Quadratic Functions

When I received the "Deepening Quadratic Understanding" worksheet, I thought I already had a solid knowledge base of what it meant for a function to be quadratic.  Of course we all know it forms a parabola, but there are other important relationships that sometimes go unnoticed.

My group in class decided to find the significance between quadratics and their second differences in the output.  One of my peers suggested using a Excel spreadsheet to organize our findings, which was much faster than plugging in data points by hand and finding the differences between points (one great use of technology).  With lots of data quickly organized, it was easy to see a relationship between our functions and their second differences in the outputs.  We noticed that the second differences were always twice the leading coefficient.  The name "second differences" also made us think about second derivatives of the functions.  We found that the second derivative was also equivalent to the second differences and the value of 2a (when "a" is the value of the leading coefficient).  This makes sense because in a quadratic function, the degree is always 2.  So 2 is multiplied by the leading coefficient, a, as it is when taking the derivative.  Graphically, taking the first derivative will give us a linear equation, which is represented by a consistent change in slope, just like the first differences increase or decrease by a constant amount.  The second derivative names the slope of the line, and is represented graphically by a constant horizontal line and in a table, gives a constant second difference.

Here is a short example of what our excel sheet looked like,

As you can see, the value of the second difference was always 6, which is also equal to 2a because for this example a=3.

Working through these exercises is important for a few reasons.  First, it is important to work through exercises on your own time to expose yourself to important concepts that you might take for granted.  New discoveries about old concepts might seem insignificant if you already know how to work with the concepts, but these new things might be the link that students need in order to understand the concept fully for the first time.  Secondly, doing these exercises highlighted the importance of working with groups.  In groups, individuals can benefit from working with others who can contribute different ways of thinking (I, for one, would never have thought of organizing our data on an excel spreadsheet, but it was far more efficient than my method of doing it by hand.)  These exercises, enriching our ideas of quadratic functions, were not only helpful in understanding quadratic functions, but also understanding processes of thinking, and working towards enlightening solution.

Many students ask, "When are we going to use this?" whenever they are confronted with a difficult math problem, or whenever they feel like challenging the teacher.  Dan Meyer has created ways to involve students more in their math learning by connecting math problems to everyday items, such as a water tank, then asking students what they would like to know about the water tank.  This relates mathematical formula to an object that they are familiar with.  I believe that if teachers utilize this method in their classrooms, students might become more interested in and engaged in what they're learning.  Additionally we can expand this concept to accompanying math problems with student's drawings, instead of just relying on the numbers and formula.  If students make pictures to illustrate what the problem is asking it might help them realize that the problem is not just a math problem with numbers and symbols, but it connects to the wider world.  An example of this could involve the measurements of a picture frame.  When students draw out the frame, they might realize that it is simpler than they thought and it is relatable.  Yes, it still involves math, but now it is also relevant to students' everyday lives.