Boolean expressions on the TI-84 for Piecewise functions

Here I will try to explain boolean expressions. Boolean expressions are expressions that evaluate to “true” or “false”. “True” is like a value of 1 and “false” is like a value of 0 (zero).

SCREEN05Suppose you have (x>-1.0881) as your boolean for Y1. If X is greater than -1.0881, the expression can be replaced with a “1”. In turn, 1 \times (1.5x + 3) = 1.5x + 3 , so a value of true causes Y1 to plot.

If X is equal to or less than -1.0881, this same expression can be replaced with a 0. Y_1 does not plot in that case. But since X \leq -1.088 falls in the domain of Y2, then it is Y2 that will plot instead.

Suppose now I have two functions on my graphing calculator:

Y_1 = (1.5x + 3)(x < -1.0881)
Y_2 = (2x^2 - 1)(x \geq -1.0881)

I have written the two booleans in such a way that only one of the statements can be true at a time. Notice that if the boolean for Y1 us true, then the one for Y2 is not. That means for X values less than -1.0881, only Y1 will plot. For all other values, only Y2 will plot, because its boolean will now be true and Y1‘s false. Ultimately, we obtain the graph:

SCREEN03This is produced by a piecewise function which can be expressed using the following standard notation:fxpiece

Others have used different techniques for piecewise functions. One which seems to possess certain advantages would have been to place all details on to Y_1 as the sum of both “pieces” where each “piece” is multiplied by the boolean of the restriction, as follows:

Y_1 = (1.5x + 3)(x < -1.0881) + (2x^2 - 1)(x \geq -1.0881)

While being slightly unwieldy (being sure to run past the end of the display), it saves the user from having to flip between Y_1 and Y_2 using the \uparrow key.

There was also a useful idea where you could disable plotting for Y1 and Y2; and, for a new function Y3, enter: Y1 + Y2 using the VARS key to obtain the Y variables. The result is a single function that does not require flipping between functions with the \uparrow or \downarrow, and proving continuity is much easier. And, the function is more like a piecewise function rather than two separate functions.

Algebra Tiles

I once thought algebra tiles were stupid. But these days, I believe that they are essential for young children up to Grade 10 to understand how the factored quadratic is expanded, and to actually “get” the FOIL method when it’s introduced.

It is especially revealing when you understand the root of the word “quadratic” comes from “quadrat”, which is a device used to measure land area. Imagine buying a tract of land (x – 3) meters long by (x + 4) meters wide. Such a property must be rectangular, and I think for consistency sake, so should your algebra tiles. The placement of the tiles should reflect the dimensions of the rectangle. That is, an observer should be able to make out the factors of the quadratic in the finished product.

I actually don't own any algebra tiles, so I just drew a picture conveying the general idea. The square is the square of x; the lines each represent x, to make 8x; where the lines cross each represent one more number added to the constant term. 15 crosses thus make the number 15. This looks like a great idea for teaching how binomials can be multiplied to kids who haven't seen it before. This graphc was scrawled out in MS-Paint.

If you are like me, you probably don’t own these things, but to teach expansion of binomial factors to kids, you can certainly draw squares and lines.

Multiplying a pair of binomials will generate a quadratic upon expansion, of the form Ax^2 + Bx + C.

Along the horizontal, we can suss out a measurement of x + 5 units in length in the first illustration; along the vertical we see a dimension of x + 3 units. Draw the lines so that they continue past the square and extend so that they cross all of the lines going in the other orientation. Counting the number of crosses will give you the constant term (C); counting the total number of lines will give the coefficient before the x term (B); and counting the number of squares give the coefficient of the x^2 term. Knowing that MS Paintbrush does an ugly job when using the same colour, I used a different color for the horizontal lines. These different colors can become handy when the binomial contains a subtraction. For example:

This is an improved graphic, and does illustrate how the black represents subtraction.

This second illustration shows how -2x may be represented by algebra tiles.  We see the middle term go to 3x - 2x = x. But what about the crosses? We need to make up a rule whereby if the crosses are of different signs (that is, different colors), the count is to a negative number (in this case, six crosses make -6). Conversely, if they are of the same sign (both negative or both positive), the count is to a positive number.