# How to explain the motivation of parentheses in addition, subtraction and multiplication?

My kid, 5 years old, knows addition, subtraction and multiplication now, of course, in a basic level.

Also he understands that parentheses means "whichever inside shall be computed first".

When I try explain the rules of "taking off the parentheses", he seems thinks "so long as we can compute, why take it off?"

For example, $$10 - (5-3) = 10 - 2 = 8$$, why shall it be done as $$10 - (5-3) = 10 -5 +3 = 5 + 3 = 8$$?

• One possible place where this could be explained is when 'multiplying by making fives/tens' or other similar calculations where one way is easy to remember and another less so: 3(5+1) = 15 + 3 = 18 vs. 3(6) = 18 (memorized) 3(10+3) = 3(10)+3(3) = 30+9 = 39 vs. 3(13)=39 (memorized) Understanding this leads to the standard base-10 algorithm for multiplication. Any time a student asks for why they should learn something that they can do the 'easy way,' give them a scenario when the new way IS the 'easy way'. – Opal E Feb 26 '19 at 2:24
• "why take it off?" - because one can. Sometimes it makes sense, sometimes not. Without variables, it is never necessary. – Jasper Feb 28 '19 at 6:52

Why take off the parentheses?

Because sometimes "taking off the parentheses" results in an easier to calculate expression. For example,

$$123456789-(-9876543210+123456789)$$ is easier to evaluate by using $$123456789+9876543210-123456789=9876543210$$.

• Or vice versa: 1 + 2 + 3 + 4 + … + 98 + 99 + 100 = 50(101) – Rusty Core Oct 25 '19 at 17:10

Your question relates to a 5-year old child. This should be taken into account (e.g., no point in talking about algebraic manipulations). Parentheses are here to help us; and if they do, there is no reason to remove them. I would still tell a child that we usually prefer expressions that have as fewer symbols are possible, and explain that sometimes, parentheses are unavoidable, but if your 5-year old likes the expression (3+5)+2 better than 3+5+2, this is OK. A dialog like "we first add 3 and 5 and then, whatever we got we add to 2" is quite valuable. Rather than trying to eliminate parentheses, I'd rather explore the value of an expression with and without them.

The following aren't the same thing:

1. The meaning of an expression
2. An option for computing the value of the expression
3. A rigidly imposed sequence of instructions to be obeyed to compute the value

The following example may help you more than your student:

Define x = 1 if there are infinitely many twin primes, and x = (- 1) otherwise.

If what is inside parentheses "shall be computed first", then how are you going to compute the value of the following expression: (1 - x) + (1 + x)?

The instructions, if taken literally, require that we begin by resolving the twin primes conjecture, but nobody has achieved that. Nor is it necessary. Both roads go to the same result: 2.

As for your question "why shall it be done as 10 − (5 − 3) = 10 − 5 + 3 = 5 + 3 = 8", I think we can omit the how it "shall be done" command, and focus on recognizing that a - (b - c) = (a - b) + c is true for various kinds of numbers a, b, and c.

Let us suppose we are asked to determine the value of: (a - (b - c)) - ((a - b) + c).
We're back to a situation like the twin primes conjecture above, because that value can be determined to be zero, without any need to know the value of a, b, or c. Thus, there is no need to begin by computing (b - c) and (a - b). There may also be no possibility of computing (b - c) and (a - b) if we aren't provided with any information about the numbers a, b, or c.

Parentheses could be introduced in connection with operation symbols so that we write an expression like (a + b). If the plus symbol "+" is always accompanied by parentheses, then ambiguities won't arise. However, it will quickly seem to be a boring chore to write all parentheses, and that will motivate some consideration of when and how to omit parentheses without introducing ambiguities.

The idea of parentheses can be conveyed by starting with a similar notation in linguistics that allows us to distinguish between sentences that have the same surface structure, but different deep structures or tree diagrams.

For example:

1. The fruit flies like a banana. (Here "flies" is a verb and "like a banana" is an adverb phrase)

2. The fruit flies like a banana. (Here "fruit" is an adjective in the noun phrase "fruit flies")

You can do something similar with the following sentence: "British Left Waffles on Falkland Islands." (I used initial capitals on most words in order to avoid having to choose between the past tense verb "left" and the political noun "Left.")