23 votes
Accepted

Is it meaningful to add a number to itself a fractional number of times?

For the product $a\times b$, I intentionally don't use the phrase "add $a$ to itself $b$ times", but rather I prefer something like "start with zero and add $b$ (copies) of the number $...
Nick C's user avatar
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11 votes

Is it meaningful to add a number to itself a fractional number of times?

Frame challenge: I think your verbiage "adding (whole number) to itself (whole number) times" is misleading and incorrect and exhibits an off-by-one error. Think about the example $(4×1)$. ...
shoover's user avatar
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8 votes

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?

I think having a student play with examples is a great approach, and that seems the easiest route to having them see that this is true. If seeing multiple permutations of factors all giving the same ...
Nick C's user avatar
  • 9,436
8 votes

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?

For up to three dimensions you could use the concept of "area of a rectangle" or "volume of a rectangular cuboid". Use only numbers (no variables) and let the student pick a ...
Jasper's user avatar
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7 votes

Is it meaningful to add a number to itself a fractional number of times?

The Common Core State Standards "definition" of multiplication is that $A \times B$ represents the number of units when you have $A$ groups, each $1$ group containing $B$ units. For instance ...
Steven Gubkin's user avatar
5 votes

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?

Some great answers were provided here, but I wanted to add something which I think is essential: to emphasize to the student that it is a very good thing to question this! Why am I saying that? ...
Amit's user avatar
  • 339
4 votes

Is it meaningful to add a number to itself a fractional number of times?

Putting this all together, I've been able to explain, for example, that 3×4 would be (3+3+3+3) or (4+4+4), i.e. "3 added to itself 4 times" or "4 added to itself 3 times." No, 3 ...
hkBst's user avatar
  • 141
3 votes

Is it meaningful to add a number to itself a fractional number of times?

I think it is important to consider your audience. Are these kids going to immediately jump to asking about "adding 4 to itself $\frac{1}{2}$ times?" I'd guess not. I think this is a ...
David Steinberg's user avatar
3 votes

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?

My advice is "and". List the principle. But NOT in a symbolic (ab) manner. Just "you can change the order when multiplying." [Realize you will need to repeat it, several times. ...
guest's user avatar
  • 41
3 votes

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?

Most of the answers here have been geometric. Here is an answer for three or more factors that is algebraic. It requires the student to know that multiplication is associative (more understandable to ...
Steve's user avatar
  • 1,514
2 votes

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?

Many college students have a 2nd issue with a problem like this. If they see 2(3 · 4), they sometimes want to distribute, and write 2·3 · 2·4. Yikes. This one requires seeing the difference between ...
Sue VanHattum's user avatar
  • 20.3k
2 votes

Is it meaningful to add a number to itself a fractional number of times?

Go back to your egg-carton analogy. Then, move it to the area of a rectangle as number of unit squares Cutting the rectangle in half, either horizontally or vertically, produces the same answer.
Rendu's user avatar
  • 131
2 votes

Is it meaningful to add a number to itself a fractional number of times?

You can try going back to the way you may have first taught them the concept of addition: combining collections of things -- "if you have 3 apples, and I give you 4 apples, how many apples do you ...
Barmar's user avatar
  • 369
2 votes

Is it meaningful to add a number to itself a fractional number of times?

I think you have to teach progressively. Even with bright kids, advanced kids. Even when not really establishing mastery, but giving exposure. So, let the kids have more exposure to fractions ...
guest troll's user avatar
1 vote

Is it meaningful to add a number to itself a fractional number of times?

Half times something is the same as half of it. When students are used to that, and also used to the distributive law, then you can put them together like $2\frac{1}{2}\times a$ is $a + a + \frac{1}{2}...
Peter's user avatar
  • 131
1 vote

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?

Introduce them some of euclid's axioms. For example: Things that are equal to the same thing are also equal to one another . If equals are added to equals, then the wholes are equal (Addition ...
An_Elephant's user avatar
1 vote

How to explain to a ~$12-16$ year-old student who is weak at maths, that $\ 38 \times 27 \times 14 = 27 \times 14 \times 38$?

Exercise: Following are several products of the same 3 numbers, but in different orders. Calculate these products using a calculator. 14 x 27 x 38 14 x 38 x 27 27 x 14 x 38 27 x 38 x 14 38 x 14 x 27 ...
Dan Christensen's user avatar

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