Answer of the week

Last week I asked that if you multiply 7 by some number, is it possible to make the answer to be no more than 7. And, how would a third or fifth or seventh grader answer the question.

I asked two kids this question. One had studied fractions and decimals, but still only found the easy answers that if you multiply 7 by 1 or 0, then the result is no more than 7. He seemed to be misled by the word 'number' - thinking of only whole numbers.

The other child had studied fractions some but not decimals. He immediately found those two easy ways and was of a definite opinion that there are no other ways to do it.

I hope you realize that there is an infinite number of ways to do this.

Basically we're looking at an expression x*7, or 7x, or 7 multiplied by some number x. So can you make 7x equal something no more than 7? Say, can you make 7x equal 5? Can you make 7x = 2? Can you make 7x = 3/4 ? Can you find a number x so that 7x = -12 ?

These little equations should look pretty familiar to anyone who has studied algebra. And the answer is yes to all those questions - AND the way to find the number x is pretty easy too:

If you want to find a number x so that 7x = 3/4, just divide both sides of this equation by 7. So your x is 3/4 ÷ 7 which is 3/28.

Here's a truth:
No matter what number a I have on the other side (whether a is big or small, fraction or decimal or irrational number), I can always divide that number a by 7 and find the answer:

7x = a
x = a/7.

In other words (in algebra terminology): If we write f(x) = 7x, we can say that the range of function f is all real numbers.

So my little question is an important TYPE of question for even future algebra studies.
SO many things in mathematics CONNECT with others!

I will try to make another question for you soon.

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