another point.

Plot (0,3) on the graph. Then, from (0,3) rise -3 (go down 3) and run 1 (go right 1). Next, connect the points to draw a line.

Take a Quiz

Graph QuizWhich of the following is the correct process for

graphing the equation ?24

3−= xy

Oh So Smart!

Absolutely! You chose the correct sequence of steps to plot the line .

1. First, plot the y-intercept (0,-2)

2. Second, use the slope to “rise 3 and run 4”

3. Finally, connect the points with a line. (Remember a line extends forever in both directions, so arrow heads are required!)

24

3−= xy

Not Quite!

It is very tempting to start at the origin (0,0) and plot the slope from there. But remember, the origin isn’t always a point on a graph. In this case, the line does not pass through the origin so you cannot use it as your starting point! The y-intercept (0,-2) should be your starting point.

Still confused, return to examples & explanation.

I got it!Take the quiz again.

Congratulations, you have successfully completed Part 2! What is your next step?

Stop (I will complete the next section(s) another time)

Continue to next section(Part 3- finding the equation from the graph)

Review the previous section(Part 1- calculating slope between two points)

ReviewSo far you have learned…

• how to calculate the slope between two points using .

• that the y-intercept is the point, (0,b) where a graph of a line crosses the y-axis.

• using the slope to “rise & run” from the y-intercept will give another point on the graph, and by connecting the two points the line can be drawn.

12

12

xx

yym

−−

=

Determining the Equation from the Graph of the Line

Now, using what you have learned previously, you will learn how to determine an equation from a graph.

Finding the Equation in Slope-Intercept Form

1. Identify the two points plotted on the graph.

(-4,-1) and (2,8)

2. Using the formula from Part 1-Calculating Slope from Two Points, find the slope of the line.

3. Look at the graph to determine where the line intersects the y-axis.

The line & y-axis intersect at (0,5)

4. Substitute into the slope-intercept formula

42

18

++

=m12

12

xx

yym

−−

=42

18

−−−−

=m6

9=m

2

3=m

52

3+= xybmxy += Take a Quiz

Equation Quiz

Which equation matches the graph of the line:

53 += xy 23 −−= xy

12

1+−= xy12 +−= xy

You’re Great!

Yes, the two points plotted on the graph are (-2,2) and (6, -2). By calculating the slope, you find:

and the y-intercept is (0,1). So, by substituting the slope and y-intercept, you find the equation:

12

12

xx

yym

−−

=26

22

−−−−

=m8

4−=m

2

1−=m

12

1+−= xy

Uh Oh!

Review the steps necessary to find the equation of the line:

1. Identify the two points plotted on the graph.

2. Using the formula from Part 1-Calculating Slope from Two Points, find the slope of the line.

3. Look at the graph to determine where the line intersects the y-axis.

4. Substitute into the slope-intercept formula

Still confused, return to examples & explanation.

I got it!Take the quiz again.

Congratulations!You have successfully completed this section. Think back to the barn

example. Let’s determine the equation for the slope of the left pitch of the barn roof, assuming the peak of the barn roof is the point (0,0). Since the roof peaks at (0,0) we can substitute the y-intercept is y=mx+0. From the peak, for every 8” down (rise of -8), there is a run of 12” back (run -12). Let’s substitute the slope:

Now simplify the equation: 012

8+= xy 0

3

2+= xy

012

8+

−−

= xy

xy3

2=

xy3

2=

x-axis

y-axis

Congratulations, you have successfully completed Part 3! What is your next step?

Stop-I’m the Slope Master! (I have completed all three sections)

Review the previous section(Part 2- solving for y and graphing the line)

Review the first section(Part 1- calculating slope between two points)