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If you were to explain what "consistent" means to a middle schooler in elementary Algebra would you say it is the fact that a rule continues to hold for more general cases?

With the example of the rule for positive exponents continues to hold for zero and negative exponents.

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I would say consistent means that you cannot derive a contradiction from (whatever it is you're talking about); in the case of extending a rule, the extension of the rule would be consistent if it holds in more general cases. –  Hayden Jun 28 at 22:55
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For a middle school student, maybe start with something like this: You ask your mother if you can have a cookie, and she says no. Then if you have a brownie, she can't say that you went against what she said. Or you ask you mother if you can go outside, and she says you can if your father doesn't say "no", so you go outside without asking your father (because by not asking him, he hasn't said "no"). Consistent means you haven't technically broken any rules. Then move on to math things. –  Dave L. Renfro Jun 30 at 13:47
    
@DaveL.Renfro what would be the first math thing you would use? –  Ice Boy Jul 3 at 8:38
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I think your exponents example is a good one, and I pretty much always used the laws for positive integer exponents to suggest the extension to negative and rational exponents. However, I think one should be careful about over emphasizing formal reasoning and proof, especially at the middle school level. In fact, I always considered the work with extending the rules of exponents at this level being mostly a way of re-discovering the rules if you forgot exactly how they went. Incidentally, you may find it useful to google the phrase "principle of the permanence of equivalent forms". –  Dave L. Renfro Jul 3 at 12:44

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I do not know if this has to do with extending rules. For instance, you cannot extend properties of prime numbers to all integers, even though we hope elementary number theory is consistent.

Consistency in mathematics has to do with how a given statement $a$ relates to a set $X$ of other statements. $a$ is consistent with $X$ provided there is no proof of $\neg a$ using the statements in $X$.

$X$ is self-consistent if you cannot prove the opposite of one of its statements using the others. This is equivalent to saying you will never be able to prove both a statement and its negation by assuming only the statements in $X$.

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Thank you for the clear explanation, but how can this be shown at the elementary Algebra level studied in middle school at the age of 13 years old? –  Ice Boy Jun 29 at 7:55

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