Posts
The whole family has gotten into the act on the What is 10% off? issue.
Even Ed is now writing math word problems.
This is serious.
Here are Ed's two from this afternoon:
1.
When you're training for a 10K race, you run a series of 1K laps. Your first lap takes you 4 minutes. Each subsequent lap is 10% slower than the last one.
What is your total time for the 10 laps?
2.
If on average you run 1K in 4 1/2 minutes, how long will it take you to run a 10K race?
I realize these two questions are logically contradictory, but I'm not going to worry about that for now. Ed says sports are a great source of word problems, and he's right.
I decided today to start giving C. the same problem written as a percent & as a fraction.
Then, having fixed on this plan, I decided to throw in a whole-part problem (something he's never done before) to boot:
1.
finding the parts when the whole is given [note: I labeled the problem with these words]
Christopher wants to buy a $50 video game for 20% off.
By what dollar amount is the price reduced? ___________
What will Christopher pay? ___________
Draw and label a bar model of the problem.
Then write the equation and solve it.
2.
finding the parts when the whole is given
Christopher wants to buy a $50 video game for 1/5 off.
By what dollar amount is the price reduced? ___________
What will Christopher pay? ___________
Draw and label a bar model of the problem.
Then write the equation and solve it.
3.
finding the parts when the whole is given
Christopher wants to buy a $50 video game for 1/5 off.
By what dollar amount is the price reduced? ___________
What will Christopher pay? ___________
Draw and label a bar model of the problem.
Then write the equation and solve it.
This is one of those moments where you see exactly how valuable an experienced teacher at the top of his/her game is to kids learning math. Or to kids learning anything.
Because I've worked my way through so much of Saxon, Singapore, & "Russian Math," I have pretty good pedagogical content knowledge. For instance, I now know what "part-whole" versus "whole-part" problems are, a concept I'd never heard of before.
But I still lack "kids-learning-math" knowledge.
I don't have a good sense of the proper use of contrast and comparison in instruction (i.e. having C. do the same problem framed as percent and fraction - good idea or not?); nor do I have a sense of how long it should take for a student C's age to learn these things, which means that when C. doesn't seem to be learning what I'm teaching I can't tell whether he needs more practice or I need to teach differently or both.
I'm making all my mistakes with my own kid.
Still.
By the end of this summer - preferably by the end of tomorrow - he is going to know what 10% off is or I am going to die trying.
hyperspecificity in autism
hyperspecificity in autism and animals
hyperspecificty in the rest of my life
hyperspecificity redux: Robert Slavin on transfer of knowledge
Inflexible Knowledge: The First Step to Expertise
Devlin on Lave
rightwingprof on what college students don't know
percent troubles
what is 10 percent?
birthday and a vacation
Subscribe to:
Post Comments (Atom)
18 comments:
Looking at the very first question, I think your son might have an instinctive notion of the logarithm.
If the number of races were represented by x, and the total time taken so far by y, then the resulting graph would be rather logarithmic in growth, wouldn't it? And there's no limit to y as x approaches infinity, because the total time taken would never converge (10% reduction is too little).
Besides, even if you weren't thinking about Zeno's paradox and infinite series, doing the first problem conveniently requires the use of exponents. (You could of course, calculate every race if you wanted to...)
Well not to divert this off-topic. But unlike many of the other questions, you could exploit this question for something beyond ratios.
How are the questions contradictory anyway?
I just mortifyingly realised that I may have misidentified someone. But still, it only takes a bit of tweaking to make the first question a precal problem!
"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%.
Steve--
I like your comment. It focuses on the relevant point--10% of what.
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.
"...memorization and mental math skills ..."
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.
[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.
"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."
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?
"I tried 5/7 on my TI-30X IIS and get 0.71"
I had the FIX setting. Sorry. Now I am floating but I sill have to guess that the entire sequence 714285 will repeat.
Well the general handheld calculator I suppose. I supposed an improvised calculator (like the one on google) would be able to give fractions, with some minor tweaking.
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 ...
How are the questions contradictory anyway?
The first question implies that your speed isn't constant; you get slower as the race goes on.
The second question, as written, pretty much implies that your speed is constant across every km you run.
I think.
If the next lap is 10 percent slower, does it mean that the time is 10 percent greater, or does it mean that the speed is 10 percent less?
Well, seeing as 0.9 / 10 < 1/11, I'd be more inclined to go with the former.
If the next lap is 10 percent slower, does it mean that the time is 10 percent greater, or does it mean that the speed is 10 percent less?
Am I writing these things wrong?
(Still difficult for me to write clearly about math, I fear.)
The idea is that each lap is 10% slower than....hmmm. Not sure than what.
Than the immediately preceding lap or than the first lap?
yikes
Eeek, re-reading my comment, I think I wanted to write "latter".
But now that I actually do the problem to find it out, I realise that "former" would be a better option. If it is literally 10% slower (the latter option), then the second lap would have a speed of 13.5 km/h, etc. (the first one was 15 km/h). The third lap 12.15 km/h. So the time increases with the pattern of 4 minutes, 4.444(...) minutes, 4.938(etc.) minutes, for the 1st, 2nd and 3rd laps.
Which is a messy question. So a better question would have the time increase by 10% each time instead I suppose.
"Than the immediately preceding lap or than the first lap?"
Well, I interpreted it as each immediately preceding lap, since that what was what the problem seemed to describe. But if a middle schooler could do this problem, then he could do compound interest problems.
August 3, 2007 4:59 PM
Did you have success?
If not, I would ask if your child has had the 'aha' moment of understanding what a percent truly is. For us, the grokking resulted from Ed Zaccaro's "Primary Grade Challenge Math" when E.Z. took an object and divided it into 100 pieces...and the pieces were not chopped up identifiable pieces of the object...they were round spheres as if the object had been treated like a lump of clay and extruded into 100 equal pieces. No array of squares, no wedges of pie, just 100 equal pieces. That followed by the literal definition of percent => 'per' = 'for each' and 'cent' = 'one hundred' and some visualization of percents of other irregular objects nailed it for my visual/spatial kiddo.
Manipulating the symbols and using mental math are additional skills as is knowing that the word 'of' means 'mutiply'. I had good results using Singapore Math afterwards. If you need seperate sheets on some of the prerequisite arithmetic skills, Houghton Mifflin has wkshts here:
http://www.eduplace.com/math/mw/g_6.html Click on leveled practice; it takes some searching if you don't have the text, but it is worth the time if you can use free problem sets or you have a v/s learner who gets more out of reading than listening. I found the reteach sheets are clear to the child and the practice and h.w. sheets useful for exercises.
Post a Comment