I have two high school-aged daughters and I teach an introductory level principles of microeconomics class. One of the disconnects I see between high school math and college-level economics classes is students have a difficult time applying basic math concepts. Students can DO the math, but they can't APPLY the math outside of a 'math' class. I focus here on math, because that is where I see the disconnect most often. I admit this is a biased observation, as I am sure that other disciplines share similar frustrations with other subjects.
My contention, based on almost 20 years of observation, is that seeing how math can and will be used later, through somewhat realistic examples, will help students retain basic concepts until 'later' occurs.
So here is my first foray into what might turn into a series of posts I've tentatively titled "What in the world will I ever use this for?" (WWWTF--or something like that) If you have other suggestions, I would be glad to hear them in the comments.
I am intentionally keeping this simple and leaving out a lot of explanation. You can think of this as an example you might find in a high school level math textbook.
Who knows, this might catch on and help educate.
Math Concept: Linear equations and the intersection between two lines
Equation of a line: One way to write the generic equation of a line is: Y=mX+b. This is called the slope-intercept form of a line and represents the relationship between an independent variable (X) and a dependent variable (Y)--that is Y depends on X. The slope of the line (m) tells us the rate of change in Y when X changes by 1 unit. For example, if the slope of the line is 2 (m=2) then every time X increases by 1, Y increases by 2. The constant (b), sometimes called the y-intercept, tells us how big Y will be when X=0.
Intersection between two lines: When two lines cross, they will cross at only a single point (unless they have the exact same slope). This is called the point of intersection. The point of intersection happens when the two Y-values for the lines are equal to each other and teh two X values are equal to each other too. An example will help. Here are two lines:
Line 1: Y=2X+3
Line 2: Y=-3X+8
There a number of different ways to find the point of intersection. Here I will describe what I think is easiest when both lines are in slope-intercept form.
Because at the point of intersection both Y's and both X's are equal, we can set the right hand side of line 1 equal to the right hand side of line 2:
Now we just need to rearrange terms to solve for X. Adding 3X to both sides and subtracting 3 from both sides gives us:
Dividing both sides by 5 gives us:
Now that we have X, we can find the Y value for the point of intersection by plugging X=1 back into either Line 1 or Line 2 (if we did the math right we should get the same answer for both):
Now you ask, what in the world will I ever use this for?
Application: Predicting price in the U.S gas market
Economists believe two 'laws' hold when people are buying and selling things: The Law of Demand and the Law of Supply.
Buyers want to buy more stuff at lower prices and less stuff at higher prices. This is called the Law of Demand. For example, at higher gas prices, drivers buy less gas, and at lower gas prices, drivers buy more gas. Economists represent the relationship between the price of gas (call it P) and the quantity of gas drivers buy (call it Q) using a demand function. Based on data from the U.S. government, the monthly demand function for gasoline in the U.S. (per person) can be represented by a linear equation.
Demand Function: P = 5.3- 0.14 Q
Some questions to think about:
- What is the slope of the demand function for gasoline (hint: rearrange the demand function into a form similar to Y=mX+b where y is represented by P and X is represented by Q?
- How does the slope of the demand function relate to the Law of Demand?
If the demand function represents what buyers want, we also need to represent what sellers want in a market. Sellers want to sell less stuff at lower prices and more stuff at higher prices. This is called the Law of Supply. Continuing the gas example from above, economists represent the relationship between the price of gas and the quantity of gas sellers want to sell using a supply function. For example, suppose the monthly supply function for gas in the U.S. is:
Supply Function: P = 0.27 Q
Some questions to think about:
- What is the slope of the supply function?
- How does the slope of the supply function relate to the Law of Supply?
Economists can use demand and supply functions to predict gas prices in markets and to predict how much of something will be bought or sold in a market.
To predict the price in the gas market, we need to find a price where the quantity of gas that buyers want to buy is exactly equal to the quantity of gas sellers want to sell. We call this price the equilibrium price.
Using the demand and supply functions above, we can answer the following questions:
What is the predicted equilbrium price in the U.S. gas market?
What is the predicted quantity of gas bought and sold in the U.S. per person per month?
Suggested answers below the jump.
Setting the demand function (P=5.3-0.14Q) and the supply function (P=0.27Q) equal to each other, we get:
Adding 0.14Q to both sides we get:
Dividing both sides by 0.41, we get:
This quantity represents the amount of gas (in gallons per month) that will be bought in the U.S. To find the price of gas, we can substitute Q=12.93 back into either the demand function or the supply function. If we solved the two equations correctly, we should get the same answer.
Using the demand function, the predicted price of gas per gallon will be:
Using the supply function, the predicted price of gas will be:
So the predicted equilibrium price and quantity of gas in the U.S. gas market will be 12.93 gallons of gas bought per person at a price of $3.49 per gallon.
We can also see this solution using graphs of the demand and supply functions.