Another Proof of the Sum of the First n Positive Integers

We have discussed how Gauss was able to devise  a clever way to add the first 100 positive integers at a very young age in a few minutes. We generalized his method and have also seen the link between the sum and the area of a triangle. In both discussions, we have shown that the sum of the first n positive integers is \frac{n(n+1)}{2}.

In this post, we discuss another geometric proof of the problem above.  We start with a specific case adding the first 6 positive integers, and the proceed to the general case.

We can add the first 6 positive integers manually or just by counting the number of unit squares in the figure above. The figure shows the geometric representation of the sum of the first 6 positive integers. Adding manually is not difficult if the given is small. However, we need a strategy for larger numbers. » Read more

Month in Review – April 2012

Hi everyone. We didn’t have Weeks in Review for the whole month, so here is the summary of posts for April 2012.

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Happy Weekend!

Guest Post: Realizing The Benefits Of Digital Textbooks – Without A Tablet In Sight

There’s no question that digital textbooks are the newest craze when it comes to classroom technology. Thanks to the success of the newest iPad, the contracts signed between Apple and major publishers, and the cost advantages of digital versions over time, teachers and administrators have been increasingly making the switch from traditional to electronic textbooks.

According to early studies, such a transition is in the students’ best interest: a pilot program carried out in Riverside, California found that 78 percent of digital textbook users scored “Proficient” or “Advanced” on a standardized math test. This compares with a statistically significant 59 percent figure among students who studied with traditional means. While this is only one study, it still generally supports Apple’s contention – that digital textbooks are not only easier to use and cheaper to purchase, but they also do a better job of stimulating children to learn.

But for many educators, these conclusions do little to spur their interest in digital textbooks. These educators don’t like the upfront costs, the flashiness of the system, and the simple fact that teaching is not done with pen and paper. They don’t believe that a piece of technology can alone make students learn better; rather, they maintain that it is the teacher’s prerogative to instill that understand in their classroom. » Read more

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