Engaging students: the difference of two squares

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Dale Montgomery. His topic, from Algebra II: the difference of two squares.

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Application/Future Curriculum (science)-

You can use difference of squares to find a basic formula to be used in any problem where you drop an object and want to find what time it will take to land. This physics concept will be of interest to your students considering any mechanical science and a useful tool to introduce problem solving by manipulating equations.

Take any height h. If you were to drop an object from this height then it could be modeled with a distance over time graph using the equation

(h- 9.8/2) t^2.

By applying difference of squares you get the expression

[\sqrt{h}+\sqrt{4.9}] t) \times ( [\sqrt{h} - \sqrt{4.9}] t).

Then by setting this expression equal to 0 and manipulating you would get that
t = \pm \displaystyle \frac{\sqrt{h}}{\sqrt{4.9}}.

I like a situation like this because it allows you to give them linking knowledge about quadratic equations. Most students may not have been exposed to this type of physics yet.  However, it is a requirement, and having this knowledge will help them in that class. On top of that it helps with equation manipulation and answering the question, “Does my answer make sense.” This question needs to be asked since it is possible for a student to get an answer of negative time. All of these skills combined with the new topic of difference of squares make for a multifaceted problem. This would probably not be great for day 1 of difference of squares, but I could see it as an engage for the continuance of the lesson.

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Curriculum:

You can use the idea of graphing to show that difference of squares works. This is a good way to give visual representation to your students who need it. If you compare the factoring of x^2-9 to the graph of y=x^2-9 and finding the roots of that graph, you can show that they have the same solutions. It is not that novel, but this visual can just help the idea click into students’ minds.

ParabolaGraph

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Manipulative

A manipulative that I got the idea for from http://www.gbbservices.com/math/squarediff.html is using squares to show the difference of squares. This is done quite easily as shown in the picture below. This could be done along a lesson on difference of squares. Maybe this would follow easily from a factoring using algebra tiles. The image below is fairly self explanatory and would really help if made into a hands-on manipulative that kinesthetic learners could make great use of.
DifferenceOfTwoSquares

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