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so here again with Bill McCollum one of the authors of the Math Standards of the Common Core and I thought today we could think about how the the Common Core thinks about negative numbers and like always I guess it would be interesting as an overview, Bill, you can tell us um how how the Common Core thinks about it especially relative to your sense of previous standards. Well..negative numbers are introduced in a way that is fairly straightforward and unconventional the only comment I would make is that because we've had that emphasis on the number line throughout starting from there I'm putting a whole numbers and by fractions on the number line it's a it's a natural thing to extend that number line where some point and think about what would happen it got numbers that were less than zero and so I think this set up a number line with us all up I Sunday a representation at the member system picture in good shape for representing negative numbers we do start out with this idea that you are negative numbers to have some sort of and motivation exactly what qualities in service that scared about um thinking about parties that have opposite direction tho values I'm such as your temperature below to the wrong things like that right I often say that I have three children a 5-year-old a three year old and a negative three-month-old a my real real-life context clues to the at this point in my life and and because the very high level way that it seems to be split is that in 6th grade you do what you just talk about you you think about the number line you you you associates a meaning with negative numbers like things long temperature thanks to the right or left or something things above and below something to these are all the sixth grade standard right over here and then in at in 7th Grade you actually start to compute with right and your I mean the operations with negative numbers um addition is really a very natural um extension addition on the underlying with positive numbers a when if you have a very solid mutual understanding but additional numbers on the number line additional positive numbers is about putting links together at a edits foundational level that that anytime you want to add two numbers is just like measuring to a certain point on the line when you're measuring it for a good distance to add a second number then it's a pretty easy chip from there to understand that with native number to just go in the opposite direction but it's still a model %uh adding numbers right where you put lakes together except the lights now have a have a positive or a negative writer kinda was more like one dimensional vectors or something exactly that exactly if I could probably use the word vector in seventh grade but it's good it's a good foundation thinking when kids get the high school are probably about vector addition right and you know one one question exist I miss has been curious and even before the concourses and I've always been curious that you know the the idea haven't you know that I have a number line is obviously I'm you choose to children when they're alright but quite young and and a notice what I've turned out that the vid cousins there is a negative numbers don't seem that daunting to them you know specific us decide if directionality you going to the left now that many to the left what's your sense of why his previous standards and also the comic or wait until six great you know why not introduce it in first-grader second well you care I mean the main reason it I think I i think at a sort of just as basic idea of going below zero kids will not really think about you know they're all these videos are capable working with kids in much earlier grades with negative numbers um the main reason not to I would say is that there's a lot of work to be done with fractions its people get find difficult and part of the principal a problem call was to focus on what was most important thing at grade level in the league if that understanding the time to mature and develop I'm not clutter of early grade levels with things that were not necessary at night by a double so I think the main reason to hold office to make sure you have a really solid understanding a positive numbers particularly rack your fractions firm before you stop caring about negative numbers right now that's it there's no harm in having it come up earlier as an extension popping as long as it's not getting any in in in the way of a major work it very I see the senses a.m. you're right I I've observed this with with people earning a fractions is one of those big things in elementary school that your many middle school student even don't have for mastery of it and so you're saying you know especially when their first getting exposed ideas fractions dude in and also simplest way possible so they don't have to worry about you know were hard by the nato numbers and and whatever I mean no sometimes these might think your DVD topics like negative numbers um become an excuse not do the hard work up a great it's a little bit like you know when you're supposed to be getting bitchy helping you find any excuse but bum I mean it so I think we didn't wanna provide those excuses we wanted to say look at the far post a raise case your kid who is getting addition down the focus here great injured by but getting modification infractions down when you know once you done that anything stop the about negative numbers I see is it's like that I think now what is a little bit difficult but the people start to get a hang up his winnings are multiplying all the rules multiplying positive and negative numbers those are very mysterious to some people yup yup yep absolutely and just I guess that at the time we have %remaining% you just pick looking at a couple EastEnders that but don't tell me when I when I first look at the Standard & you know this this idea that recognize opposite sign for the number as indicating locations on opposite sides of the other number I'm probably make sense of a coru単a positive three and you have -3 just like that then there's this kind of implicit are emphasis on recognizing that the that the opposite over numbers is that so you have seen some items and even from the we put on Academy were you know we use the word opposite in I remember the first time that I review look at those items or what thats that's not my natural word to use are you know so with the negative on this number what's black so what's your sense of this you know using the word opposite ha how important it is that is it for people to outnumber that we are well as with all of these language issues if you understand what you mean it doesn't matter what already it is but light which can cause misunderstandings and misunderstanding that can arise from or eating or negative his kids get to college and we'll see negative X mmm and don't think that's native cora alright right chipped a and across Texas native a negative and positive mindset thats gets that word negative used indicate taking the officer classes with the word negative as many actually negative here at the very bottom but he said that the point xxx the main reason from now you know that said if you understand negative x could also be positive then there's no I'm using the word I wouldn't want teachers are instructional materials to speak that way because I think using at work in court this misconception used to use negative into you would prefer that pursuant to more familiarity with the word you know the opposite in a mathematical context at least we were talking about sign is Rai I opposite side and I and goes but you want to go to replace the officer to Parliament negative number of 78 number the parking officer putting that native sorry I'm could right bro number changes its sign right now that that that that make sense the other thing that happens in in six grade is and I guess what is this happening generally aren't we there looking at you know SNC look not only plotting on a on a kind of a one-dimensional number line but flooding in two dimensions is that happening in general in sixth grade at that point and this is a speech senator has that happened already in fifth grade only be parting in corner playing with just the positive quadrant before in fifth grade he wakes getting back to the other up a four-point gain sang israeli preparation for graphing functions I'm for analytic geometry it's a white paper reaching for Obama but you want kids to start getting used to the idea but you can plot are up to conventional coordinate system where I did for me to be negative and my sis is that is al.'s relative to what I remember from school that is earlier than what was to be done arm you know I think it varies I didn't have a solid sense of what the average was I'm and again I guess I would say a I bad it's not you don't want to get carried away with his plate level mmm for example we're not plotting functions arming I I anything except the first quarter and we're not really very naughty going to get a great seven let me start I'm thinking about proportional relationships which is sorta thats you know no credits so to the function concept but was still grapple reporting relationships as relates it been positive qualities just came out first walking I see I C and then there's more is kinda under Settings order that I'm it's a wonderful things in general design principles of the Common Core is you'll see standards in a great double what you really just like thin streams live beginning on that then wiping out into larger rivers in later grade levels and that's an example this is a seed standard if you like arm people aren't always used to thinking that way that stands at RCS terribly only got have to do like every day right now where it's not really that it's just that it's just me you see ID you might as well do it at the same time we've introduced the negative part of a member Lauren model introduced negative I'm axes in a two-dimensional system as well yup make sense and I'm see also in six grade ordering in the idea absolute value is gonna distance to the the right to the left of over up zero and then and of course in seventh grade that's when it really is just all about computation and it's about me it's about extending the operation it's only operations your very cool thanks a bunch okay thanks