Third grade grandchild had this for homework. Can someone explain the intent here?
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.
Here are the two previous pages from those materials (a pre-publication version found with a Google search):
And here is the page containing the homework problem in question:
The intent now seems pretty clear. Students know that you can join groups by adding them, but in the case where the groups are equal in size (e.g. five bags, each with nine goldfish), you can also use multiplication to join the groups.
This is an excellent example of a question which leaves both children and adults baffled. This is a truly meaningless question. Let's start with the fact that you add and multiply numbers whereas you join together sets (groups) of objects. The clause "multiply to join groups" may mean "use multiplication to find the total number of objects when joining together groups of items", in which case I guess that the expected answer is "only if all the groups are of equal size". But the fact that I am speculating about the intent of the question says it all.