What activities can my algebra class do to help understand different types of functions?
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dxs (
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October 11th, 2016
I’m helping with a college-level algebra class, and our class is set up around functions. We use them to introduce different algebraic concepts. For example, this week we’re looking at linear functions, which required us to get a better understanding of graphs and tables. Next week we’re looking at quadratic functions, which will require us to factor (i.e. where does the graph intersect the x-axis?). Get the picture?
I want them to be able to understand how the functions work, though. For linear functions, we had a “race” where everyone had a difference pace (represented by slope), and some people began early or started later (y-intercept), etc.
>>>What kind of activity can we do so that they understand what quadratic functions are like? Exponential? Inverse? Square root? Logarithmic? Absolute Value?
Humorous answers welcome. They can spark ideas.
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Quadratic Drop something off the building and time it. S = ½ * a * t^2. Calculate the height of the building from the time. Or do the opposite. Drop a water balloon and predict the splat time.
Exponential… Zeno’s Paradox. Walk halfway across the room, then half way then half….
Oooo! The Fibonacci numbers and the ratio of the two neighboring elements. When n=infinity the ratio is phi 1.61803… The Golden ratio. It shows up in nature, (pineapple, daisy, head hair…) and architecture Calculate it using different methods.
snaz
I like @LuckyGuy‘s choices. My own is less inspired: use side elevation photographs of various suspension bridges transposed onto graph paper (or a projected X-Y axis with some kind of scale) and back-figure the equations that created each particular parabola.
Interesting ideas. Thanks!
@CWOTUS‘s idea can be expanded upon to show something about parabolas that most people do not know. Just like circles and squares, all parabolas have the same shape, though they differ in size. If you take pictures of the same bridge at different elevations, the parabolas will appear narrower. All that is being done is changing the scale.
Here is the mathematics behind what I am saying, though it may be a little advanced for the algebra class.
Suppose the parabola has coordinates (x,y) on the photograph and is part of the parabola y= x^2. At a higher elevation, the same part of the parabola may be (x’, y’) = (x/10, y/10) on the photograph.
y = 10y’ and x=10x’. Substituting into y= x2 we get 10y’ = (10x’)2. 10y’= 100(x’)2, y’=10(x’)2, which is a narrower parabola than y=x^2. You would get similar results if you changed the unit of measure from millimeters to centimeters.
You can pour boiling water into a mug and plot the temperature as it decays over time. You can solve for it and show how it decays. Exponential decay
You could start to illustrate for them the old story about a poor adviser to a Chinese emperor who asked to be paid only in rice placed on a checkerboard. He agreed to work for the emperor for sixty-four days, just over two months. He had the emperor agree to pay him only one grain of rice on the first square of the board for his first day’s work, and then each day he was to double what was on the previous day’s square.
Have the students work out:
A. How much rice was paid out on a given day of the month, and
B. What’s the formula to figure that out, and
C. Provide an estimate of the total payments to the nearest ton, and
D. For extra credit – because of the season – have them write a short essay about how the emperor solved his debt crisis.
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