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Tagged: Order of Operations
This topic contains 6 replies, has 5 voices, and was last updated by Bill McCallum 2 years, 6 months ago.

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This standard reads: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. Everything I have read in trying to interpret this standard indicates that, mathematically, there cannot be brackets or braces in a problem that does not have parentheses. Likewise, there cannot be braces in a problem that does not have both parentheses and brackets. Using this information, I would believe that 5th graders must interpret nested expressions involving parentheses, brackets, and/or braces. As a matter of fact the PARCC assessment blueprints and specifications indicate that there will be test items involving nested items using parentheses and brackets. However, in the OA progresions (on page 32) it is noted that “Grade 5 work should be exploratory and expressions should not contain nested grouping symbols and should be no more complex than (8 + 27) + 2.” So, I am confused. So, should 5th graders experience evaluating expressions involving nested grouping symbols? If so, to what degree? And what about the mathematical correctness of using braces without brackets and parentheses, and brackets without parentheses?
Sorry for the delay in replying, somehow this one got lost! I really think this is just a miscommunication. We should be keeping things simple at Grade 5, and there was no intention here to suggest nesting of symbols. It was simply a matter of being agnostic about which grouping symbols to use. Later on students learn a hierarchy of symbols, but here they just learn the idea of bracketing things off (or bracing them, or putting them in parentheses). In practice it will probably always be parentheses, and we should probably just have said that.
I am trying to figure out at what point students would be expected to work with a a hierarchy of symbols. Unless I am missing it, that isn’t explicitly called out in the 6 – 8 standards.
Yes, you are right, this isn’t called out explicitly. And, in fact, the main point for students to understand eventually is nesting of grouping symbols, rather than the hierarchy itself. That is to say, if you can parse a complex expression with parentheses nested 2 or 3 deep, then that is the main point; it doesn’t particularly matter if you go parentheis, bracket, brace, as I was taught. It was really this idea of nesting that I wanted to get across, and mainly the idea that it doesn’t have to happen in Grade 5.
This whole subject is an area where the standards are pretty agnostic. Reading the conventions of mathematical notation is important, but conventions themselves are not mathematical concepts. So, the ability to read nested grouping symbols is implicit in ASSE.1–3, for example, but not explicitly mentioned.
Mathematically, there cannot be brackets or braces in a problem that does not have parentheses. Likewise, there cannot be braces in a problem that does not have both parentheses and brackets.
I found this website when I search the sentences above in GOOGLE.
I did so because I found those sentences in some instructional materials. I could also find the exact same quote in several educational websites. School districts and State Departments of Education equally repeat these guidelines, but they are simply incorrect.
While there is a widely accepted hierarchy for grouping symbols, nothing in MATHEMATICS forbids the use of brackets of braces in the absence of parenthesis. The expressions below may not be the most elegant, but they are perfectly “legal” mathematically speaking:[x+1]
(3[2^4+1] +17)^2
[2x+3]*[x+1]
2x + [x+1]^2 + (y2)
([2x+3]*[x+1])^2
(2^3 + [3^2 + √(2) ])Yep. You won’t find any disagreement here. Read the posts above yours in this thread and you will see your point confirmed at least twice. The original post in this thread suggested a “correct” order in which to use grouping symbols, and all of the replies to it suggested that such an order – even if it exists by convention – is neither important nor the point of 5.OA.2.
Yep, you are both right!

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