Watch Out for Parentheses 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.