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A Good Fraction Problem...or Even a Problem At All?

In selecting problems each week for MathNEXUS, I search through problems I have given over the past 38+ years to students. Often, I groan and wonder how in the world I chose a particular problem.....

To illustrate, go back in time with me to the early 1970's when I was teaching seventh grade students. One of my pedagogical gimmicks was to offer special problems tied to pending holidays. Here is my choice one year...a problem to reinforce their work with fractions...


Take the first 1/2 of thin and the first 2/3 of ant. Add the first 1/4 of king and the first 1/3 of set. Now add the first 3/4 of give and then the last 3/4 of ring. What November word do you have?
To improve clarity, the problem probably should have be written as follows...
Take the first 1/2 of "thin" and the first 2/3 of "ant". Add the first 1/4 of "king" and the first 1/3 of "set". Now add the first 3/4 of "give" and then the last 3/4 of "ring". What November word do you have?
Now, in retrospect, I can see many difficulties with this problem:
  1. Given an object, what is the meaning of the "first 1/2"? That is, when cutting an object in half, is there an implied order...first...second...?
  2. Getting technical, is "th" actually 1/2 of "thin"? That is, don't the letters t and h "dominate" the letters i and n (if nothing else measure the size or ink-usage)?
  3. And, I remember some of my students getting creative (something I encouraged)...and they would divide the word horizontally rather than vertically, so that their "first 1/2" of the word "thin" becomes a vague collection of marks and dots. Now, they knew what was wanted and could have solved the problem as posed, but liked to remind me how inane some of my problems were...
  4. Finally, does solving this problem help a student understand the concept of fractions...or if they solve it, does that mean they understand the concept of fractions? Probably not...!
And now, if faced with teaching seventh graders again, would I give the same problem at this time of year....yes!

 


Hint: No hint needed...but, when looking at my Concern #3, remember the classic problem: "How is six half of eleven?" See solution commentary if you do not know the answer to this poser...

 


Solution Commentary: How is six half of eleven....Just write eleven as a Roman numeral XI and then take the "top" half" to get VI or six!