Introduction to Stick Diagrams
Our aim in introducing stick diagrams is to show how to understand and
solve equations for unknown length or number x. In the first example
below we will go from a given value of x to a stick diagram constructed or
implied by that value of x. Then we will explore a few simple examples
where we find the value of x from a stick diagram. This is the simple
and slow start of a longer story. So follow the story or chains of reason here
because at the end of this story, I or we hope you will be able to solve
equations in one and more unknowns, and beyond that, you will have a path for
tutoring or leading others to this ability. So have patience.
Step 1. From value of x to one or more stick diagrams.
Step 2 below, going the other way, is more important.
Suppose x represents the length of a stick of length 15. We will
write x = 15. We draw two parallel sticks to represent this situation in a
geometric manner.
The equation x = 15 implies x +5 = 20. We can draw the following to
represent
this equation or situation x + 5 = 20.
The foregoing shows how we can go from a value of x to an equation in x
and a stick diagram which describes that equation. That is step 1 in
our journey.
Step 2. From a Stick Diagram to a value of x, or finding an unknown length
from a stick diagram.
Imagine we are given x + 4 = 16 for an unknown length x. We want to
find x. To that we describe the equation by the stick diagram.
This diagrams indicates a stick of length x + 4 has the same length as a
stick of length16. Now observe 16 -4 = 12 or 12 + 4 =
16.
x
|
4
|
16
|
| <---
12 ---> |
<--- 4 --> |
We are almost done. Now shorten the top and bottom sticks by 4
units. This cutting off or subtraction of 4 units gives the diagram
and suggests x = 12. Let us check this suggestion.
Check: When x = 12 (or when x is replaced by 12)
x + 4 = 12 + 4 = 16
as requested.
So x = 12 is a solution of x + 4 = 16.
Another Examples of Step 2. Finding an unknown length from a stick
diagram.
Suppose x is a length which satisfies solve x + 8 = 28.
Find the length x with the aid of stick diagrams.
| Stick Diagram |
Operation |
Equation |
|
|
Given |
x +9 = 29 |
|
|
Preparation
for a subtraction
|
observe
29 - 9 = 20
or
29 = 20 + 9
|
|
|
Subtract 9 |
x + 9 = 29
9 = 9 _
x = 20 |
|
Conclusion: Must have x = 20 |
If I was solving 2x + 5 = 20 in class, I would just fill in the
table and skip the work before it. Each table consists of a diagram in
the left column, a description of what is done or given in the middle column,
and the equivalent equations in the rightmost column. At the moment, you
are required to draw the stick diagram in the solution of the equation. That
is a crutch. Later on, only the equation column is required with a few
words to explain the operations.
Let us check this suggestion.
Check: When x = 20 (or when x is replaced by 20)
x +9 = 20 + 9 = 29
So the conclusion x = 20 works.