By: Brooke Norman
Day 4
Slope
Objectives:
1- To learn what a slope of a line is.
2- To calculate the slope of a line.
3- To make a distinction between lines with
positive, negative, undefined, or zero slopes.
1-Many students will just
stare at you when they are asked about the slope of a line. Most of them know exactly what it is,
but have never thought about what it is.
Some examples they know of are a ramp for bike tricks, a hill, and even
the incline of the back of a chair.
A good way to begin an introduction to slope would be to use something
they are familiar with and relate it back to slope. LetÕs use the slope of a ramp. ItÕs easy to draw and see. Provide the students with a drawing or figure so they can
visualize exactly what you are talking about.
Above, I have constructed a
ramp with sides of 6 feet and 3 feet.
It has a constant incline, also known as slope. After you have shown the students an
example, see if they can think of more examples.
2- After you have taken some time to discuss what slope
is in general, move on to a more mathematical way of thinking of slope. Ask the students if they can figure out
the slope of the incline in the above example. Begin by letting them see that as you move to the right 6
feet, you are also going up 3 feet.
Ask them to now make this into a ratio, and you will get 3/6 or
1/2. Make sure they understand
this reasoning. Does it make
sense? Yes, because for every one
foot the ramp goes up, it goes over 2 feet. Teach them the ÒtrickÓ of thinking rise over run (rYse/run). This means that they Y-axis or the
height of the ramp will be on the numerator and the run or the X-axis goes on
the denominator. LetÕs look at
another example now. Have the
students look at the following graph and ask them how they would go about
finding the slope to this line.
Explain that slope is the change of YÕs divided by the change in XÕs. It is also stated as DY/DX (delta y / delta x). Explain that finding the change means that you find the coordinate pair for two different points. Take point 1 and point 2 for example. Point 1 consists of (X1, Y1) and point 2 consists of (X2, Y2). The change in the slope of the two points is (Y2-Y1)/ (X2-X1). Let the students know that it does not matter which point is point 1 and which one is point 2, just to make sure to not get them confused when transferring the numbers into the equation. From the above graph, ask the students to pick two points. For example, points (1, 4) and (3, 8). Ask them to calculate the slope using the formula from above. Have them label their new points to make sure everyone sees just what is going on.
X1=1
Y1=4
X2=
3
Y2=8
(8-4)
/ (3-1) = 4/2 = 2.
This
tells us that the slope of the line from the graph above is 2.
3- The next step is to teach the students the direction
of slope. In the first two
examples, the slopes were both positive. Explain that slopes are not always positive, that they
can be negative, zero or even undefined.
Tell the students to think that if they start at a point to the left and
have to Òwalk downhillÓ to get to the next point to the right, then that is a
negative slope. If they start at
the point on the left and Òwalk uphillÓ to get to the next point to the right,
then that is a positive slope.
Here is an example of a positive slope and a negative slope.
Positive Slope Negative
Slope
What exactly is a zero
slope? Explain that any line that
is horizontal has a slope of zero.
Show that it runs parallel to the x-axis and perpendicular to the
y-axis. Use the rise over run
method to show it mathematically.
Explain that the rise is 0 because it is not going up or down. Show that the run is 8. Then apply the formula 0/8 = 0.
Zero slope
What happens if the line is
a vertical line? The slope is said
to be undefined. This is because
the rise will be a number, in this case 6 and the run will be 0 because it is
not going to the right or left. This gives 6/0= undefined. We know that when we
are talking in ratios or fractions that if there is a 0 in the denominator,
then the answer is undefined.
Undefined slope
If your classroom is set up
and GSP is available, it would be a good time to demonstrate this concept. You can plot points and show the
slopes and then change the points around and have the students find the
slopes. Start by opening GSP and go
to graph and show grid. The plot
any two points you want. Construct
a line between the two points.
Highlight the line and go to measure slope. The measurement of the slope should appear and then you can
move one of the points and watch the slope change. You can also move the points around and have the students
find the different slopes. Just
make sure you hide it. When asking
for the answers, just go unhide the measurement. This can be a fun and useful tool to use in the classroom.
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