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Showing posts from June, 2013

The series of plus 1, minus 1, plus 1, minus 1

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I greatly enjoyed watching the movie below, by Dr. James Grime, also known as the numberphile (thanks, hubby!).



He takes the infinite sum (series) of alternating ones and minus ones:

1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 ...

Does this series have a SUM?

Curiously, if we place parenthesis into it this way:

(1 − 1) + (1 − 1) + (1 − 1) + (1 − 1) + ...

we clearly get ZERO as the sum.

BUT if we place parenthesis into it in a different way:

1 + (− 1 + 1) + (−1 + 1) + (−1 + 1) + (−1 + 1) + ...

we get ONE  as the sum!



WHAT IS GOING ON?


James also shows TWO WAYS of obtaining the sum of 1/2 for this series (see the video and wikipedia page on Grandi's series).

CAN YOU KEEP YOUR HEAD ON STRAIGHT ANYMORE?



To get some SANITY into the situation, we need to look at what is true: what is the actual DEFINITION for a SUM of an infinite series?

That has to do with the partial sums. We need to look at the partial sums (that is, the sum of the first 2 terms, the sum of the first 3 terms, …

Free worksheets for variable expressions

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I have created yet one more free worksheet generator. This one makes problems that give the student an expression in words, such as the quotient of 7t and 5, the difference of x and 8, divided by 2, or the quantity 8 plus 2t, cubed, and ask the student to write a mathematical expression to match that.


These worksheets suit best grades 6, 7, and 8, including pre-algebra and algebra 1 courses.

They specifically address the Common Core Standard 6.EE.2:
Write, read, and evaluate expressions in which letters stand for numbers. However, this skill is also practiced in grades 7 and 8, or in pre-algebra and algebra 1 courses.

When using the actual generator, you can control the number of problems, workspace, border around the problems, and additional instructions.

Here are some quick links for ready worksheets. Refresh the worksheet page to get another of the same kind, until you are happy with the problems and layout.
One operation; 12 problems per sheet
Two operations; 9 problems per sheet
T…

Negative times a negative is positive, ONCE AGAIN

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In the video below, Mike shows a fun, geometric way for kids to see that a negative times a negative is positive.



I had not seen this exact proof before (or if I have, I had forgotten) - so it was cool to see it. Another cool thing was how it connects with the distributive property.

This illustrates two things I love about math:
You learn some principle (such as that negative times negative = positive). Later, you will see that same principle illustrated and proven in other contexts, over and over and over.

This is what Mike did for me right here: prove a principle I already knew, once again. When this happens to your kids (they see a principle reproven), it helps them really BELIEVE and remember that particular principle of mathematics.
I also saw how this principle I already knew CONNECTS with another principle of mathematics, this time with the distributive property. This is also what you should strive for in math teaching: to connect things students already know with new thing…

FUN measuring activity

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Miss MathDork has this really fun measuring activity - but it's not only fun, it ALSO teaches CONCEPTS, which is SO important in math.

Just like my hubby said last night... if you only focus on memorizing the formulas of math, you might get by for a little while, but eventually you come to a stopping point and it all falls apart. It's kind of like having lots of "head knowledge", which won't work. You have to get  "heart knowledge" - a true understanding of math and its principles and WHY it works. That's the only true way.

So anyway, I feel Miss MathDork is on the right track and really achieves that with her activity.  She also has a poster for you to download.



Go take a look!


This girl loves Mathy the mammoth!

I just found this new review of Math Mammoth grade 1 curriculum, and was tickled to find how much this girl loved Mathy my mascot!

My girls liked finding out that someone enjoyed the Mathy dot-to-dots and coloring pages they had made (find them on my home page)