Basic structure
We demonstrate how structured derivations works through some examples. Along the way we add useful features to the examples by using the structured derivation method.
The assignment is to solve equation \(3x+7=15-2x\). We can manipulate the equation by adding, subtracting, multiplying or dividing both sides of equation with same terms. The goal is to get the unknown variable (for example \(x\)) on one side of the equation and its value on the other side.
You might have used notation like this:
\(3x+7=15-2x \) | \( -7\)
\(3x=8-2x\) | \(+2x\)
\(5x=8\) | / \(5\)
\(x=\frac{8}{5}\)
Is this familiar? If we write the same example as structured derivation it would look like following:
\(\bullet\) | \(3x + 7 = 15 -2x\) |
\(\Leftrightarrow\) | {Subtract 7 from both sides of the equation} |
\(3x=15-2x-7\) | |
\(\Leftrightarrow\) | {Add \(2x\) to both sides of the equation} |
\(3x + 2x = 15 -7\) | |
\(\Leftrightarrow\) | {Calculations} |
\(5x=8\) | |
\(\Leftrightarrow\) | {Divide both sides of the equation by 5} |
\(x=\frac{8}{5}\) |
This solution includes the same steps as the first version, but in this version we have more text which explains the solution and some new symbols.
What are the basic elements of structured derivation?
The beginning of the derivation is marked by a bullet (\(\bullet\)), which is followed by steps. Every step includes three parts: terms, relation and motivation.
\(3x = 15 - 2x + 7\) | Term | |
\(\Leftrightarrow\) | {Add \(2x\) to both sides of the equation} | Relation and motivation |
\(3x + 2x = 15 - 7\) | Term |
Terms are mathematical expressions. The relation (in this case an equivalence \(\Leftrightarrow\)) tells how the terms are related to each other. The motivation explains (or justifies) why this relation holds between the first term and the second one.
Each term and motivation is written on its own line so that there is enough room for proper explanations and even longer terms. Derivations are written in two columns to make them easier to read, write and understand. Relations (here \(\Leftrightarrow\)) are written in the first column. Terms and motivations are written in the second column.
So the step above tells that expression \(3x=15-2x+7\) is equivalent (\(\Leftrightarrow\)) to the expression \(3x+2x=15-7\) since we have added the same expression to both sides of the first term. The whole solution is constructed by adding more steps with a motivation for each step.