#41




Sure, I didn't mean to suggest that was the problem. I was just wondering how his reaction might have been different  if it would have seemed more or less puzzling to him.

#42




I was well into high school before I got my hands on an old engineering textbook with a section on "estimating" with tricks about fractions. For example, each part of 1/6 is 16.67%  but 2/6, 3/6 and 4/6 are all equivalent to more commonly known fractions of 1/3, 1/2 and 2/3. So to know the percentage of all "sixths", you just need to learn 1/6 and 5/6 (at 83.3%). For sevenths, you just memorize the sequence 142857  this is the repeating decimal for sevenths and only the starting digits vary. For example, 1/7 = 0.142857, 2/7 = 0.285714, and so on. For ninths, each ninth equals 0.1111, and for elevenths, each elevenths equals 0.09090909
With memorizing these fractions, it's easy to estimate the percentage of any fraction. We could have been taught this years ago  every time we had a math test, everyone would want to know what percentage grade they got. Teachers didn't particularly score their tests out of even numbers, so if you got 47 out of 57, you could either get a calculator or do some long division. So the estimating tricks involved rounding to even numbers, reducing fractions, and then using those memorized tricks. For 47/57, it's pretty close to 48/58  reduced that to 24/29, which is close to 24/30 = 8/10 or about 80%. Or you can round to 48/56, or 24/28, which reduces to 12/14 then 6/7. Using the trick above, 6/7 = 85.7142%, so we have now established an upper and lower bound for the estimate  split the difference and you get about 82.5%  the real value of 47/57 is 82.456%. It's all simple mental math and you really only ever divide by 2 or 3. 
#43




When I took a technical drawing class in high school, the rest of the class was constantly throughout the period asking me to convert poweroftwo fractions into decimal for them. Which for a relatively small power of two (we rarely had to deal with less than X/32) is actually quite easy to do. Not a long string of numbers like 1/7th. One of them said I must spend hours memorizing every possible fraction. In fact I only knew them from dealing with them in class the same way they had done, except instead of plugging it into a calculator or asking me, I took a moment to figure out how simple it is.

#44




Quote:
Of course 1/3 is more than 1/4..and 30% is more than 40%. Seriously though, I had a fellow student in Drafting/Design and Technology ask me what half of 1/8 is, I told her 1/16 and she insisted that couldn't be true because 16 is twice as much as 8. There is a certain logic to her ignorance. 
#45




And that is why math education should include more than just knowing how to get an answer, but lots and lots of practice, converting math facts into charts, graphs, and other representations, and rote memorization of at least the core math facts required by the decimal system (such as the addition and mulitplication tables up to 10). Your acquaintance could probably have worked through figuring out that 1/16 was the right answer (like Plato teaching the slave to square 2), but in real life most people's math is snap impressions, and it would be helpful to the person and society if people could make better conclusions on such things on the fly (and recognize on the fly when something needs further thought).

#46




As a kid who stuggled in grade school in with math, graphs and charts did not help me. I felt like I was being asked to memorize pretend systems in hieroglyphs that didn't have anything to do with reality. I needed the apple and marble demonstrations. Once I saw the practicality of math, the solidness of it, I believed it in. I even snuck into the kitchen to test the measuring cups. Before that I was a skeptic.

#47




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I ate a restaurant once where a full order of four potato pancakes was $3.99, while a half order of two was $1.75. Quote:

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