Ms. Lemily's Lesson: Looking Closer at Math Excellence (With Elves!)

Last week, PrichBlog shared some of the “Finding Solutions” account of Christa Lemily’s eighth grade students working on figuring out whether classic Corvettes make good investments. The question can be answered with several different mathematical strategies, so it's a great example of the fluent, flexible problem-solving called for by Kentucky's academic standards.

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:

1. Maybe you used addition.
2. Maybe you multiplied.
3. Maybe you’ve worked or played with numbers enough that you just know what happens as you double a small number. 
4. 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.
5. 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 
   

Fortunately, we don't need to unpack that formula to see the point about mathematical options.

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:

Under Kentucky's standards, Ms. Lemily’s students should be ready to work out those numbers by multiplying percentages, figure out that Corvettes are better as transportation than as investments, and notice that the graphed results do not look like a line. Kentucky's math standards for grade 8 call for students to be able to:

•  "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]

From the “Finding Solutions” description, Ms. Lemily's students may have reached that standard and also be closing in on some high school expectations, like being able to:

• "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)."

Do notice that the standards are not asking students to remember every formula they see in a math class. The goal is for them to recognize the kind of change involved and construct a function that works for that situation. That's one reason the Prichard Committee report came with a subtitle about how “Standards Push Students Toward Real-Life Problems.”

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:

One more thought. An even deeper goal in our standards is for students to master these key mathematical practices:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

The work happening in Ms. Lemily's classroom embodies those practices, and equips her students to use all eight in high school, in college, and across their careers. It's an impressive step up in how Kentuckians teach and learn math!