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Common Core Standards: Multiplikation & Division negativer Zahlen

so I'm here with Bill McCollum one of the authors of the math common core standards and I thought it would be interesting in the last year we talked about have a common core things in general about negative numbers in the sixth and seventh grade I thought would be interesting the dive a little bit deeper in this video on the idea of multiplying and dividing a good numbers which is in some way something rightly straightforward for a lot of students but loved toons never really get the intuition they becomes memorize these negative times the negatives a positive but you know well that up you know this is not the only way to throw tear locke says it's raising the mathematics where you can come up with some concrete metaphor some sort of context which I know explains why rule works the way it goes intensive real wannabes I have to say I've never seen a great metaphor for explaining the rules model plane positive and negative numbers now some other sort of easy to get for example K two times -5 that's like well it in killing 85 it's okay if I passed 85% made a pair but once you get from today rules for example if I want to even understand understand why the album will rule works the other way around it's hard to explain one you real reason that we have the rules we have this because we want the properties of operations to still work the way they used to work when the key property here is the distributive law to just stop fooling around with body things using its beautiful I tried something you know to times five-plus -3 Inc rental cities and then you say well you know I know what type of native tree isn't scary right so the answer is two textures that form okay but now suppose I distribute this app well that is true about five plus two times -3 so I tried a few times dated 3 have to be I am well lemme see two times fighters 10 I so and the axis post to be four to two times in 83 had better be -6 for this to work out in okay already got that little white you can do the same thing if you put in if you see right now it stopped thinking about a negative kill on fight but negative where you can play the same game with bad arm an you will force yourself to conclude but negative through time -3 had better be set it's gonna work and so that's really where these rolls come from right and that's important because when kids get the college then they're seeing things like she times X plus Y or yes Himes by watch TV or something in and then they really need to know that the stupid rule works it know what they did over there XO wired he is positive or negative it still has to worry I'm sorry that's why you warm their rules K make sure to distribute waterworks and in I don't know what to wear you know it's a take kids in seventh grade about this I think I think you should tell me this its it's a pretty sophisticated abstract sort of approach to thinking about the number system is that I want members is going to be a system that follow certain basic properties but that's really where those rules from he's me and what about introducing something like and Andy this is a I just want to get your your thoughts on the if you know if we if we bestow up but about something like itself two times -5 if you said right he said let's do negative two times negative 5 jook when you doing negative something your well that a adding about me that we're actually at that from and um sort of and I guess what I would say is that works if you already have the understanding %uh multiplication by an by negative cue as subtracting the same thing if I'm thinking about it I'm so to me that's could be a little bit circular but if it works it works I mean anything that helps you sort of get a grip on these things another way to do this which is very similar to what you're saying is to think about table bomb a multiplication table a where you have like you know if he has won three times to three times three sorry I was out you know 369 and the fact that I'm going to extend a stable down into the native three times the rate think I'm -1 in at here and if you want that table to have the same rate or marinade namely that I'm you get the right up by three and everything to get alaska goes down by three anything to gain you'll be forced to conclude that three times native cue is by native States and you can play the game where they might be a multiplying by is a negative number and then you will be forced to conclude that I'm it the native plant tomatoes will have to be possible it'll be the same sort of idea but arm but I you'll be York the table being very good he say it so I think that's actually a good way to work with kids and that's very mathematically on us to Nate McLouth if you think was really lying behind your intuition here yeah it was the intuition every time I had one um take the number that's being multiplied by three I have re to the answer right buybacks that's the structure that I'm building in here well why isn't true yeah that strip as in distributive law three projects plus one is 3x what's trace our again in the AM it comes down to that saying it's not okay arm basic structure warning 18 appreciate that basic structure but it's not actually pushing it white um total into their consciousness as much as is as much as the first explanation I K right right now that the the both of you really interesting a big you know might my takeaways if I'm if I'm actually block working with the about myself some threads that would be a really fun conversation at you know just like here's why does this make sense to look at what the let the students right kinda I think my physician run a couple by her story for the playing with mathematics mom rather than trying to come up with some public at a metaphor about taking money he out a bank for like having Nick using at or something like that it's like well let me know losing days anyway I arm about its its it's playing with the numbers as real objects this is really what you want study in sixth grade if you want people to view the number line is a system if you those numbers as objects that they can work on and this is like playing around with numbers and um I think the public if you would like to yeah yeah yup absolutely and on the division side of things I mean how hop once again I'm sure there's the the rule which is very similar to what you get with multiplication the rules obviously unique extend the most gay lyle's easily by just doing division are not upon my return at are you can also view is that however it was related to a modification I'm if I love you POV cue arm as the number that when I mobile quite like you I get p so it's the it's a solution yeah from yeah somewhere that but it takes a second may want to meet members and their but but like -3 over to by occur okay and I am well whatever that number is two times that number is -3 by here okay but now I know my mobile professionals light to is positive native fish native to this number on the left here baby negative arms in other words made it clear to bed at like you can its native three right an right for this is writer that buy a giftly over time i jus I didn't I wouldn't know that from my mobile patient here and so that's why did he that's why make it appear procuring has to be negative wanna dprk and you can play you the are that that that that's really interesting You're no that's that's you know I was if it was -3 over -2 well say like by definition a woman already accused times that number could lead to negative rating because that's what I mean you know if actions can be interpreted as like and stop yes and then you gotta ok I'm getting a negative number the signs my different this was clearly negative right to this one must be positive for this is actually the same thing as now you know i mean la to give us a lot of people but let's leave the club they learn rules about moving native signs around baylon rules about canceling negative science all with all these things both sorta we an easy way to remember what you are fine as long as you have some understanding a to pin it down some you'll see kids during really complicated hey if they have it so there's no money about skipping bike on a member how it works I the I know makes you worry yeah only as a minute ago I had a very good writer with more and more a matter of these 4 p.m. onyx with my weights work it's not grounded some understanding yea yeah Witcher yeah very fragile the concert I guess it over Division one with us this is a robust metaphors if you know if you have negative the same -6 divided by to you go -6 on the number line as the North 3 are and if you divide this yes action by two you're gonna you divide into two sections are negative really I guess I think this is a good light will be there are 93 585 batteries not understand in conventional terms so late model plane by a positive number is fairly easy to work with and dividing by putting it up it's fairly easy to work with its it's when you start providing a model plane by native number that yeah you I some away i'm looking up gets a little bit SAR thanks were great with this was this was really interesting text much bill you