Math Misconceptions And Considerations 8.F.4 8.F.5
3 + 4 = 34 Look closely at errors in students’ work (formative assessment) to help you reflect and make instructional decisions to suit all students’ needs.
The slope of a vertical line is undefined. The slope of a horizontal line is zero. Both of these cases might be considered “no slope.” However, if we say something has “no slope” then how do we know if a line is vertical or horizontal? The term, “no slope” is imprecise and unclear and should be avoided. It is better to say the slope is undefined or zero so the line of the graph is clearly described.
MISCONCEPTION:
WHAT TO DO:
y = -2 The slope-intercept equation is y = mx + b. In our equation, y = -2, there doesn’t appear to be the “mx” portion; therefore, the equation does not appear to have a slope. When we graph the equation we have a horizontal line. To find the slope we divide the rise by the run. According to the graph, the rise is zero and the run is some non-zero number. Slope = The three in the equation above can be replaced with any non-zero number. Regardless of what number we place in that spot the slope = 0 because zero divided by any number is zero. This makes sense because the definition of slope is a line’s steepness…a horizontal line doesn’t have any steepness.
x = -2 The slope-intercept equation is y = mx + b. In our equation, x = -2, finding the “mx” part is difficult; therefore, students really aren’t sure if there is a slope or not. When we graph the equation we have a vertical line. To find the slope we divide the rise by the run. According to the graph, the rise is some non-zero number and the run zero. Slope = The three in the equation above can be replaced with any non-zero number. Regardless of what number we place in that place, the slope is undefined because we cannot divide a number by zero.
Some linear functions are proportional relationships while others are not. A linear function in the form of y = mx is proportional; the graph goes through the origin. A linear function in the form of y = mx + b is only proportional exactly at the point when b = 0; y is proportional to x at this exact point.
PROPORTIONAL RELATIONSHIP: This function shows a proportional relationship because y always remains proportionate to x. Each time x increases by one, y decreases by two-thirds. In actuality, this equation could be rewritten as
because the y-intercept is zero.
NOT A PROPORTIONAL RELATIONSHIP: This function does not show a proportional relationship because y is not always proportionate to x. Each time x increases by one, y decreases by two-thirds and then increases by two. This is not proportional because the yintercept is two.