Elementary representations of functions


Analyzing and representing patterns and functions is one of the building blocks of algebra. The functions we will be working with for now are sequences: where there is one picture and number of pieces at the first step, a related picture at the second step, and so on.  Our first goal is to represent these patterns using pictures, tables, graphs and equations.  Picture representations are the most concrete of these, and equations are the most abstract.

Example 1: Twos Tower (all examples taken from the 5th grade text: Investigations in Number, Data and Space, book: Patterns of Change)

Here is a pattern made of colored tiles: the red tiles show the first step, then the blue tiles are added on, then the green tiles.
two red tiles, surmounted by 2 blue tiles, surmounted by 2 green tiles

Representing this with a picture, we could draw the colored picture above, or a set of pictures like this:
a 2x1, a 2x2 and a 2x3 set of tiles
The mathematical information we will analyze about these patterns is the number of tiles used at each step.  We make a table showing both the number of new tiles (the step size) and the number of tiles so far:

Step Number
 Number of new tiles (step size)
 Number of tiles total so far
1
 2
 2
2
 2
 4
3
 2
 6
4
 2
 7
5
 2
 10

The next step in abstraction in understanding the functions is to make a line graph using the information from the table.  Both of the number-of-tiles columns can be seen as a pattern or function, where the step number is the input variable, and the number of tiles is the output variable. Here are graphs of  both step size and total number of tiles as functions of the step number:
step size graph    total tiles graph
One thing we can recognize from the graph is that the total number of tiles goes up in a straight line, which happens because the step size is always the same (so it goes up by the same amount for each step).

We can connect each of these patterns to an equation.  The step size equation is just: step size = 2.  That may look a little strange to you, but it is fine for an equation to be just a number.

The equation for the total number of tiles is Total =2n.  This makes sense with the table (even numbers) and graph (step size and slope), but the best way to explain it is to look at the picture, and think of the array diagram for multiplication:
2-tower as 2x3 array

Example 2: Squares
Consider the picture showing how to make a bigger square by adding on to a smaller one.  This is an interesting example--one where you can see a pattern in the step size too  (the new tiles, which we are calling step size, are the L's that are being added on)

sequence of nested squares
Table:
Step number
 Number of new tiles (step size)
 Number of tiles total so far
1
 1
 1
2
 3
 4
3
 5
 9
4
 7
 16
5
 9
 25

Graph:
graph of odd numbers     graph of square numbers
Discussion: the step size graph is a straight line.  Its equation is:  step_size = 2n-1 (notice that the step size goes up by 2 each time).

The total tiles graph is curved.  The step size gets larger, and the total tiles graph gets steeper and steeper.  When the step size increases, the total tiles graph isn't straight, and it gets steeper as it goes up.

The equation for total tiles is Total = n2
Notice that all of the total tiles pictures are squares.  That’s why n2 is called “n-squared” because they are all squares:

squares as an array

Example 3: Stairs

a 4-step staircase ,made of square tiles, where each step is a different color


Table:

Step Number
 Number of new tiles (step size)
 Number of tiles total so far
1
 1
 1
2
 2
 3
3
 3
 6
4
 4
 10
5
 5
 15

Graphs:  
Graph of n      graph of total number of tiles

Discussion: the step size graph is a straight line.  Its equation is:  step_size = n (notice that the step size goes up by 1 each time).

The total tiles graph is curved.  The step size gets larger, and the total tiles graph gets steeper and steeper.

To find the eqation, we look at the picture. 
two stairs put together to make a rectangle
If you put two step-4 stairs together, you get a rectangle.  The size of the rectangle is 4x5 or n(n+1): In two step 4 stairs, there are 20 tiles, and in two step n stairs there are n(n+1) tiles. So, in one step 4 stairs there are 20/2 tiles, and in one set of step n stairs there are n(n+1)/2 tiles.

Key ideas: