Slope

In this exploration you will review how to calculate the slopes of lines in the coordinate plane. This exploration will reinforce the concepts in Lesson 0.1 of Discovering Advanced Algebra: An Investigative Approach.

Sketch

This sketch shows the points (x1, y1) and (x2, y2) on the line through points A and B. The subscripts mean that these are two distinct points of the form (x, y).

You can move (x1, y1) and (x2, y2) along the line, or you can drag points A and B to change the line itself. Notice that the coordinates of (x1, y1) and (x2, y2) are shown and are updated dynamically if you change the positions of the points.

Recall that the slope of a line is defined as change in <i>y</i> divided by change in <i>x</i>. You can understand this definition by using a slope triangle. Press Slope Triangle. The change in y is the length of the leg marked vertical change, given by y2 – y1. The change in x is the length of the leg marked horizontal change, given by x2 – x1.

You can display the value of the slope by pressing Slope. Press Reset Slope Triangle to move points (x1, y1) and (x2, y2) back to between points A and B. You may have to press this button twice. Press Start Over to move points A and B to their starting position.

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Investigate

  1. Drag point A so it has coordinates (0, 4), and drag point B so it has coordinates (5, 2). How can you compute the slope of the resulting line? Press Slope to check your answer.
  2. Press Start Over. Drag point A so it has coordinates (0, 1) and drag point B so it has coordinates (2, 4). How can you compute the slope of the resulting line? Press Slope to check your answer.
  3. Adjust points A and B so that the line passes through the point (4, 7) and has slope 0.6. What are three more points on the line?
  4. Adjust the points so that the sketch can help you solve these proportion problems:

    a. 12/16 = 9/a

    b. 12/16 = b/10

    How can the slope triangle help you to understand the solutions to these problems?
  5. You can use this sketch to study the connection between inches and centimeters. Recall that 1 inch is equal to 2.54 centimeters.

    a. Adjust points A and B so that the graph can be used to convert inches to centimeters.

    b. Which axis represents inches and which represents centimeters?

    c. 8 inches =   ?  centimeters

    d. 3.5 inches =   ?  centimeters

    e. 4 centimeters =   ?  inches

    f. 10 centimeters =   ?  inches

    g. How does the slope of this line relate to the questions you just answered?