tag:blogger.com,1999:blog-8231784566931768362.post3525355562442113603..comments2023-09-09T08:21:55.454-04:00Comments on MathNotations: If a hen and a half can lay...Dave Marainhttp://www.blogger.com/profile/13321770881353644307noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-8231784566931768362.post-21291851346027100892012-03-31T14:18:57.506-04:002012-03-31T14:18:57.506-04:00I think it is smart. The deeper issue is why does ...I think it is smart. The deeper issue is why does it make mathematical sense to multiply those units:<br /><br />The fundamental rate which we're assuming is constant here is eggs per hen per day or (E/H)/D = E/(H•D). Thus E = k(HD) We used to say that E varies JOINTLY as H and D. Do texts still do that?<br /><br />To determine k, we sub in values:<br />3/2 = k(3/2)(3/2) or k = 2/3. Now we can replace H and D by 3 to obtain E = (2/3)(3)(3)=6. Voila!Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-89865450941345714632012-03-31T14:01:46.393-04:002012-03-31T14:01:46.393-04:00I would teach it as "we're multiplying by...I would teach it as "we're multiplying by hens and days, and get eggs. Set up the ratios appropriately to cancel units." I'm not saying this is smart, but it's my general method!orangemathhttps://www.blogger.com/profile/05099727076265177042noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-53714599353188545052012-03-31T11:51:36.733-04:002012-03-31T11:51:36.733-04:00That's wonderful, Joshua. It is not algorithm...That's wonderful, Joshua. It is not algorithmic, appeals to intuition and is mathematically powercul and generalizes to many variables. You and I know how iften students rely on procedures which causes a loss in confidencr in trusting one's instincts. I tend to use a TABLE approach with 3 columns for those who need a visual ir more structure. <br /><br />Even some younger children are capable of reasoning: <br />Hey, there are twice as many chickens and twice the time so they can lay 2•2 =4 times as many eggsDave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-8231784566931768362.post-12131418436974709002012-03-31T10:23:11.960-04:002012-03-31T10:23:11.960-04:00I would point out that there are TWO things that h...I would point out that there are TWO things that have changed, and encourage them to deal with the changes one at a time.<br /><br />What happens when you double the number of hens?<br />What happens when you double the number of days?Joshua Zuckerhttps://www.blogger.com/profile/04689961247338617418noreply@blogger.com