Math Magic: Engaging Young Minds with Whole Numbers (1970s)

[Music] work with the operations on whole numbers is at the core of the elementary school mathematics program Readiness for this begins in the preschool years as children learn to use one toone correspondence good morning boys and girls morning how are you fine better take our lunch count now so you have to think did you bring your lunch or you bring your milk or what okay why don't we start out if you need to [Music] buy lunch lunch at school please stand [Music] up will you count with me please 1 2 3 4 5 6 7 8 now if you brought through a bunch so you need to buy would you please stand up milk would you count with me one two 3 4 now if you brought lunch and you brought milk so you need to buy nothing please stand up [Music] research has shown that Proficiency in counting and work with sets facilitates the learning of addition and subtraction and most kindergarten and first grade teachers plan activities which will strengthen this background we know from research that the greatest sources of pupil difficulty with all operations are due to lack of knowledge of basic facts and lack of understandings about the operations we are constantly aware of the need to teach basic facts more effectively some Studies have shown that clarifying the inter relationship of the operations facilitates understanding thus these first graders are focusing on the relationship of addition and subtraction how about doing it with some more people why don't we have Maryanne and Brenda and shoda and Alan please stand now everyone tell me how many people are standing up I want to change that I would like some more we'll ask these two boys to stand up Mark and [Music] Charles we had four + 2 = good well six I don't know if that let's see what we can do with six why don't we have two people sit down again we don't stand up sit down and now we're back to okay thanks you may sit down see this chain of beads I have do you see how long it is how many beads are in it Mike four four beads I don't know if I want this to be four beads long think I like it longer I'll add this many Johnny how many is it six okay how many is just here two two and if I add four plus two I'll have six could someone come and write that equation on the board for me chalkboard [Music] Steve it's going to start with four that we had plus two equals six thanks Steve you know sometimes I change my mind this is probably too long this is probably too long why don't I just take these two off again now where how long is it again Daddy is this what it was like before could someone write that equation of what I just did Maran if you remember to start with six- 2 = good Maran let's read that one together 6 - 2 = how many did I add in this equation I had four and then how many more did I add Charlie two down here I had six and then how many did I subtract Johnny how many did I subtract two so I added two and I subtracted two let's try another one I think I'll just take three who can tell me how many do you want me to add John case could you tell me how many you'd like me to add to this okay just half half them right here is that right Jen now how many Susie good come and WR that addition equation mustle notice the use of varied materials now the teacher has pupils write equations or math sentences on the board maybe could write she went on to have them use individual beads and make up equations Mike do you want to tell us what you did how many did you start with Mike three three and I added three and I have six thanks Mike who wants to write that equation on the board that might Branda maybe you could go over here and try it started with three equal thanks Brenda and Mike then when you had six what did you do um I had three I three and you were back to three would someone like to write that equation daddy now look at those equations here's the one I started with when I began with and I added two I got and then I started with six and I subtracted two and I was back to four just sort of doing something and then undoing it you know I have some equations up here too and I'm have the same one on the flanel board 4 + 2 = 6 6 - 2 = 4 can you see some numbers in there that are the same you seeing these two any that are the same Charlie that's the same these two okay what did you do here what does that say plus two and um minus two Charlie could you join the two that are the same with that piece of felt the same adding then you're subtracting the same number any more than are the same CH what else is the same how about do those two else anything else the same what number is that child practice work is also directed at strengthening awareness of the relationship between the two operations in this lesson with third graders several forms of arrays are included as aids to understanding multiplication notice the application of the commutative property as they rename the number somebody said somebody named this a particular way somebody before and I can't remember whom How Could You Name It 3 * 8 3 * 8 why 3 * 8 yeah because there's there's three sets of eight where are the three sets of eight on on the building okay there's one up another one a third one how else might you name it yes um 8 * 3 why because there's eight and there's three sets all just the opposite the reverse of it all right how else yes 4 * would you be likely to just look at that and name that as 4 * 6 would you name it that way why not why wouldn't you name these as 4 * six or why would you maybe you would why would you where are the six rows oh okay so you could name it that way any other way 4 * six any other way 12 * 2 why because there are 12 sets of two where on the Windows there are two one on top of the other and 12 of those everybody say it all right and if you name it that way what's another way that you could name it then yes two * okay any other way any other way there are other ways look at the blocks on your desk you have been in groups of 24 right there see if you can think out any other ways that you could arrange those to make 24 to show 24 differently and any other way to name 24 other than the ways we have on the board I see somebody who has a pattern over here that's possible let's see how she names it now I don't say that you have 6 * 4 there though I say that you have something else I see that you have that as one factor can you go on from there and name it now another way everybody look at I'll make a very quick drawing of what she has on her desk it's not going to be a very good picture of these arrays but would you agree that's what you have on your desk all right and I said I could see another way that she could name that using three but different from any way we have over here anybody else see it * two [Music] agreed okay see if you can find another way that uses three factors somebody just suggested this too how do you know it's too much right that's the same as saying 12 * 4 because that's four but we already know that that's true therefore that can't be true okay how [Music] else yes or no you're disappearing with this house everybody agree that this is another name for 24 everybody raise your hand if you think this is another name for 24 raise your hand if you don't believe it's the name for 24 why don't you think it's the name for 24 5 * 4 is what if you multiply 20 * 4 what would you get 20 * [Music] 480 and so if we named it this way we'd be all right we take five times 4 plus four but not in a group pattern you know what we've been doing we can also be shown over here on this chart I guess I'm a little tired of hearing the blocks and that's why I want you to look over here right now put the blocks up at the top of your desk and just leave them okay watch what I'm going to do up here what am i showing there yes show 24 24 everybody agreed but I'm showing 24 in here who will come up and prove to me that that's 24 however you can prove it to me do it out loud so I can hear what you're doing 6 8 10 12 13 16 18 20 22 okay how many is it anybody else got a good guess on how many it is like 35 Pro it you got if you can follow where the dots are I'll put my fingerprint in there there's 35 we still have some more so it's not 35 can somebody tell how many it is now yes 42 42 let's everybody finish off here's 35 36 37 38 39 40 41 42 how else could you have told that that is 42 or how else can you tell now that's 42 yes you can take um like six down the side and seven acoss six there and and seven there and six s are 42 very good how else could you tell yes bottom down all right go the opposite way how else could you tell I thought for a minute somebody was going to do that before what happens when you just count across here and you say 1 2 3 4 five six seven what do you know about how many there are in each row seven seven are each line across that way then this must be seven 14 21 [Music] 21 42 all right or how else could you do it you can do it counting by sevens how else can you do it counting by six by going across the this way let's do it 18 24 36 42 42 all right but you know you had trouble in there how could you use this idea over here using this algorithms used to record work with operations cause difficulty for many children the use of multiple approaches allowing pupils to reach Solutions in many different ways enables each child to find a way he understands allowing him to work on his own level underlying each of these is the distributive property of multiplication with respect to addition use of a diagram also facilitates computation of the product there are certain strategies which can be used across all four operations with whole numbers as well as with other rational numbers research shows that to facilitate achievement retention and transfer instruction in mathematics must be meaningful the use of materials is an essential base for developing meaning and understanding [Music] n [Music] n [Music]