To explain that, let’s start by spending a minute with the Brothers Grimm and their tale of “The Shoemaker’s Elves.” The first night, the elves turned one piece of leather into a pair of shoes so fine that, after selling them, the shoemaker could afford to leather for two pairs. The second night’s two beautiful pairs brought in enough money to buy leather for four. The third night? You know the answer: enough for eight pairs.
So, how did you figure out that it was eight? Here are five respectable options:
- Maybe you used addition.
- Maybe you multiplied.
- Maybe you’ve worked or played with numbers enough that you just know what happens as you double a small number.
- You probably didn’t use percentages, but you could. If the story was trickier, with two pairs yielding revenue to buy leather for three, multiplying by 150% each morning might be a good choice.
- You probably didn’t use an exponential function either, but if the growth was 20% each night, and you wanted to know the result after 10 nights, you might end up using a formula like
Instead, let's swing back to Ms. Lemily's class at South Warren Middle and their question about Corvettes. If the shoemaker story is about growing (appreciating) value, the car story turns out to be about the opposite: depreciation or value going down. On average, Corvette convertibles lose 15% of their value the first year, and between 8% and 10% each year after that. With a $60,000 car, that’s going to produce values like this over time:
- "Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.” [Emphasis added]
- "Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another," and
- "Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table)."
Overall, South Warren Middle School's strong focus on the math standards seems to be paying off. With Ms. Lemily in the lead, South Warren was one of the first middle schools in the country to join the work of the Mathematics Design Collaborative, and as of last year, their K-PREP scores impressively outpaced the state average for most groups:
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.