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Mathematical Monday ~ Fractions

J likes fiddling with my math manipulatives. He has discovered how to add fractions with my fraction discs and strips.

1/4 + 1/2
Change 1/2 = 2/4. Now 1/4 can be added to 2/4. 

Recently, he asked me how 1/3 can be added to 1/4. "Thirds cannot be changed to fourths and fourths cannot be changed to thirds," he said. I told him to change both to something else. What about twelfths? I asked. He was soon able to tell me that 1/4 is equal to 3 twelfths and 1/3 is equal to 4 twelfths. We did not have our manipulatives with us then, but the pictures below illustrate what I have just discussed.

1/3 + 1/4
Change 1/3 to 4/12 and 1/4 to 3/12. Now 4/12 and 3/12 can be easily added.  
I think J's earlier contact with the fraction resources has helped him to visualise fractions. I was too excited to stop teaching altogether and so I proceeded to discuss multiplication of fractions with him. What is 1/2 of 1/2? I asked. J ventured a few guesses, but was not able to get the answer. Undeterred, I drew the following picture. The purple part was drawn first.

Me: Can you see half here? (point to shaded purple part)
J: Ah ha.
Me: Now, what is half of this?
J: Oh! 1/4. I knew it!
Me: 1/2 of 1/2 is also the same as 1/2 x 1/2. You can also get 1/4 by multiplying 1 by 1 in the numerator and 2 by 2 in the denominator.

We tried 2 more examples and J got the correct answer. He was also able to draw pictures similar to what I have drawn above.

When I taught product of 2 fractions in school last time, I folded a piece of paper into 2 equal parts and shaded one part to represent 1/2. I further folded that 1/2 into 2 equal parts to show that 1/2 of 1/2 is 1/4. Some pupils did not understand how the folding was related to 1/2 x 1/2. Perhaps the pictorial representation is clearer.

I guess I am a geek, but Maths is exciting. :)


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