Ordering numbers in multiplication equations has been a headache for a long time. As a 2nd grade student and a Elementary School teacher myself, it was always the same. It is called the "Sandwich Rule" . Number of objects in each group x Number of Groups = Total Number of Objects.
Quite interestingly, although this rule is so common in educators in Japan, it is not necessarily clearly stated in their world widely famous national standards. According to the English Translation for the Japanese Mathematics Curricula in the Course of Study, 2nd grade D. Quantitative Relations states; (2) Student will be able to represent the cases in which multiplication may be applied by using mathematical expressions and will be able to interpret those expressions. A Japanese teacher in Tokyo told me that teachers assume the students' understanding of the multiplication situation by which numbers students are using for multiplier and multiplicand. Since teachers stress about the number order, students should be able to demonstrate the proper number expression. For example, three children are at the park. Each child is holding 2 balloons. How many balloons are there in all? If you are Japanese 2nd grader, the answer is 2 x 3 = 6. Your teacher will love you because you put the numbers in the right order. Your answer proves your ability to identify what kind of number your are dealing with, therefore you can apply multiplication concept in your own life. A teacher can analyze it by just a glance.
Whereas American styles vary. Every time I teach the multiplication unit, I ask, research, and google math lessons and concepts. Some text books favor the Japanese way, some are strongly against it, and some don't care much. It has been a frustration because no one seems to explain why they use number orders the way they use although they have strong opinions about it. No wonder kids get confused while their teachers are not sure...
One bright piece of news is that Common Core State Standards Mathematics stated clearly for the guidance. Grade 3 Operations and Algebraic Thinking. 1. Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7. Hooray! It is totally opposite to what I am familiar with, one way or another. The committee decided to put it in this way because of how we use English language, in my opinion, which is okay. I can deal with it.
But when you see the question like this; A baby whale is 3 meters long. A mother whale is 2 times longer than her baby whale. How long is a mother whale? What would you do? 3 x 2, right? 2x 3 just doesn't make sense. Here is my theory. Objects in each group and the base number (baby's whale) seem like the same category while number of groups and ratio (2 times longer) could be similar. So, I suppose American students are required to manipulate numbers in multiplication in case by case. My concern is how smoothly flexibly they can accept the different ways of thinking as the grades going up. Do they have a strong enough foundation?
I am open to any ideas, especially, the newly coming Common Core. In fact, I share one of the worksheets I have created below. Of course, Step 1 used to be Step 2.... Anyway, I would like to hear some clarification. I want to be confident and consistent. Keep our conversation alive!
Multiplication Word Problem Analysis
Name ___________ Date________
Page/ #

Step 1
# of Groups

Step 2
# of objects
in each group

Step 3
Total # of objects

Step 4
Equation

Labeled Answer

1






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