Hitting the Slopes
Are you ready to hit the slopes?
OBJECTIVE: To understand the concept of the slope of a line and the math behind it
When you look at the graphs of linear equations, you'll observe that the lines slant in different directions. The slant may be upward from left to right, downward from left to right, perfectly horizontal or perfectly vertical. This slant is called the slope of the line.
THE FOUR TYPES OF SLOPES
Move one or both dots in the applet below to discover the four types of slopes.
Now that you've discovered the four types of slopes, can you match the type of slope with the graph?
For each new line shown, choose the correct type of slope from the four choices at the right.
Repeat at least 10 times.
Now let's do more than just clicking the mouse!
The next activity asks you to match the given type of slope with an appropriate graph. Are you up to the challenge?
Repeat at least 10 times.
A common mnemonic to remember the concept of the slope of a line is
RISE OVER RUN.
Use the slider at the bottom of the next applet to dynamically illustrate how 'rise over run' works.
The ramp shown below is an example of a simple machine called an inclined plane.
Move the white dots to come up with a ramp design that is not only aesthetic, but also efficient.
QUESTION: How do you think is the concept of the slope useful in designing such a simple machine as a ramp? Briefly explain.
Now it's time to introduce the Math behind the 'rise over run' concept.
Mathematically, 'rise over run' refers to the ratio of change in y to change in x. In symbols,
where m = slope and (x₁, y₁) and (x₂, y₂) are the coordinates of two points.
The applet below gives a value of the slope at the upper left-hand corner. Your task is to move the dots in such a way that the slope of the line passing through the two dots matches the given slope.
Repeat at least 10 times.
TIP : Keep the lower dot fixed. Then move the upper dot to the position of the lower dot and apply 'rise over run.'
where m = slope and (x₁, y₁) and (x₂, y₂) are the coordinates of two points.
The applet below gives a value of the slope at the upper left-hand corner. Your task is to move the dots in such a way that the slope of the line passing through the two dots matches the given slope.
Repeat at least 10 times.
TIP : Keep the lower dot fixed. Then move the upper dot to the position of the lower dot and apply 'rise over run.' Let's now reverse the process!
From the graph, solve for the slope using the formula:
.
Repeat at least 10 times.
.
Repeat at least 10 times.Ready for a steeper challenge!
No more graphs, just the coordinates of two points.
It's the same banana, guys! Just different varieties.
Repeat at least 10 times.
Finally, let's go for the cliffhanger!
On the upper left-hand corner of the applet below, you'll find a linear equation in standard form: Ax + By = C.
Your task is to find the slope from the given form: m = -A/B.
Just that! We'll skip the rest of the activities for a future trip to the slopes.
Repeat at least 10 times.