8 replies on “Multiplication Issues (from Khan Academy)”

They seem to think that every number is the product of two distinct numbers. I’d look for a way for them to see this visually—perhaps the classic of having them make a box go 6×6 dots and counting the dots, then do the same thing with a 12×3 box.

It’s also a great chance to start to show them rarely is math about the “one way”—there are many things you can multiply together to get 36, and in the future, they’ll find that there are multiple ways to solve just about every problem they will encounter.

My mind goes to the idea that 36=2×3×2×3 but I honestly am not sure how that would “help” in this situation.

You could show that grouping the prime factors differently results in the two problems.
36 = (2×3)x(2×3) = 6×6
and also
36 = (2x3x2)x3 = 12×3

6×6=12×3 is nice, because you can see it by making 6 groups of 6, and then dividing each group in 2 to get 12 groups of 3.

But what about 9×4? we can’t see that from 6×6, but we can see it from 12×3. How?

Also a nice setting for some number strings:
12×3
6×6
2×18
4×4
8×2
16×1
9×2
3×6
1×18
What do you notice?

And here I thought of combining “pairs” of six together and getting three groups of 12 🙂

“Half and double” is a great strategy for mental math, and show powerful number sense when used with understanding. We called it “P_____’s Theorem” in my sixth grade class this year (after a student who repeatedly brought it up as a strategy.)

I often challenge kids with “but how can ______ = __, since _____ equals the same thing!” It definitely calls for a deeper understanding.

Conceptually, I think that using arrays/dots/grids would help the student visually see all the ways to make 36. The big idea is that there are multiple ways to arrange or group 36 items.

More abstractly, I think the prime factorization of 2x2x3x3 allows for some great opportunities to explore. My hope is that students could discover all of the different factor pairs (or triples?) by using different combinations of 2, 2, 3, and 3. However, there must be an understanding that numbers can have more than one pair of factors before you get to the point of using the prime factorization.

Dots themselves are an abstraction of shapes or objects that a young child might easily grasp as something they’d share. Who wants dots? Not I, but I would like candy or even stickers of stars. Something that they can tangibly see (maybe even hold) and something you can easily abstract is a good object to use for these type of problems. For tutoring older individuals I use the example of you are at the grocery store, how do you buy apples so that you have enough for each person in your family to eat two a week? It is a simple thing they can readily see and grasp in context. Then from there you can move on to answering the student’s question even further.

If you’ve been using stars, you can put in the middle of them the numbers and then label the groups of stars as well. If I had the blocks or objects in a way that the child could rearrange, I’d show him how you can manipulate the arrangement of blocks to get each set you want. Then you can break those groups into their primes like mentioned above and show how by prime manipulation you can find all the factors, because the real understanding you need is that there are more than one solution to the problem and that they can be taken from many ways. You should probably follow up with another number like 24 or 18 to test the student’s understanding of the concepts.

Sometimes I call him, father, My mother sometimes calls him “dear” and my sister calls him “dad”. His boss calls him Joe”. All of these are names of the oldest person in our home. How many more names can you make for the number 36.

## 8 replies on “Multiplication Issues (from Khan Academy)”

They seem to think that every number is the product of two distinct numbers. I’d look for a way for them to see this visually—perhaps the classic of having them make a box go 6×6 dots and counting the dots, then do the same thing with a 12×3 box.

It’s also a great chance to start to show them rarely is math about the “one way”—there are many things you can multiply together to get 36, and in the future, they’ll find that there are multiple ways to solve just about every problem they will encounter.

My mind goes to the idea that 36=2×3×2×3 but I honestly am not sure how that would “help” in this situation.

You could show that grouping the prime factors differently results in the two problems.

36 = (2×3)x(2×3) = 6×6

and also

36 = (2x3x2)x3 = 12×3

6×6=12×3 is nice, because you can see it by making 6 groups of 6, and then dividing each group in 2 to get 12 groups of 3.

But what about 9×4? we can’t see that from 6×6, but we can see it from 12×3. How?

Also a nice setting for some number strings:

12×3

6×6

2×18

4×4

8×2

16×1

9×2

3×6

1×18

What do you notice?

And here I thought of combining “pairs” of six together and getting three groups of 12 🙂

“Half and double” is a great strategy for mental math, and show powerful number sense when used with understanding. We called it “P_____’s Theorem” in my sixth grade class this year (after a student who repeatedly brought it up as a strategy.)

I often challenge kids with “but how can ______ = __, since _____ equals the same thing!” It definitely calls for a deeper understanding.

Finally, (sorry for the long reply) I posted this last week as a “Math Teacher Parent Flashback”

http://findingemu.wordpress.com/2012/07/27/flashback-friday-half-and-double/

Conceptually, I think that using arrays/dots/grids would help the student visually see all the ways to make 36. The big idea is that there are multiple ways to arrange or group 36 items.

More abstractly, I think the prime factorization of 2x2x3x3 allows for some great opportunities to explore. My hope is that students could discover all of the different factor pairs (or triples?) by using different combinations of 2, 2, 3, and 3. However, there must be an understanding that numbers can have more than one pair of factors before you get to the point of using the prime factorization.

Dots themselves are an abstraction of shapes or objects that a young child might easily grasp as something they’d share. Who wants dots? Not I, but I would like candy or even stickers of stars. Something that they can tangibly see (maybe even hold) and something you can easily abstract is a good object to use for these type of problems. For tutoring older individuals I use the example of you are at the grocery store, how do you buy apples so that you have enough for each person in your family to eat two a week? It is a simple thing they can readily see and grasp in context. Then from there you can move on to answering the student’s question even further.

If you’ve been using stars, you can put in the middle of them the numbers and then label the groups of stars as well. If I had the blocks or objects in a way that the child could rearrange, I’d show him how you can manipulate the arrangement of blocks to get each set you want. Then you can break those groups into their primes like mentioned above and show how by prime manipulation you can find all the factors, because the real understanding you need is that there are more than one solution to the problem and that they can be taken from many ways. You should probably follow up with another number like 24 or 18 to test the student’s understanding of the concepts.

Sometimes I call him, father, My mother sometimes calls him “dear” and my sister calls him “dad”. His boss calls him Joe”. All of these are names of the oldest person in our home. How many more names can you make for the number 36.

1 + 35

2 + 34

6×6

37-1

72 x 1/2