Help with teaching & learning fractions
In this article I discuss, first of all, why fractions are so difficult to learn. It has to do with two major factors. One being that there are so many "rules" to learn that children get mixed up with them and confuse them. And secondly, fraction arithmetic needs to be taught basing it on visual models so that students will get a firm grasp of the CONCEPTS, before simply memorizing the various "rules." Lastly you'll find links to my free fraction videos, and to my self-teaching fraction books.
Why are fractions so difficult to learn?
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.
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:
I can't thank you enough for the creative method you've developed to teach fractions. Yesterday I worked with my granddaughter on equivalent fractions based on your video. We've worked on them before and she could do them but she didn't really understand the concept. while we were doing the very first example she said, "This is more fun than I thought it would be, usually it's pretty boring!" Working with the pie diagrams and the idea of "splitting" helped her visualize the concepts. Within 5-10 minutes she was clicking off the examples. I'm very grateful.
If you think through pictures, you will easily see the need for multiplying or dividing both the numerator and denominator by the same number. But before voicing that rule, it is better that kids get lots of 'hands-on' experience with fraction pictures they can draw themselves. They can even have fun splitting the pieces further, or conversely merging pieces together. They may find the rule, or you may tell them about it - and it will make sense. If they later forget the rule, they can always think back to splitting pieces, and re-discover it.
Another example is the topic of addition of unlike fractions. The teacher can show how the individual fractions need to be split into further pieces so that they are all same kind of pieces. You don't need to discuss "least common denominator" at this point. The teacher can simply use pictures or manipulatives. Then, the kids will do the same with manipulatives, or by drawing pictures. After a while, some kids might discover the 'rule' of what kind of pieces the fractions will need split into. And in any case, they will certainly remember it better when they have been able to verify it themselves with numerous examples.
I'm not saying that the rules are not needed - because they are. You can't get through algebra without knowing the rules for fraction operations. But if 10 years from now the child has maybe forgotten the fraction rules, hopefully he will have retained the simple fraction pictures and is able to "do math" with the pictures in his mind, and not consider fractions as something he just "cannot do".
Do you need help with fractions?
First, check out my free fraction videos:
Fraction topics, part 1 (mixed numbers, equivalent fractions, addition, subtraction, comparing)
Fraction topics, part 2 (simplifying, multiplying, and dividing fractions; fractions to decimals; ratios and fractions; a fractional part of a group of objects)
Then, these two inexpensive worktexts just might help even further! They both explain fraction operations with a visual model of a pie, and let you practice sufficiently with picture exercises so you will understand the concepts. Fraction math is full of all kinds of rules, which are easily forgotten. Students fare so much better if they can learn to visualize these "pie pieces" in their minds while doing easy, mental fraction operations - and then the rules will start making sense.
A self-teaching worktext for 5th grade that teaches fractions and their operations with visual models. The book covers fractions, mixed numbers, adding and subtracting like fractions, adding and subtracting mixed numbers, adding and subtracting unlike fractions, and comparing fractions.
Download ($3.50). Also available as a printed copy.
A self-teaching worktext that teaches fractions using visual models, a sequel to Math Mammoth Fractions 1. The book covers simplifying fractions, multiplication and division of fractions and mixed numbers, converting fractions to decimals, and ratios.
Download ($5.75). Also available as a printed copy.
See also a list of online fraction learning games.