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Jennifer CorforteMemberFebruary 13, 2020 at 11:50 am
ABSOLUTEY! In as early as Grade 3, we work on the meaning of each factor in a multiplication equation. Teachers, district wide, have established with our students that factor #1 is always “number of equal groups” and factor #2 is always “number in each group” and the product tells us “number in all.” Students understand that 3 x 4 represents something different than 4 x 3 even though their products are the same (Commutative Property for Multiplication).
When we begin division, we discuss that every multiplication equation has two “related division equations.” Instead of the traditional “fact families” which have 4 equations, we only highlight 3 equations as a “family.”
For instance, the multiplication equation is 3 x 4 = 12 (meaning “3 groups of 4 is 12″) has the two following<b style=”font-family: inherit; font-size: inherit;”> related division equations:
#1: 12 divided by 3 = 4 (meaning “I have 12 in all and want to make 3 equal groups, the quotients tells me how many in each group.“)
#2: 12 divided by 4 = 3 (meaning “I have 12 in all and want to place 4 in each group, the quotient tells me how many groups I can make.”)
We continue using these same ideas as we study the area model and partial products/quotients for multiplication and division in grades 4 and 5, always relating it back to their Grade 3 studies.
When we get to division of decimals with 1- and 2-digit whole numbers, we think of the divisor as “number of groups.” When we switch to a decimal divisor we discuss how it “hurts our brains” to think about 0.4 groups, for instance. So, re-enter the divisor as “number in each group.” This is easy to show when the dividend and divisor are inside their basic facts, such as 2.4 divided by 0.4. We can use a number of strategies such as the number line, skip-counting by 0.4, repeated subtraction, etc.
Eventually, we present an example where they realize it would require an exorbitant number of skip-counts…(ain’t nobody got time for that! haha!) So, we suggest…“What if we could somehow ‘change’ this divisor and go back to our ‘number of groups’ type of thinking?” Using the Identity Property for Multiplication along with their knowledge of Multiplying by Powers of 10, we show how to transform a HARDER problem (14.4 divided by 0.04) into an EASIER problem (1440 divided by 4). Now, students use strategies such as partial quotients (most of our students) or the area model (some of our lower students who still like the visual of 1440 square units, 4 rows, how many in each row?).
All of this we can accomplish! Here is where my question comes in. How much further does a 5th grader need to go? Do we need to annex zeros past the decimal point if we are not doing long division? Or is this something we can reserve for Grade 6?
Again, sorry for the novel. This would clearly be better suited for a conversation, but I’ll take what I can get when you are in Canada, and I, New Jersey. I appreciate the response and would love to hear your thoughts now that I’ve clarified where our students are conceptually about division.