begin here!!

2 bananas * 2 bananas =

2 bananas * 2 bananas = 4 bananas^2

2x * 2x = 4x^2

2ft * 2ft = 2ft^2

m Unit1 * n Unit2 = m*n * Unit1*Unit2

Consult any algebra book.

Also, algebra does not go in NQS. It's a fundamental tool underlying all of advanced mathematics and science.

I say that

2 bananas * 2 bananas = 256^4789564532211 *bananas^.00000564321 *meters/s/s *2 retangle rhinoceroses relaxing on a rather rough rock reading rare recipes

Consult any algebra book and you will find a way to prove that the above is correct.

It's not surprising that someone who has never studied the subject would reach that conclusion.

Elementary math might as well be magic to the willfully ignorant.

Wow! The *nerve* of me to use math in a science forum!

well we will just wait to see what your doctors tell you.

of course you said he wrote the wiki article maybe he just doesnt realize that you didnt want him to explain it so well.

so that it could remain an unknown , so that you can continue to teach children the way that you think things are in reality.

of course if he tells you something other than what you believe exist in reality you can always get a second opinion.

its not that you use math , its that your math is inconsistent
with correctness.

I don't teach kids the verbal stuff. I teach them the unambiguous mathematical notation - which you disagree with.

Of course, if the good Doctor says I'm wrong, I'll admit it, but it's a minor issue.

In the mean time, can you tell us what "kwh" means?

Says you - the guy who has never taken algebra and refuses to open an algebra book.

What does "kwh" mean?

http://en.wikipedia.org/wiki/Kilowatt_hour



above you claim to know what kwh means.

and when I said that

1000 watts * 50 hours = 50,000 watts

I was correct , but you said it was wrong.


1000 * 50 realy does equal 50,000 !!

its simple math but your algebra says its not correct and that is the problem with algebra.

you say that

1000 watts * 50 hours = 50kwh

and that is wrong..

its

1kw * 50 hours = 50kwh

Paul,

1000 watts * 50 hours = 50kwh
That is correct.

1kw * 50 hours = 50kwh
That is ALSO correct.

This is because 1kw means 1000 watts.

Now, can you tell me what 'kwh' means? It's a simple question.

Let's ignore the rest and focus on what we agree about:

1kw * 50 hours = 50kwh

What does "kwh" mean here? You can say it. We already know. But you ought to say it - for your own good.

As a mathematical answer, I have no problem with this, but:

(4 bananas^2)/4 = 1 banana^2, I think.

Would I be spoiling an academic discussion if I were to ask what, in the real world (whatever that is), is the distinction between: a banana and a banana^2?

Is it possible that a banana^2 is a banana * a banana? and if so, is that different from 2 bananas - in a non-mathematical, real-world sort of way.

Part of the beauty of algebra - which is a foundation of all higher, analytic math (geometry, trig, analytic geometry, calculus, probability, statistics, differential equations, linear algebra, numerical analysis, etc.) - is that interim results don't need to have a physical interpretation.

What's important is that what you have on both sides of an equal sign are actually equal - and that can't happen if the dimensions are inconsistent.

OK, OK, I know - the real world is a mathematical world - go back to sleep, that man!

Paul, you agree that

1kw * 50 hours = 50kwh

What does "kwh" mean here?

Paul,

Can you just google 'kwh' and tell us what it says?

That's not even algebra.

Visual representation of polynomial multiplication:
http://www.youtube.com/watch?v=7TBlQNnHVTw

Still waiting to hear back what 'kwh' means.

What does 'kwh' mean?

Good link, TFF.
I could with that sort of explatation for the more complex stuff.

Bill, is there anything specific that you're looking for?

There is a hole in my education where the basis of maths should be. As a result, any efforts I make to master the practicalities are hampered by lack of foundations.

You will appreciate that this is an impediment to grasping scientific concepts, especially in view of my advancing years. Learning is a lot slower process than it was, so I don't kid myself that I shall master calculus, but anything that helps to move in that general direction is a bonus.

1 banana = 1 banana

if a banana is 1 banana in lenght then
1 banana * 1 banana = 1 banana squared.
or
1 banana * 1 banana = 1 square banana.

if you are using meters as the length then you would have

when calculating area
1 meter * 1 meter = 1 meters squared

and

2 meters * 2 meters = 2 meters squared

and

3 meters * 3 meters = 3 meters squared

or

1 meter * 1 meter = 1 square meters

and

2 meters * 2 meters = 4 square meters

and

3 meters * 3 meters = 9 square meters


we have no symbol to use for square meters so TFF used
the ^2 which is not correct and he wrote his answer out
as four bananas squared when he solved his equation!!!

2 bananas * 2 bananas = 4bananas^2 (four bananas squared)

he should have used
four square bananas
or
two bananas squared

that is what this is about.

I know its confusing but the confusion comes from the lack of a symbol to use when writing out a formula or a result of a formula.

it was when he wrote out what his result meant that I realized he thought that squared was the same as square.

Start with this: http://www.youtube.com/watch?v=XUc9Uwh3ibc
http://www.youtube.com/user/videomathtutor#p/c/2CC06557AC6512FD

Here's a few of mine:
http://www.youtube.com/user/TheFallibleFiend?feature=mhee#p/search/2/wvisikhVQNo

Tips for solving math problems:
http://www.youtube.com/user/TheFallibleFiend?feature=mhee#p/search/3/10gYLWs6aSY

Then try this:
http://www.youtube.com/khanacademy#p/c/7AF1C14AF1B05894

Thanks, TFF.

Duty calls, but I'll have a look at those links a bit later.

A banana is not "one banana" in length, because "banana" is not a length. (We could define a banana as some arbitrary length, but we would need to remember that real bananas are generally not the same standard length.)

Usually, the length of a particular banana will be in a standard unit of length.

We would not compute the total length of a bunch of bananas by multiplying bananas by bananas, even if we do use the "banana" as a standard unit of length.

Paul wrote: "it was when he wrote out what his result meant that I realized he thought that squared was the same as square."

I'm still not convinced I'm wrong on that. I only said that I might be wrong - and I still think that I *might* be wrong. Certainly Dr Math disagrees with me and I acknowledge he knows a lot more about it than I do.

I don't have a problem - none at all - with being wrong. My problem is one of consistency.

Regardless of what anyone thinks: 4m^2 is equivalent to a 2x2 grid, with each cell being a 1mx1m.

We read this as "four meters squared" in normal discussion in the same way that we might read
4m^3 as 4 meters cubed and
4m^5 as 4 meters to the fifth

That is, in normal mathematical notation, the exponent ONLY applies to the presenting element (in this case, the 'm').

If Dr. Math (and Paul) are correct in that
"4 meters squared" actually means (4 meters)^2 = 16 square meters

then you have a case where this regular case means something different than what we normally mean when we use mathematical notation (and MOST technical people use the technical notation).

I don't have a problem with common usage being different than technical usage. If that were the issue, then this is not worth discussing. But Dr. Math did not give any indication that this was "just the way lay people talk about it."

On the other hand, wolfram alpha agrees with me, which is odd, because ostensibly, I thought they were the same group of people behind both of them.

I have sent an email to request clarification. He has responded to me quickly on two previous occasions relating to other matters, but I have not gotten a response on this yet. I'm not complaining; merely noting. The guy probably gets hundreds of emails daily; he can't answer them all.

then why did you write your answer to your equation the way you did?
2 bananas * 2 bananas = 4 bananas ^2 (four bananas squared)

the way you wrote your answer is sudgesting that bananas have a length?

if bananas dont have a length then why wasnt your answer

simply 4 bananas

why did you write

4 bananas^2 (four bananas squared)

no , a 4x4 grid = 4M^2 with each cell being a 1mx1m.

a 2X2 grid = 2 squared not 4 squared

4 meter length * 4 meter length = 16 square meters

2 meter length * 2 meter length = 4 square meters

2 meter length * 2 meter length = 2 meters squared

YES

thus the dangers of not having a square symbol and the necessity of this discussion in its own discussion.




we shouldnt use ^2 for both square and squared

because 4^2 and 4^2 look the same

but 4 square and 4 squared are different

these would work


254 np ■

223 np ▀

8 np ◘

but they wont work in this forum

what we really need on this forum is a smiley for square
4 ft * 4 ft = 16 ft

Writing:
2 bananas * 2 bananas = 4 bananas ^2 (four bananas squared)

IF a "banana" is a unit of length then "bananas squared" would be a unit of area. (I'm not saying that bananas are standard units of length - I'm positing a hypothetical.)

2ft * 2ft = 4ft^2

length * length = area

Suppose I had a "standard banana" and I wanted to find the length of my office in standard bananas.

I count 15 across.

The equation is not
Length of office= 15 bananas * 1 banana

The equation is
Length of office= 15 * 1 banana

If I wanted to know the AREA of my office, suppose it is 15 bananas in one direction and 20 bananas in the other direction. The formula would be:

Area of office = 15 bananas * 20 bananas = 300 bananas^2

(The square ONLY being applied to the bananas and NOT to the 300.)

write that out

use the smiley for square

ie...

4 ft

or

4 ft

both the above have the same value.

2ft * 2ft = 4ft^2

Two feet times two feet equals four feet squared.

I could also write it as:
Two feet times two feet equals four square feet.

According to Dr. Math (and you) these mean different things. I don't recall ever having heard a single technical person using these as anything other than synonyms.

I will use the for 1 sq ft






notice there are 16 square feet in 4 feet squared

the above is 4 feet SQUARED you can tell by counting
the number of linear feet on each side of the SQUARE

the below is 2 feet SQUARED you can tell by counting
the number of linear feet on each side of the SQUARE




notice there are 4 square feet in 2 feet squared


see the difference?

lets check this

the below is 1 foot SQUARED you can tell by counting
the number of linear feet on each side of the SQUARE



notice there is 1 square foot in 1 foot squared

Bill S. Another good source to learn about math is The Khan Academy. There is a lot of stuff in there. The mathematics section starts at a simple level and builds on up through Calculus and Linear Algebra. You should be able to find something
at any level you need there.

So try the Fiends links and mine. You should be able to find something that will help you.

I may be running a bit late getting back to you, but Turner Classic Movies had a Buster Keaton festival a few days ago and I am still working my way through the ones I recorded. I just finished "The Navigator". Anybody who is interested should be aware that Buster Keaton was one of the funniest comedians in the movies, and they are still just as funny as when he made them.

Bill Gill

I see the difference. I have never seen it mean that previously. This would be inconsistent with every other technical expression.

In writing the notation 4ft^2, every technical person would recognize the latter, 2x2 grid (or equivalent).

If Dr. Math is correct, then when speaking the words, we would have to be inconsistent with every other technical expression.

thats why we need a symbol for square because communication between several tech's could result in big problems.

if it already hasnt.

ie..

one tech calls another to ask how large of an area something is.

his worded reply was four square feet , the caller writes down 4ft^2

and orders enough steel to cover 16 square feet.

It's possible, though I've never heard it used that way.

*USUALLY* we are at a whiteboard and we write down the notation or we are sending emails where we also use the notation.

There are occasional cases where someone is talking and I don't know what they're driving at, so I ask them to write it down. I can't think of any examples at the moment, but I'm pretty sure this is not among them. It's extremely infrequent, because usually we're talking in front of the formulae where there is no ambiguity.

I have not seen kallog weigh in on this. I'm curious to know whether he has ever heard or seen it used as
4 meters squared means (4 meters)^2 instead of 4 (meters^2)

Ah...I can think of one case, actually. The term Kilobit means something different to communications engineers and computer designers or system admins. Kb to a comm engineer means 1000 bits; Kb to a system administrator means 1024 bits. I'm not saying there aren't some inconsistencies in terminology between fields, but I have never known the interpretation of "squared" to be among them.

Khan academy is outstanding. There is a wealth of other stuff on the net. My oldest daughter had gotten straight As in math up through HS and a 5 on her AP BC. Her first math class in college was linear algebra (or maybe it was differential equations ... I don't remember exactly) and she was making Cs and Ds. This was really odd, because linear algebra is USUALLY a cake-walk after you've already had calculus.

Anyway, she stopped going to class and started instead to watch MITs OpenCourseware videos, http://ocw.mit.edu/index.htm . Here grades immediately jumped back up to As. By end of semester it averaged out to a B (or maybe a B+). Anyway, there's some *really* good available now on a variety of topics if we have the patience to wade through it - from Yale, Berkeley, MIT, many others.

Would I be right in thinking that:

4m^2 = a 2x2 grid, with each cell being a 1mx1m.

but,

(4m)^2 = a 4x4 grid, with each cell being a 1mx1m.

Just a thought from a non-mathematician

I have just re-read the earlier posts, and I think TFF covered my last question en passant.

that just might work !!

you can actually see the difference , but when talking on the phone would it sound different?

if people understand the difference between square and squared the voice communication would be easy.

so I suppose if people who use math (all of us) knew the difference and knew how to express the difference in writing your sudgestion above would be perfect !!

unless the brackets conflict with other forms or methods of math such as algebra.

I'm off to the supermarket in the morning to look for square bananas. If I find any, I shall see how many I can arrange on a grid before they chuck me out. Wait for the results on SAGG.

ask him if he will sell you two bananas squared for the same price of 2 bananas.

if he agrees

you could use the tiles on the supermarket floor.

and lay out 4 bananas , 1 on each of 4 tiles in a square.

and show him that the 4 bananas are actually 2 bananas squared.

and you can get the bananas at a bargain price.

LOL

Exactly right. Are you ever over on chess.com?

Exactly right.

4m^2 = a 2x2 grid, with each cell being a 1mx1m.
As long as you understand that 4m^2 = four square meters

and

(4m)^2 = a 4x4 grid, with each cell being a 1mx1m.
As long as you understand that (4m)^2 = four meters squared

I think I last played chess about 40 years ago. Now I find there are so many things I would like to do, and so little time to do them in, that I have to make hard choices, like keeping away from chess, resisting the temptation to learn Welsh and cutting down my numbers of cacti, because there is just not time to look after them properly.

1 blender

1/4 cup oatmeal

1/2 cup water

blend 2 minutes

1 frying pan

put the blend in the pan

cook n stir until thickened

return to the blender

blend for 2 minutes

add water and sweetener as desired

drink the potion.

this might just give you that extra time needed.

Good suggestion, Paul, but I'm not into cooking.

BTW, you didn't say if I should add cacti to the mix, or where I might have to spend any time thus gained.

you can find a similar product in the organic section in grocery stores or supermarkets.



can you eat cacti?




on this side of the grass?

LOL

I figured an active chess player would use the term en passant. In any case, I know where you're coming from, but I still play on occasion. We keep a set at work on a table at the end of the hall - and when someone walks by, she makes a move. Takes several days to play a game. I also sometimes play at the coffee shop - online at chess.com.

Yes, the fruit of many cacti are very good. I believe you can also eat the pads of various Opuntia Spp. although I have not done that.

A friend's hamster consumed the greater part of a Lophophora williamsii, without any ill effects. I believe this is because they have to be dried before they release their mescaline. (Lophophoras, that is, not hamsters)

There is a rumour that the hamster attained great size and lived for years under a paving slab, but I suspect that story is apochryphal

Just to bring the thread back on subject I should say that the slab under which the hamster is reputed to have lived had:

Length = A m
Width = B m
So, area = AB m^2

My wife says I should also give the hamster's dimensions in "teeth squared", but she doesn't appreciate the niceties of science.

TFF, In your Blogg you quote a verse from the Rubaiyt of Omar Khayyam. Which of Fitzgerald’s translations is that from? Many decades ago I learned the first translation, and I still remember enough of it to know that your quote is not from that one,

that is good information , you never know when you might
need to find something that is edible in the future.

I think they are also a good source of water , if not couldnt you place a large clear plastic bag over the cactus and capture the perspiring water that way?

or do cactus perspire?

sounds right to me , although TFF will most likely find fault in it , I can understand what you are saying.


so the communication is clear knowing that you are aware that
sq area = linear length * linear length

Just to make sure we agree on what is going on with the math under discussion, and because I haven't posted anything for a few days, I will do a quick review of the proper math notation.

Given: X = a * b = ab
This means that the quantity X is equal to a multiplied by b. The 2 forms are equivalent.

Given: X = a * b^n = ab^n
This means that the quantity X is equal to a multiplied by b raised to the n power. Again the 2 forms are equivalent.

Given: X = a^n * b^n = (ab)^n
This means either that a is raised to the n power and b is raised to the n power and then the 2 results are multiplied or a is multiplied by b and the result is raised to the n power. The 2 forms provide the same result.

Notice that there are in fact more ways to indicate raising a value to a power. This is what I have found to be one of the more common ways. It comes from computer programming where a common way to write an equation to be evaluated by the program is to use the ^n form. In fact if you provide appropriately formatted variables in a program the statements above would be successfully performed by many programming languages.

And in regard to how to handle units. When performing mathematical operations on units of measure such as meters (m), grams (g), Joules (J), etc the proper way is to use the same mathematical operations on the units as on the metrics that go with them. For Example:

Acceleration has the units of m/s/s = m/s^2 (meters per second squared). That is meters divided by time squared. Notice that the ^2 applies only to s, if I wanted it to apply to m/s I would need to contain (m/s) in parentheses. So m/s^2 is quite different from (m/s)^2. The first way squares s the second way squares both m and s.

The way that things work out you can determine the speed that something will attain under acceleration by the formula:

v=at Where v is velocity, a is acceleration and t is the time the acceleration takes place.

Example:

Given a = 9.80 m/s^2 (9.80 meters per second squared)
This is approximately the acceleration of a falling object at the Earths surface.

What is the velocity of a falling object after 10 seconds?

Acceleration has the units m/sec^2.

So: v = (9.80m/s^2) * 10s

Notice that I enclosed the 9.80/^2 in parentheses to indicate that the 10s is not a multiplier of s^2, but is applied to the whole expression.

Now rearrange it a little bit to separate the units from the metric.
v = 9.80 * (m/s^2) * 10 * s

10 multiplied by 9.80 = 98
(m/s^2) multiplied by s = m/2
Therefore: v = 98m/s

And that is how you handle units in science.

Bill Gill

I don't know which translation. I've read several different versions. I assume the quote you're talking about is about the Inverted Bowl which has been in all the copies I've read. I have three copies and keep one on my desk, but I just got comfortable and have no intention of standing up just now.

So long as a is a constant, everything you said is correct.

great explanation, Bill; one that even a "hitch-hiker" like myself can understand.

I'm just hoping that no one comes along and argues with it and leaves me confused - as usual.

I am interrupting to say that this topic is the funniest thing I have read for ages. It was the introduction of the hamster that made me write, when I got my breath back!

I don't understand WHAT YOU ARE ON ABOUT with the maths, but as a hitchhiker too may I say I am shocked at the casual treatment of the banana. No one in Oz treats a banana in such a brusque manner..... since the cyclone in Queensland wiped out all the banana plantations they are priced higher than ever before!! They got to $19.99 a kilo in some places and they were the little squidgy ones not fancy ones.!!! No one would be ill-treating bananas here at the moment!

I await the next instalment of whatever this topic is.

Wow! Chalk up one success! I've made a few attempts to introduce the term "hitch-hiker" in this context. This is the first time anyone has picked it up.

BTW, Whatever Australians do with bananas; does anyone know what they do with "pigs"?

Bill S--- Re: pigs and Australia---- Do we want to know?

I am sorry to rain on your parades but as a QM zealot I need to clear up some things

1.) Was anyone actually observing said banana's because that may change the outcome.

2.) Do we know the position or the velocity of said bananas because clearly we can't know both

3.) Can we establish that said banana's are not infact entangled


If we can establish all of that I can tell you whether or not the said bananas are infact here or at some other random location in the universe.

And to Bill S the Australian females generally marry those Australian "pigs".

Oh! BANANA'S---- indicating possession (or possessed bananas)---, OR--- Did you mean more than one banana--- as in one banana, two bananas?

one banana two bananas three bananas four
four bananas make a bunch and so do many more !
tra la la la la la la
tra la la la la la la

http://www.youtube.com/watch?v=2uxlmFrDL5I

They keep their washing on the line; same as everyone else.



With Rev passing racist remarks, and you being sexist, I'm beginning to wonder what's happening to this forum.

I missed Orac's nasty remark. Sorry I was sexist--- but I've actually missed how I transgressed in that way. No excuse I know. I was misled by the possessive banana's.

Rev and I are approaching ancient (him more than me - eeek now I'm being ageist!) Our age should confer some excuse or reason ---- maybe?

I am sorry paul you can't count the banana's it's impossible it's covered by the "Indistinguishability Postulate" of QM and its perfectly clear



See perfectly clear you can't count them.

We are all individuals ..... I'm not!

Four bananas, five bananas, six bananas, seven,
Do you think we'll ever know if Rev believes in "Heaven"?
Eight bananas, nine bananas, that seems quite a lot;
It started out as algebra, but now it's gone to pot!

Since you got GOD and bananas in the same post lets go there

Did you ever see the Intelligent Design video that banana's were proof of GOD's intelligent design. Apparently noone told him they were genetically modified by man :-)

Funny as hell

http://www.youtube.com/watch?v=Y4yBvvGi_2A

Be reasonable! This is a thread about algebra.

1 hand + 1 banana + 1 idiot = intelligent design.

That's a straightforward mathematical proof.

"Algebraic Thinking in Adult Education"
National Institute for Literacy, Algebraic Thinking in Adult Education, Washington, DC 20006
http://lincs.ed.gov/publications/pdf/algebra_paper_2010V.pdf

The authors put it bluntly, "Mathematics proficiency is needed to satisfy formal academic requirements for advancement as well as to meet the genuine skill demands of home and work."

Before he retired, the previous principal of the HS where I tutor asked the dean of the local engineering school what he most needed from the HS students. His response: "A mastery of algebra. We can teach them calculus and the rest, but we need them to know algebra when they come through the door."

More from the NIL report:

"Even among those who have finished college and are starting their careers, there is a sobering lack of practical mathematical ability...In fact, 30 percent of students earning two-year degrees and 20 percent of those earning four-year degrees have only basic quantitative literacy skills. These students were unable to complete such tasks as calculating the total cost of office supplies or estimating whether their car has enough gas left to make it to the next gas station."

Here's another view:
http://www.npr.org/templates/transcript/transcript.php?storyId=101298505

Very briefly, he thinks algebra is necessary to use spreadsheets to their full capacity and to do modeling.

Devlin has a very nice blog at:
http://profkeithdevlin.wordpress.com/

I agree with his opinion. Some people think you can just give kids a calculator and that solves the problem. That's silly. I often find kids with fancy, very expensive calculators who 1) don't know how to use them correctly - even for simple functions, and 2) don't know how to tell if they have something approaching a correct answer.

Thats pretty good bill S

every word rhymes along with the song.

common sense dictates that properly using calculators would be the more desirable thing to teach students.

1) when you graduate from all of your schooling , you will not be allowed to use a paper and pencil to do calculations.

2) calculators are never wrong as long as you enter the correct data into them.

3) a combination of how to properly use a calculator and how to properly use equations would deliver more efficient and productive work.

but lets face the facts , theres money in filling a students day with useless courses that he will never use in a job or in his life.

when you think about it , its not the students fault that he cant perform math , its the teaching establishment and the methods and tools that the teachers have and are allowed to teach with that are at fault.

these methods and tools ensure that students will learn at a slower pace , thereby increasing the time period that the students must attend a course of study in order for him or her to receive a diploma.

lets carry this one step further.

if certain useless courses were done away with or not dragged out to infinity over years.

students could be being taught useful courses such as physics
in fact by the time a student leaves high school he could already have 4 years towards his engineering degree.

The problem is that people don't understand enough to apply the equations. I use paper and pencil (or whiteboard which amounts to the same thing) all the time. In real problems, the actual equations need to be derived from given information and the rules of algebra.

Any student who actually understands algebra can easily use the calculator correctly. The problem is that people - all people DO - make mistakes and LOTS of them. This is why one needs to know if an answer makes general sense.

I can see your point, Paul, and I agree with quite a lot of what you say. I don't know about the US, but I know the education system in UK needs a good shake-up. However, as one who had a poor grounding in basic maths, who has had to try to make up the ground later, I can say with certainty that there is nothing like having good foundations on which to build; without them it's hard going.

I have contributed a lot of silly nonsense to this thread, but that's just because I'm that sort of person, not because I think the basic discussion is unimportant.

if the foundation isnt solid anything built on it also isnt solid.

one thing I remember from my grade school days was the repetition.
I dont think that repetition is the key.

I think that as TFF stated the ability to work an equation is a extremely important part of math.

instead of repeatedly making students work equations they should be given word problem situations and they must decide which equation that they should use to solve the particular problem situation.

this would build math skills to a greater extent and the student would walk away with an ability to solve everyday math problems.

just sitting in a class and working 100 adding and subtraction and multiplication and division equations
wont really help as much as a repetitious need to decide which equation to use.

not to mention the boredom that causes many students who learn quickly to loose interest quickly.

solving problems is fun!

the repetitious solving of equations is not.

I agree that word problems are necessary.

Drills are also necessary for some things, although I think this can sometimes be taught through games.

It's not sufficient to learn formulas; one has to know how to combine formulas and manipulate them in other ways.

Nor are feet!!!

Kilowatts used per hour

Would that be to maintain temperatures for growing bananas, or for keeping feet warm?

Check: http://www.wolframalpha.com/input/?i=kwh

And then: http://www.wolframalpha.com/input/?i=watt

in each case, look at the section labeled "basic unit dimensions"

Now, can you tell whether kwh means kw * h or kw / h

its an amount , not bound to any time period.

you can use 50 killowatt hours in 1 sec or in ten years.

for instance the (kWh) time period can be used to determine how long a appliance can operate off of a stored 50 kWh of power.

a refrigerator that uses 280 watts.

would use 280 watts each hour.

how many HOURS could you operate the refrigerator using a 50kWh battery.
50,000 watts / 280 watts (per hour TFF) = 178 hours


if you had an appliance that uses 180,000,000 watts
you could only power that appliance for 1 second with
50 kWh of stored power.