6 Fractions and Ratios

Notes

Lessons below may revise or consolidate fraction skills and concepts with operations on lengths. In the process, algebraic descriptions of operations are indicated not as requirement for fraction mastery but as enrichment option. Methods for comparison, addition, subtraction, multiplication and division are all developed or justfied below by raising terms to transform a general case into simpler cases. Algebraic view of those operations, click here.

Step-by-Step Skill Development

While all methods are likely to be well-known in the mathematical sense, the methods themselves may be new to many students and teachers. These raising term methods may overwhelm some students but help others. Instructors and tutors should emphasize fraction sense and skill mastery by rote when and where explanations of why methods work do not help. Partial or full comprehension may be left to older or gifted students, and to students who need to understand the development or origins of methods before using them.

What is a Fraction: Meaning of a Fraction - A whole number counts how many ones, A fraction counts how many parts of equal value. Algebraic Description included.

Fraction Multiplication I. What is a Unit Fraction of a Unit Fraction? What is a half of a third? What is a unit fraction of a unit fraction? A unit fraction has one in the numerator. Algebraic Description included.

Fraction Multiplication II. Unit Fraction of a Simple Fraction: what is a half of two thirds? What is a quarters of seven tenths?

Fraction Multiplication III. : What is a Fraction of a Fraction: what is seven quarters of three tenths?

Equivalent Fractions: What is the difference between two quarters and one half? What is the difference between six eights and three quarters? What is the difference between an apple and four quarters of the apple? The thought that there is no difference, that different fractions may describe the same amount, quantity, leads to the idea of equivalent fractions. Examples of fraction simplifying and equivalence included.

Multiplication Algebraic Development. The first, easy case treats the case where the denominator of one fraction is a divisor of the numerator of the second fraction The general case follows by raising terms in the second factor to apply the easy case.

Mixed Numbers and Equivalent Fractions: We may describe a distance as 3 half meters or we may describe the same length as 1½ meters. Physically, there is no difference in the distance, only its descriptions. The descriptions 3/2 meters and 1½ meters both have the same value physically. So we declare the numerals 3/2 and 1½ to have the same value, or to be equivalent. Likewise 3 fifties (half-hundreds) is the same a 1½ hundreds. There is another instance where the fraction 3/2 and the mixed numeral 1½ may be identified as adjectives for the same count or measure.

Comparison of Fraction and comparison of Mixed Numbers: Algebraic Description included of the first included.

Fraction Addition I: Easy case of like denominators - the easy case Algebraic Description included.

Fraction Addition II: General Case of unlike denominators. the general case follows from raising terms (as little as possible) to use the easy case.

Examples to show "raising terms" similarities between comparison, addition and subtraction of fractions.

Fraction Addition III: Methods for adding and subtracting Efficiently - Questions and Problems: (a) How is the list method used to obtain a least common denominator = the least common multiple of a pair of denominators? (b) How can the prime number decomposition (also known as factorization) be used to calculate the LCM and GCD of a pair of whole numbers? (c) How can Euclid's algorithm be used forwards and backwards to calculate the GCD (the normal result) and then the LCM of a pair of whole numbers? (d) Employ the M-method to find the sum of eighteen 21st and nine 14-ths? Say which of the latter is more than the other, and find how much more.

Fraction Multiplication IV: Efficient Ways to Multiply Fractions: Learn how to calculate a few products of fractions with and with the cancellation methods described below for "efficient" multiplication, or more precisely efficient or easier simplification after (cross) cancellation of common factors.

Fraction Division, Compound Fractions and Reciprocals: A Physical Introduction to Fraction Division: Explanations, twice-over, are given in the next lesson.

Fraction Division Methods Explained. See the previous page for an introduction of the fraction division methods or formulas below.
- Two Step Development Option - Explanation in two smaller steps, first with the easy like denominators case and second with unlike denominators (Raising terms in dividend and divisor fractions turns the second case into the easy case).

How to do Arithmetic with Rational Numbers, that is signed fractions, signed whole numbers and signed mixed numbers. Here a model for introducing arithmetic with real numbers.

A. Ratios And Fractions
Similarities and Differences

Fractions may be identified with two term ratios, and vice-versa as well, sometimes. Two fraction are equal or equivalent when and only when the corresponding two term ratios are equal or equivalent. But fractions can be added, subtracted, multiplied and divided while the same operations are not defined for two- and multiple term ratios. While we may call a fraction, a ratio or a rational number, ratios are different. Triple term ratios exist, but triple term fractions do no exist. Three and more -term ratios cannot be identified with fractions.

The following lesson cover the properties of two term and multiple term ratios.

Fractions As (two term) Ratios and Fractions Versus Ratios: Fractions are often called ratios, and vice-versa. But the vice-versa only holds for two term ratios. This lesson identifies fractions with two-term ratios and contrasts the properties of fractions and two-term ratios. (Ratios cannot be added, subtracted or compared, but like fractions, the terms in ratios can be raised or lowered).

Implied or Derived Ratios - New Fractions and Ratios from Old: If two fractions are equivalent then their reciprocals are also equivalent. Likewise if a pair of two-term ratios are equivalent, interchanging the first and second terms of each ratio in the pair leads to a pair of equivalent ratios. Beyond that, more equivalent ratios can also be generated from a pair of ratios. Food for thought: How may equivalent fractions or ratios may be formed from the relations ad = bc?

Multiple Ratios: Multiple Term Ratios - Three Term Ratios to be precise. We read the triple ratio a : b :c as a to b to c. We further write

a : b: c :: A: B: C

to say two triple ratios a : b: c and A: B: C are equal or equivalent when and only when

a
b

A
B

and

b
c

B
C

there are other ways to say when two triple ratios are equal or equivalent.

Note: Triple ratios or triple proportionalities occur between the sides of similar triangles. More generally, multiple ratios or proportionalities occur between the sides of similar triangles.

The discussion of ratios or multiple ratios is best understood besides a discussion of proportionality.

Inner Versus Outer Terms - small point: In the discussion of equality of ratios a : b = A: B written in that order, the inner terms are small b and big A while the outer terms are small a and big B. In contrast, if we rewrite the equality as A: B = a : b, we find the inner and outer terms are interchanged. However, the equality requires the product of the inner and outer terms be equal, that is aB = Ab. That equality is not affected by rewriting a : b = A: B as A: B = a : b, and the resulting swap of inner and outer terms

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills.

Mathematics Skills For Ages 3 to 14

Skills with take home value

Basic skills include time-date-calendar Matters; money matters; map, plan and scale diagram matters;counting, measuring and figuring; decision making with logic and likelyhood; being careful and being aware of the domino effect of mistakes; reading and writing with precision.

Is your child able to add, subtract and multiply amounts of money, work with fractions, work with clocks and calendars, work with maps and plans, and measure length, weight-mass and volume? Schools may promote your son or daughter without providing basic skills in reading, writing and arithmetic.

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.