As many teachers and parents know, learning the various fraction operations can be difficult for many children. It's not the concept of fraction that is difficult - it is the addition, multiplication, subtraction, simplifying, etc. - various operations that you do with fractions.
And the simple reason why learning the various fraction operations proves difficult for children is the way they are typically taught in school books. Just look at the amount of rules there are to learn about fractions:
1. Fraction addition - same denominators Add the numerators, and use the same denominator
2. Fraction addition - different denominators First find a common denominator by taking the least common multiple of the denominators. Then convert all the addends to have this common denominator. Then add using the rule above.
3. Finding equivalent fractions Multiply both the numerator and denominator with a same number
4. Mixed number to a fraction Multiply the whole number part by the denominator and add the numerator to get the numerator. Use the same denominator as in the fractional part of the mixed number.
5. (Improper) fraction to a mixed number Divide the numerator by the denominator to get the whole number part. The remainder will be the numerator of the fractional part. Denominator is the same.
6. Simplifying fractions Find the (greatest) common divisor of the numerator and denominator, and divide both by it.
7. Fraction multiplication Multiply the numerators, and the denominators.
8. Fraction division Find the reciprocal of the divisor, and multiply by it.
IF children simply try to memorize these without knowing where they came from, they will probably seem like a jungle of seemingly meaningless rules. By meaningless I mean that the rule does not seem to connect with anything about the operation - it is just like a play where in each case you multiply or divide or add or do various things with the numerators and denominators and that then should give you the answer.
Fraction math can then become blind following of the rules, tossing the numbers here and there, calculating this and that - and getting answers of which the kids have no idea if they are reasonable or not. And of course, it is quite easy to forget these rules, or remember them wrong - especially after 5-10 years.
The solution: use manipulatives and visual models (pictures)
Instead of merely presenting a rule, as many schoolbooks do, a better way is to teach children to visualize fractions, and perform some simple operations with these visual images or pictures, without knowingly applying any given 'rule'.
If a child is able to visualize fractions in his mind, they become more concrete - not just a number on top of other number without meaning. Then the child can estimate the answer before calculating, and evaluate the reasonableness of the final answer, and perform many of the simplest operations in his head.
Of course textbooks DO show fractions with pictures, and they DO show one or two examples of how a certain rule connects with a picture. But that is not enough! A better way is to make kids do lots of problems with fraction manipulatives - and DRAW fraction pictures for problems. That way they will form a mental visual model and can think through the pictures for simple problems.
Some simple examples are equivalent fractions and simplifying fractions from my books.
See also this video, which shows a visual method for equivalent fractions: that of splitting the pieces further into a certain number of new pieces:
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