Adding decimal fractions – Grade 4

[Duane]

4.NF.5 asks students to add two fractions with respective denominators 10 and 100. Is the intention that students only add common fractions (converting if necessary) as shown in the example 3/10 + 4/100, or that students also add decimal fractions, e.g. 0.3 + 0.04?

Also, the NF Progressions (p.8) seems to suggest that students should encounter expressions such as 0.3 + 0.04 but convert them to common fractions instead before adding them. Is that correct? It seems a slightly round-about way of handling it.

Thanks!

[Duane]

I just noticed another Standard that is related and was queried by Brian Cohen on October 5, 2011, on the NF blog page. After reading Bill’s response to Brian I still am slightly unsure whether 4.MD.2 means that students should add decimal fractions in decimal fraction notation (e.g. 8.5 m + 1.2 m) or whether the decimal fractions should be converted to common fractions first (e.g. 85/10 m + 12/10 m).

My best guess is that students can perform at least addition (what else is required?) of decimal fractions presented in decimal fraction notation. There should also be teacher emphasis on how decimals are a type of fraction (e.g. saying “eight and five-tenths” for 8.5). Is this correct?

Further, using addition would satisfy 4.NF.5, though 4.MD.2 requires all four operations. What is the expected limit for operations of decimal fractions in decimal fraction notation in Grade 4?

[Bill McCallum]

The thread that Duane is referring to is here. As Duane says, the Common Core views 0.2 and 2/10 as different names for the same number. So adding 0.3 and 0.04 is the same thing as adding 3/10 and 4/100. In order to add them you have to convert 3/10 to 30/100. If you write this down as
$\frac 3{10} + \frac{4}{100} = \frac{30}{100} + \frac{4}{100} = \frac{34}{100}$
then people might call it fraction addition, and if you write it down as $0.3 + 0.04 = 0.30 + 0.04 = 0.34$ then people might call it addition of decimals. But if you ask a child to explain either one of these, the explanation is exactly the same: “I have 3 tenths and 4 hundredths, and to add them I have to express them in the same units, and I know that 3 tenths is 30 hundredths so the sum is 34 hundredths.” Thus, when you are teaching for understanding, the distinction between fraction addition and decimal addition melts away. They are only different operations requiring “conversion of decimals to fractions” if they are each taught as blind procedures.

Of course, once students have a secure understanding of the underlying meaning, we are not going to expect them to keep repeating the explanations. At some point, when they acquire fluency, they can just “see” the answer. This is similar to adding 30 and 4, where students initially think explicitly in terms of 10s and 1s, but at some point have a sufficiently robust understanding of place value that they just “see” that 30 + 4 = 34.

[Duane]

Thanks for your reply Bill and I agree that a verbal explanation of how to add fractions makes clear the links between the two different formats and reveals students’ understanding. My question though is what written symbolic operations are required in Grade 4? In 4.MD.2 students are asked to “use the four operations to solve word problems… involving simple fractions or decimals…” Presumably students are to write down their thinking and not say their thinking aloud all the time. Are they expected to write and solve expressions such as “3.4 + 1.2” and “5.6 – 2.3”?

 

[Bill McCallum]

Yes, that’s correct, written expressions involving decimals are certainly included.

[Author]

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