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CONTENT/DEPTH Question: Grade 5 Decimal Division
I apologize in advance if this is confusing, but I coach 5th grade teachers and cannot find any literature as to the depth of this standard. My question is very specific related to decimal division in Grade 5. Since long division is not a requirement of the grade 5 standard for decimal division, we use more conceptual strategies like the area model and partial quotients. The standard states decimal division to the hundredths place. Our students love partial quotients and this does not pose a challenge so long as there is no remainder.
<b style=”font-family: inherit; font-size: inherit;”>However, how far do we need to conceptualize decimal division to hundredths in Grade 5 with this strategy? In particular, cases where we need to annex zeros past the decimal point. Once you start annexing zeros past the decimal point it feels more like long division.
For instance…14.4 divided by 32. Our students think of the decimal dividend as a whole number of decimal parts (in this case, 14.4 = “144 tenths”) and then use partial quotients to divide with a 2-digit divisor knowing that the quotient will be the number of “tenths.” However, using partial quotients you get a remainder. Is the partial quotients strategy meant for annexing zeros past the decimal point? Is this an expectation of 5th graders? If so, what does this look like? We place our partial quotients above the dividend, but now, lining up place value matters. Should these type of problems be reserved for Grade 6 and the standard algorithm. (see attached photos).
ALSO, once students get this quotient of 4.5, they must come back to the idea that this is actually “4.5 tenths” or “0.45.” We can use properties and base 10 blocks to demonstrate what this looks like, but is this the expectation of the 5th grade decimal division standard?
I reached out to the Common Core to see if they could answer my question but I was unable to get any answers. I know this is very nitty gritty, but I just don’t know how to coach my teachers as to where they can stop. Any guidance/thoughts/opinions would be appreciated.
Thanks so much!
jc
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3 Replies
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This is an extremely interesting question!
I had to go and look up the specific Grade 5 Common Core standard to make sure and I’ll paste it here for everyone else to read as well:CCSS.MATH.CONTENT.5.NBT.B.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Something that I’m wondering first, before we get to how in-depth or specific we need to get with dividing decimals, is:
How well do the students understand the two types of division (partitive and quotative)?
How well do the students understand dividing fractions?
To me, both of these ideas must be very solid prior to diving into operations with decimals since a decimal is an abstract (and very constrained) version of a fraction.
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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.
Thanks, Jennifer
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Would this be a good time to jump into ratio tables? That’s what I used for my sixth graders, and it helped them get a better grasp on orders of magnitude through it.
Does that make sense?
