Whole Numbers and Introduction to Algebra Written by Sue C. Little North Harris College Offered with thanks by Mr. Roberts Clement Middle School, RUSD Chapter One

Place Value and Names for Numbers

The position of each digit in a number determines its place value Ones Hundred-thousandsHundred-billions Ten-billionsBillionsHundred-millions Ten-millionsMillions Ten-thousandsThousandsHundredsTens

A whole number such as 35,689,402 is written in standard form. The columns separate the digits into groups of threes. Each group of three digits is a period. MillionsThousandsBillionsOnes Ones Hundred-thousandsHundred-billions Ten-billionsBillionsHundred-millions Ten-millionsMillions Ten-thousandsThousandsHundredsTens

To write a whole number in words, write the number in each period followed by the name of the period. thirty-five million, six hundred eighty-nine thousand, four hundred two Ones Hundred-thousandsHundred-billions Ten-billionsBillionsHundred-millions Ten-millionsMillions Ten-thousandsThousandsHundredsTens

The name of the ones period is not used when reading and writing whole numbers. Also, the word “and” is not used when reading and writing whole numbers. It is used when reading and writing mixed numbers and some decimal values as shown later. Helpful Hint

The place value of a digit can be used to write a number in expanded form. The expanded form of a number shows each digit of the number with its place value. 4,786 = Standard Form Expanded Form

Adding Whole Numbers and Perimeter = 7 addend addend sum

Addition Property of 0 The sum of 0 and any number is that number = 8 Commutative Property of Addition Changing the order of two addends does not change their sum = 2 + 4

Associative Property of Addition Changing the grouping of addends does not change their sum. 3 + (4 + 2) = (3 + 4) + 2

Descriptions of problems solved through addition may include any of these key words or phrases: Key Words ExamplesSymbols added to 3 added to plus 5 plus more than 7 more than total total of 6 and increased by 16 increased by sum sum of 50 and

A polygon is a flat figure formed by line segments connected at their ends. Geometric figures such as triangles, squares, and rectangles are called polygons. rectangle square triangle

Finding the Perimeter of a Polygon The perimeter of a polygon is the distance around the polygon distance around the polygon.

Subtracting Whole Numbers Subtraction is finding the difference of two numbers.

Subtraction Properties of 0 The difference of any number and that same number is = 0 The difference of any number and 0 is the same number = 7

Descriptions of problems solved by subtraction may include any of these key words or phrases: Key Words ExamplesSymbols subtract subtract 3 from difference difference of 8 and less 12 less take away 14 take away decreased by 16 decreased by subtracted from 5 subtracted from

Rounding and Estimating Rounding a whole number means approximating it.

23 rounded to the nearest ten is rounded to the nearest ten is rounded to the nearest ten is

Rounding Whole Numbers to a Given Place Value b Step 1. Locate the digit to the right of the given place value. b Step 2. If this digit is 5 or greater, add 1 to the digit in the given place value and replace each digit to its right by 0. b Step 3. If this digit is less than 5, replace it and each digit to its right by 0.

Making estimates is often the quickest way to solve real-life problems when their solutions do not need to be exact.

Multiplying Whole Numbers and Area

= 5 x 4 = 20 5 fours factor product Multiplication is repeated addition with a different notation.

Multiplication Property of 0 The product of 0 and any number is 0. 9  0 = 0 0  6 = 0 Multiplication Property of 1 The product of 1 and any number is that same number. 9  1 = 9 1  6 = 6

Commutative Property of Multiplication Changing the order of two factors does not change their product. 6  3 = 3  6 Associative Property of Multiplication Changing the grouping of factors does not change their product. 5  ( 2  3) = (5  2)  3

Distributive Property Multiplication distributes over addition. 5(3 + 4) = 5   4

There are several words or phrases that indicate the operation of multiplication. Some of these are as follows: Key Words ExamplesSymbols multiply multiply 4 by  3 product product of 2 and  5 times 7 times  6

Area 1 square inch 11 11 5 inches 3 inches Area of a rectangle = length  width = (5 inches)(3 inches) = (5 inches)(3 inches) = 15 square inches = 15 square inches

Dividing Whole Numbers

Division is the process of separating a quantity into equal parts. quotient dividend divisor

Division Properties of 1 The quotient of any number and that same number is   The quotient of any number and 1 is that same number.

The quotient of 0 and any number (except 0) is 0. Division Properties of   The quotient of any number and 0 is not a number. We say that are undefined (u). are undefined (u).

Here are some key words and phrases that indicate the operation of division. Key Words ExamplesSymbols divide divide 15 by 3 15  3 quotient quotient of 12 and 6 divided by 8 divided by 4 divided or shared equally $20 divided equally among 5 people 20  5

How do you find an average? A student’s prealgebra grades at the end of the semester are: 90, 85, 95, 70, 80, 100, 98, 82, 90, 90. How do you find his average? Find the sum of the scores and then divide the sum by the number of scores.

Exponents and Order of Operations

An exponent is a shorthand notation for repeated multiplication is a factor 5 times Using an exponent, this product can be written as exponent base

This is called exponential notation. The exponent, 5, indicates how many times the base, 3, is a factor. exponent base Read as “three to the fifth power” or “the fifth power of three.” is a factor 5 times

4 = 4  4 = 4  4  4 = 4  4  4  4 = is read as “four to the first power.” is read as “four to the second power or four squared.” is read as “four to the third power or four cubed.” is read as “four to the fourth power.”

To evaluate exponential notation, we write the expression as a product and then find the value of the product. = = 243

1. Do all operations within grouping symbols such as parentheses or brackets. 2.Evaluate any expressions with exponents. 3.Multiply or divide in order from left to right. 4. Add or subtract in order from left to right. Order of Operations

Introduction to Variables and Algebraic Expressions

A combination of operations on letters (variables) and numbers is called an algebraic expression. Algebraic Expressions 5 + x 6  y 3  y – 4 + x 4x means 4  x and xy means x  y

Keywords and phrases suggesting addition, subtraction, multiplication, or division. AdditionSubtractionMultiplicationDivision sumdifferenceproductquotient plusminustimes divided by added to subtracted from multiplyinto more than less than twiceper increased by decreased by of for every totallessdouble slower than faster than