Third grade grandchild had this for homework. I don't even know the intent here?

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    Probably part of the idea is we can't add unlike objects. For example, adding a car and a truck means what? Unless of course we take cars and trucks as examples of automobiles so then adding a car and a truck makes perfect sense. As far as multiplication goes, I leave it to the experts here. Personally, the idea of identifying multiplication and addition as necessarily tied to counting or grouping problems has the unfortunate side-effect of saying addition and multiplication are necessarily those counting processes. But, they're not. Arithmetic exists independent from these heuristics. Imho. – James S. Cook Sep 12 at 1:40
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    It is unfathomable why someone believes that is a good homework question. Probably most mathematicians can't make any sense of it so how is a 3rd grader supposed to? – Bill Dubuque Sep 12 at 2:30
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    I would like to see questions 1-13. Perhaps they have more information or context about how the student has been taught addition and multiplication and what it means to "join groups." – ruferd Sep 12 at 12:10
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    I agree with ruferd. We cannot answer this, because we have not seen the previous material which may have explained what "join groups" means. – Gerald Edgar Sep 12 at 13:27
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    Is this even correct English? – Dominique Sep 13 at 10:10
up vote 3 down vote accepted

I am not too familiar with the Common Core State Standards Initiative (whose standards I assume the question above is intended to follow), but according to this introduction to the standards for Grade 3,

Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size.


Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.

To me, it seems that the emphasis is on "equal-sized groups," "same-size units of area," "identical rows," and "identical columns." The child's teacher could have emphasized this in class.

If so, then perhaps one "valid" answer to this higher-order thinking question is "You can add objects together if they belong to the same group. You can multiply groups of objects if they are of the same size."

For example, say that students are riding in $3$ buses: one bus has $30$ students, another has $30$, and another has $32$. How many students are there in total?

The answer is not $30+30+32+3$, that is, the number of buses is not added because buses are not students.

The answer is not $30\times 3$, because not all the buses have exactly $30$ students.

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    @user10216038: Regarding "What does it mean to Multiply Groups", I'm pretty sure Joel intended to describe a situation such as the following: Exactly $5$ groups of marbles such that each group contains exactly $3$ marbles represents (when brought together to form one group of marbles, without omitting or including other marbles) a group of $5(3) = 15$ marbles. And given that the context of this is math education, I think the shorthand phrasing he used was fine. And I agree with ruferd that this question is probably taken out of context, although personally I don't like the question. – Dave L Renfro Sep 12 at 15:48
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    Why one has to backtrack and decode this nonsense to figure out what the creator of the question meant? Are third-grade kids supposed to do this themselves? It is a rhetorical question. And by the way, slapping the "Common Core" label on a question does not make it Common Core, because Common Core is a set of requirements, not a curriculum. But parents think this nonsense IS Common Core. – Rusty Core Sep 12 at 16:48
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    "Why one has to backtrack and decode this" - because one was not present in class. – Jasper Sep 12 at 19:14
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    @Jasper No, it is because education goes through fads, and one of the current fads is "higher order thinking", so some dude created a murky question labelled just that. – Rusty Core Sep 12 at 21:28
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    Between the feedback here and some independent research of "Common Core" I think I understand the intent, although I still couldn't answer the problem as written. This seems to be math redefined by non-math people. Instead of "equations" it's now a "division sentence" or a "multiplication sentence". see: Thanks all! – user10216038 Sep 13 at 14:49

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