math explanation? help me with the words!

[MeganW]

I realize that I should be able to figure this out, but I have thought about it way too much and now I'm totally confused!

 

My kids read a multiplication sign as "___ groups of ___ items".  (For example, 4 x 5 they read as "four groups of five items".)  

 

For division, they do the opposite:

20 / 4 = ?, they say "twenty divided into four groups - how many are in each group?"

 

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How  do I apply that to more items to multiply?  Or can you?

4 x 5 x 2 x 8

 

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What would the similar phrases be for fractions?  (I'm getting mixed up when we divide fractions.)  

 

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[kateingr]

Such great questions! Wording makes such a difference in how kids process these concepts. 

 

For division, they do the opposite:

20 / 4 = ?, they say "twenty divided into four groups - how many are in each group?"

 

For division, what your kids are saying is one possibility, but you could also phrase it as "twenty items divided up into groups of 4." Both are possibilities, depending on the situation. You can also think of it as repeated subtraction--how many times can I subtract 4 from 20? It's clunkier, but also the foundation of our long division algorithm. 

 

My kids read a multiplication sign as "___ groups of ___ items".  (For example, 4 x 5 they read as "four groups of five items".)  

How  do I apply that to more items to multiply?  Or can you?

4 x 5 x 2 x 8

 

The "new groups" start fresh with each new multiplication sign. So, there are 4 groups of 5. And then 2 of those (4x5) groups. And then 8 of those (4x5x2) groups. For example, this could occur if you had: 

 

4 erasers in each box. 

5 boxes in each carton

2 cartons in each crate

8 crates in each pallet

 

How many erasers are in the pallet? 4x5x2x8

 

What would the similar phrases be for fractions?  (I'm getting mixed up when we divide fractions.)  

 

Fractions get tricky, but it's the same general reasoning. 

 

For multiplication, 1/2 x 1/3 can be thought of as "What is 1/2 of 1/3?" It sounds different than the whole number language, but it's really along the same lines. Essentially, you're asking, if I have 1/3 of something, how big would a "one-half group" be? It's a little easier to see if you mix whole numbers and fractions: 4 x 1/3 means four groups of 1/3.

 

For division, say you have 1/2 divided by 1/3? As in whole numbers, you're asking to take 1/2 and divide it into "groups" of 1/3. Or, you can think of it as taking 1/2 and asking how many 1/3s fit in it.  

 

 

[Kiara.I]

Well, for repeated multiplication....

 

You could do something like:

4 groups of 5 items each, then 2 rows of that, then 8 layers tall.  It's convoluted, though.  But you could do it with manipulatives.  Once.  And then never, never again.....  ;)

 

I'm not sure how you'd produce a similar phrase for dividing fractions...

[Monica_in_Switzerland]

For dividing fractions, maybe start with a concrete image and go from there.  Also, choose fractions where you are going to get a "pretty" answer the first few times.

 

Ex.  Start with 1/2m of rope.  Divide by 1/6.  So how many 1/6ths can you get from 1/2m?

1/2 div 1/6 = 3  <----3 is easy to visualise, compared to 1/2 div 1/3 = 3/2.  It's hard at first to visualise one and half thirds.  

 

So a sample problem might be, I have 1/2m ribbon.  I need 1/6m to make a bow.  How many bows can I make?  

 

Once you are used to working with problems that work out *nicely*, switch to a problem with a less intuitive answer.

I have 1/2 gallon of gas.  My lawnmower needs 1/3 gallon to fill the tank.  How many times can I fill the tank with what I have? (1 1/2 times)

 

As already mentioned, always use *of* to read a multiplication sign with fractions.  1/12 OF 3/4.

[mom2bee]

I realize that I should be able to figure this out, but I have thought about it way too much and now I'm totally confused!

 

My kids read a multiplication sign as "___ groups of ___ items".  (For example, 4 x 5 they read as "four groups of five items".)  

 

For division, they do the opposite:

20 / 4 = ?, they say "twenty divided into four groups - how many are in each group?"

 

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How  do I apply that to more items to multiply?  Or can you?

4 x 5 x 2 x 8

 

How do you approach 1 + 2 + 4 + 8?

 

Multiplication and addition are binary operations. That means that no matter how many factors or addends there are, you must compute them exactly 2 at a time and eventually you will have only two factors/addends and you perform the operation on those two sub-totals and you get your final product/sum.

 

Since you know that addition is combining groups of the same thing so you say to yourself that

1 + 2 + 4 + 8 is just combining FOUR groups of the same thing so lets get started.

1 + 2 = 3. | So we REPLACE 1 + 2 with its numerical synonym, "3" and 1 + 2 + 4 + 8 becomes

3 + 4 + 8  | and 3 + 4 = 7 so we REPLACE 3 + 4 with its numerical synonym, "7" and 3 + 4 + 8 becomes

7 + 8        | and now we have ONLY 2 addends so after we perform addition this time, we will have the sum, so here we go: 7 + 8 = 15

15

The final result is 15 so we know that 1 + 2 + 4 + 8 is a "long way" of saying "15"

Of course, this is math, we can commute the addends if we are numerically aware/fluent with math facts you could do  1 + 2 + 4 + 8 to get 5 + 10 and still get 15

 

Similarly, since we know that multiplication is scaling the size of groups by a factor we can do the problem you asked about like this.

4 x 5 x 2 x 8

4 x 5 = 20 so we replace 4 x 5 with 20 and get from 4x5x2x8 to

20 x 2 x 8 | and we know that 20 x 2 = 40 so we replace 20 x 2 with 40 and get

40 x 8 | again, at this point we have exactly two factors so once we multiply here, we will have the product of 4x5x2x8

40 x 8 = 320

320

 

To understand when you would have some real life situation that could be modeled by  this problem, think about individually wrapped snacks in boxes. Pretend that in your family you absolutely love Little Debbie Honey Buns. Now, you buy them from the dollar store so there are

4 honey buns in a box,

5 boxes of honey buns can fit a single shelf in your home,

2 shelves in each of your kitchen cabinets are for honey buns ONLY and there are

8 cabinets in your kitchen.

 

How many Honey Buns do you after a "Snack Stocking" expedition? The answer is 4x5x2x8 or 320.

 

Now you can always just "calculate it out" going in order one-by-one, and if you know your times table then you WILL get the right answer, but if you are clever you can rearrange the problem to be a little easier to see the answer to by doing (5x2)x(4x8) and getting (10)x(32) and then it'll be easier to see that the answer will be 320.

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What would the similar phrases be for fractions?  (I'm getting mixed up when we divide fractions.)  

 

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[cindydanleo]

Thos explanations were awesome. I have to print this thread out. I am terrible at math and have been dreading teaching past 4th grade. Why didn't they teach this way 30 years ago? I just had to remember my times table...but had NO idea what it all meant. When I got to higher math....same thing...I just followed the formulas but had no idea what I was doing.

[MeganW]

This is SO helpful!!  Thank you all so much!!!  

[Incognito]

When multiplying by fractions, it might help to understand that you are really dividing (because fractions are division).  

 

1/2 x 1/3  = 1/6  Then show what happened with a cookie (well two cookies, so you and your child are each doing it).  You divide the cookies in half.   Use just 1/2 of the cookie, since you are starting with 1/2.  Show that you divide it into 3 groups, and just want one of those groups (that is the x 1/3).  Then compare what you have of the cookie, to a whole cookie.  Notice it is 1/6 of the cookie.   Understand, then eat.  Repeat until fraction multiplication makes sense (or you run out of cookies).

 

 

[SporkUK]

For multiplication, I teach that we're making something [x] times as big rather than in groups because I want to avoid the false concept that I was taught that multiplication is simply repeated addition which can cause trouble in more advanced maths as it did for me. I often use the visual of stretching something to make it bigger like dough or cloth. It happens to give the same answer as repeated addition for some numbers which makes it a quick trick for repeated adding which is fun to discuss and useful but I purposefully try to avoid putting the two together when teaching. 

 

So for a string of multiplication, we're making 4 5 times as big, then we'll make that twice as big, then we'll make that 8 times as big. 

 

For multiplying fractions it is the same - we'e making something [x] as big as the original - it just happens to be 1/2 as big which makes it smaller than than a whole number that makes it bigger just like multiplying by 0 makes it 0.