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Common Core Standards: Math

Grade 8

Expressions and Equations 8.EE.B.6

6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Joe Weakleg is in a bicycle race—one of those long, grueling marathons that are on TV for hours and hours on Sunday afternoons. This part of the race is uphill and it seems to Joe that the more he pedals, the steeper the hill is getting.

After the race, he takes a step back. (Well, actually a whole lot of steps.) He stands back and looks at the road from a distance. As it turns out, the road he was on was one long straight line.

Besides learning that he needs to train better before attempting another one of those races, Joe learns that if points are collinear—or on the same straight line—then the slope between different parts of that line is the same. If it's pretty steep at the beginning, then it retains that same "steepness" until the very end.

Students should know that mathematically, this means the line's slope stays the same. They can use similar triangles to solidify their understanding of this concept.

In the diagram above, ΔABE and ΔACD are similar, which means that all their corresponding sides are in proportion. The ratio of CD to BE is the same as that of CA to BA and DA to EA. So while the distance from A to B isn't the same as from A to C, their steepness is the same.

If students need an example, use the triangle's vertices as points. For instance, we can find the slopes of the line segments if A is at (0, 0), B is at (4, 3), and C is at (12, 9). Using the slope formula, we can calculate the slope of AB, AC, and even BC.

So regardless of the distance, all three line segments have the same slope. Again, students should be aware that slope represents the steepness of a line and that negative slopes are possible.

Finally, it's time to introduce them to linear equations in standard form. Students should know that lines can be expressed in the form y = mx where m is the slope because given an x coordinate, all we need to do is multiply it by the slope to find the y coordinate. Once students are comfortable with this, make sure to tweak the formula slightly to get y = mx + b, where b is the y-intercept.

If students get confused, make sure to pedal back to where they were last on track. Once they've flexed and trained their mathematical muscles enough, they should be closer to Lance Armstrong than Joe when it comes to linear equations. Although after a marathon's worth of B.O., they probably don't want to stand too close to either one of them.

 

Drills

  1. Which of the following points is on the line with equation y = 2x?

    Correct Answer:

    (3, 6)

    Answer Explanation:

    When finding the points on a given line, all we need to do is plug the x coordinate into the linear equation and see if the right y pops out (or plug in both y and x values and see if the equation holds up). In this case, the y value is always double the x value. Here, we know that the equations resulting from (A), (C), and (D) aren't true. The only one that works is (B) because 6 = 2(3) is a true statement.


  2. "Collinear" is used to describe which of the following?

    Correct Answer:

    Points that are on the same line

    Answer Explanation:

    "Co" means, "together." "Linear" means, "a straight line." So "collinear" means, "together on a straight line." That's your English lesson for the day. Now back to math.


  3. Points A, B, C, and D are collinear. Which of the following must be true?

    Correct Answer:

    AB and CD have the same slope

    Answer Explanation:

    If the points are collinear, the segments aren't parallel because a line can't be parallel to itself. We can't claim to know anything about congruence or addition of line segments either because they have to do with length, not slope. The only possible answer is (A) because segments on the same line have the same slope.


  4. Why are both y = mx and y = mx + b equations of a line?

    Correct Answer:

    The b represents the y-intercept and can be left out when the y-intercept is zero

    Answer Explanation:

    If students are ever unsure, they can fix the values of m and b and graph lines to observe the difference. While m is the slope, b represents the y-intercept. Only when the y-intercept is zero (meaning that b = 0) can we simplify y = mx + b to y = mx + 0, or more simply, to y = mx.


  5. What is the slope of the line y = 2x – 8?

    Correct Answer:

    2

    Answer Explanation:

    Since the equation is y = mx + b, the slope is m, the number right before the x. In this case, our line has a slope of 2. Answers (C) and (D) are taken from the b value of -8, which is the y-intercept and (A) incorrectly adds a negative sign.


  6. What is the y-intercept of the line y = 10x?

    Correct Answer:

    0

    Answer Explanation:

    If there's no b at the end, it's because the y-intercept is 0. The only way for a line to not have a y-intercept is if the line is perfectly vertical, and such a line would take the form x = c. Changing the equation y = 10x to y = 10x + 0 highlights the fact that b exists, but equals 0.


  7. What is the y-intercept of the line y = 3x – 12?

    Correct Answer:

    -12

    Answer Explanation:

    The y-intercept is the number at the end, with the sign before it. If we take the general form y = mx + b, we can see that b has to be negative for the plus to change into a minus. In other words, b is not 12, but -12.


  8. Which is the equation of the line through the origin that has a slope of ⅘?

    Correct Answer:

    y = ⅘x

    Answer Explanation:

    A line going through the origin means that we have the point (0, 0) as part of the line. We can plug this into the different equations to check, or just realize that the origin means a y-intercept of 0, which changes y = mx + b to y = mx + 0 = mx. Answers other than (D) all have y-intercepts other than 0.


  9. What is the equation of the line with a slope of -6 and y-intercept of 2?

    Correct Answer:

    y = -6x + 2

    Answer Explanation:

    In the equation y = mx + b, m is the slope and b is the y-intercept. If we substitute m = -6 and b = 2, we'll have y = -6x + 2, which is the right answer. The other options either switch the positions of m and b or use the wrong signs.


  10. Which is the equation of the line that goes through (0, 0) and (4, 2)?

    Correct Answer:

    y = ½x

    Answer Explanation:

    The line moves up 2 for every 4 over, so the slope is ½. Since it goes through the origin, its y-intercept is zero. If we substitute these values into the standard form, we should end up with (D) as the answer. All other options come from switching the x and y coordinates or using a y-intercept other than 0.


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