# I don’t think they mean that

Thanks to John Golden (@mathhombre) for the find.

From the Common Core State Standards Progressions document on the 6—8 Expressions and Equations standards:

The “any order, any grouping” property is a combination of the commutative and associative properties. It says that sequence of additions and subtractions may be calculated in any order, and that terms may be grouped together any way (p.5).

### 4 responses to “I don’t think they mean that”

1. Thanks Christopher, do you have a suggested fix? I’m guessing you are worried about the second part, “terms may be grouped together in any order,” which could be interpreted as saying something like 3 – 5 + 4 can be calculated either as (3 – 5) + 4 or 3 – (5 + 4). What I had in mind when I wrote this is that the terms in this case are 3, -5, and 4, but the internet does not support me in this usage. Which is weird, because nobody calls 4 a factor in 3÷4. But never mind. Maybe the easiest fix is to say “addends may be grouped together in any order”. And then perhaps add a note that at some stage students learn to see a sequences of additions and subtractions as a sequence of additions with negative terms.

2. I have an opinion!

Maybe the expression 3 – 5 + 4 was created to represent the situation “start with 3 blocks, remove 5 blocks, add 4 blocks”. In this context, the order that you add or remove blocks doesn’t matter. I think this is what is meant in the expression above – in fact, I think it’s interesting that you say “additions and subtractions may be calculated in any order” (notice the plural, implying that you are speaking of an action rooted in some sort of context and that the numbers have a physical representation), as opposed to “addition and subtraction may be calculated in any order” (which are mathematically defined operations devoid of a specific context – and, by the way, would be a false statement since these properties do not hold for subtraction).

So, maybe the idea is that operations of adding and removing within a context can be done in different orders to make calculations more efficient (which I think is true). If so, then maybe this ‘any order’ comment is trying to say that students are beginning to notice the patterns that are eventually crystallized as the commutative and associative properties. But at this level (the 6th grade), students are still looking at the + and – symbols as operations rooted in a context – it’s not until standards 6.RN.5 and 7.RN.1 that negative numbers are introduced (also, initially, through a context) and students begin to interpret ‘3 – 5’ as ‘I’m combining the number 3 with the number negative 5’ instead of ‘I have 3 somethings and I take away 5 somethings’. So, I think the line about ‘seeing sequences of addition and subtractions as sequences of additions with negatives’ is absolutely necessary – otherwise, the property doesn’t hold.

One last thought – maybe the problem is with the order of the first sentence. I disagree that ‘any grouping, any order’ is a property – I think it’s a pattern that students begin to notice and that we encourage. And I disagree that it’s a combination of the commutative and associative properties – I think the associative and commutative properties are the names we’ve given to this particular pattern in the world of math devoid of context, once we’ve established that a sequence of subtractions is the same as a sequence of additions with negatives.

So… there’s that. Hope it makes sense.

3. The quick solution is to leave subtraction out of it altogether. It’s unclear to me what is gained by including it explicitly and then excluding it implicitly with the use of the word “term”. The sentence is both correct and useful if it focuses only on addition. You could add a follow up parenthetical that makes it explicit that this applies even when the numbers involved are negative.

mathy‘s approach is more ambitious, I think. He is suggesting that “addition” and “subtraction” refer not to the formal mathematical operations but to their concrete analogs. I would be surprised if that’s what you had in mind, and I think forging the relationship between this approach and the commutative and associative properties of addition is going to require more nuance than I think you want to take on in this marginal note.

So my vote is to leave out the reference to subtraction, insert note re: negative addends. (BTW, I would argue that “addend” is the additive analog of “factor” rather than “term”.)

4. Thanks, Chris, this is helpful (just remembered to check back on this thread).