It's amazing how simple questions from young students can often uncover unexpected gaps in (at least my) knowledge, or at least the ability to explain.
Today, a student asked me why she can "forget the bracket":
$$ x+(x+5)=x+x+5 $$
Elementary school student's idea of brackets is I have to calculate this before anything else and thus the student thinks that perhaps $(x+5)$ is a entity of its own, not to be touched (since you can't really add $x$ and $5$).
My approach was to
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Demonstrate on natural numbers (i.e. proof by example) that no amount of bracketing will change the result with addition to deal with this specific example.
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Explain that $(x+5)$ and $-(x+5)$ can be thought of as a special case/shorthand of $c(x+5)$ (because multiplying a bracket by a number is something the student's mind automatically recognizes and knows how to do) and thus $(x+5)$ "really" equals $1(x+5)$ and $-(x+5)$ "really" equals $-1(x+5)$, hopefully ensuring the student wont make a mistake in the future.
However, I am not convinced that I succeeded fully at providing a good mental process for dealing with brackets in her mind. Thus I am asking:
How do/would you explain brackets? What is the best/generally accepted way (if there is one)?
Best Answer
Perhaps it's worth taking a step back and reminding your students how they learned to do addition.
If I have a pile of X jellybeans, and another pile of X jellybeans, and another pile of 5 jellybeans, does it matter which order I put them together?
hmmm... jellybeans :)
Edit: in response to comment, please feel free to replace jellybeans with an alternative confection of your choice.