Functions: Variant Notations
In this series, we’ve done a review of mathematical notation, with an eye on how each notation helps or hinders student learning. In the prior posts, we started to explore function notation, which in middle school and high school shows up as , and examined its use. In this post, we will continue the thread of looking for notations that have the power of lambda calculus but are suited to middle school use. Our goal here is to come up with a notational scheme for functions that is suitable for composition and that can gradually morph into the standard notation students will encounter in text books in later grades. This exercise is motivated by the hunch that a light-weight notation for functions could usefully inform the way that expression, equations and variables are viewed and understood.
Basic Functions: functions of the kind shown below.
These correspond to the basic arithmetic operations from elementary school, in a special way. Adding a number, subtracting a number, multiplying by a certain number, and dividing by a particular number.
Linear Functions: The composition of one or more basic function results in a linear function. For example:
It may not be obvious that all of these are linear functions, though the third one should look the most familiar. It takes the number coming in, multiplies it by 2 and then adds 3. This one corresponds directly with the y= m x + b form of the linear function that students learn in middle school. Here, the slope is 2 (that is, positive two) and the y-intercept is 3 (positive 3). The second one, first subtracting 5 and then multiplying by 2, corresponds to the x-intercept form of the linear function – not always taught in middle school – it’s the line through (5,0) with a slope 0f 2. In contrast, the third one shows the line through (0,3) with a slope of 2. In the figure below, we show more linear functions:
On the left, we have a function that corresponds to the point-slop form of the line; here it is the line that goes through the point (5,3) with a slope of 2. In general, the composition of any two linear functions is itself linear. The second function (multiply by 2 and then add 3), composited with the third function (multiply by 5 and then add 1) gives rise to the fourth function (multiply by 2 and then add3, then multiply by 5 and then add 1), and this is indeed linear. It could be re-written equivalently as a multiplication by 10 followed by an addition of 16. (Note that I’m not slowing down this account by substantiating my claims here – I’m just playing with a notation and showing some of its power here.)
Polynomials: In standard notation, a polynomial in x may look as follows: .
This could be shown as:
Each of the vertical constructions that feed into the +-bar is one of the terms (nomials) of the polynomial. Each of these terms shares an input, and this input corresponds to the “x” in the standard notation. The boxes like ” ^ 5″ show exponentiation, the boxes like ” * 2 ” show multiplication.
There is another way to show polynomials, named after William George Horner, which sidesteps exponentiation altogether. Below is the picture for :
In this rendition, you see each of the coefficients used as one input into a section of two boxes. This section multiplies and then adds. Each section gets its other inputs from the previous section, and from the input to the entire box. Once you see how this works, you can see that it will work for polynomials of any degree and with any coefficients. There is a patterned-ness to this that, though strange at first, becomes easy and powerful after a while.
More on this in the next post.