Survey Says.....

After watching the Heisman awards show the other night, I started thinking about my classes recent attempts at verifying the validity of surveys. The way that I see it, the Okie QB should not be holding the hardware right now if Superman got the most first place votes, and I am not a Gator fan at all, as the picture should prove! -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

At any rate, our class has determined that a valid survey should represent exactly 100% of the group that was surveyed. That seems to make sense, as, if there were five people asked to respond, you should have five responses, not 4 or 12 ;-}-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

However, we determined that this all becomes a little trickier if we blend the way that the results are recorded into a mush of fractions, percents, and decimals. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Now the challenge ensues. See if you can figure out if the following survey could be valid. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

All of the teachers at CCE were asked to provide one response each to the following question; How could CCE best improve the CCE experience for our students?------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

0.22 of the teachers replied that Mr. P. and Mr. D. should not be allowed to wear Steelers football clothing.--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1/5 of the teachers replied that Mrs. H. should hold no more than 12 tech meetings a week.-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

30% of the teachers stated that Mrs. Phillips should lobby for overtime pay for time spent scoring diagnostic tests. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

0.125 of the teachers said that Coach Hall should have a Character Ed. class for the teachers.------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1/8 of the teachers said that Mrs. Phillips, the math teacher, should write all lesson plans for all math teachers.------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

3% of the teachers said that lunch should be catered by P.F. Changs.------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1/4 of the teachers said that there should be no more surveys ever.-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Please look for combinations that "simply must go together". Are there any amounts that are equivalent? Would it help to know that 0.125 could be read as one hundred twenty-five thousandths or 12 and one half hundredths or 12 1/2% ? ----------------------------------------------------------------------------------------------------------------------------------------

Well, is it mathematically valid? -----------------------------------------------------------------------------------------------------------------------------------------------------------------

Peace and Happy Holidays!

T-Cubed ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Sorry for all of the dashed lines. This post scrunches all of my hard returns together leaving no space between the lines.

Please have some flexibility!

READING DECIMAL NUMBERS

Imagine playing a game with the following three decimal number cards:

0.4 = Four tenths

0.45 = Forty-five hundredths

0.375 = Three hundred seventy-five thousandths

You might be asked to put these in order from least to greatest. You might be asked to add these all up to find a final sum. You might be asked to shade in a 10 X 10 grid in a way that represents the amounts shown. In fact, students will be asked to all of the above.


A few simple things will always makes matters work out better:

First, since out students are so tuned in to comparing fractional amounts through conversion to equivalent percents, why not convert decimals to percents? But how?

Well, it's easy enough for most kids to see that 0.4 (four tenths) could be written as 4/10, and most know that we could multiply 4/10 X 10/10 to get 40/100. Once we have a denominator of 100, we have a percent (parts per hundred). So, 0.4 = 40%.

For 0.45 (forty-five hundredths) we just think 45/100, and we have our percent. 45%

Now for the challenging one, not really, if we look at .375 (three hundred seventy-five thousandths) we could write 375/1000. If we divide 375/1000 by 10/10 we get 37.5 /100 or 37 1/2 percent. so, if you were shading in a 10 by 10 grid, you would simply shade thirty-seven and one half blocks.

If I was asked to add these decimals, I would rewrite them so that they all have the same number of digits on the right hand side of the decimal point. 0.4 = 0.40 = 0.400 = 0.4000000000000000000000000000000000000000000.

0.45 = 0.450 etc.

I'd end up with:

0.400
0.450
+0.375

From there, I would add the largest place value (the tenths) .4 +.4 +.3 = 11/10 or 1.1
Next, I would add the hundredths .00+.05 +.07 = 12/100 or .12
Finally, I would add the thousandths .000 + .000 +.005 = 5/1000 or .005

1.1
.12
+ .005 = 1.225 or 1 22/1000 or 1 22.5/100 or 122 1/2 %



Have Fun :-}

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 :-}

Homework That ROCKS!

The homework shown above (you can click on each image and it will enlarge) really shows what we are looking for in terms of effort in and out of the classroom!

The first thing that I notice is that ALL of the work is completed on separate paper. Bigger space means more ability to think outside of the tiny "box" provided on the page itself, and this is not optional. All students should be completing work on separate paper.

The second thing that I look for is work that was revised, enhanced, or merely recorded during our class review sessions. These sessions take about 20 minutes, and they are only valuable if the strategies shown are recorded. Again, this is not an option. Even if you had the same answer, you can record alternative strategies.

The last thing that I noticed was the level of detail in the review work. This student was so focused on our discussion that she recorded her answer in the same "surfer voice" that I was using during our review ("The length of Mrs. Phillip's porch is totally like 16 feet!") .

This work is totally rad and awesome! WORD!

Mr. R.

AKA T-Cubed

I should have mentioned that leaving a reply is pretty easy. All you have to do is click on the "comments" tab, type your reply (please don't use your name if you are a student), and then click on "anonymous" before you hit "publish". The post will come to me, and if it is appropriate, I will post it to the page.

Beijing Olympic Subtraction Example: Cool!

Take a look at this work from our Beijing Olympic Subtraction problem, and see if you are on the right track.

The work shown is very well constructed and efficient.

Please let me know what you think:-}

Splitting Up Work Makes Life Easy!

I wish that I could make this uploaded work a bit more clear, but it still is great to emphasise my point.

This work comes from our last Mini-Quiz, and it is a great example of how organizing your work in a very simple way can lead to a great outcome. K did an amazingly simple but powerful thing when she drew lines to separate her problems. She made it easier for her brain to focus on the tasks individually. Separate spaces for separate thoughts. Simple but very important.

I also noticed something else really cool. K did not use ultra-small writing. While I like saving trees as much as the next teacher, I know that simply writing bigger and using more space often creates better results.

I also love love love love love love the expanded notation in the addition problem, the open number line in the subtraction problem, the generic rectangle in the multiplication problem, and the misconception corrected in the division problem. This work ROCKS!!!!!!

What is subtraction anyway? Difference or...?

We will be starting our investigations in the area of number sense tomorrow, and our unit starts with a look at subtraction. You know, where you find the difference between two numbers.

What if I changed that to read "find the distance between two numbers"? Does that sound any easier?

Here's my point: If I ask a kid to subtract 95 from 100, I hate to see the kids that are driven by traditional thinking as they line up their numbers in columns and subtract to find the distance. Wouldn't it be easier to just think about how far it is from 95 to 100? You can use an open number line if you wish as well, for all of you visual learners out there. Hopefully, you'll just think about the distance and know that the distance is five.

Many teachers that I talk to say "That's great for easy small numbers, but I don't want my kids to rely on that for bigger numbers!" I think that you might want to rethink that stance if you've taken it already.

Using an open number line or "counting up" works well with any numbers, and it prepares kids to make estimates far better than traditional thinking. We all know that "in the real world" folks will use calculators to deal with big numbers anyway. So why not build some audacious number sense now.

$1,000,000 - $257, 665 can be solved as follows:

257,665 +35 = 257,700 +300 =258,000 +2,000 = 260,000 +40,000 = 300,000 +700,000=

Bingo 1,000,000

So, if I add the numbers in bold that I "added on" I get a sum of $742,335 which is the distance between the two numbers. The beauty is that I did it in my head for the most part:-} Another beautiful thing is that there are an infinite number of ways to "add on". Kids choose the number combinations that they like. I would, of course, make sure to lead them into using landmark numbers (usually numbers that end in 0, 00, 000...) as targets to add up to.

Peace, and try it...

T-Cubed P.S. Do you know the man in the picture?

United Streaming, Fay, and Other Far-Out Stuff

Recently, a very intelligent cohort of mine called me to tell me that she had found, on United Streaming, an hour's worth of five-minute snippets of real folks telling how they used math in their jobs. She described the very varied types of individuals profiled. All of the interviews sounded great! I cannot wait to watch these interviews.

Then, T.S. Fay toppled a huge Oak tree onto my house and I lost phone service, cable, Internet, and electricity. At that moment, 3:00 A.M. 08-21-08, I started to think about math! Some of my thoughts: Is there still a $5,000 hurricane deductible? How much do carpenters charge per hour? How much does plywood cost per sheet? Exactly how hot will it get in my house without AC? How long will my generator run on one five-gallon tank of fuel?

Math may not always be pretty, but it is always around us!

Peace,

T-Cubed

1st Thought Was?

I am going to write out a math situation, and then I am going to ask you to do two things. First, I want you to tell me what words you think are the key(s) to solving the problem. Second, I want you to tell me what your first physical step (or first couple of steps) would be to solve the problem (grab a calculator, line up the digits...). ELA FOLKS, THERE IS NO WRONG ANSWER HERE!

Gidgi and Kay-T want to go to a concert with there best friend Roger. The tickets are $37.50 each. Together they have saved $92.75. Do they have enough money? Do they have too much money? Will they get change back? Will they need more money? Will Roger enjoy the show? Will this blog ever end?

OK, I'll Explain The Garb!

Many people have inquired about my state of dress in my photo. Well, here's how it went down. I went to Kingsley Plantation, a former cotton plantation (owned at one point by an African-American woman!). I was visiting the slave quarters that were originally constructed of tabby, a mixture of oyster shell, sand, water, and lime. One of the small buildings was being restored to its original condition which is quite different than the weathered look that the buildings have today. Part of the restoration process involves letting new tabby dry at a slow rate, and that involves wrapping the drying tabby in burlap.

To be short, I found some burlap on the ground. I made my daughter wear it for a gorgeous shot. She took a photo of me, and then the Park Ranger screamed at us for bothering the burlap. Really, it was just resting on the ground.

That's it!

Doubling: A Great Skill & A Million Different Ways To Get There!

Doubling numbers is a powerful math skill. Doubling can help solve a multitude of problems be they addition, subtraction, multiplication, division...

I'll try doubling 13. 13, 26 (two 10s + two 3s), 52 (two 20s + two 6s...40 +12 or 40 + 10 +2), 104, 208, 416, 832, 1664, 3328 (had to think about that one for a moment!), 6656, 13,312, 26,624...

Try to think about why some numbers are easier to double than others. What strategies make the tougher ones manageable?

Try doubling 16 until you run out of MENTAL strategies....What have you got to lose?

Peace,

T-Cubed

Getting Ready For A Brave New Math World

As I sit here, a few days before the kids roll into class, I am thinking about math issues that 10-11 year-old kids would really engage with. I am asking math geeks from all over the planet to send me one idea that they think that the kids would really get a kick out of (preposition, see, I am not an English teacher!).

I know they love to gather info on the size of feet, circumference of heads..., but I really want to start out the year with a meaningful challenge that will generate buzz. I want them to text someone about how cool their work seems.

We are working on building number sense (4 basic operations). So, I would love suggestions in those areas.

Money? "Green" issues? Tech milestones (how much will I have to pay if I go over my texting limit by 1000 texts?)?

Help?

By the way how do you subtract 269 from 412? (with no paper, it took me 8 seconds) What you got for me?

More about 25 please!

Wow! I am sitting in a cyber-cafe in the mountains thinking about 25!

What is it larger than?
What is it half of?
What is half of 25?
What if it were a percent, what would it look like?
What landmark numbers is 25 close to? How close?
What's a landmark number?
If it was money, what could I buy?

Now you try.........

Peace,

T-Cubed

25 huh? Tell me more!

A challenge follows. Please take a few minutes and jot down everything that you know about 25.


I'll be back (to be read in an Austrian accent with malice).

Peace,

T-Cubed

Subtraction by Addition?

Ask somebody to think of subtraction as the distance between two numbers. Then ask them to find the distance between 464 and 782 by adding up (you may want guide them into using landmark numbers...usually numbers that end in 0 or 00 or 000...). 782 - 464 becomes: 464 + 6= 470 470 + 30 = 500 500 + 282 = 782 In summation, we found the distance to be 6 + 30 + 282. Adding from left to right gives place value sums of 200 (2 is the only hundreds place value digit)+ 110 (the sum of 30 +80) + 8 (6 + 2) or 318. This may look complicated, but all can be done as simple mental math, and the concept works well with all number combinations.

Please remember that you can start thinking this way using really small numbers to help understand. We all know that 10-4 = 6, but if you think about the distance between 4 and 10, it becomes easy to see how we can add-on to 4 until we get to 10 (4 + 6 =10)

Peace,

T-Cubed

Adding and Thinking From Left to Right!

Add a number from left to right! $1,248 + $6,921 becomes 7000 + 1100 + 60 + 9 or $8169 (read as 81 hundred sixty-nine dollars) Place value-place value-place value.................................... Start with the place value that has the most value. If you make a mistake, it will likely be in a lower place value and not in a higher place value. Would you rather be off by a few bucks or a few thousand bucks?

Peace,

T-Cubed