Glory to God in the highest; and on earth, peace to people on whom His favor rests! - Luke 2:14

The Joy of a Teacher is the Success of his Students. - Samuel Chukwuemeka

# Fractions, Decimals, and Percents I greet you this day,

First, read the notes. Second, view the videos. Third, solve the questions. Fourth, check your answers with the calculators.

I wrote the codes for the calculators using JavaScript, a client-side scripting language and AJAX, a JavaScript library. Please use the latest Internet browsers. The calculators should work.

Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me. If you are my student, please do not contact me here. Contact me via the school's system.

Samuel Dominic Chukwuemeka (SamDom For Peace) B.Eng., A.A.T, M.Ed., M.S

## Objectives

Students will:

(1.) Define natural numbers.

(2.) Define whole numbers.

(3.) Define fractions.

(4.) Define rational numbers.

(5.) Define decimals.

(6.) Define percents.

(7.) Write whole numbers as rational numbers.

(8.) Write mixed numbers as rational numbers.

(9.) Determine if a given rational number is a repeating decimal.

(10.) Determine if a given rational number is a terminating decimal.

(11.) Write fractions as decimals.

(12.) Write mixed numbers as decimals.

(13.) Write terminating decimals as fractions.

(14.) Order rational numbers from least to greatest.

(15.) Order rational numbers from greatest to least.

(16.) Solve real-world problems involving rational numbers.

(17.) Add fractions with same denominators.

(18.) Subtract fractions with same denominators.

(19.) Add mixed numbers with same denominators.

(20. Subtract mixed numbers with same denominators.

(21.) Simplify variable expressions involving "Like" fractions.

(22.) Solve linear equations in one variable involving "Like" fractions.

(23.) Check the solutions of linear equations in one variable involving "Like" fractions.

(24.) Solve real-world problems involving "Like" fractions.

(25.) Add fractions with different denominators.

(26.) Subtract fractions with different denominators.

(27.) Add mixed numbers with different denominators.

(28. Subtract mixed numbers with different denominators.

(29.) Simplify variable expressions involving "Unlike" fractions.

(30.) Solve linear equations in one variable involving "Unlike" fractions.

(31.) Check the solutions of linear equations in one variable involving "Unlike" fractions.

(32.) Solve real-world problems involving "Unlike" fractions.

(33.) Write percents as fractions.

(34.) Write fractions as percents.

(35.) Determine the percent of a given number.

(36.) Solve algebraic expressions involving percents.

(37.) Solve algebraic equations involving percents.

(38.) Solve real-world problems involving percents.

## Skills Measured/Acquired

(1.) Use of prior knowledge

(2.) Critical Thinking

(3.) Interdisciplinary connections/applications

(4.) Technology

(5.) Active participation through direct questioning

(6.) Student collaboration in Final Project

## Vocabulary Words

natural number, counting number, whole number, fraction, numerator, denominator, proper fraction, improper fraction, rational number, integer, mixed number, decimal, terminating decimal, exact decimal, repeating decimal, recurring decimal, irrational number, real number, complex number, add, sum, subtract, difference, multiply, product, divide, quotient, "like" fractions, "unlike" fractions, variable expressions, linear equations, equivalent fractions, same denominators, different denominators, common denominator, least common multiple (LCM)/least common denominator (LCD), proportions, equations, greatest common factor (GCF)/greatest common divisor (GCD)/highest common factor (HCF), prime factorization, left hand side (LHS), right hand side (RHS), percent, percentages, ratio, arithmetic, arithmetic operators, augend, addend, minuend, subtrahend, multiplier, multiplicand, factor, dividend, divisor, positive, negative, nonpositive, nonnegative, constant, number, variable, term, equal, equality, ratios

## Definitions

A natural number is any positive integer.

It is also known as a counting number. It is a number you can count.

It does not include zero.

It does not include the negative integers.

It is not a fraction.

It is not a decimal.

A whole number is any nonnegative integer.

It includes zero and the positive integers.

It does not include the negative integers.

It is not a fraction.

It is not a decimal.

An integer is any whole number or its opposite.

Integers include the whole numbers and the negative values of the whole numbers.

A fraction is a part of a whole.

It is the part of something out of a whole thing.

It is also seen as a ratio.

It is also seen as a quotient.

The numerator is the part.

It is the "top" part of the fraction.

The denominator is the whole.

It is the "bottom" part of the fraction.

A proper fraction is a fraction whose numerator is less than the denominator.

An improper fraction is a fraction whose numerator is greater than or equal to the denominator.

A mixed number is a combination of an integer and a proper fraction.

A decimal is a linear array of digits that represent a real number, expressed in a decimal system with a decimal point; and in which every decimal place indicates a multiple of negative power of 10.

A terminating decimal is a decimal with a finite number of digits.

A terminating decimal is also known as an exact decimal.

A repeating decimal is a decimal in which one or more digits is repeated indefinitely in a pattern or sequence.

A repeating decimal is also known as a recurring decimal.

A non-repeating decimal is a decimal in which there is no sequence of repeated digits indefinitely.

A non-repeating decimal is also known as a non-recurring decimal.

A rational number is any number that can be written as a fraction where the denominator is not equal to zero.

You can also say that a rational number is a ratio of two integers where the denominator is not equal to zero.

A rational number is a number that can be written as: $${c\over d}$$ where $c, d$ are integers and $d \neq 0$

A rational number can be an integer.

It can be a terminating decimal. Why?

It can be a repeating decimal. Why?

It cannot be a non-repeating decimal. Why?

Ask students to tell you what happens if the denominator is zero.

An irrational number is a number that cannot be expressed as a fraction, terminating decimal, or repeating decimal.

When you compute irrational numbers, they are non-repeating decimals.

A real number is any rational or irrational number.

It includes all numbers that can be found on the real number line.

A complex number is a number that can be expressed in the form of $a + bi$ where $a \:and\: b$ are real numbers, and $i$ is an imaginary number equal to the square root of $-1$.

Variable Expressions are expressions that contain variables.

Linear Equations are equations in which the highest degree of the variable in the equation is 1.

Equivalent Fractions are two or more fractions that have the same value when they are expressed in their simplest forms.

Common Denominators are the common multiples of the different denominators of unlike fractions.

Least Common Denominator is the least of all the common multiples of the different denominators of unlike fractions.

Prime Factorization is a method used for finding the least common denominator of unlike fractions, in which each denominator is broken down into a product of prime numbers. This means that each denominator is split into a product of prime factors.

The basic arithmetic operators are the addition symbol, $+$, the subtraction symbol, $-$, the multiplication symbol, $*$, and the division symbol, $\div$

Augend is the term that is being added to. It is the first term.

Addend is the term that is added. It is the second term.

Sum is the result of the addition.

$$3 + 7 = 10$$ $$3 = augend$$ $$7 = addend$$ $$10 = sum$$

Minuend is the term that is being subtracted from. It is the first term.

Subtrahend is the term that is subtracted. It is the second term.

Difference is the result of the subtraction.

$$3 - 7 = -4$$ $$3 = minuend$$ $$7 = subtrahend$$ $$-4 = difference$$

Multiplier is the term that is multiplied by. It is the first term.

Multiplicand is the term that is multiplied. It is the second term.

Product is the result of the multiplication.

$$3 * 10 = 30$$ $$3 = multiplier$$ $$10 = multiplicand$$ $$30 = product$$

Dividend is the term that is being divided. It is the numerator.

Divisor is the term that is dividing. It is the denominator.

Quotient is the result of the division.

Remainder is the term remaining after the division.

$$12 \div 7 = 1 \:R\: 5$$ $$12 = dividend$$ $$10 = divisor$$ $$1 = quotient$$ $$5 = remainder$$

A constant is something that does not change. In mathematics, numbers are usually the constants.

A variable is something that varies (changes). In Mathematics, alphabets are usually the variables.

A mathematical expression is a combination of variables and/or constants using arithmetic operators.

A mathematical equation is an equality of two terms - the term or expression on the LHS (Left Hand Side) and the term or expression on the RHS (Right Hand Side).
This implies that we should always check the solution of any equation that we solve to make sure the LHS is equal to the RHS.
Whenever we solve for the variable in "any" equation, how do we know we are correct? CHECK!

The solution of an equation is the value of the variable which when substituted in the equation ensures that the LHS is equal to the RHS.

The solution of an equation is also known as the root of the equation or the zero of the function.

The solutions of an equation are the values of the variables which when substituted in the equation ensures that the LHS is equal to the RHS.

The solutions of an equation are also referred to as the roots of the equation or the zeros of the function.

A percent means something out of $100$.

A ratio is a comparison of two quantities.

A proportion is the equality of two ratios.

## Fractions

A fraction is a part of a whole.

It is the part of something out of a whole thing.

It is also seen as a ratio.

It is also seen as a quotient.

The numerator is the part.

It is the "top" part of the fraction.

The denominator is the whole.

It is the "bottom" part of the fraction.

A proper fraction is a fraction whose numerator is less than the denominator.

An improper fraction is a fraction whose numerator is greater than or equal to the denominator.

Like Fractions are fractions with the same denominator.

Unlike Fractions are fractions with different denominators.

### Examples - Introduction to Fractions

For the following scenarios, identify the common denominator.

(1.) $\dfrac{2}{3}$ of the students in Mr. C's class are females and $\dfrac{1}{3}$ are males.

Common denominator = $3$

(2.) $48$ of the $50$ states in the United States are in the continental US while $2$ of the $50$ states are not.

Common denominator = $50$

(3.) $9$ in every $10$ leaders in Nigeria are corrupt (hence the poor masses suffer terribly) while $1$ in every $10$ Nigerians are probably not.

Common denominator = $10$

(4.) Peace bought a large pizza and divided it into $12$ equal parts for her $12$ children. Each child got a part.

Common denominator = $12$

For the following scenarios, identify the different denominators.

(5.) $\dfrac{2}{5}$ of the students in Mr. C's class are females. $\dfrac{1}{3}$ of those females do not like math.

Different denominators = $5 \:and\: 3$

Round to =