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Examples I’ve seen of division in 6th grade, both with and without decimals, have nicely terminating quotients. Do students in 6th grade need to learn about remainders, for whole numbers and decimals? Do students in 6th grade need to learn about extending the zeroes in whole numbers and decimals in order to divide? Should students in 6th grade be solving the problem 14.6/3? How should they interpret the quotient?
Students learn about whole number remainders in Grade 4:
4.OA.A.3. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
In Grade 6 they study division of fractions, which would include 14.6/3. They are not required to know about infinite repeating decimals until Grade 8, so they might express the answer as a fraction rather than a decimal, as in $4 \frac{26}{30}$. In the Common Core finite decimals are treated as a way of writing certain sorts of fractions, namely those that can be written with denominator 10, 100, and so on. There is no explicit requirement that they express this problem as division with remainder, but it seems a natural extension of their Grade 4 work to be able to say that 14.6 = 3 x 4 + 2.6, and to interpret both the quotient and the remainder in a context.
It also seems to me that, although knowledge of infinite repeating decimals is not required until Grade 8, simple examples such as 1/3 = 0.333 … could appear earlier. But it is not necessary, since you can always just use fraction notation.
Thanks so much for responding to this question. I have been wondering about this as well. However, would you mind elaborating on missbaldin’s question “Do students in 6th grade need to learn about extending the zeroes in whole numbers and decimals in order to divide”? Thanks
Yes, but I wouldn’t describe it that way. Students should understand that a decimal is a fraction, so dividing decimals is dividing fractions. For example, to calculate 2.16 ÷ 0.3, they would know that 2.16 = 216/100 and that 0.3 = 3/10, which is also 30/100. If they were asked to calculate 2.16 ÷ 3 they would use this understanding to rewrite this as (216/100) ÷ (30/100) = 216 ÷ 30. I think this is what you mean by extending the zeros … and of course I wouldn’t expect them to got through this reasoning every time, once they understood it.
I would like to add a thought about “once they understood it.” Until recently, I would explain briefly and then show them examples. I didn’t realize I had trained them to zone out until I showed the examples. Sometimes I would question them to make sure someone understood it, but the main take-away was procedure. Then they would ask, “Do I multiply or add? Where do I put the zero?…” Now when they ask questions like that, I help them reconstruct the conceptual explanation. Now the main take-away is “what would be a sensible step?”
I’m scratching my head about Bill’s response to kirkkimb’s query as it relates to 6.NS.2 and 3. Those standards require fluency with the standard algorithm for multi-digit decimals. Bill’s response is more of a “reasoning via common fractions” approach instead of an extension of existing skills with the standard algorithm developed in Grade 5.
My interpretation of “extending the zeros” with the standard algorithm is shown in this link for dividing 121 by 8:
https://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/8d3dfd8f-9610-413b-9952-8db75f1b71a4.gif
(hopefully this link will persist for a while – I have no other way to show it)
In other words, continuously extending the decimal places of 121 to thousandths (121.000) to assist with recording a decimal remainder. Is this part of what is expected for fluency with the standard algorithm for multi-digit decimals?
My name is Erneice Jackson and I am a student in the Masters of Mathematics Education program. I am preparing for a project on Fluency through grades 6-8. I saw that Standards 6NSB2 and 6NSB3 contain the word “fluency” in the wording. Who would contact to get more information on why students should be fluent with these two skills? Why now? How does being fluent in these skills help with real life situations? Please contact me at ejackson1@hazelwoodschoold.org.
My name is Erneice and I am a student in the Masters of Mathematics Education program. I am preparing for a project on Fluency through grades 6-8. I saw that Standards 6NSB2 and 6NSB3 contain the word “fluency” in the wording. Who would contact to get more information on why students should be fluent with these two skills? Why now? How does being fluent in these skills help with real life situations? Please contact me at ejackson1@hazelwoodschoold.org.
An answer for this question isn’t determined by the Standards because Standards do not specify a particular standard algorithm for each operation. Moreover, specifying an algorithm doesn’t necessarily specify how it’s notated. That said, one might want students to “fluently extend” zeros because doing so can help with lining up places in quotient and dividend, as can using graph paper.
I am looking for clarification about 6.NS.B.2 and b.3: is 6th grade the first time students are expected to be taught division using the standard algorithm or is it actually introduced and practiced in another grade level?
I am unclear how fluency is expected and mastered enough so that state testing asks them to divide a 5 digit number by 2 digits. Especially since this is not a major standard in 6th grade, nor even a supporting standard. It is an additional standard.
Same with decimal division: Is this the first time students are expected to learn standard algorithm division of decimals, fluently?We always struggle with this, as our 5th grade tells us they don’t teach standard algorithm at all, as it is a 6th grade standard (as 6th grade mentions it explicitly). I think people interpret CCSS differently, and I would like to understand the standards very thoroughly. The progression documents were not clear. Thank you for your help!!
The standards say in which grade a certain skill or concept should be achieved and is fair game for assessment. That doesn’t necessarily mean you don’t start working on that skill or concept until that grade. The standard algorithm for division is a case in point. It is the culmination of a long progression of strategies and algorithms that stretches over many grades. In the Illustrative Mathematics curriculum students start seeing the standard algorithm in grade 5, but are not expected to have it down until grade 6. I think it would make sense for your 5th and 6th grade teachers to get together on this!
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