tag:blogger.com,1999:blog-7691251033406320222.post29721889537997565..comments2024-03-26T04:19:38.862-07:00Comments on kitchen table math, the sequel: the hard kind of percent problemCatherine Johnsonhttp://www.blogger.com/profile/03347093496361370174noreply@blogger.comBlogger36125tag:blogger.com,1999:blog-7691251033406320222.post-21109383861189314802008-02-15T12:13:00.000-08:002008-02-15T12:13:00.000-08:00SusanS--the shooter was a graduate of NIU, and was...SusanS--the shooter was a graduate of NIU, and was attending graduate school at the University of Illinois. The media is reporting that he had been "off his meds" and was behaving erratically for the last several weeks. No real motive has yet emerged as to why he targeted NIU in general, and that class in particular. <BR/><BR/>Our neighbor is a senior at Northern; he's okay, but his mother is (naturally) pretty badly shaken up. So many of our university's students have friends and family at NIU, so our student body is pretty shaken up as well.Karen Ahttps://www.blogger.com/profile/06132085963945807124noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-71066302875427632082008-02-15T08:37:00.000-08:002008-02-15T08:37:00.000-08:00I would like to add an increment to the discussion...I would like to add an increment to the discussion of teaching percent problems.<BR/><BR/>(1) The discussion of partitive and so on seems like jargon on the not-helpful side, but I have not read all the old posts, so .... <BR/><BR/>(2) but still, I think it is a mistake to think that there are really three kinds of percent problems, at least for beginners. For beginners I like to start at the beginning every time (which obscures the difference among the types of problems until the end game), which is a setup like this:<BR/><BR/>Let P be the price of a shirt. The total cost is P plus tax, which is a fraction of P. We know the total is 52.50. We also know the fraction, the tax rate, to be 0.05, so the tax is P x 0.05. The situation is then<BR/><BR/>total = 52.50 <BR/> = P + P x 0.05<BR/> = P x ( 1 + 0.05 )<BR/> = P x 1.05<BR/><BR/>and now P = 52.50 divided by 1.05, which I'm guessing will be done with a calculator ...?<BR/><BR/>This is what I call starting from the beginning. If you knew the price P, but not the total, the total is thus calculated. (It seems like the hardest problem would be to find the tax rate if that were not known, since you have to remember to subtract 1 -- you want 0.05, not 1.05).<BR/><BR/>There seem to be three critical prerequisites. First, do students understand how and when to introduce a variable ("Let P be the price of the shirt"); second, do they understand distribution, which is critical in going from <BR/><BR/>P + P x 0.05<BR/><BR/>to<BR/><BR/>P x 1.05 ;<BR/>and third, do they understand that undoing a product requires division?<BR/><BR/>If any of these three issues are shaky, it would seem that you need to do everything you can to shore up that understanding immediately. You can use percent problems as a setting for all three issues, but your priority should be teaching use of variables, distributivity, and dividing to solve T = P x 1.05 for P.<BR/><BR/>The poster, Catherine Johnson, set up the problem like this:<BR/><BR/>"let x = price of shirt<BR/>1.05x = 52.50<BR/>52.50 ÷ 1.05 = x<BR/>50 = x"<BR/><BR/>I think that introducing 1.05 directly (aka the shortcut) should only be done when students have done so many problems, and seen the use of distribution yield 1.05 from 1 and 0.05 enough times that they do it themselves at the outset.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-77621441737705069012008-02-15T06:53:00.000-08:002008-02-15T06:53:00.000-08:00Karen,I know. I saw it before all of the Northern ...Karen,<BR/><BR/>I know. I saw it before all of the Northern IL stuff started happening. It might not be as funny now, but I think all of us have wanted to do that to our computers. Odd how all of the observers just watch and even go back to work. <BR/><BR/>I'm in the Chicago area, too. It makes you want to keep the kid home from college, as well. Jeez.<BR/>What is going on? They're reporting that the shooter was a "stellar student."<BR/><BR/>Susan S.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-86824526437031772592008-02-15T06:23:00.000-08:002008-02-15T06:23:00.000-08:00SusanS--I just watched the video. Thank you for a...SusanS--<BR/><BR/>I just watched the video. Thank you for a much-needed laugh. My heart is breaking as a result of the latest campus shootings at Northern Illinois; it hit just a little too close to home.Karen Ahttps://www.blogger.com/profile/06132085963945807124noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-57528557203822596932008-02-14T20:15:00.000-08:002008-02-14T20:15:00.000-08:00I probably shouldn't weigh in here because I just ...I probably shouldn't weigh in here because I just got back from teaching a four hour math class, but I can't resist it.<BR/><BR/>Remember your algebra 1/b *a = a/b = a(1/b)<BR/>With numbers on a more elementary level, consider 3/4.<BR/>With a ruler, draw a line 3" long.<BR/>Without mmeasuring try your best to divide that line into 4 equal part. Now measure each part with the ruler. Each part will meaure 3/4 of an inch. So 1/4 of 3 is 3/4.<BR/><BR/>Now draw a line 1" long and divide it into 4 parts. 3 of those parts is 3 * 1/4 or 3/4. That's the second way of looking at the fraction 3/4. <BR/><BR/>Either way 3/4 is a definite number on the number line.<BR/><BR/>As for the 3 problems, I would say they all very nicely can be set up as proportions and solved the same way.Hypatiahttps://www.blogger.com/profile/07259464372533861186noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-11112892389888532192008-02-14T19:29:00.000-08:002008-02-14T19:29:00.000-08:00So now we've got (8 /3 ) /5.That is: (8 divided by...So now we've got (8 /3 ) /5.<BR/>That is: (8 divided by 3) divided by 5.<BR/><BR/>So how do we handle this?<BR/><BR/>We go back to the earlier discussion of division:<BR/>q = m /n means there is some number q where m = q * n.<BR/><BR/>So 8 divided by 3 means there is some number q where 8 = q * 3. q is 8 /3.<BR/><BR/>And now, we divide that q by 5.<BR/>so NewQ = q / 5.<BR/>But that just means q = 5 * NewQ.<BR/><BR/>So the original division of 8 by 3 means there was some q that when multiplied by 3 gives you 8. And now we find that there's some newq that when multiplied by 5 gives you 8 / 3.<BR/><BR/><BR/>So 8 = q * 3. And q = 5 * NewQ.<BR/>So 8 = 5 * NewQ * 3. Rearranging,<BR/><BR/>8 = 5 * 3 * NewQ.<BR/>8 = 15 * NewQ.<BR/><BR/>NewQ = 8 /15.<BR/><BR/>We could also have done it as <BR/>Q = (8/3) / 5.<BR/> (8 /3 ) / 5 means there is some number q such that <BR/>8/3 = Q * 5.<BR/><BR/>And 8/3 is a quotient that we can rewrite too.<BR/>Q_2 = 8/3.<BR/>That just means there is some number Q_2 such that<BR/>8 = Q_2 * 3.<BR/><BR/>So we could have said:<BR/>Q_2 = Q * 5<BR/>and <BR/>8 = Q_2 * 3 = Q * 5 * 3.<BR/><BR/>so 8 = Q * 15.<BR/><BR/>What's Q again? It's the quotient of 8/ 3 /5.<BR/><BR/>That's the same as saying<BR/>8 = Q * (3 * 5).<BR/><BR/>this is not easy stuff, and I probably could have done it better. Anyone else want to try?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-26445668576745628352008-02-14T19:09:00.000-08:002008-02-14T19:09:00.000-08:00--.... I really had problems (and may still have p...--.... I really had problems (and may still have problems) understanding why fraction multiplication means that you multiply the numerator by the numerator and the denominator by the denominator -- what is going on there????<BR/><BR/>let's solve the first part first.<BR/><BR/>2/3 * 4/5 <BR/>is the same as saying<BR/>(2 divided by 3) * (4 divided by 5) =<BR/>(2 / 3) * (4 / 5) <BR/>don't think of these as fractions, think of them as numbers with operations in between. We could re group this as <BR/>((2 / 3) * 4) / 5<BR/>and we could regroup this again as<BR/>((2 * 4) / 3 ) / 5<BR/><BR/>so that's the same as seeing that first we multiplied the top of the fractions together and then we divided by one bottom and then divided by the other.<BR/><BR/>that's why you multiply the tops--because the associativity property and commutative properties let you do this.<BR/><BR/>And once you see that you've got <BR/>((2 * 4) = 8, what's remaining is<BR/>8 / 3 / 5<BR/>(cont).Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-53701106892212522572008-02-14T14:12:00.000-08:002008-02-14T14:12:00.000-08:00She says that to the best of her knowledge no one ...<I>She says that to the best of her knowledge no one knows how to teach "critical thinking," "higher order thinking," and, I assume, "knowledge transfer" and "generalization,</I><BR/><BR/>But that doesn't stop them from trying. In the second grade.<BR/><BR/>This is off topic, but if you need a laugh about now, try this link. It's worth it. I can't watch it without tears.<BR/><BR/>http://glumbert.com/wii/view.php?name=baddayoffice<BR/><BR/>Susan S.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-87407949458230900062008-02-14T14:09:00.000-08:002008-02-14T14:09:00.000-08:00"Does he know 9/10 * 10/9 = (9 * 10/9)/10 because ..."Does he know 9/10 * 10/9 = (9 * 10/9)/10 because 9/10 is the same as 9 divided by 10, and you can change the order of operation of 9/10 * 10 to be 9 * 10 / 10, etc?"<BR/><BR/>I remember thinking that the basic identities were all pretty trivial and stupid.<BR/><BR/>a*b = b*a<BR/><BR/>1*a = a<BR/><BR/>Big Deal!<BR/><BR/>However, you need to see these identites in use. You need to see them over and over again in real applications. Everything has to be seen in terms of the basic identities.<BR/><BR/>Just the other day I had to have a talk with my son who had the following to solve:<BR/><BR/>15/4 * 32/5<BR/><BR/>He was going to multiply 15*32 and 4*5. He couldn't see that he could rearrange it to 15/5 * 32/4, reduce, and then multiply.<BR/><BR/>a*b = b*a<BR/><BR/>It IS a big deal, and it's not trivial.SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-41345963104951792992008-02-14T13:58:00.000-08:002008-02-14T13:58:00.000-08:00Does he know 9/10 * 10/9 = (9 * 10/9)/10 because 9...<I>Does he know 9/10 * 10/9 = (9 * 10/9)/10 because 9/10 is the same as 9 divided by 10, and you can change the order of operation of 9/10 * 10 to be 9 * 10 / 10, etc?</I><BR/><BR/>I'm going to say 'no' to that.<BR/><BR/>I remember spending a VERY long time wrestling with slightly more elaborate versions of this back when I first started teaching C. math....<BR/><BR/>But I'm afraid I can't reproduce those traumas now.<BR/><BR/>Let's see.... I really had problems (and may still have problems) understanding why fraction multiplication means that you multiply the numerator by the numerator and the denominator by the denominator -- what is going on there????<BR/><BR/>I felt that invert-and-multiply as the means of dividing a fraction by a fraction worked by magic.<BR/><BR/>That finally got cleared up when I came across a homeschool page showing the division of a fraction by a fraction as a manipulation of a complex fraction.<BR/><BR/>Obviously, to change the denominator to 1 you had to multiply the denominator fraction by it's reciprocal.<BR/><BR/>That was a revelation.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-19302388589144756022008-02-14T13:57:00.000-08:002008-02-14T13:57:00.000-08:00Let me add that it's OK to have "cross-multiplicat...Let me add that it's OK to have "cross-multiplication" come out of your mouth if your brain is really thinking "identity".SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-61084829742427070802008-02-14T13:53:00.000-08:002008-02-14T13:53:00.000-08:00"...he's hung up on cross-multiplication."Warning!..."...he's hung up on cross-multiplication."<BR/><BR/>Warning! Warning! Warning!<BR/><BR/>There is no such thing as cross-multiplication!!! It's not an identity.<BR/><BR/>OK. I'm getting carried away. We've been here before. You know I think it causes way too many misunderstandings. The classic is when a student tries to solve:<BR/><BR/>X + 2/3 = 4/5<BR/><BR/>Students see those two fractions on either side of an equals sign and their brain screams CROSS-MULTIPLICATION!SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-41773165771549217642008-02-14T13:52:00.000-08:002008-02-14T13:52:00.000-08:00I don't think 1.05x = 52.50 is an example of the s...<I>I don't think 1.05x = 52.50 is an example of the shortcut method. You need to write it that way to make the equation true and to solve this percentage problem algebraically.<BR/><BR/>The shortcut method would apply to problems like this. A shirt costs $25.00. The sales tax is 12%. What is the total cost. 25 x 1.12.</I><BR/><BR/>That makes sense.<BR/><BR/>I can say the same of 25 x 1.12, though. <BR/><BR/>Until C's 5th grade math teacher showed me this I'd never thought of it.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-43077285933926102492008-02-14T13:50:00.000-08:002008-02-14T13:50:00.000-08:00does he know that 90/90 is 1?yesdefinitelydoes he ...<I>does he know that 90/90 is 1?</I><BR/><BR/>yes<BR/><BR/>definitely<BR/><BR/><I>does he know that if he computes it? Does he know that 9/10 * 10/9 =<BR/>9*10 / 10*9</I><BR/><BR/>yes<BR/><BR/>Interestingly, I find that the commutative property seems to be somewhat "natural" or "obvious" to students....<BR/><BR/>I think.<BR/><BR/>The identity properties aren't so obvious (that's my impression).<BR/><BR/>90/90 = 1 is "less obvious" than 90*10 = 10*90.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-79159545547296320632008-02-14T13:48:00.000-08:002008-02-14T13:48:00.000-08:00I find it useful to emphasize that "percent" is an...<I>I find it useful to emphasize that "percent" is an abbreviation of "per centum" or "per one hundred".</I><BR/><BR/>You know what??<BR/><BR/>That is a great idea.<BR/><BR/><I>Thank you.</I><BR/><BR/>I find this whole realm fascinating (obviously).<BR/><BR/>I've been reading Vicki Snider's book advocating the development of a science of teaching (which would mean actually paying attention to the science of teaching we already possess).<BR/><BR/>She says that to the best of her knowledge no one knows how to teach "critical thinking," "higher order thinking," and, I assume, "knowledge transfer" and "generalization."<BR/><BR/>I think she's right (although I have yet to read Arthur Whimbey's books. Whimbey claims you can teach people to think.)<BR/><BR/>This is why I <I>must</I> get around to writing up the cumulative practice study. <BR/><BR/>That study set out to see if mathematical problem solving could be taught.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-27064674835404483032008-02-14T12:59:00.000-08:002008-02-14T12:59:00.000-08:00"I'm guessing this goes back to the "oneness" of 1..."I'm guessing this goes back to the "oneness" of 100 percent"<BR/><BR/>I find it useful to emphasize that "percent" is an abbreviation of "per centum" or "per one hundred".<BR/><BR/>Thus 5% is "five per one hundred", which is often written as "5/100". To cement this, you might want to discuss "mill levies" in property taxes. 19 per mill is the same as 1.9 percent.<BR/><BR/>This helps to illustrate that the definition of "percent" is not a unique thing, but that it exists on a spectrum. Specifically, it means that a percent is just a fraction.<BR/><BR/>ps. There is a character for "per mill" -- ‰.Doug Sundsethhttps://www.blogger.com/profile/01848091504066560951noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-87727001801737819452008-02-14T12:57:00.000-08:002008-02-14T12:57:00.000-08:00oh, back to the "does he know 9/10 * 10/9 is 1"doe...oh, back to the "does he know 9/10 * 10/9 is 1"<BR/><BR/>does he know that 90/90 is 1?<BR/><BR/>does he know that if he computes it? Does he know that 9/10 * 10/9 =<BR/>9*10 / 10* 9, and 9* 10 = 90, and 10*9 = 90, and wow, 10*9=9*10 because it's always true in the natural numbers that A*B = B*A?<BR/>Does he know 9/10 * 10/9 = (9 * 10/9)/10 because 9/10 is the same as 9 divided by 10, and you can change the order of operation of 9/10 * 10 to be 9 * 10 / 10, etc?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-9824008800486512872008-02-14T12:48:00.001-08:002008-02-14T12:48:00.001-08:00what is cross multiplication?what is cross multiplication?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-40911876243732090322008-02-14T12:48:00.000-08:002008-02-14T12:48:00.000-08:00maybe you should let me EDIT it first! :) at least...maybe you should let me EDIT it first! :) at least to put together the decimal and percentage bits...Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-61006419919936905532008-02-14T12:44:00.000-08:002008-02-14T12:44:00.000-08:00[let x = price of shirt1.05x = 52.5052.50 ÷ 1.05 =...[let x = price of shirt<BR/>1.05x = 52.50<BR/>52.50 ÷ 1.05 = x<BR/>50 = x<BR/><BR/>price of shirt = $50<BR/><BR/>I have to include this for people like me: the reason you take 1.05x is that it is a shortcut.]<BR/><BR/>I don't think 1.05x = 52.50 is an example of the shortcut method. You need to write it that way to make the equation true and to solve this percentage problem algebraically.<BR/><BR/>The shortcut method would apply to problems like this. A shirt costs $25.00. The sales tax is 12%. What is the total cost. 25 x 1.12.Instructivisthttps://www.blogger.com/profile/01652458042291988959noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-13301768473583637262008-02-14T12:41:00.001-08:002008-02-14T12:41:00.001-08:00Cross-multiplication is incredibly useful but it i...Cross-multiplication is incredibly useful but it is a real "conversation stopper."Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-79314947700518776552008-02-14T12:41:00.000-08:002008-02-14T12:41:00.000-08:00This is difficult because the 100% number is not g...<I>This is difficult because the 100% number is not given. But, hopefully, you see that:<BR/><BR/>1.05 * X = 52.50</I><BR/><BR/>Right -- that's one of the things that is VERY hard to see.<BR/><BR/>I'm guessing this goes back to the "oneness" of 100 percent....as well as to the fact that in daily life we rarely talk about percentages greater than 100%. (At least, I rarely do.)<BR/><BR/>Chris is <I>starting</I> to see this reasonably often. (As I say, I didn't see it until his 5th grade teacher taught it to me!)<BR/><BR/><BR/><I>Remember that I like to use formulas, not proportions or charts. Maybe it's because I automatically see an added 5% tax as a multiplier of 1.05 times the 100% number. Depending on the problem, I either see the percentage all by itself or added/subtracted from one.</I><BR/><BR/>Right - last night I was trying to get C. to "see" this --- to see that we're talking about a formula or an equation, not a chart or a proportion.<BR/><BR/>I'm also TRYING to get him to see that the formula is the next step in the proportion.<BR/><BR/>Needless to say, he's hung up on cross-multiplication.<BR/><BR/>I've got to get Allison's explanation pulled together & put up front.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-62173980002658231942008-02-14T12:23:00.000-08:002008-02-14T12:23:00.000-08:00Wow! That might be your longest post.I haven't had...Wow! That might be your longest post.<BR/><BR/>I haven't had a chance to read it all, but I am interested. I vaguely remember having some trouble with percents, but after some point, I never had trouble again. I will think about it some more, but it seems that all of my problems went away when I focused on figuring out what number is the 100% number. <BR/><BR/>"A shirt sells for $52.50 including a tax of 5%. What was the original price of the shirt?"<BR/><BR/>This is difficult because the 100% number is not given. But, hopefully, you see that:<BR/><BR/>1.05 * X = 52.50<BR/><BR/>Remember that I like to use formulas, not proportions or charts. Maybe it's because I automatically see an added 5% tax as a multiplier of 1.05 times the 100% number. Depending on the problem, I either see the percentage all by itself or added/subtracted from one.<BR/><BR/>If you think of the equation above, you could have a problem statement where any one of the three numbers is unknown.<BR/><BR/>We got into this a little bit a while back on KTM. You can also have trickier problems, like:<BR/><BR/>A store discounted a $100 jacket by 30%, but it didn't sell. They decided to take 20% more off of the price. What is the final sale price?<BR/><BR/>Of course, the big question is what is the 100% number for the second 20% off, the original retail price, or the discounted 70% price?SteveHhttps://www.blogger.com/profile/03956560674752399562noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-45168011680618190242008-02-14T12:17:00.000-08:002008-02-14T12:17:00.000-08:00ARRGH! I made a mistake, and probably confused you...ARRGH! I made a mistake, and probably confused you more! <BR/><BR/>See, this is why ALL teaching should be Direct Instruction, with SCRIPTS that you don't deviate from.<BR/><BR/>Here goes:<BR/><BR/>fractions are NUMBERS. They exist on the number line. <BR/><BR/>Decimals are a special type of fraction: they have a 100 in their denominator.<BR/><BR/>PERCENTAGES are numbers, represented as a decimal, but they are only used to refer to a MULTIPLE of a number. they are "ths" of something. so 5% of x is 5/100ths of x.<BR/><BR/>5% means 5% OF SOMETHING. A decimal is just .05, but 5% refers to .05, or 5/100ths OF something.<BR/><BR/>Now, what is multiplication? It's grouping a bunch of numbers together conveniently so you don't have to add them.<BR/>So, 24 = 3 * 8 is the same as 8 + 8 + 8. We say that there are 3 8's. The three is the number of multiple 8s we've got.<BR/><BR/>And when we say we've got "3 of these 8s", we see the OF, and we think , oh right, we could add 3 groups of 8, or we could just multiply 3 * 8.<BR/><BR/>same with percentages. they are multiples of something. 5% of 100 is saying "we've got 5/100ths of 100" and we know that when you've got a number of somethings, you can multiply to figure out the total count of the somethings.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-55553813832068586152008-02-14T12:14:00.000-08:002008-02-14T12:14:00.000-08:00PISSED OFF TEACHER has a post on geometry teachers...PISSED OFF TEACHER has a post on geometry teachers in h.s. I've been meaning to get a link to.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.com