Quadratics - What Do They Model?

  Have you every wondered what the a, b, and c do in the quadratic equation?  What are they other than coefficients?  Personally, I never actually gave it a thought.  I had to identify them so I could use them in the quadratic equation to find the roots.  When I studied math, that was never a question that came up.  We accepted it.

I've been exploring the idea that quadratics can represent area but during my search to find more information, I stumbled across this Geogebra applet that is awesome.   It allows students to play with the values of a, b, and c individually so they can see what happens as each coefficient is changed.  I admit, I had so much fun playing with the applet that I had to show it to others.

The thing is, I had a really hard time finding the information I was looking for.  After a nice long search, I found that Plus Maths organization has a list of 101 uses for quadratics which has lots of details to explain the formula, who came up with it, and what it represents.  The information has been spread out between two parts.

Part 1 begins back in Babylonia with their record keeping on tablets.  The first example is simply the area of a farm and what happens when you double the length of the side.  Then the author moves on to the Greeks,  the Pythagorean Theorem, and the Golden Ratio.  From here, the Greeks and conic sections are discussed and includes illustrations showing how each conic section is cut out of the original cone.

Part 2 continues the history lesson but brings it up to modern times with more modern applications. It begins with a discussion of Galileo and his accomplishments.  The connection between his discoveries and acceleration or stopping distance both of which use the quadratic.  From there the author shows the connection to ballistics.   Then as far as historical references, they moved on to Newton, his contributions and his works relation to quadratics.  They moved to Bernoulli and air travel.  The last two topics include chaos and mobile phones.

Every example includes the mathematics involved and explains why and how its a quadratic.  It would be easy to take the information out of this set of articles to create lessons in class or assign students to work on a presentation using the information provided.

Teaching Quadratics

  I am getting ready to teach quadratics in about a week and a half.  I"m looking for new ways to teach it so I don't do the same old same old and maybe make it more interesting.

I have two different apps on the iPads I can have students use to practice factoring but I'd like to do something other than teach it in my standard way.  So I did a web search and came across a few nice ideas.

First is Tic Tac Toe: Quadratic Factoring which uses a Tic-Tac-Toe grid to help factor quadratics.  It comes with a complete lesson plan to use and includes all of the necessary papers.  I am not familiar with this method but it looks like it is worth learning and sharing with students.

I believe that it is important to share several ways of doing anything so students find a method that works for them.  Back to the lesson, it includes objectives, a powerpoint to present the method, guided practice sheets and independent practice sheets along with both formative and summnative assessments.

I have to teach completing the square so again I did a search and ended up at the same place as before looking at a Hip to be (completing the ) square.  I only know of one way to do it and it is long and tedious so I needed help finding material that might make it a bit more exciting.  This has a way I've never done it before.  It actually uses a 2 by 2 square and might actually be easier than the way I learned.  Cool.

In fact, I did a search of the cpalms site and there were 188 lessons that popped up when I typed quadratics in the search.  188 complete lessons!  The lessons cover transformations, graphing,  the easy vertex form, real life applications and all sorts of other aspects of quadratics.

I found a lovely Dan Meyers basketball task introduce quadratic/parabolas to the class.  This looks like  a cool hook to use to get the students interested.  This came from a a website that has ideas for introducing the topic, including the vocabulary.  I love the idea of borrowing other people's suggestions to use in my own classroom.

This site, Better Lessons,  has a nice entry and exit ticket in addition to ideas on the order in which to teach the topic.  It has a lovely video with instructions on doing a think, pair, share to discuss the ideas from the video.  It has everything needed to teach this lesson.

Ok, I am set, now all I have to do is sit down and plan. 

Common Core Nuggets: Quadratic Zeros

This app is another one in the common core nuggets series.  It specifically works on helping students factor quadratics to find zeros or start with the zeros to find the equation. The app has three parts, each focusing on a slightly different skill to meet the standard.

The directions are rather sketchy but with a bit of play, I figured out how to work each section.

The first section focuses on finding the zeros of a quadratic with a leading coefficient of 1.  The app gives you an equations such as f(x) = x^2 + 4x + 3. You are expected to factor it, set the factors equal to zero and find the zeros.  Then you move the green sliders around till they are on the proper numbers on the number line.  Hit the red show/next button to show where the actual graph is.  If you are correct, the green dots and the graph line up.  If not, you see where they are they should be.



The second section focuses on finding the zeros of a quadratic equation with a leading coefficient of something other than 1. You follow the same process as in part 1 looking for the zeros, set the green dots and check.  If you are correct, the green dots match up with the graph.

The third part gives you the zeros and you figure out the equation that goes with the zeros.  You are expected to put in the values for a, b, c from the equation you calculated and if its right, the graph goes through the two points, if you are wrong, it just gives you the next two points.  It does not show you where the graph should be.

As I stated earlier, the directions on how to use the app were rather sparse but I used the play around with it until you figure it out.  Much like what I tell my kids when I can't answer a question on how to do something in an app. Overall I do like it and plan to use it.