I am right a fraction of the time!

Be you a student, a teacher or an interested guardian, please ask yourself if you could come up with any more profound statements about the fractions 2/3 and 3/4. Also, please challenge yourself to create accurate models of each as well. To the mathematically uninterested, this may not seem like a grand accomplishment, but to a well-trained math geek, this is poetry!

Yep, all of this was done by a ten-year-old! Could you have done this at ten? Yes, I am quite proud of my students.

Well, chime in now. What do you think?

Peace and have a great holiday!

T-Cubed

I am about to make Mr. Pinchot really mad!

No, the cat will not anger Mr. Pinchot, unless it makes him think of the Mt. Nittany (spelling?) Lions that his beloved Penn State uses as a mascot! You see Penn State lost to Iowa and ruined their perfect record and their chances of an NCAA football national championship. However, none of that is going to make Mr. Pinchot as angry as he will be when he finds out that I actually published a math piece using the sophisticated math terms "chop-chop" and "double-double"! He absolutely hates these terms even though they are based on sound math principles.

The term "chop-chop", as all of my students know, refers to creating an equivalent fraction by reducing the numerator and denominator at the same rate. Usually, the "chop" implies dividing by 2/2, but we could "chop" by any fraction with a value of 1 whole, n/n, 10/10, 4/4...We would create a fraction that has an equivalent value.

For example: 12/16 divided by 2/2 = 6/8, and 6/8 divided by 2/2 = 3/4 ! Oh, I say chop-chopping is a fine way to create new and equivalent fractions!

You could probably guess that he also hates "double-double", which is just multiplying by 2/2 (the whole number 1). This, again, creates a new and equivalent fraction! For example part deux; 3/4 X 2/2 = 6/8, and 6/8 X 2/2 = 12/16..., but we don't have to settle for using 2/2 all the time. We could go crazy and use 13/13 or 3.5/3.5 to multiply by. We could take 1/2 X 3.5/3.5 and get 3.5/7, and that IS equivalent to 1/2. Boooyahhh!

The key is to remember that what we are really doing is multiplying or dividing by 1!

Please be sure to ask Mr. P. about Penn State, and please tell him that we have been chop-chopping a bunch!

Some of the digits? No, Sum of the digits!

Hello again. Once more, a student has shown how working "smarter" makes for a better and more efficient math student!

In this example of our last homework, this student relied heavily on the "rules for divisibility" in order to make answering the divisibility questions a snap. In general, I am opposed to anything that smacks of being a math "trick" as an aid to completing math problems. However, the rules of divisibility are all based on sound math principles, and these rules do lead to more efficient and more accurate work. We rely heavily on these rules each day as we try to find common factors for the numerator and denominator of fractions. Right now, 5th grade students (and really 4th grade students) should know the "easy to use" divisibility rules for 1, 2, 3, 4, 5, 6, 9, and 10. The rule for 8 is also useful, but the rule for 7 is really too complex to use at this stage.

If you don't know the rules, please contact any worthy math teacher, student or the Internet for help:-}

I also have to comment that the Prime-Factor tree diagram shown is also a huge step towards understanding higher math. I also love PFTs , because they are lateral thinking exercises, as there are many paths to the correct answer. Again, ask a wise student to explain. The kids love doing these "puzzles" too!

Thanks for reading,

T-Cubed

Problems In Our "Times"

On a recent test, students were asked to find an expression the would be equivalent to 25 X 41 .

I'd be willing to bet that you read that as 25 "times" 41.

The students were given four choices as a correct answer:

A) (20 x 40) + (5 X 1)
B) (20 X 40) + (40 X 1)
C) (25 X 40) + ( 25 X 1)
D) ( 25 X 40) + (26 X 1)

Now, before you put on your number crunching hat, consider the rereading the "times" sign in the original problem. If you read the X as "groups of", you become aware that we are looking for some combination that is equivalent to 25 groups of 41 or 41 groups of 25.

Do you see a choice that makes 41 groups of 25? I do! Decompose the 41 into 40 and 1. Does that help? The answer is C, and I'll C U L8TR.

T-Cubed

May seem so simple, but......WOW!

Sometimes the simple things in life are better! This student used simple multiplication problems to help solve his/her division situations. I am so pleased to see students working "smarter not harder"! Obviously, he/she really understands that division depends on a real understanding of the relationship between multiplication and division. After all, in division you start with a known product and a known factor, and you simply try to figure out logically what the unknown factor is. 745 divided by 16 is the same as 16 X n = 745 .

So, why not start out with 16 X 1 = 16 ? It's easy and it leads directly to figuring out 16 X 10 = 160 , and 16 X 2 = 32 , and 16 X 20 = 340 , and 16 X 40 = 680... (being able to double and halve is sooooo important!). By the way, I'd probably throw in a 16 X 5 = 80 , as it is just half of 16 X 10!

At any rate, this work really does show great "smart" thinking! Even with the error in problem number two, this really is great mathematical thinking. Can you find and fix the error in problem number two? I bet the student that did this will see it quickly and be able to revise this problem easily!

T-Cubed

Multiplication and Division RELATED? :-}

The two scans above, starting with the top scan (uh, not the pug), represent some of the most efficient strategies around when it comes to multiplication and division. Perhaps, the best thing about these strategies is that most of the calculations can be carried out mentally and do not rely on the rote memorization of many math "facts". Truly, this is how most mathematicians think, smarter not harder!

Parents, students, fellow countrymen, and all other living beings, please challenge yourself to see if you can find the beauty in this kind of thinking. You may just find your inner math geek after all!

As a footnote, I wish that I could say that I "hired" a student to write this for me, but that would not be honest. I wrote this myself! Bad hand writers of the world unite! We shall rule.

PLEASE COMMENT!

T-Cubed

Revsions: Not easy to do, but worth it!

Remember, you can click on the images to show them at full size.

These revisions are very well done. They really show an understanding of what was done incorrectly, and the new work shows that the concepts in question are, indeed, now understood.

I especially liked the explanation in the "money" problem that sated that the student in question "Didn't add back the 3 (cents) that I rounded to." This shows a pretty complete understanding of the error that was made. I can also accept the "blonde moment" explanation for multiplying by 7 instead of 6, as that is just a lapse in concentration, and it's funny!

I hope this shows that really revising problems is a bunch of work, and that it is not something to rely on as a way to overcome a lack of participation in the class or at home. Rather, revisions are a way to refocus on a few concepts or errors that were not quite clear at test time.

What do you think?

Click on "comments" below. Type your comments. Type in the funky letters in the funky letter box. Click on anonymous. Click publish. wait for me to approve and post your comment :-}