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For K-3 students, perhaps it is not acceptable to introduce multiplication by zero as a property or definition. Instead, the child may think about multiplication as, e.g., repeated addition.

Examples of the "repeated addition" conception: $3 × 2 = 3 + 3 $ and $ 4×3 = 4 + 4 +4$.

My questions:

How will students conceive of $ 3 × 0 $?

How should early elementary/primary curricula deal with multiplication by zero?

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    $\begingroup$ I think the graphical approach gets the idea across. You could represent $3\times 1$ by a row of three dots; likewise $3\times 2$ is two rows of three dots; etc. If you asked them to guess what $3\times 0$ was, some would probably tell you you'd have no rows so no dots, so the answer is zero. Humans are very good at inductive logic, even if we aren't aware of it. $\endgroup$ – Cameron Williams May 21 '15 at 19:39
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    $\begingroup$ Alternatively (following up on @CameronWilliams suggestion) one could regard $3 \times 1$ as "three rows with one dot in each row", $3 \times 2$ as "three rows with two dots in each row", and then $3 \times 0$ would be "three rows with no dots in them" so there are zero dots. $\endgroup$ – mweiss May 21 '15 at 19:52
  • $\begingroup$ Thanks very much. I think that the alternative is more clear. We may use sets instead of rows. For example $ 3 × 2 $ as 3 sets each contains 2 elements. For $ 3 × 0 $ it shuld be 3 empty sets. Thanks again. $\endgroup$ – Abdallah Abusharekh May 21 '15 at 20:19
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    $\begingroup$ No offense meant, @AbdallahAbusharekh, but that's a very mistaken idea. Don't introduce the idea of sets to children. Visual aids are best for children. $\endgroup$ – Cameron Williams May 21 '15 at 22:21
  • $\begingroup$ @CameronWilliams, I get your point, but perhaps he's just thinking of unstructured collections rather than rows. $\endgroup$ – DavidButlerUofA May 21 '15 at 22:31
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Turning my comment into an answer as per request:

I think the graphical approach gets the idea across. You could represent $3\times 1$ by a row of three dots; likewise, $3\times 2$ is two rows of three dots; etc. If you asked them to guess what $3\times 0$ is, some would probably tell you you'd have no rows so no dots, so the answer is zero. Humans are very good at inductive logic, even if we aren't aware of it.

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    $\begingroup$ I do have to say that in my mind I usually interpret $3 \times 2$ to mean "3 groups of 2" to me rather than "2 groups of 3". But I know others interpret it the other way. Still, it's worth keeping in mind with your children. $\endgroup$ – DavidButlerUofA May 22 '15 at 0:45
  • $\begingroup$ Also, a further idea you could add to your answer is to think of it this other way around. $1 \times 3$ is three groups of 1, $2 \times 3$ is three groups of 3, so working backwards, $0 \times 3$ would be three groups of none, so none in total. $\endgroup$ – DavidButlerUofA May 22 '15 at 0:47
  • $\begingroup$ @DavidButlerUofA good point. I don't remember how I thought about it exactly as a kid, I just know it was one of our two ways. These days I think it's pretty interchangeable for me. $\endgroup$ – Cameron Williams May 22 '15 at 0:54
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I'm sticking with food math.

You have 4 friends and need 2 candy bars per friend, so 8 bars. Similar, if say, 1 bar per friend, you have 4 bars total. If you have 0 bars per friend, it's 0. But if it's 0 per friend, no matter how many friends you have, you still need 0 candy bars.

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