Slope

Slope

Slope is where arithmetic meets geometry. We know that when we count in 8’s, we get higher faster than if we count in 3’s.

A line that goes up 8 units for every one horizontal unit gets higher faster than the line that goes up 3 units for every one unit across.

Slope is a number that represents how quickly a line segment goes up a grid. Slope ‘8’ means that it goes up in 8’s. The grid may or may not be drawn. What matters are the coordinates.

We can easily see that some line segments are steeper than others. Here’s some slope projects to tune in to slope!

Classroom stations: Which of these quadrilaterals are parallelograms? Justify your answer.  Identify the Parallelograms

You can read all about slope on mathisfun here or watch this introduction to slope here.

Move the line segment to create a certain slope (Skill #1)

Use the applet to create line segments of the following slopes:

  1. Slope AB = 5
  2. SlopeAB = -3
  3. Slope AB = 0
  4. Slope AB = 1
  5. Slope AB = -1
  6. Infinitely steep
  7. Slope AB = \frac{5}{2}
  8. Slope AB = -\frac{1}{4}

Find the slope of given line segments

https://www.mathsisfun.com/geometry/slope.html

A slope of 2 means, 2 up for one across. This is the same ratio as 10 up for 5 across. We calculate slope by dividing rise over run. If it goes down (left to right), the slope is negative.

On this applet, if the slope is not an integer value, enter your calculation as a fraction, eg 11/3.  Remember to include ‘-‘ for a negative slope.

 

Use Slope and y-intercept

On the ‘appliance repair‘ problem in unit 5, we saw that the cost of the repair technician was 60 dollars per hour, plus a 70 dollar call out fee.

We start at 70 dollars for the call out fee, and add 60 dollars for every hour.

Therefore, for 5 hours, we have 70 + 60 \times 5 = 370.

Notice we could say, 60 \times 5 + 70.

That is, 60 \times \text{ hours}+70.

Or, let’s write y for the cost, and x for the hours.

    \[y=60x+70\]

Every linear equation has this form. A ‘starting number’ and a ‘rate’ that the y values go up in. The ‘rate’ is the slope.

https://www.mathsisfun.com/equation_of_line.html

In unit 5, you were asked to figure out this formula by looking for the rule between the x coordinate and the y coordinate.

On this applet, we see the points plotted as a line. The slope m is given, and the coordinates of the y intercept are marked.

Enter the equation of the line. If correct, you will see your green line match the given blue line.

Calculating Slope using the Coordinates (Skill #2)

The formula for finding the slope of a straight line between (or through) two points is:

    \[m=\frac{\text{rise}}{\text{run}}=\frac{y_2-y_1}{x_2-x_1}\]

One point has coordinates (x_1,y_1), the other has coordinates (x_2,y_2). It doesn’t matter which one is which, as long as whichever point is first on the top line is also first on the bottom line.

On the applet below you can click ‘show triangle’.


Unit 6 Menu

Next Page: Linear Functions using two points or slope and point