Guidelines
Fraction arithmetic should be taught in preparation for solving Linear Systems.
Here is an outline for teaching fractions. We focus on addition because that is the hardest to teach:
- Define prime number & prime factorization into a prime string
- Define Lowest Common Denominator as the mnemonic CoPS (Covering Prime String). "Covering" means that the prime string is has enough primes to span its factors. I find that this terminology captures the LCD algorithm better than the traditional term LCD. The diagram below shows that if you line up the factors vertically, COPS prime string does cast a "shadow" that "covers" all relevant primes.
find the COPS for 6, 8, 20 cops = 2 * 2 * 2 * 3 * 5 ========|===|===|===|===| | | | | | 6 = | | 2 * 3 | 8 = 2 * 2 * 2 | 20 = 2 * 2 * 5 - introduce your favorite mnemonic for negative arithmetic, or use the DSL-SAK algorithm below.
- algorithm & mnemonic for fraction addition - this mnemonic is admittedly lame right now, so if there's a better one, please feel free to add it: (1) COPS (2) find missing child primes and (3)merge them with their "family???". Finding missing child primes refers to completing the input fractions as to a full cops string. Merging refers to multiplying out the tops & bottoms, then adding the tops together.
DSL-SAK pronounced diesel-sak (lame). This is an algorithm for negative number arithmetic.
| DSL-SAK | test | verb | final sign |
|---|---|---|---|
| explanation | do the nouns have the same or different signs? | add or subtract the positive nouns accordingly | assign the final signs accordingly |
| case 1 - DSL | nouns have Different signs | Subtract the positive nouns (big noun minus small noun) | the Larger numbers' sign is final |
| case 2 - SAK | nouns have the Same signs | Add the positive nouns | Keep the original sign (remember both nouns have the same sign) |
Flopposites
Flopposites is the effect of a sign change of an exponent. The name is a contraction to help kids remember the 2 steps: the fraction flip (vertical movement across the fraction bar) and opposite sign change in exponent.
Flottom
Flottom is the unraveling of a nested fraction. The term stands for "flip the bottom". For example 1/(b/c) = 1/1 * c/b
We use it to help students remember how to divide fraction, eg. (2/3) / (4/5) = 2/3 * 5/4
Exercises
remember to keep the numbers small. The purpose is to become familiar with the algorithm, not fluency in large number arithmetic. At this point, students may reference their multiplication tables for factorization and multiplication. They do not need to memorize it.