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Watch Out for Parentheses 1


Alignments to Content Standards: 5.OA.A.1

Task

Evaluate the following numerical expressions.

  1. $2\times5+3\times2+4$
  2. $2\times(5+3\times2+4)$
  3. $2\times5+3\times(2+4)$
  4. $2\times(5+3)\times2+4$
  5. $(2\times5)+(3\times2)+4$
  6. $2\times(5+3)\times(2+4)$

Can the parentheses in any of these expressions be removed without changing the value the expression?

IM Commentary

This problem asks the student to evaluate six numerical expressions that contain the same integers and operations yet have differing results due to placement of parentheses. It helps students see the purpose of using parentheses. Asking if the parentheses could be removed points out that sometimes we use parentheses to emphasize the grouping of numbers only, as in part (e). In this case, the parentheses can be removed without changing the value of the expression.

This type of problem helps students to see structure in numerical expressions. In later grades they will be working with similar ideas in the context of seeing and using structure in algebraic expressions.

The task could be used for either instruction or assessment.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail. 

This particular task supports the demonstration of Mathematical Practice Standard 6, Attend to precision. Students evaluate six numerical expressions that contain the same integers and operations but differ in the placement of parentheses. It is essential that students attend to the placement of the parentheses to compute these expressions accurately.  During this exercise, students experience first-how the value of an expression is dependent upon parentheses placement.  As a result, they learn to appreciate, understand, and use symbols more precisely.

 

Solution

  1. We follow the usual order of operations and multiply before adding:

    $$2\times5 + 3\times2 + 4= 10+6+4 = 20$$

  2. Before multiplying the first 2, we complete the operations inside the parentheses using oder of operations: $$2\times(5+3\times2+4)=2\times(5+6+4) = 2\times15=30$$
  3. We first complete the addition in parentheses and then follow the usual order of operations: $$2\times5+3\times(2+4)= 2\times5 +3\times6 = 10+18 = 28$$
  4. We first complete the addition in parentheses and then follow the usual order of operations: $$2\times(5+3)\times2+4=2\times 8\times 2 +4 = 32+4 = 36$$
  5. In this case the placement of parentheses does not change the value of the expression. We can remove them and see that we get the same expression as in part (a): $$(2\times5)+(3\times2)+4=2\times5+3\times2+4 = 20$$
  6. We first complete the addition in parentheses and then multiply: $$2\times(5+3)\times(2+4)=2\times 8\times6 = 96$$

The five expressions aside from part (e) evaluate to different results. Therefore, we cannot remove any of the parentheses without changing the value of the expression, since doing so would give us one of the other expressions in the list. For example, if we remove the parentheses in (b) we get the expression from (a). The placement of the parentheses forces us to complete the computations in a different order than we would according to the standard order of operations.

Jayesh says:

about 2 years

I think they should take out the solutions and when students are done with the problem, they can submit their answers and people from Illustrative Mathematics can check it. When the people are done they can send a email to the account holder telling them the students grade and what they can work on.

Jayesh says:

about 2 years

This is just a suggestion.

Jayesh says:

about 2 years

I think they should take out the solutions and when students are done with the problem, they can submit their answers and people from Illustrative Mathematics can check it. When the people are done they can send a email to the account holder telling them the students grade and what they can work on.

Jayesh says:

about 2 years

This is just a suggestion.

Kristin says:

over 5 years

Thanks to Heather Brown, we have changed this task to better illustrate this standard. The original version of the task still represents important ideas for students transitioning between numeric and algebraic expressions, so it has been moved to the cluster "6.EE.A Apply and extend previous understandings of arithmetic to algebraic expressions." See

http://illustrativemathematics.org/illustrations/1136

Heather_Brown says:

over 5 years

On June 3, 2012, Bill McCallum posted on his blog (Complete Draft progression for cc and oa) the following statement.

“Nested grouping symbols” means grouping symbols within grouping symbols, like 5[3 + 2(9-2)]. The progression suggests limiting 5.OA.1 to expressions without nested grouping symbols, such as 3(4+5).

Doesn't this problem include nested grouping symbols?

Kristin says:

over 5 years

Thanks for pointing this out, Heather. I agree that given the progression document, it is too complex for 5.OA.1, and think this task fits better with the cluster 6.EE.A. I'll have the author take a look to see if it needs to be adapted a bit before we move it.

tangeenglish says:

over 5 years

I can't find any examples/ illustrations of OA using brackets and braces. My state frameworks don't have them nor do I see them here. I would love to see one. Is there any particular reason why?

Cam says:

over 5 years

Nope, there's no (deliberate) reason -- the standard 5.OA.1 explicitly mentions the ability to use parentheses, braces, and brackets. It would of course be safe to use any existing task with parentheses and replace corresponding pairs of parentheses with either brackets or braces. I suspect that the reason for their absence in tasks is simply that the unspoken convention is to resort to braces and brackets only when they would help the visual processing of an expression, e.g., when there are sufficiently many parentheses in a task that it's not easy to pair off corresponding opening and closing parentheses. At the level of intricacy of algebraic expressions we see at the grade 5 level, it's much less likely to run into situations where they are deemed necessary.

In any case, thanks for the observation -- I'll keep my eye out for tasks in the future where we can diversify our punctuation!