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Making Math Complicated

Number-sense is great...and can be useful!

So, you would think that ARCO's guide for Teacher Certification Tests (1996) would certainly capitalize on the power of number-sense when solving multiple-choice problems. Not true...

For example, the manual poses this problem: The fraction 12/11 is between each of the following pairs except
[A] 2/3 and 4/3
[B] 9/11 and 11/9
[C] 1 and 2
[D] 0.9 and 1.1
[E] 11/12 and 12/12

Think...how would you solve it?

The book suggests that you first draw a number line, change 12/11 to both 1 1/11 and 1.09, locate it on the number line...placing the fraction version above the line and the decimal version below the line.

Now, for each option, locate each pair of numbers on the number line and connect them by a horizontal bracket.

Visually, you should discover that the bracket for option [E] does not include the dot for 12/11, and thus [E] is the answer.

Just to be sure, the text solves it a second way as well. Change all decimals to fractions, and then cross multiply each option pair with 12/11. To illustrate, for option [A], 2/3...12/11...4/3 after two cross multiplications leads to 22 < 36 < 44. Thus, [A] is not the desired answer, etc.

Now, I would guess that you looked at 12/11, used number-sense to conclude that it was bigger than 1, and thus [E] had to be the answer.

Just one more example of showing math as being dependent on algorithms with little understanding... even when modeling for prospective teachers! Ugh...