Math Thinking

When I was a first semester college freshman, over 25 yrs ago {shuddup}, I enrolled in college calculus.  I was the ONLY freshman in that class.  And I was one of TWO students in the class of forty who was taking the class for the very first time.  That means that 38 upperclassman in that class had gotten a D or lower, or dropped/withdrawn from the class mid-semester the first time around.  I have many, many stories from that class; and it was one of the two classes that I did repeat in my undergraduate years.  Thankfully, I was able to take it from a different professor the second time around, who realized that not all of us who have to take the class are mathematically inclined.  So he focused lots on the theory behind the formulas and the practical applications, so we were able to understand more of what we were doing and why.

Because I needed to take more mathematical classes in order to fulfill my liberal education requirements, and because I could not take a "lower" math that normally would be taught before calculus {such as algebra or geometry}, I was severely limited on what options were realistically available.  However, I was able to take "math thinking", a course which focused on those very things that I could understand and find interesting.

I loved that class.  In fact, most of the math that I use now comes from my understanding of principles taught in that class, oh so long ago; so that even if I cannot remember formulas, I remember the theory behind them~~and that is really what matters the most anyway.

Which brings me to the picture of dominoes seen here...

In this set of dominoes, we have double blanks on up to double fifteens.  That's sixteen of each type of domino; but there are not 256 tiles {16x16}~because there is only ONE domino that has two pips and eight pips.  So you cannot count that same domino twice, once in the twos and once in the eights; there is only one.  If you count that domino in the twos, then you cannot count it again in the eights.  So that is why I have sixteen dominoes under the row of fifteen pips, fifteen dominoes under the row of fourteen pips, fourteen dominoes under the row of thirteen pips, and so on.

It's like when you learn your times tables.  If you are learning your sixes, you already know what one times six is, two times six, three times six, four times six, five times six...because you learned those when you learned your ones, twos, threes, fours, and fives.  That's why the times table on the back of composition notebooks had an angled diminishing graph that looked like the picture you see here.

So how many dominoes are in a set that includes double fifteens?   136.  {16+15+14+...+1  or plug in the formula that involves something like {{N{N-1}}/2}+N, where N is the number of double dominoes}

Hope that makes sense and that I phrased it all correctly.