Expanded Notation and Order of Operations

[steveoc]

The Grade 5 CCSS states expanded notation should be represented like 3 × 100 + 5 × 10 + 7 × 1 + 8 × (1/10).

On the other hand 5.OA.1 only addresses parentheses, brackets, and braces for the order of computation.

Shouldn’t we express the above as (3 × 100) + (5 × 10) + (7 × 1) + (8 × 1/10) until students have been taught that one multiplies before adding?

Am I missing something?

• This topic was modified 11 years, 7 months ago by steveoc.

[Bill McCallum]

Students have been using the order of operations since Grade 3. There’s a footnote on 3.OA.8 that says

… students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

Order of operations is a convention about how to read numerical expressions, rather than a property of operations, so it doesn’t receive the same prominence as properties of operations.

• This reply was modified 11 years, 7 months ago by Bill McCallum.

[steveoc]

Thanks for the clarification. I missed that footnote when looking over the standards. Perusing the Third Grade portion of the U of A Progressions document  does indeed make that very clear.

That’s a big shift in NY as order of operations has been introduced in fifth grade.

So, the point of 5.OA.1 is to introduce alternatives to parentheses?

[Bill McCallum]

I’m not sure you need to make a big deal of order of operations in Grade 3. Just enough so that students can interpret expressions like 3×10 + 5. And there’s no rule against using parentheses if it is deemed necessary for the stage of development. In fact 3.OA.5 has an example that reads “one can find 8×7 = 8×(5+2) = (8×5) + (8×2) = 40+16 = 56. The Grade 3 footnote is about reading expressions, whereas the Grade 5 standard expects students to start using them systematically. I don’t think the point is particularly to expand to braces and brackets, although those are now part of the repertoire.

[Ellen]

Are 3rd graders expected to use parentheses when writing equations for two-step word problems?

[Duane]

I’m afraid I have to scratch my head along with Ellen. The OA Progressions on page 27 say “At Grades 3 through 5, parentheses can usually be used for such cases so that fluency with this rule can wait until Grade 6”. Yet as Bill notes, the footnote in the Standards for 3.OA.8 says that “students should know how to… [operate] … when there are no parentheses”. The two statements seem to contradict one another.

[Ellen]

Understanding that grouping symbols can be used to affect the order in which operations are carried out is a reasonable expectation for 3rd graders. I am curious why mention of the  order of operations ended up in a footnote in G3. The footnote is confusing. Please clarify the expectations for 3rd graders regarding parentheses and the order of operations.

With the example that Bill provided above in his response to Steve, the majority of 3rd graders would carry out the operations in the order in which they appear and arrive at the correct answer without knowing the order of operations. Bill, please clarify if 3rd graders are expected to interpret expressions like 5+3×10.

[Bill McCallum]

Well, the footnote is clear enough, it says Grade 3 students should know order of operations, so yes, they should be able to interpret 5+3×10 correctly. The thrust of the comment in the progressions document is that one need not expect fluency with this right away. I agree the exact boundaries are not really spelled out here. Perhaps a good rule of thumb would be to expect students to be able to deal with fairly simple expressions like 5+3×10, but not to be too aggressive about insisting they can deal with more complicated expressions. After all, it’s never wrong to put parentheses in, for either the student or the teacher. Perhaps the progressions document should be more specific here.

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