Hitting the Slopes

1
Hitting the Slopes

• An adventure on the bunny hill
• of linear equations!

Start
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What would you like to learn about?
1 Calculating the slope of the line, given two
points
2 Solving for y (slope-intercept form) and
graphing the line
3 Determining the equation of the line from the
graph
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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.

Look at an example
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Example of finding slope.

• Find the slope of the line between the points
  (0,6) and (5,10).
• Remember the formula.
• Identify which numbers represent the given
  variables.
• Substitute into the formula and simplify.
• The slope of the line extending through
• the points (0,6) and (5,10) is .

Take a Quiz
5
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
6
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.

Try calculating slope
7
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.
8
Oops!

• You are close, but lets 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, its important to rise first and then run!

Still a bit confused, take me back to the
definition.
I got it! Take the quiz Again.
9
Calculating Slope

• What is the slope of the line
• between (5,2) and (10,1)?

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

• Yes, the correct slope is .
• In this case, the slope was in fraction form,
  but sometimes this wont always be the case.
• If, for example, the slope of a line was 4. We
  should think of it as a fraction

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Uh Oh!

• Lets recall the steps to finding slope
• Remember the formula.
• Identify which numbers represent the given
  variables.
• Finally, substitute the numbers into the formula
  and simplify!

Still confused, take me back to the beginning.
I got it! Return to the quiz.
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The Graphs of Slopes

• Lets 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.

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Plotting Points Graphing Lines

• In order to graph the lines, we must first plot
  each of the points.
• Then draw a line through the points, adding
  arrows at each end to represent a line (rather
  than a line segment).
• 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.

run 5

rise 4
run 1
run -5
rise 1
rise 4
Take a quiz
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Graph Quiz

• Given the slope, click on the correct graph

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

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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.

Im still a bit confused, return to the example.
I got it! Take the quiz again.
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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
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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.

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• 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)
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Equations in Slope-Intercept Form

y-intercept
slope
slope
y-intercept
y-intercept
slope
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.
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Slope-Intercept Form

• An equation is written in slope-intecept
• form if it is of the form
• ymxb
• 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 (ymxb), negative (ymx-b or
  ymx-b), or zero (ymx)
• The y-intercept is the point (0,b), where the
  graph of the line crosses the y-axis.

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Solving for y.

• If an equation is not in y-intercept form,
  follow these steps to solve for y
• Is y positive?
• Is y by itself?
• Is y on the left side of the equal sign?
• Is the equation exactly in ymxb 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
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Slope-Intercept Form Quiz

• Which of the following equations are in
  slope-intercept form?
• a) b)
• c) d)

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

• Correct, in both equations y is
• positive
• by itself
• on the left of the equal sign
• and, in the form of ymxb

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Youre So Close!

• 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?

• Is y positive?
• Is y by itself?
• Is y on the left side of the equal sign?
• Is the equation exactly in ymxb form?

Still confused, return to examples
explanation.
I got it! Take the quiz again.
26
Not Quite

• Lets examine the four key questions again
• Is y positive?
• Yes, y is positive in each equation.
• 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?
• Is y on the left side of the equal sign?
• Yes, in both equations y is on the left side of
  the equal sign.
• Is the equation exactly in ymxb form?
• No, both equations are not in slope-intercept
  form. Which one isnt?

Still confused, return to examples
explanation.
I got it! Take the quiz again.
27
Not Quite

• Lets examine the four key questions again
• Is y positive?
• Yes, y is positive in each equation.
• 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?
• Is y on the left side of the equal sign?
• Yes, in both equations y is on the left side of
  the equal sign.
• Is the equation exactly in ymxb form?
• No, both equations are not in slope-intercept
  form. Which one isnt?

Still confused, return to examples
explanation.
I got it! Take the quiz again.
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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

• Is y positive?
• Is y by itself?
• Is y on the left side of the equal sign?
• Is the equation exactly in ymxb form?

Still confused, return to examples
explanation.
I got it! Take the quiz again.
29
Slope-Intercept Form

• Remember, an equation is written in
  slope-intecept
• form if it is of the form
• ymxb
• 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 (ymxb), negative (ymx-b or
  ymx-b), or zero (ymx)
• The y-intercept is the point (0,b), where the
  graph of the line crosses the y-axis.

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Graphing Lines from equations in Slope-Intercept
Form

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

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 an example
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Example of Graphing

• 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).

• 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.

• 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

Take a Quiz
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Graph Quiz

• Which of the following is the correct process
  for graphing the equation ?

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Oh So Smart!

• Absolutely! You chose the correct sequence of
  steps to plot the line .
• First, plot the y-intercept (0,-2)
• Second, use the slope to rise 3 and run 4
• Finally, connect the points with a line.
  (Remember a line extends forever in both
  directions, so arrow heads are required!)

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Not Quite!

• It is very tempting to start at the origin (0,0)
  and plot the slope from there. But remember, the
  origin isnt 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.
35

• 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)
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Review

• So 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.

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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.

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Finding the Equation in Slope-Intercept Form

• Identify the two points plotted on the graph.
• (-4,-1) and (2,8)
• Using the formula from Part 1-Calculating Slope
  from Two Points, find the slope of the line.
• Look at the graph to determine where the line
  intersects the y-axis.
• The line y-axis intersect at (0,5)
• Substitute into the slope-intercept formula

Take a Quiz
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Equation Quiz

• Which equation matches the graph of the line

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Youre 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

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Uh Oh!

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

• Identify the two points plotted on the graph.
• Using the formula from Part 1-Calculating Slope
  from Two Points, find the slope of the line.
• Look at the graph to determine where the line
  intersects the y-axis.
• Substitute into the slope-intercept formula

Still confused, return to examples
explanation.
I got it! Take the quiz again.
42
Congratulations!

• You have successfully completed this section.
  Think back to the barn example. Lets 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 ymx0. From
  the peak, for every 8 down (rise of -8), there
  is a run of 12 back (run -12). Lets substitute
  the slope
• Now simplify the equation

x-axis
y-axis
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• Congratulations, you have successfully completed
  Part 3! What is your next step?

Stop-Im 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)