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In the following multiplication example,

$$\begin{align} 34\;& \\\underline{\times\;\; 7\;}& \end{align}$$

first one would multiply the units digits, producing the partial product $28$ as follows:

$$\begin{align} \color{maroon}{2}\;\;\;\!\>\!\\ 34\;& \\\underline{\times\;\; 7\;}& \\\color{maroon}{8}\;& \end{align}$$

How would this partial product operation best be described?

$$ ( 4 \times 7 ) \qquad\text{or}\qquad ( 7 \times 4 ) $$

Similarly, how would the next partial product operation, which adds the carry, best be described?

$$ ( 3 \times 7 ) + 2 \quad\text{or}\quad ( 7 \times 3 ) + 2 \quad\text{or}\quad 2 + ( 3 \times 7 ) \quad\text{or}\quad 2 + ( 7 \times 3 ) $$

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    $\begingroup$ I suppose it would depend on what your country's department of education says. Perhaps you should try to get a copy of their official textbook? $\endgroup$
    – JRN
    Jul 18, 2018 at 23:59
  • $\begingroup$ Thanks everyone for the edits and for your thoughtful answers and comments. $\endgroup$
    – Marc Zehngut
    Jul 20, 2018 at 14:36

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This answer is very culture-dependent. If the primary language used in your classroom is English, then since in English we read left-to-right and top-to-bottom I would pick the way where these two directions agree. If the language you primarily use in the classroom differs from English in this way, then change the directions accordingly.

Following your example, in the first step, reading downward the order is $4$ then $7$, so write $(4 \times 7)$. In the second step, the order is $2$ then $3$ then $7$, so write $2+(3 \times 7).$

Really though, pending a study into this that I imagine is way too specific to ever get funded, I don't think it matters too much to a students' understanding what order you choose.

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    $\begingroup$ In fact in Israel, where I am currently residing and tutoring, even though language is written right to left, math equations are still written left to right. $\endgroup$
    – Amy B
    Jul 19, 2018 at 9:56
  • $\begingroup$ I never realized this until I read this question and answer, but for me, I do the exact opposite of top-to-bottom and left-to-right, despite that being the way we read English.. I start with the 7 and think (7x4) and the move left towards (7x3)+2. I don't know if that's how I internalized so many years ago or if that's the way it was explicitly taught to me. $\endgroup$
    – ruferd
    Jul 19, 2018 at 12:04

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