Equations for Decision-Making

You have likely used equations as informal mental models for decisions.

We are going to make that process more formal by writing it down using mathematical notation.

Question Posing

An equation is a good way to pose a question.

We use equations when we have two different ways to express the same quantity. When we set them equal to each other we can learn about the relationships between the variables.

Formula vs Equation

We often say equation when we mean a formula or expression.

This is a subtle difference, a formula often means to use the variables on one side to calculate a resulting value.

The equation (in this context) expresses the same thing two different ways and then uses algebra to gain some insight.

Formulas

Formulas allow us to insert quantities to calculate other quantities.

  • speed multiplied by time equals distance
  • amount multiplied by price equals total cost
  • mass times the speed of light squared equals energy

Equation and Question Examples

We often use a formula where we already “know” the answer and then solve for one of the variables in the formula.

  • How long will it take me to travel to my destination?
    • speed multiplied by time equals distance to destination
    • solve for time
  • How much of something can I afford
    • amount multiplied by price equals money I have available
    • solve for amount
  • how many hours do I need to work to earn X amount of money?
    • X may be the cost of an item you want to buy
  • now + added = next (stock and flow model)
  • start leaving time + packing + driving + walking + sitting down = class start time

Equation Recipe

  • Identify two quantities that your question requires to be equal
  • Set them equal to each other
  • Use formulas for one or both of these quantities that contain your known quantities and your unknown quantity
  • Solve for the unknown quantity
  • Examine your equation for its implications

Tree Example

Water Example

If we have a known amount of water, how high will that water be in a given shape?

V = L \cdot W \cdot H

Imagine we have a known volume of water, V, and we pour it into a box with a known length and width.

By solving for the height, H, we know how high that water will fill the box.

Energy Example

We can compare the measured energy a building uses and the energy we expect to use from the thermostat and the weather.

E = U A \cdot \textrm{HDD}

E is the energy read from a bill, UA tells us how energy efficiently the building holds thermal energy, and HDD is a measure of the weather.

By solving for UA, we learn how well the building holds heat and compare that to our expectations.

Friends of Equals

  • \gt greater than
  • \lt less than
  • \ge greater than or equal to
  • \le less than or equal to
  • \stackrel{?}{=} is it equal

Making Equations from equations

Since both sides are equal you can do math on two equations. You can add them, you can subtract them, and you can multiply and divide them (if they don’t equal zero).

Proportional Reasoning

We will use a lot of proportional reasoning in this class, but using an equation is different.

Be clear about what technique you are using to solve a problem to ensure you aren’t using an inappropriate one.