In this blog, we have talked about mathematical selfies from a
global perspective by comparing what different students have shown and said
about particular topics. We’ve talked about selfies from a meta perspective and
considered selfies as a concept independent of content. For this installment of our continuing story,
we offer an example of one student’s complete submission on the topic of functions
including definitions, domain and range, and linear examples. The student’s submission illustrates her
strong understanding of functions and provides a way to see through her
eyes. Enjoy your excursion into
Caitlin’s view of functions.
These two images illustrate functions. The definition of a function is “a rule that takes… inputs and assigns each… exactly one output. The output is a function of the input” (textbook, pg 2). In my selfie on the left, the length of the handrail (in feet) is a function of the numbers of steps. At stair #1, the handrail is zero feet long. With every stair, the handrail length increases by 1 foot. Therefore a good equation would be: L = s, where L is length in feet and s is the number of steps. In the picture on the right, speed is a function of gas (and therefore pedal pressure). The harder you push down on the pedal, the faster the car goes. A rough estimate would be to say that for every inch you depress the pedal, the car goes 20 mph. Therefore, an equation for this function would be: S = 20p, where S is the speed in mph, and p is the inches the pedal is pressed down.
On the left is my original function selfie. The domain is the limitation of the input, which in this case is the # of stairs. In this image, there are 13 stairs, so the domain would be [D: (0,13)]. The range is the length of the handrail, which, according to the equation, would be the same as the range, so [R: (0,13)]. On the right, the piano’s input are its keys, and the output is the corresponding note. There are 88 keys on a standard piano, so the domain would be [D: (0,88)]. If you look at this function as representing only one key being pressed at a time, the range would be the same as the domain, [R: (0,88)]. However, if you see a chord as being one single note, then you could have any number of combinations. A good estimate might be around 1000, so range would be [R: (0,1000)].
These selfies are examples of vertical and horizontal lines. A vertical line is a line where x = k, with k being a constant, and the slope is undefined (textbook, pg. 40). A horizontal line is a line where y = k, with k being a constant, and the slope is 0. My selfie on the left shows both vertical and horizontal lines. The posts stretching from left to right have a slope of zero (they never increase or decrease). The vertical posts are going from top to bottom, and have an unchanging, undefined slope. On the right is an image also illustrating both horizontal and vertical lines, and the principle is the same. The photo is tilted slightly, which does not give an accurate representation, but the line going from left to right is straight and never changes slope (slope = 0). The vertical lines go from top to bottom and never change either.
These
pictures illustrate increasing lines.
An increasing line is one with a positive slope (textbook, pg. 39). In the left
image, the handrail is increasing linearly, and has a positive slope. We are
able to tell this by the direction the line runs (it increases from left to
right). In the right image, the brick wall is increasing from left to right as
well, and has a positive slope. An equation for either would be y = mx + b,
where m is positive.
These
images show a decreasing line.
Decreasing lines are lines with a negative slope (textbook pg. 39), and
decrease from left to right in a plane. On the right, the right half of the bird
house roof is a decreasing line. It moves downward from left to right. In the
picture on the left, the handrail moves downward from left to right as well.
Both the birdhouse roof and the handrail have negative slopes. If we wrote an
equation, it would look like this: y = mx + b, where m is a negative number.