Math: basic fractions not Algebra

"I don't know what the hell I'm doing." ~Frustrated student.



The text book one of my students uses. Currently he is doing fractions...and has a test that we were studying for. This is quite common even up to the college level, reviewing fractions because the students forget the rules. I would prefer to teach fractions using algebra but in this case we don't have the time it takes to develop the concepts...I had one hour. So I used the 5 basic concepts
to bring home the point that the rules did not change we can't add the numbers unless they are SAME. And Pointed out the numbers have two parts the "how many" part and the "what kind" part. With fractions the numerator tells how many and the denominator (de nome of it) tells what kind. Have to be the same KIND before we can add.



So basically we did an hour on "fractions". Here we are adding two mixed numbers. We talked a lot about multiplying by 1 and that "one" could be 2/2, 3/3 etc. And how numbers have to be same before we can add, subtract or divide them. Felt sorry for him because we had to spend an hour basically on fractions by himself. It would have been more fun if he had another student there and we could mix it up and actually do some algebra but we didn't have time. He did say he didn't mind because "you have a way of explaining it that's easy and kinda fun, my teacher makes it confusing and hard."

That's because his teacher is teaching the same way we have been teaching for decades, never mind that it doesn't work, using rules and memorization instead of concepts and algorithms based on those concepts. He teaches the way he was taught to teach...so all the way to college, students forget what they learned in 4th grade about fractions. If we taught English this way nobody would speak the freaking language...and math is a language why not teach it that way?



Rather than convert the mixed numbers into improper fractions, I showed him how to add and subtract using the associative property of addition. I didn't tell him that; we just "added up the parts." First the whole numbers then the fractions...if the fractions ended up being improper we just added one to the whole number...7 1/2 + 6 3/4 for example. 7 + 6 = 13 then 1/2 + 3/4 = 5/4 so 13 + 5/4 = 14 1/4.

"Wow! That's easier."

7 3/8 + 9 1/4 = 7 + 9 + 3/8 + 2/8 = the easy way.

59/8 + 72/8 = the hard way.

We had to go over common denominators, equivalent fractions, and improper fractions. It was not tremendous fun but he did enjoy it because he knew he was going to ace the test, now that he understood "what the hell he was doing."

Note how the blocks are set up to show that 3/4=6/8=9/12=12/16.

I keep threatening to build a fractions page at Crewton Ramone's House of Math...but so far I haven't.

And BTW if you are on Maui, or anywhere on the planet for that matter I can do tutoring with you.

Food: important for the math brain.

Let's start with what not to eat it's a shorter list...lol...

I stole this from an email I got but the info is basic and I have seen it around the web, I think she basically compiled a list of data she gathered herself. She left off foods that have heavy food coloring particularly red...

Find Crewton Ramone on Face Book.

Her name and a link to her site:'

http://www.greattastenopain.com

Sherry Brescia


These foods that are so deadly and harmful that eating
(and drinking) them on a regular basis will certainly chop
years off your life, and guarantee that the years you have
left will be filled with sickness, disease and very likely
obesity.

I call them the Evil Eight.**

I hope this list helps to burn into your brain that in the
same way you can't trust government, politicians, drug
companies and insurance companies with your health, you
cannot trust Big Food manufacturers either. They create
their concoctions to make money. And they know one thing...
if it tastes good, people will eat it.

Information is power. So here's the information you need to
avoid illness and enjoy good health.

Here are the Evil Eight:

1) Soda
========

Oh, I know... boo hoo. But the truth is, ANY soda -- diet
or regular -- is acid in a bottle (or can). It's SO acidic
that it takes 32 CUPS of water to neutralize the acid in
just ONE can of soda in your body.

Having so much acidity puts a tremendous strain on your
kidneys, which filter acid out of your bloodstream and put
it into your urine.

Your body also uses its own alkaline minerals (calcium
from your bones and magnesium from your muscles) to
neutralize the acid. Eventually that can lead to
osteoporosis, weak muscles, tooth decay and brittle, porous
bones that fracture easily.

Plus soda contains the worst form of sugar on the planet --
high fructose corn syrup (more on that further down).

Don't think diet soda is any better -- because here are
the dangers of...


2) Artificial sweeteners
=========================

Simply put, they're poisons. No other way to say it.

Here are just some of the documented side effects of
aspartame (known by the brand names Equal or Nutrasweet):

Eyes:
-Blindness in one or both eyes
-Decreased vision and/or other eye problems such as
blurring, flashes, tunnel vision
-Eye pain; Dry eyes/decreased tears

Nerves:
-Seizures
-Headaches/migraines
-Dizziness
-Confusion/memory loss
-Severe drowsiness and sleepiness
-Numbness of the arms and legs
-Slurring of speech
-Tremors/trembling

Psychological/Psychiatric:
-Severe depression
-Irritability/agitation
-Anxiety/phobias (fears)
-Personality changes

Chest:
-Heart palpitations/rapid heart beat
-Shortness of breath
-High blood pressure

Digestive:
-Nausea
-Diarrhea, sometimes with blood in stools
-Stomach pain/pain while swallowing

Skin and Allergies:
-Itching without a rash; Hives

Endocrine and Metabolic:
-Loss of control of diabetes
-Thinning or loss of hair
-Low blood sugar (hypoglycemia)
-Severe PMS symptoms

The most severe effects of aspartame poisoning are:
-Death
-Irreversible brain damage
-Birth defects, including mental retardation
-Ulcers
-Aspartame addiction and increased craving for sweets
-Hyperactivity in children
-Severe depression
-Aggressive behavior
-Suicidal tendencies
-Cancer


Think Splenda is just splendid? It's no better.

Studies are associating Splenda with a growing list of
medical problems that include:

- Flushing or redness of the skin; rashes
- Anxiety; panic attacks
- Nausea; bloating; diarrhea; vomiting
- Stomach cramps
- Depression; becoming withdrawn
- Feeling forgetful; memory loss
- Headaches
- Feeling irate, impatient, hypersensitive, moody
- Chest and body pains
- Seizures

If you want to use a non-calorie sweetener, use Stevia.
It's been around for centuries and has a proven track
record of safety.


3) High fructose corn syrup
===========================

High fructose corn syrup (HFCS) is directly associated
with the following conditions:

- Kidney disease
- Osteoporosis
- High cholesterol
- Raised insulin levels (leading to diabetes)
- High uric acid levels (which can result in arthritis)
- Obesity

And the newest one: Mercury exposure

Mercury is a poison to your brain and nervous system. It
has been linked to fertility problems, memory and vision
loss, and trouble with blood pressure regulation. It can
also cause chronic fatigue and neuro-muscular dysfunction.

Mercury exposure has also been linked to autism, heart
disease, Parkinson's disease and multiple sclerosis.

The problem here is HFCS is found in just about every type
of product on the market including:

Soda
Ketchup
Salad dressings
Sports drinks
Cough medicines
Jellies and jams
Yogurt
Ice cream
Cereals
Snack chips and crackers
Breads
Candies
Syrups
Pickle relish
Non-dairy whipped topping
Granola bars
Lunchables
Packaged pastries, snack cakes and cookies

Can you believe that food manufacturers would put this
poison in our foods? Well, believe it. It's called
profits...

So what should you do? Read labels and find alternatives.
Organic products typically don't contain HFCS.



4) Hydrogenated oils
=====================

Hyrdogenated and partially hydrogenated oils contain trans-
fatty acids, which are fats that are not found in nature,
and that your body CANNOT metabolize. Instead, they just
accumulate, clog your arteries, and cause inflammation,
cell damage and heart attacks.

These bad fats are found in deep fried foods, margarine,
baked goods, powdered and artificial coffee creamers, and
most processed (packaged) foods.

Look for the words "hydrogenated" or "partially
hydrogenated" on package labels. If you see them, do NOT
buy the product.


5) Nitrite cured meats
========================

Cured meats made with sodium nitrite (or sodium nitrate)
include hot dogs, lunch meat, bacon, ham and corned beef.

When you eat nitrites, they can be converted into
nitrosamines (which are potent cancer-causing chemicals) in
your body. The primary types of cancers associated with
nitrite consumption are colo-rectal, stomach and pancreatic.

In addition, children born to women who ate a lot of cured
meats during pregnancy are from two to three times more
likely to develop a brain tumor than those born to mothers
who did not eat cured meats.

Another study showed processed meats increased the risk of
pancreatic cancer (the deadliest of all cancers) by 67%.

Nifty, huh?


6) Processed foods
===================

Processed "foods" include all packaged foods (in boxes,
cans, jars, bags or frozen cartons) that contain
ingredients other than a basic food. Although the appeal
behind them is convenience and indefinite shelf life, they
are extremely unhealthy and loaded with chemicals and
preservatives that your body was never meant to ingest.

Besides the chemicals they contain, processed foods are
harmful to your health for these reasons:

* They provide no usable naturally occurring nutrients;
they use up loads of digestive energy, and then basically
turn to nothing but waste.

* They contain no water to transport nutrients or carry
away wastes from your body, like fruits and vegetables do.
Instead, they add to the waste pile, making you very toxic.

* Your body's hunger feeling is triggered by a need for
nutrients from real food -- not just something to occupy
space in the stomach. Since processed "foods" don't
nourish you, even though your stomach may be physically
full, the body thinks it still needs more food because it
hasn't gotten nourished from what is in the stomach.
That's why you can polish off an entire family-size bag of
Doritos then feel hungry again an hour later. No nutrition.

It's a recipe for obesity.


7) Farmed salmon
=================

Farmed salmon are crammed into little pens and fed things
they were never meant to ingest like soy, poultry litter,
and hydrolyzed chicken feathers. As a result, farmed
salmon is lower in vitamin D than fresh caught and higher
in contaminants such as carcinogens, PCBs, and pesticides
like dioxin and DDT.

Rule of thumb: Eat wild-caught fish, not farm-raised.



8) Tap water
=============

All water is NOT created equal. Since our bodies are
alkaline, our water must be alkaline too. Thanks to acid
rain and the chemicals added by municipalities, that's not
always a given.

Regular (unfiltered) tap water is acidic and contains
toxins like flouride and chlorine. Avoid tap water (this
includes water fountains) at ALL possible cost, no matter
what you may read in the news or your local politician may
tell you.

The best sources of alkaline water are distilled water or
filtered water (using reverse osmosis, Brita filters or
similar filtering).

There you have them -- the Evil Eight.

(c) copyright 2010 Holistic Blends

=============

**The FDA has not evaluated all of these statements. The
contents of this email are not to be considered medical
advice and are for educational purposes only. If you are
experiencing health challenges, always consult your doctor
for medical advice and follow it even if it contradicts the
contents of this email.**

=====================================

Crewton Ramone's House of Math

10 Year Olds do the Math

These are a couple of exceptionally bright 10 year olds who came to a two hour session for enrichment not because they are having trouble.... If you would like more info about Math Tutoring on Maui go here.

They start the class with a simple whole brain warm up exercise where they have to build the addends for 7 thru 10. They have to use both hands simultaneously and they race me. I also have a tray. Right now they cannot beat me but soon the students will surpass the master...


Here is a shot of them using the blocks. They are racing to see who can get the problem st done first. We did the basic one step percentage problems, there are 3 kinds:
20% of 60 =?
24 is 60% of what number?
What % of 70 is 21?

We did a bunch of problems for each kind, then we did a mash up where all three kinds were in the problem set. It's always more fun with two or more students because we can race and get more problems done in less time.


Just for fun I taught them how to convert base ten to binary but forgot to snap a picture or two then we moved on to percentages. Here you see the problems and sketch of the solutions plus a matrix for the second problem so they can arrange their thoughts. They were using the blocks to figure these out at first then moved on to drawing them.


We spent a lot of time on binary conversion and percentages and they begged me to do algebra. That's not an exaggeration. So for the last 40 min or so we did some factoring. Started off easy and worked our way up. Here the student is doing a fast sketch and the writing the symbols...



What you don't see is the time we spent modeling these problems with the blocks. The base ten blocks are like training wheels, and these two wanted them to come off as soon as possible. Here we went directly to symbols no sketches no blocks...



You can tell how confident they are by asking "do you want an easy one or a hard one?" By the end of it they were saying, "give us some hard ones!!" Of course there are no hard ones when you understand what you are doing. (BTW x²-18x - 81 cannot be factored, but x² - 81 can. Look for a lesson on the difference of two squares soon.) They have fun and get a sense of satisfaction because they know much older kids who have a hard time with these...I will remind you that these are 10 year old boys.

This is a bad shot but shows economy of symbol. We don't need to write (x-9) twice. We talked about square numbers and what that meant. Soon they will be evaluating these for various values of x. Ie f(x)=5 or 10 or 2 etc...and they will see the result is always square. When it was over they thanked me profusely. One of them who was coming for the first time admitted that he thought it was going to suck but instead he had a good time and couldn't wait for the next session.

Neither can I.


Crewton Ramone Does Algebra With 4 and 5 Year Olds.


More at Crewton Ramone's House of Math.

Social Media Management

Here is a website I have been working on. Has more to do with math than you might think.

Internet marketing is a numbers game.

Social Media is growing exponentially in that realm. People are figuring out how to use social media in commerce.

Enter The Social Media Manager.


Maui Social Media Manager.

http://www.jfsag.com/social-media-manager.html


Note: the website is still very much under construction.

I have been using these principals to promote Crewton Ramone's House of Math, and guess what? It works. I get clients off my Facebook page and I have managed to get to the top of quite a few search engine listings...type in "math tutor Maui" or "Maui math tutor" for example.

But I also rank highly for "Mortensen math" and actually higher than the Mortensen Company for "Mortensen Math Tutor" which makes me laugh.

Now I am going to start using this knowledge to help some local business on Maui...and make a penny or two for myself while I'm at it...

More sensory input = more understanding.

The mathematics expresses reality numerically...music: the beautiful expression of the Mathematics...




Advanced use of the mathematics and music to create virtual manipulatives to increase understanding. The blocks are primitive in comparison but the principle is the SAME.

Greek Alphabet kids will sing and dance to...

Catchy.

Math is fun.

I'm pretty much boycotting this blog until somebody donates a buck. Over 1000 views (in the USA alone) and not one in a thousand has a paypal account...hmmm...so until somebody donates a buck you get piffle, just like you do in regular school.

Make sure you check out my face book page.

There is some interesting stuff there as well.

Getting A Feel For Numbers

This is a simple exploration of ten. You can do it with all the numbers, but here a five year old is getting some basic information about 10. Ten is two fives, we have already built tens out of nine and one, eight and two etc...so we "know" a ten can be made of two fives, now we can see it can also be made out of five twos....



Just to make sure we put the fives on the twos and see they are exactly the same...



Then we lined them up and counted and compared.



We can do this with twelves and fifteens etc...

This is counting that sets up for multiplication...note we aren't even using symbols yet, just the blocks.

Using Algebra To Teach Basic Operations.

The basic operations are usually defined as addition, subtraction, multiplication and division. Algebra, because it is generic math can get the basic concepts across quickly and easily. Notice that I used the term "generic," much less intimidating than "abstract mathematics".

There are many kinds of base ten blocks that you can purchase, but base ten blocks are not all the same and some are much more useful than others. First off, make sure they are base ten blocks, I have seen block sets where they are base seven and even base three...where seven of the x are same as an x² or just three y are the same as the y² while still quite workable doing the larger problems becomes somewhat difficult because it takes too long to count out the units. This is very much akin to getting lost in computation...many students lose track of the problem they are doing because even if they understand the concepts they can't do the math or it takes them a long time to do the math and they forget what they were doing in the first place.

Here we see a seven year old student doing simple sums with the blocks handy in case he gets stuck...he didn't get stuck. Note there are a few problems where he has to figure out what x is, and to do so he will have to do some subtraction even though all these problems seem to be "addition."
For 7 + x = 15,

He doesn't try to figure out the answer by counting forward he knows two ways to solve it. One he "subtracts" 7 from 15 by using addends...seven needs 3 to be ten (taking the seven out of the ten) and 3 + 5 = 8 or "using up" the five, he can take 5 out of 7 which leaves 2, and 2 needs 8 to be ten...so the answer is 8 either way.

Next we build some simple rectangles in algebra. For the the first problem he got out 3x and a nine (the nine block) but that didn't make a rectangle. He looked at me for help and I just smiled. "Make me a rectangle..."

He puzzles over it for a moment and then you can see the light bulb go on. "I need three 3's!" Now counting the sides is EASY 3 across and x + 3 up...the next one looks "hard" so he skips it and does the last one. I think this is because we spent some time playing with nines and he hasn't had much experience with sevens.

You don't see it here but he got out 9x and then got out nine fours and was able to see that the rectangle was 9x + 36 and the sides were 9 across and x + 4 up.

9x + 36 = 9(x + 4)

Then I showed him that he could have used 4 nines and we talked a little about 9 being the "common factor", but mostly this lesson was about division for him...making 36 "fit" on the 9x...if the whole thing is 36 and one side is 9 what does the other side have to be? Four. Which is why he got out fours...



He did the same thing with 7x + 21 (the problem in the middle, solution pictured and built but the answer isn't written yet)...took him several tries before he got out three's...I asked him why he didn't just get out sevens...he thought for a moment and then went, "OH, now I get it...!"


Next he got to build his own problem and we see that "bigger is funner." I had two top trays out, which meant he had 3x² and 20x to work with so he used all of them. BIGGER IS FUNNER. He spent some time building it and making it so he could use up all the x. Then he fit in the eights and I told him to count it.

"What do you mean?" he said coyly.

"Tell me the whole thing and the sides....he did and then I told him to write it down.



Remember the child is seven, his writing skills are not well developed, first he wrote it too small then he wrote it too big for purposes of the picture but I think he did just fine...it was taking him quite a while to write it and I wanted to move on to the next problem...

The mind wants to work very rapidly, counting out individual units from the various manipulative sets takes too long. Sometimes even counting out tens or x's takes too long. So while sets that have base ten blocks with tens and individual units are quite useful they become tedious if you want to use them to demonstrate many problems in a row or even just a few larger problems. By larger I simply mean problems that will have a lot of pieces to get out. Showing 60% of 80 for example or even a simple problem like (x+6)(x+7) will require getting out 42 units...you already have to get out 13x, counting out 42 units takes a lot of time, and the student can often get lost (and bored) because of it; however, if all you have to do is get out an x-square, 6 sevens (or 7 sixes) and you can grab 7x and 6x, the problem can be built very rapidly.

In this case he had the rectangle

2x² + 20x + 42

And he had to find the sides. Later, when he knows his multiplication better, he will see that the factors of 42 will be a big help in solving this problem quickly, but for now he takes the 20x and makes a rectangle out of that...it took some trial and error to get 6x on top and 14x (two sets of 7x) so that they would fit...that was a lot of counting and adding. Then he still wasn't sure he had it right because he could not tell if 6 times 7 was 42 because he didn't know it. You can see him doing computation on the side...he has 14 three times and the sevens drawn out...again lots of math going on here to figure out if he has 42 or not. 30 + 4 times 3 or 30 plus 12, and then the long way where he has to add seven plus seven plus seven etc until he gets to 42. If he had to count out individual units this problem would have taken MUCH longer and he may not have finished it at all.

2x² + 20x + 42 = (x + 7)(2x + 6)

(In the picture the 2x is cut off.) With a few simple pictures, you can actually SEE what it is I'm talking about. You can see that it becomes easy to teach counting and addition, and multiplication and division using algebra because the child has to count out and add up the blocks in order to build the rectangles needed to solve the problems. At this point you begin to see why a picture is worth a thousand words, and if a picture is worth a thousand words having your hands on the blocks and then being able to draw them is worth a thousand explanations. Further, this is the way the subconscious mind works, using pictures NOT symbols and we have come to realize over the last 100 years that most learning takes place in the subconscious mind not in the conscious mind.

Students who benefit from base ten blocks most are the kinesthetic learners, because they literally can get their hands on algebra and work with the problems. They can see and touch the problems in a very real sense. They can see that x times x is an x square, they can touch it...they see and touch the distributive theory of multiplication because they can see an x+3 is just a number with two parts (a binomial) and if I have 3(x+3) (spoken three times x plus three), I get 3x+9 and they can see it.

Getting this problem out is still easy even if the blocks you have are all individual units, but it's still easier to get out 3 threes...and by the way we can note than 9 is a square number when we build it this way. As soon as you do a problem like 5(x+8) counting out the 40 units is going to slow things down and actually take away from the lesson and the concept that the 5 multiplies the x and the 8. Manipulative sets that use rods to represent numbers are again quite useful but will have serious drawbacks when attempting to demonstrate algebra or square numbers and square roots because most often 3, 3-rods do not make a square. Simple concepts like x + x = 2x are easy for any student who is just learning to count. Base ten manipulatives or base ten blocks can be used for so much more than just counting or to teach addition. When used properly very young students can lean to count and add WHILE they learn algebra and vis-versa, and counting, adding and subtracting lead to multiplication and division.


"We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about 'and'." ~Arthur Stanley Eddington, as quoted in A Dictionary of Scientific Quotations" (1991) by Alan L. Mackay, p. 79


Lastly for fun he got to build whatever he wanted. Lots of multiplication went on here as I made him count by nines and fives and tens.


The tens are two fives so we counted pretty high by fives, and we counted all those nines (the teal colored one that make up the roof). Using the "trick" "one less and what does it need?" 6 nines is 54, which means 5 is one less than six and six needs four to be ten...for 14 nines it's two less and what does it need? 14 minus 2 is 12 and 4 needs 6 to be ten so (14)(9) = 126 We also counted by adding nines...

BTW you can check me out on FACE BOOK now...and get up dates etc...and of course get more at Crewton Ramone's House of Math

Pi to a Million Places...

Click for 1 MILLION places....


3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679

Sing along:

http://z0r.de/1976

JFSAG.


Check out my Math Quotes Page.

Here is the Education Quotes Page.

And JFF:

Platypus Keytar.

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