Observing Student Connections
 Introduction | Investigating Functions | Problem Reflection #1 | Connecting to Logarithms | Problem Reflection #2 | Classroom Practice | Observe a Classroom | Your Journal

Here is how the teacher set up the task: "Conduct a thorough exploration of the equation y = 2x - x - 3. Then prepare a mathematical report of your findings that demonstrates your grasp of mathematics."

Here is how one student responded:

Ramon: To think about y = 2x - x - 3, I started by using my calculator to make a table of values for when x is positive, 0, and negative. I had to switch to a graph to find y for x = -100 and x = 100 because I got an error on my calculator. First, here's a table that I made after I tried x = -100:

Ramon: When I checked x = -100, I noticed that the point is (-100, 97). But that means that 97 = 2(-100) - (-100) - 3 = 2(-100) + 97. So, I discovered that 2(-100) = 0.

Please take a moment to review Ramon's work. Did you notice that his misconception may be the result of rounding? His calculator is limited to 14 significant digits.

The next day, after Ramon excitedly pointed out his "discovery," Mr. Costa suggested that the class explore Ramon's conjecture. Mr. Costa also posed this question: "If Ramon's conjecture is true, at what value of x does 2x begin to equal 0?"

Celia graphed the equation on the class computer and traced the graph until she found that at (-27.5, 24.50000000527), y did not equal x - 3, or "3 less than x." A few students also found the intersection of the curve with x = -100 and changed the window on their calculators to a range very close to Celia's value.

Mr. Costa then led a discussion that helped students realize the limitations in the number of significant digits used and displayed in various graphing technologies. The teacher took the opportunity to review the connection between scientific notation and e notation on the calculator screen, explaining that "e" did not stand for "error" but rather "exponent," with an implied base of 10.

The students discussed the question of when so many significant digits might be necessary. Two examples they thought of were scientists working with a microscope to measure parts of a cell, and astronomers who generally use very large numbers.

As the discussion progressed, Carrie noted that the graph seemed linear in the second quadrant. Several students offered reasons why that would not be true. Ramon said that "linear" implies a constant slope, a constant change in y per one unit change in x. Some students then checked the slope between two points by evaluating x = -4, x = -5, and x = -6, and found that at least that portion of the graph was not linear.

Jim stated, simply, "There is a variable as an exponent in the equation y = 2x - x - 3; it doesn't fit the form of a linear function. It seems like it would be doubling, not growing at a constant rate, because of that x in 2x." Then Jim zoomed in on the range around x = -5 and showed that the differences in y for successive whole number values of x were not linear.

As the discussion began to wind down, Ramon suddenly declared, in great excitement, "I know why it seems linear in the second quadrant! It looks like a graph of y = -x - 3 after about x equals -10 because the '2 to the x power' part gets tinier and tinier. There are too many zeros in the decimal to make a difference on the graph."

Mr. Costa reminded the students to also think about the portion of the graph in the first quadrant. Students shared their observations about the rapid growth rate of y, and discussed limits and positive infinity. The students then summarized their understanding in their project papers. One student even built on the discussion of a limit to reconsider the value of 2x in the second quadrant, suggesting that it gets "closer and closer to 0 as x gets more negative"

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