advice re: fractions, decimals, percent

from the Comments thread to: What is 10%?


from Steve H:

"What is 10% off? issue."

I've just been going over this with my son. I keep asking him 10% of WHAT NUMBER, exactly? I want him to always know what that amount is. I don't call it the whole because it might be confusing for problems that ask for 10% off of 50% off. The classic problem is the store that tells you that you can get an additional 10% off of the 30% off discount price. The question is WHAT number, exactly, does the 10% refer to?

I also got done telling him that when you see something like 30% in a word problem, you never use the number 30 in the calculations. You have to use either .3 or 3/10. It seemed like a minor point, but I could see it sink in.

In the past, we've talked about 15% tips and how to calculate them, but he needs to see percent problems in all forms and see fractions and decimals as two different forms of the same thing.

Another good problem is to talk about the store owner who buys at the wholesale price and marks up by 100% to get the retail price. The store owner could then have a sale and mark down the goods by 30%. The question still has to be WHAT NUMBER, exactly, does the percent refer to?

Another issue I ran into was that he was thrown a little by things like 125% - percents greater than 100%.


from Joanne Cobasko:

I used many work sheets converting fractions to decimals (by doing the division-since a fraction is just a division problem) and then converting decimals to percents.

The repetition that a fraction is a decimal and a decimal is a fraction, and a percent under 100 is represented by a 2 digit decimal seems to be working well. I did this when I noticed that Saxon was not providing the simple algorithms to solve the fraction of a whole problems (and of course multiplication of decimals and fractions haven't been introduced so I taught that too). I have been teaching ahead of Saxons approach with the actual algorithms. My son hates drawing the pictures- but I have him draw to prove he understands. I taught him to write the equation first and solve the problem then draw the picture.

When Saxon began introducing problems looking for a fraction of a group I taught that the word "of" meant that you had to multiply the fraction (or decimal) by the whole number.

1/2 of 30 = 1/2 * 30/1 = 30/2 = 15
or
50% of 30 = .5 * 30 = 15.0

Start with the simple fractions 1/2, 1/4, 1/3, then work up to the others.

Memorizing that
1/2 =.5 = 50% or
1/4 = .25 = 25% or
1/3 =.333 = 33.3%

We are now approaching doing percents in our head such as
1/8 which is half of 1/4 so
1/8 =12.5% (1/2 of 25%) and that
2/5 = 40% because 1/5=20% so 2 of the 1/5's would be twice as much, so
2*20%=40%

I am hoping this familiarity with decimals, fractions and percents combined with memorization and mental math skills (which Saxon introduced in HS version of 5/4 and higher)will help my son to solve more advanced problems such as the ones you are now presenting to Chris.

[from Catherine: we are doing LOTS of these worksheets - and we need to do mental math, but C. isn't quite "up to that" yet...]


more from Steve H:

I see a clear difference in my son between before mastery and after mastery. Before mastery, he may be able to explain and do a problem eventually, but he doesn't fully grasp the subtleties and variations of what he is doing. After mastery, the process and understanding is automatic.

We've been working on combining plus and minus signs when you add, subtract, multiply and divide. Unfortunately, he is always trying to find a simple pattern that solves the problem. The fault with patterns is that they are based on nothing. There are lots of patterns that can be found and many of them are not helpful at all. I always try to explain things using the basic identities.

One thing we ran into the other day was where does the minus sign belong in a fraction. I told him that you can put the negative sign anywhere you want.

I told him to identify terms and always think of a term or number with a sign in front of it. If you don't see a sign, it's a '+'. I also told him that a minus sign is really a factor of -1.

if you have

3 - 1/2

Then the second term is

- 1/2

or it could be

(-1)(1/2)

or

(-1)/2

or

1/(-2)

You can put the minus sign in front of the number, like

-.5 or -(1/2)

or you can put it in the numerator or denominator. Since the fraction is just a number, you can think of the minus sign in front of everything, but you can also put it into the numerator or the denominator if you want.

He didn't like that idea.

I gave him this fraction.

(-2)/3

I then asked him what

(-1)/(-1)

equals. He hesitated and then asked, "One"?

I said OK, now multiply

(-2)/3 by (-1)/(-1)

to see what you get.

He knows how to multiply numbers with different signs, but he had to think about this. You could see the wheels turning.

I told him that whenever I look at a minus sign, I can see all of the different places I can put it or all of the different ways I can use it.

These things can't sink in without a lot of practice. Mastery provides understanding. It can't be rote. Understanding is not possible without mastery. Finally, mastery and understanding have little to do with pattern recognition.


from instructivist:

[We are now approaching doing percents in our head such as
1/8 which is half of 1/4 so
1/8 =12.5% (1/2 of 25%) and that
2/5 = 40% because 1/5=20% so 2 of the 1/5's would be twice as much, so
2*20%=40%]

This is a great way to learn mental math. I have been doing this instinctively.

Calculating tips of 15% or 20% (service mus really be good) menally should also be child's play. Ten percent of anything is easy. Add half of that and you get 15%. It's baffling that some kids struggle with this.

[1/3 =.333 = 33.3%]

There is a fancy, six-figure word that goes with repeating decimals (the bar on the repeating number or numbers): vinculum. Converting these repeating decimals to fractions is a nice algebra exercise. The number of numbers covered by the vinculum tells you if you need to multiply by 10x, 100x or whatever.

AND:

"Understanding is not possible without mastery."

That's a powerful statement. It should blow the constructivists out of the water who purport to seek "understanding" but disparage mastery with obnoxious phrases like "drill and kill."

AND:

It occurred to me that a calculator is of limited use when trying to figure out if certain fractions are repeating decimals when converted. The calculators I am familiar with do automatic rounding.

I tried 5/7 on my TI-30X IIS and get 0.71. No indication that a repeating decimal is involved. My TI-83 Plus gives me more but also rounds without showing the group of repeating numbers.

I see this as another reason why long division is important. How would calculator-dependent students see that the sequence 714285 repeats, I ask NCTM?

[Catherine again: I've informed C. that he will be doing long division worksheets shortly, and he will be doing them to fluency]


from le radical galoisien:

There is a rough method of deriving a fraction from any arbitrary decimal.
For example, 2/7 is 0.285714286 (etc.) 1 divided by that decimal is 3.5. That is 7/2, or the inverted form of 2/7 ...

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