Wednesday, April 25, 2012

MLCS: A lesson with algebra

I thought about titling this blog, "where's the math?" because that's a question I get often when instructors first look through our lessons.  There aren't x's up and down each page, but there is definitely math on every one.  Since today's lesson was such a fun one and it has so much algebra, I thought I'd share the lesson pages and explain the approach in more detail.

Our approach to algebra is different than a traditional text.  It's common practice in an algebra book to group a related set of skills together in a chapter with the goal of building that skill set, say graphing.  So a traditional chapter would have this outline:
  • The Cartesian coordinate system
  • Graphing lines
  • Slope
  • Equations of lines (y=mx+b, y=b, x=a)
  • Writing equations of lines
  • Extensions and applications

In MLCS, we hit all of these topics but use the Chicago philosophy:  do it early and often.  Day 2 of the course we learn about the Cartesian coordinate system and graph nonstop throughout the course, using graphs as visuals to understand a problem or idea better.  Slope comes up early too as we discuss rate of change conceptually, then numerically, then graphically, and finally as it relates to the equation of  line.  Students write linear models from the 2nd week of class on but we don't define them as y=mx+b yet.  When they see that form (y=mx+b), many students shut down.  But just ask them to look at a table and write a formula that will get to the y's from the x's, and they'll work just fine.  Not all students like the intuitive approach, thus necessitating today's lesson.  The lesson below can be enlarged for easier viewing.
Lesson 4.8, Chain, Chain, Chain starts by looking at alkanes which are a type of carbon chain.  We expect no understanding of chemistry for this lesson.  Any chemistry information is provided.  Students explore a few alkanes and then draw many more, listing the number of carbons and hydrogens using notation from chemistry.  They then generalize the number of hydrogens based on the number of carbons intuitively.

Next we talk about slope, trying to see it in several ways (from a table, using points, from the rate of change in the physical situation, and finally from the graph).  That leads into the idea of writing a formula, using the idea of y=mx+b.  The result confirms the intuitive result.

We graph the points we create, achieving a line.  But where does the line start?  In this case, the y-intercept (0, 2) means when there are 0 carbons, there should be 2 hydrogens.  That doesn't make sense in real life.  This relates to the concept of domain.  So we start the graph at the point (1,4) and extend it from there. 

Next we work on finding various numbers of carbons and hydrogens given one or the other.  Plus, the idea of what type of number the H's will be is discussed.  The number of hydrogrens is always even.  Students have to explain why.

Summarizing and generalizing the process, student practice writing the equation of a line given a point and slope or two points.  We also cover vertical and horizontal lines.  This is the theoretical part of the lesson.

Closing the lesson, we connect the new skills to the original context, but now with alkenes.  Students study the pictures given and list the number of carbons and hydrogens.  The roots of the words connect to geometry and help make sense of the pictures (cyclopentene is a ring of 5 carbons with one double bond).  They generalize a formula.  We discuss how they did that.  Most like using the intuitive approach but we want students to see that the skill learned today (writing equations of lines using y=mx+b) gives the same result.  Which makes more sense to a student is a personal choice.

I ran out of time today so I had students finish the last 3 problems of the lesson in homework as well as work on MyMathLab to practice both theoretical and applied cases of writing equations of lines.

Not only is the lesson interesting, it connects so many ideas learned to date.  Students use patterns and inductive reasoning, graphs, slope, rate of change, equations, numerical ideas, and science.  Plus, we did a lesson in the second unit on the number of tables given a number of chairs in different configurations.  About 2 minutes into today's lesson and a student calls out, "Hey, this is the tables and chairs!"  So they remember what we did and can see the connection two months later.

So, where's the math?  It's everywhere.  :)

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