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.
Representing this
with a picture, we could draw the colored picture above, or a
set of
pictures like this:
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 so far total |
1
|
2
|
2
|
2
|
2
|
4
|
3
|
2
|
6
|
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.
The next step in abstraction is understanding the functions is to
makea line graph using
the information from the table. The
graph on the left shows the graph of the "Number of new tiles"
function, and the graph in the right shows the "number of tiles so far"
function.
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 to an equation:
- The step size equation is
just: step size = 2 (where n is the step number). 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). Some good ways to explain this formula (deductively) are:
- look at the picture, and
think of the array diagram for multiplication: 2 wide, and n high
- look at the picture broken out by colors: n sets of 2
- look at the picture as columns: 2 sets of n
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)
Table:
| step number |
number of new tiles
(step size) |
number of tiles so far total |
1
|
1
|
1
|
2
|
3
|
4
|
3
|
5
|
9
|
4
|
7
|
16
|
Graph:
Discussion and equation:
- 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). Nice ways to
explain this formula deductively/geometrically are:
- Look at the colored L's as two equal arms plus the
corner. For n=1 the arms are 0 long, for n=2 the arms are 1 long.
At step n, each arm is n-1 long, so the total number of tiles is 2
times the number of tiles in each arm, plus 1 for the corner: 2(n-1)+1
(which is equivalen to 2n-1)
- Look at the L's as two arms, where the horizontal arm includes
the corner, and the vertical arm doesnt. Then the horizontal arm
has n tiles, and the vertical arm has n-1 tiles, so the total number of
tiles is n+n-1 (which is equivalent to 2n-1)
- 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
tiles = n2 . To
explain this:
- Notice that all of the total tiles pictures are make a square.
That’s why n2
is called “n-squared” because they are all squares.
- Notice that the square has n rows, each of which have n tiles
in them: nxn = n2 .
- Notice that the square has n columns, each of which have n rows
in them: nxn = n2.
- Notice that a square is an array n columns wide and n rows
high, so it has nxn = n2 tiles.
Example 3: Stairs
Table:
| step number |
number of new tiles
(step size) |
number of tiles so far total |
1
|
1
|
1
|
2
|
2
|
3
|
3
|
3
|
6
|
4
|
4
|
10
|
Graph:
Discussion and equation:
- 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).
- Notice that each in each of the colored sets, the number of
tiles in the column matches the step number.
- 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.
- 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 step
n stairs
there are n(n+1)/2 tiles.
Key ideas:
- Functions can be represented in equations, tables, graphs and
sometimes pictures
- If there is a picture, it can help you figure out what the
equation is
- The step size and the total are related
- The table, graph and equation are related.