Parenthesis in equations that represent multi-step word problems

[bcohen]

Bill,

The standards ask students to “solve one- and two-step word problems… by using… equations with a symbol [or letter] for the unknown number…” beginning in second grade (2.OA.1, 3.OA.8, and 4.OA.3). In a multi-step problem, writing this equation will often require the use of parenthesis. However, parenthesis are not introduced in the standards until grade 5 (5.OA.1)… or are we supposed to introduce parenthesis in second or third grade?

The footnote on 3.OA.8 and the standard 5.OA.1 itself led me to believe that parenthesis were to be introduced in grade 5… and it seems like we would be asking too much of second or third grade students to come to understand both the convention of order of operations and our use of parenthesis to specify an exception that should precede the rest of the computation.  However, without introducing and spending A LOT of time on the order of operations and parenthesis, students could not write equations to represent many multi-step problems.

Any clarification would be appreciated.

Thanks,

Brian

 

[Bill McCallum]

Brian, these are some posts which may help:

http://commoncoretools.me/forums/topic/expanded-notation-and-order-of-operations/

http://commoncoretools.me/2011/05/29/complete-draft-progression-for-cc-and-oa/#comment-92

http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/#comment-1999

[Bill McCallum]

And one more:

http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/#comment-1609

[bcohen]

Thanks, Duane.  I’m glad I’m not the only one who sees some tension between some of these standards.  Some of this was helpful, but I’m still struggling with making determinations for what should be taught/assessed when.  In this comment Bill frames the tension that I see:  “…On the one hand, there is no explicit requirement to use parentheses until Grade 5.  On the other hand, there’s a footnote on 3.OA.8 that… suggests that parentheses might well be used much earlier…” (http://commoncoretools.me/2012/04/02/general-questions-about-the-standards/#comment-1609)
 
Then, discussing 3.OA.8, Bill says: “The thrust of the comment in the progressions document is that one need not expect fluency with this [order of operations] 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.” (http://commoncoretools.me/forums/topic/expanded-notation-and-order-of-operations/#post-1188)
 
So here’s a specific concrete question meant to feel out the boundaries.  This is a NY sample question for 4.OA.3:
“Students from three classes at Hudson Valley Elementary School are planning a boat trip. On the trip, there will be 20 students from each class, along with 11 teachers and 13 parents.
 
Part A: Write an equation that can be used to determine the number of boats, b, they will need on their trip if 10 people ride in each boat.”
(http://engageny.org/sites/default/files/resource/attachments/math-grade-4.pdf)
 
Recognizing that you could not write an equation, the expression would be (11 + 13 + 3 × 20) ÷ 10, or an equivalent expression.  Does this fall within the expectations of 4.OA.3?
 
Thanks,
Brian

[Cathy Kessel]

I was puzzled until I looked at the item which has a slot for the answer as b = _______ . The problem states explicitly that b represents the number of boats needed.

Putting (11 + 13 + 3 × 20) ÷ 10 in the slot doesn’t give an expression that’s equal to b unless you assume that 4/10 of a boat is going to carry 4 people. (However, putting that expression in the slot does give an equation.)

On the next page (p. 5), the answer key states that b is 84/10 and also that b is 8 R 4. However, you interpret those (is 8 R 4 equal to 84/10?), that seems to be different from the solution (which is 9). But, using the transitive property of equality (see p. 90 of the Standards) together with the answer key, yields 84/10 = 8 R 4 = 9.

I think the test developers need some comments.

A possible solution (less computing and no letters):

11 + 13 + 3 × 20 = 10 + 1 + 10 + 3 + 6 x 10 = 8 x 10 + 4, so 9 boats are needed.

There’s a similar example on p. 30 of the OA Progression.

[Bill McCallum]

Here’s what we know from the standards: students should be able to read and write numerical expressions without parentheses in Grade 3, and they should be able to read and write numerical expressions with parentheses in Grade 5.

The question Brian and Duane are discussing is: what is the pathway from Grade 3 to 5 with regard to numerical expressions? It’s a good question and a good discussion, but the standards are largely silent on the answer. Remember, standards are not curriculum. One curriculum might decide to introduce parentheses in Grade 3, another in Grade 5. Yet another might follow Brian’s suggestion about 4.OA.3 and introduce them there. That’s certainly a point at which their usefulness becomes evident. But not every multistep word problem leads to an equation with parentheses, and anyway I don’t read the standard as requiring that every single word problem be accompanied by an equation, merely that students should have experience writing equations for word problems.

The point is, all this discussion takes place within the space of flexibility allowed by the standards. The standards require that there should be a pathway, but don’t say what it is.

[ghanlonnyc]

I understand that the standards are not curriculum, nor should they be.  One difficulty here in NY is that the very high stakes tests this year are allegedly “aligned to Common Core.”  Yet the NYS Common Core curriculum has not yet been written (the promise is that a half year curriculum material will be available this summer – 3 or 4 months after the test is administered!)  Understandably teachers are confused and frustrated….. 

Some of us are taking the pathway of introducing parentheses in Gr. 4 since another sample problem from the state requires students to represent a multiple step problem with the equation (240-32)/4.  That we think is reasonable for Gr. 4.  The problem Brian and Duane sent is the second in that same sample for Gr. 4 but we think it is more appropriate for Gr. 5.   Hoping the test developers are on the same path!

 

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