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CCSS.Math:

if we have two groups and in each group I have four so that's one group of four and then here's my second group of four we already know that we could write this as two we could write this as 2 times 2 times 4 which is the same thing which is the same thing as 4 plus 4 notice I have two fours here I have 4 plus another 4 which is going to be equal to well if I have 4 plus 4 or if I have 2 groups of 4 either way I'm going to have a total of 8 things and you see that right over here we have 1 2 3 4 5 6 7 8 things what I want you to do is pause the video now and try to group these same 8 things but to droop it in other ways so that you can represent 8 as the product of whole numbers here I've represented 8 as the product of 2 & 4 2 times 4 is 8 see if you can represent 8 as the product of other whole numbers or as whole numbers in different ways grouping it in different ways so I assume you've paused the video so let's try it out ourselves so one thing we could do we could view this instead of as two groups of four we can view 8 as four groups of two so that's one group of two two groups of two three groups of two four groups of two so we could write down we could write that 4 times 2 4 times 2 is equal to 8 and we could view this is the same thing it's literally four twos we have one two three four twos each of these have two in them so we're going to say 1 2 3 4 2 s 2 plus 2 plus 2 plus 2 is equal to 8 these are both equivalent 4 times 2 literally 4 groups of 2 that's the same thing as taking four twos and adding them together notice we have two twos right over here we added them together one two here we have four twos and we are adding them together one two three four we take our four twos and we add them together how else could we represent eight well we literally could view it as eight groups of one so let's do that so eight groups of one would look like this that's one group of one two three four five six seven eight so we could write this down as 8 times 1 eight times one is once again equal to eight and if we wanted to write this down as repeated addition well this is the reate ones so 1 plus 1 plus 1 plus 1 plus 1 plus 1 let's see this one two three four five six seven eight one two three four five six seven eight one plus one plus one plus one plus one plus one plus one plus one is equal to eight now you might be a little stumped well what's another way of getting to eight well you could literally view it as one group of eight one group of eight so let me do it that way so this is just one one entire group of eight the whole thing the whole thing is a group of eight so we could write this down as one let me scroll over to the right a little bit we could write this down as one times 8 and one times eight is equal to eight and how would we view that well we're not at we only have one eight now we don't have to add that one eight to anything else so we could literally write this down if we wanted to to do it the way we've done the last few we could literally write it down so it's just we just have one eight well one eight is clearly going to just be equal to eight so now let me ask you another question so far we've been focused on each of these groups but what if we actually view this as four groups of eight so what if we actually viewed it as let me let me clear this what if we actually viewed this as four groups of eight then how many things are we actually going to have so let me make this very clear so we have one group of eight two groups of eight three groups of eight and four four groups of eight so we would view this as four times four times 8 or which is going to be the same thing as eight eight plus eight plus eight plus eight for eight what is this going to be equal to and I encourage you to pause the video and figure it out right now well there's a couple of ways that you could have thought about this you could have just literally just counted these or you could say well let's see eight you could skip count by eight 8 16 24 32 or you could have said 8 plus 8 is 16 plus 8 is 24 plus 8 is 32 or you could have literally just counted the triangles here