Math Worksheet Database

Several months ago, I posted an article about free printable math worksheets.  I created a couple of basics problem sets that could be printed off, and I also mentioned a couple of other websites that admittedly did a lot better at providing practice worksheets.  I quickly fell behind in keeping my selection of questions up to date, as I instead focused on keeping my blog active with new concept lessons.  Now, I want to do something a bit better and more productive.  

Instead of me having to create sheets that teachers and students can download to practice their math with, I am putting out a request to all of my readers for your help.  

Let me know in the comments below some of the sites that you know about and use where you find good practice math worksheets.  When I have a sufficient number, I will redo or add to my original worksheets blog post, but instead of providing (and having to make) the actual practice questions, I will make the page into a repository of links to other websites that all specialize in providing free math sheets.

I think that this is a way better plan that what I originally envisioned.  As much as I'd love to create all of my own work to share and make freely available to distribute, I unfortunately don't have the time necessary to dedicate to that project.  However, by making use of the power of social networking, I hopefully can put together a database of sites that are all relevant and useful, because they will already have been recommended by some of my readers!

Help me to get this database of math sites off the ground!  Leave me a note in the comments below, or on my new Facebook page, and I will compile them all and make a great resource of free practice materials for students and teachers to use.  I will also be sure to acknowledge everyone who suggests sites!  Thanks in advance for everyone's help!

Be sure to click the +1 button below, or my new Facebook Like button at the start of this post to help me get the word out!

My Facebook site is finally LIVE!

Hello and Happy New Year to all of my readers!

I'm starting the new year off on the right foot by finally getting my Math Concepts Explained Facebook site up and running!  I'll be updating that site as I add information to this blog, but I also welcome my visitors and followers to comment and ask me questions there whenever they'd like.  I will do my best to provide whatever math help I can!  Please visit and Like my Facebook page and add me to your Friends list!

Perfect Christmas Tree Math

Here's a fun little post for the holidays!  It's not going to teach any new math concepts, though it is a neat demonstration of an out of the ordinary use for mathematics.  It describes mathematically how to decorate the perfectly proportioned Christmas tree.  I can't honestly say that I have tried it myself, though I don't doubt that it might be useful!

A department store in the United Kingdom by the name of Debenhams issued a challenge to the University of Sheffield math students to devise a mathematical algorithm to determine the optimum number of ornaments, tinsel, and lights in relation to the size of the tree, as well as the perfect size of the star for the top of the tree.  Members of the University's SUMS Math group were able to come up with a set of equations that described precisely what the challenge required:

Courtesy of http://www.shef.ac.uk/polopoly_fs/1.227830!/image/formulaslarge.jpg

The students say that the calculation took about two hours to design, though to actually compute the equations is as simple as entering numbers into a calculator.  The stated purpose of the formula is to help holiday shoppers to be smarter when purchasing decorations for their tree, as with this knowledge they will be able to buy the precise number of baubels and lights rather than have a box of extras leftover once the tree is done, or else not have enough and so have to go and fight through the mall crowds to get more.  It also means that people can find the right size angel or star for their tree, rather than guessing and then potentially being way off when they actually get home.  They say that the formulas are versatile enough to apply to large trees suitable for the Royal Family, right down to smaller trees that are more common.  You can check out the original news release here.

It's probably a stretch to think that this set of equations will make it into widespread use, as decorating for the holidays is way more about spending time with friends and family and having good times, rather than looking for mathematical precision in a Christmas tradition.  However, it's a fun exercise to show young math students that demonstrates yet another useful and unexpected use of math in everyday life.

Happy Holidays to all of my followers, and all the best in the New Year!

How to Derive the Equation of a Circle

Continuing from my last post about the Equation of a Circle, here I would like go through an exercise which hopefully explains more about "why" the equation looks the way it does.  The equation of a circle isn't a difficult one to memorize, and the modifications that you can make to it to shift it either horizontally or vertically are very similar to the methods used in translating graphs of functions.  However, the theory behind the equation is very distinctively about a circle.

To begin, let's start with a point (5, 7).  Now, I ask to find points that are exactly 5 units away from this point.

To begin, it is very straightforward to deduce points that are horizontally and vertically in line with the center point.  We simply move 5 units up and down, and left and right.  This allows us to identify the points (10, 7), (0, 7), (5, 12), and (5, 2).

After these points, it seems that finding new points gets a bit more difficult.  However, it should be fairly obvious that by asking for points that are 5 units away from the center, this is essentially asking for points along a circle with radius of 5.  So then, how to find another set of coordinates (x, y)?

If you pay attention to how I have purposefully drawn the radius in the above picture, you will notice that I have inscribed a right angle triangle with a hypotenuse (radius) of 5.  This is one of the special triangles, a 3-4-5 triangle.  So then, if we then use the dimensions of this triangle, and rotate and flip it around the circle, we can come up with several more exact points that are on the circle: (8, 11), (9, 10), (8, 3), (9, 4), (2, 11), (1, 10), (2, 3), (1, 4).

As you can see, the points that we have identified are beginning to fill out the circle.  But, how can we identify even more (x, y) points?  It doesn't look like there are any points that lie on whole number integer coordinates left.  All that remains are fractional coordinates, and that seems like a lot more work than it's worth to identify more points!  Maybe we can find an expression to more accurately pinpoint ordered pairs, that will include all of those that we've already identified plus all of the ones that we can't?  Let's go back to our picture, and note the generic triangle now inscribed.

If we say that our radius of 5 is the hypotenuse of a right angle triangle, and we already know what the center point of our circle is, then we can do a bit of arithmetic to come up with generic side lengths for our triangle.  If we sweep the radius around the circle, and we always let the inscribed point be called (x, y), then we can state expressions that denote the length of the sides.  In this example, the horizontal side is equal to x-5.  I've visualized this in the following image.

Similarly, the vertical side length of our new triangle can be shown to be y-7.

Now, since we have described a right angle triangle with defined sides, we can apply the famous Theorem of Pythagoras to this triangle (a² + b² = c²).  This means that we can come up with the following equation:

And by now, you should recognize that this is the form of the equation of a circle:

Quite simply, this equation looks the way it does because it is based on the Pythagoras Theorem.  The circle equation really isn't a difficult one to have to memorize, but hopefully this demonstration has shown you the value of triangles and geometry in deriving more advanced math concepts.  Please don't forget to hit the +1 button below, and also follow me on twitter!  You can even click here to tweet about my post.

Credit to a YouTube video by DrJamesTanton where I got the idea for this post.

Equation of a Circle

Once you have worked with functions for a while, inevitably you will begin to wonder "what about circles?"  You have explored all sorts of different equations and their graphs.  Perhaps you've even gone so far as to rotate graphs sideways and learn how to manipulate those equations as well.  But despite these seemingly more involved concepts, you've yet to come across circles.  For such an apparently simple shape, why have these not been included in such rudimentary graphing lessons.

Well, circles are a little different from what you've done so far.  For starters, circles technically aren't functions.  This may surprise you at first, but recall the vertical line test.  Would a circle pass such a test?  Of course not, because passing a straight vertical line through any point on a circle (except for the tangent points on the sides) would also intersect the circle on its opposite side.  Try it and you will see!  So, if a circle is not a function, then how do we handle describing their equations?  That is a little different, but really not much harder than your typical parabola graph.  Follow along and I will explain the equation of a circle in more detail.

For starters, let me just show you the equation for a simple circle.  Consider a circle with a center at the origin (0, 0) with a radius of 1.  We call this a unit circle.  Here is what the equation looks like:

It looks simple, right?  This equation can also be modified using similar concepts to how you would manipulate a function, such as a parabola.  There is a simple change to make to the equation that causes the graph of the function to translate left or right, and a second similar change you can make that results in a vertical translation of the graph.  In this case, depending on where you want to shift your circle, that is the variable you modify.

So, if you want to center your circle at (2, 0), which is a horizontal translation of 2 units to the right, you would change the above formula to include an (x-2)2 term.  Think of it as "if x=2, the whole x term becomes 0.  So 2 is its root."  Same thing applies to vertical translation.  If you change the y term to be (y-5)2, then you can see that the whole y term becomes 0 when y=5, so this means that the circle is centered at a height of y=5.  If we combine these two translations, we shift the circle to have a center of (2, 5), and the final equation looks like this:

Hopefully you can see how easy it can be to locate the center of a circle based on its equation, and how equally simple it can be to determine the equation of a circle just by a visual inspection.  However, to fully describe a circle, there is still something missing.  We haven't looked at what the "1" means.

To fully describe a circle mathematically, the only things that you need to know about it are the coordinates of its center, as well as its radius.  You may think that you should need more information, but think of it as the mathematical equivalent of a geometry set's compass.  To draw a perfect circle with a compass, all you do is put the point down at the center, set the radius, and spin the pencil around the paper.  You only needed those two pieces of information to be able to construct a circle.

Continuing with our example, the radius of our circle is described by the 1 term.  Technically, you can think of it as a 12 term, which provides for our circle to have a radius of 1 (or a unit circle).  To change our radius to 5, we would change the 1 to a 25, because just like the x and y terms, the radius term is also squared.  If we wanted a radius of 7, we would put the term as 49.  In general, this term is r2.

If we put all of these components together, we can come up with the general form for the equation of a circle.  We can include terms that allow for horizontal and vertical translation, as well as whatever radius we want.  Let's call the horizontal shift term "a" and the vertical shift term "b", and the radius "r".  In this case, here is the general form of the equation of a circle:

The equation describes a perfect circle, and doesn't allow for any stretching or compressing along either of the axes.  All that needs to be done is determine the circle's center point and radius, and you can easily fill in the relevant values.

As an example, consider the circle that has radius of 10, and is centered at (-5, 7).  Here is its equation:

Here is the opposite form of the question: what is the center and radius of the circle described by the following equation:

Quick analysis of this by comparison to the general form allows you to find that the center point is (-12, -2) and has a radius of 12.

Hopefully you can see that graphing circles is a little bit different from graphing more familiar functions, though still similar enough that the math concepts we've learned so far can easily be adapted to apply here as well.  In my next post, I would like to explain a little about just "why" the equation of a circle looks the way it does.  For teachers, this is also a good exercise to have students explore when first learning how to graph circles.  

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