linear

Class: 9th grade prob and stats. Topic:  Linear regression.  Enter: the Laundry Data.

The data sheet seemed to spark a LOT of curiosity.  In retrospect, I wish I would have given them some time to Notice and Wonder about the detergents.  Probably I’d also add some estimation first about these bottles of detergent rather than just handing them the data.  I should have known better.

Still, an interesting discussion ensued about ounces of detergent and loads of laundry.  We plotted the points on Desmos and wanted to choose two of them to create our linear model.  I teach three sections of this class and all three classes picked different points to make their equation.

One class picked (50, 33) and (200, 140), and after determining that they needed to find the slope in order to write the equation of the line, I posed that question to them.  How would we find the slope between these two points.  Crickets.

I want to note that a good minute of silence and eye-contact avoidance went by before one brave student spoke up.

S:  You FOIL them.

Me:  Can you explain what you mean by that?

S:  (coming to the board) You multiply them like this.

IMG_5025

Me:  What do we think of what S just wrote up here?  (at least 8 hands shot up in the air)

Me:  Please put your hands down and let’s discuss this.  What I like about what S just did here is he got us started somewhere.  He was willing to take a guess and risk being wrong.  Before S showed us his idea, no one was willing to volunteer their method.  Now that S has broken the silence, lots of you seem ready to discuss.  Thank you S for starting us somewhere.

After this student broke the ice, we came up with about 4 ways to determine the slope of this line and about 8 ways overall to figure out the equation of the line between these two points.  In the past, I would have said to this student, “No, we don’t FOIL, who has another idea?”  Now I know that allowing this student to explain his method does multiple things.  First, it helps the other students practice patience and courtesy when listening and responding to this student whose solution they know is incorrect.  Second, it is a great opportunity for students to engage in SMP #3: Construct viable arguments and critique the reasoning of others.  Third, it provides an opportunity to praise the value in providing the wrong answer.  So much of math class for these students has been about getting the right answers.  I’m glad this teachable moment came about for students to learn from the wrong one.

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Surgery for Function Operations

My college algebra course boasts one of the driest textbooks on the planet. It’s one of those versions that has exercises from 1 to 99 for each section…brutal.   Can you relate?
The topics for college algebra are very standard and cover little more than what students should have encountered recently in their algebra 2 course. I therefore decided that this class would lend itself quite nicely testing out the theory that a high-level, rich question questioning can be facilitated from a traditional, drill-and-kill style textbook.

Previously, I recall that Operations on Functions was a particularly awful topic for both me and my students.  The textbook presents this concept in exactly the way you might think:

f(x) = [expression involving x]  and g(x) = [similar expression involving x]

Find f(x) + g(x), f(x) – g(x), f(g(x), f(x) *g(x), f(x)/g(x)…f(snoozefest)…you get the point.  It’s boring, they’ve done it before, and there’s not much high-level thinking involved.

Fortunately, it’s fixable by asking new questions from the same problems.  For example, have students choose a pair of functions from the book.  We have 99 choices after all!  For example, something quadratic and something linear,  like f(x) = x^2 + 1 and g(x) = 2x+4.

Here come the questions:

  • Which of these function operations are commutative and which are not?  How do you know this?
  • Does this work for all functions, or just the ones that you chose?
  • For what values of x are the non-commutative function operations equal?
  • What do you notice about those values of x for the different operations?
  • Can you prove any of your results?
  • How do the graphs of these new functions compare to the original graphs?

Compositions of functions are the most fun!  Here come some more:

  • For which values of x is f(g(x)) > g(f(x)) for your specific functions?
  •  Do your results hold true if both functions are quadratic?
  • Both linear?
  • How are the graphs of f(g(x)) and g(f(x)) related to both f(x) and g(x)?
  • Don’t forget about f(f(x)) or g(g(x))! How do those relate to our original functions?
  • What about g(g(g(x))) and g(g(g(g(x))))?
  • What do you notice happening each time we compose the function with itself again?
  • Can you generalize your conclusions based on the number of compositions and tell me what g(g(g…g(x)…)) would look like?
  • What do you notice about each of these compositions?
  • What do you notice about their graphs?

A personal favorite of mine is:  If 4x^2 + 16x + 17  =  f(g(x)), what could f(x) and g(x) have been?  This works really well with whiteboards and partners.

I might have students throw out any questions that they find interesting.  In fact, I’ll bet we can come up with at least 99 questions more intriguing than the ones given in the textbook.  Then let them choose which one(s) pique their curiosity.   Now hopefully we’ve taken the time that they would have spend doing 1-99 from a book and turned it into time better spent.

 

 

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