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Hey dogmom, if you're feeling patient, I want to look at the 25x4=4x25 thing from the pics of you thread.
(And if you feel really patient, I'd be totally up for having a continued conversation in PM or email or whatever.)
Anyway:
When you multiply by 4, you're basically saying "I have 4 of these." In this case, I have 4 things worth* 25 each.
*(doesn't have to be about worth. could be about size or weight or time or anything really. I'll just say worth for now.)
Or, you could look at it the other way, and say you're multiplying by 25: "I have 25 of these." From this perspective, you have 25 things worth 4 each. (I'm abandoning the idea of cents here, just focusing on the numbers for the moment.)
But you said you think in pictures. If you're up for it, will you get a piece of paper? Lined paper is good: you can start writing numbers down the side of the page:
so you have 25 of something. 25 lines, 25 boxes of candy, 25 alldeaf posts, I don't care what. But there are definitely 25 of them.
Then draw a vertical line next to your numbers
Then draw more vertical lines, dividing up each row into 4 sections
Then at the top of each section, write the numbers 1, 2, 3, 4.
So now you have a bunch of boxes. you can look at them as being in groups. You can take the perspective that each column is a group (so you have 4) or you can think of each row as a group. (so you have 25) If there are 4 groups, each group has 25 things in it. (or a value of 25 or however we want to interpret that.) If there are 25 groups, then each group has 4 things in it.
I want you to count the total number of things, in all the groups, altogether. I want you to do it twice (So you'll need two sheets of paper.) Everything up to this point you do the same on each paper.
But now you start writing numbers in the boxes. First start counting across 1,2,3,4, then drop to the next line and continue 5,6,7,8, etc. You should get to 100 by the bottom corner.
On the other paper, count downwards like in the picture.
Again, you should get to 100. You can look at one picture as saying "if I have 4 of these things, each thing being worth 25" and the other one as saying "I have 25 of these things, each thing being worth 4." and either way, altogether, you have a total (total worth, total boxes, however you want to think of it) of 100.
Again, you don't have to think in terms of worth. You could say 25 containers, each with 4 boxes inside, or 4 containers, each with 25 boxes inside. Either way, there are 100 boxes. Or cents. Or pieces of chocolate
Or even without writing down all those numbers, no matter how you look at it, the number of boxes is the same. So no matter what two numbers you multiply, you could draw a grid like this and look at it both ways. No matter what the two numbers are one way of multiplying (I'll say axb) has to be the same as multiplying the other way (=bxa). Like 2x3 and 3x2 or any numbers you want to pick.
(And if you feel really patient, I'd be totally up for having a continued conversation in PM or email or whatever.)
Anyway:
When you multiply by 4, you're basically saying "I have 4 of these." In this case, I have 4 things worth* 25 each.
*(doesn't have to be about worth. could be about size or weight or time or anything really. I'll just say worth for now.)
Or, you could look at it the other way, and say you're multiplying by 25: "I have 25 of these." From this perspective, you have 25 things worth 4 each. (I'm abandoning the idea of cents here, just focusing on the numbers for the moment.)
But you said you think in pictures. If you're up for it, will you get a piece of paper? Lined paper is good: you can start writing numbers down the side of the page:
so you have 25 of something. 25 lines, 25 boxes of candy, 25 alldeaf posts, I don't care what. But there are definitely 25 of them.
Then draw a vertical line next to your numbers
Then draw more vertical lines, dividing up each row into 4 sections
Then at the top of each section, write the numbers 1, 2, 3, 4.
So now you have a bunch of boxes. you can look at them as being in groups. You can take the perspective that each column is a group (so you have 4) or you can think of each row as a group. (so you have 25) If there are 4 groups, each group has 25 things in it. (or a value of 25 or however we want to interpret that.) If there are 25 groups, then each group has 4 things in it.
I want you to count the total number of things, in all the groups, altogether. I want you to do it twice (So you'll need two sheets of paper.) Everything up to this point you do the same on each paper.
But now you start writing numbers in the boxes. First start counting across 1,2,3,4, then drop to the next line and continue 5,6,7,8, etc. You should get to 100 by the bottom corner.
On the other paper, count downwards like in the picture.
Again, you should get to 100. You can look at one picture as saying "if I have 4 of these things, each thing being worth 25" and the other one as saying "I have 25 of these things, each thing being worth 4." and either way, altogether, you have a total (total worth, total boxes, however you want to think of it) of 100.
Again, you don't have to think in terms of worth. You could say 25 containers, each with 4 boxes inside, or 4 containers, each with 25 boxes inside. Either way, there are 100 boxes. Or cents. Or pieces of chocolate
Or even without writing down all those numbers, no matter how you look at it, the number of boxes is the same. So no matter what two numbers you multiply, you could draw a grid like this and look at it both ways. No matter what the two numbers are one way of multiplying (I'll say axb) has to be the same as multiplying the other way (=bxa). Like 2x3 and 3x2 or any numbers you want to pick.
Botti TRUE
also to tuatara and respectyoda
