Hundred-thousands. Millions. Ten-thousands

Transcription

1 Place Value, Names for Numbers, and Reading Tables The digits used to write numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Unit 1 Lesson 1a Natural numbers are positive numbers only {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }. The numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 are called whole numbers. The position of each digit in a number determines its place value. The number 12,345 is written in standard form. A group of three digits in a whole number is called a period. ** The three dots ( ) indicated that the numbers continues meaning indefinitely. The digits are written in standard form. Hundred-billions Ten-billions Billions Hundred-millions Place Value Chart for Whole Numbers Ten-millions Millions Hundred-thousands Ten-thousands Thousands Each group of three digits is called a period. Hundreds Tens Ones Commas separate the digits into groups of threes, starting from the right. Example 1: Find the place value of the digit 8 in each whole number. a. 547,098,632 b. 1,659,984,036 c

2 Writing a whole number in words and in standard form To write a whole number in words, write the number in each period followed by the name of the period. (The ones period is usually not written.) This same procedure can be used to read a whole number. Example 2: Write each whole number in words. a. 241 b. 108 c. 52,160 Writing a whole number in standard form To write a whole number in standard form, write the number in each period followed by a comma. Example 3: Write each whole number in standard form. a. thirty-nine b. two hundred forty-five A comma may or may not be inserted in a four-digit number. c. three thousand, seven-hundred one d. six million, twenty-seven thousand, eight hundred twelve Writing a whole number in expanded form Expanded form a number written as a sum of all digits multiplied by their place value. Example 4: Write 257,016 in expanded form. 2

3 5 + 6 = 11 addends Adding and Subtracting Whole Numbers sum Example 1: Identify the addends and the sum = 25 Numbers that are added together are called addends. The sum is the result of adding two or more numbers. Unit 1 Lesson 1a Addition Property of 0 the sum of 0 and any number is that number Example: = 11 and = 11 Commutative Property of Addition changing the order of two addends does not change their sum Example: = Associative Property of Addition changing the grouping of addends does not change their sum Example: (4 + 5) + 3 = 4 + (5 + 3) Subtraction properties of 0 The difference of any number and that same number is 0 Example: = 0 The difference of any number and 0 is that same number Example: 9 0 = 9 Example 2: Which property is used here? a) = c) = 9 b) = 0 d) 3 + (4 + 2) = (3 + 4) + 2 e) 35 0 = 35 To add whole numbers Arrange the numbers vertically so the digits in each column have the same place value. Start at the right and add the digits in each column, regrouping (carrying) if necessary. Example 3: Add. a) b) 47, ,898 3

4 c) d) Subtraction is finding the difference of two numbers. minuend 11 6 = 5 subtrahend difference The first number in a subtraction is called a minuend. It s the number you are subtracting from. The subtrahend is the second number in a subtraction. It s the number you are taking off. The difference is the answer you get when subtracting one number from another. To subtract whole numbers: Arrange the numbers vertically so the digits in each column have the same place value. Start at the right and subtract the digits in each column, regrouping (borrowing) as needed. o What is borrowing? Borrowing is when a digit in the second number is larger than the corresponding digit in the first number. Example 4: Subtract. Check each answer by adding. a = b = c = d = 4

5 Example 5: Subtract. a b c d. 23,901 8,064 Perimeter Perimeter of a Polygon The word perimeter of a polygon is the measure around or distance around a polygon. The sum of the lengths of the sides of a polygon. A polygon can be described as a flat figure formed by line segments connected at their ends. Some examples of polygons: Example 6: Txtbk Go to page 28 and complete questions 51, 53, and 57. 5

6 Rounding and Estimating Unit 1 Lesson 1b Rounding a whole number means approximating it. Rounding Whole Numbers to a Given Place Value 1) Locate the digit to the right of the given place value to be rounded. 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. 3) If this digit is less than 5, keep the digit in the given place value and replace each digit to its right by 0. Example 1: Round to the nearest ten. a. 67 b. 342 c. 725 Example 2: Round to the nearest thousand. a. 34,992 b c. 437,200 Example 3: Round to the nearest hundred. a b. 76,831 c. 978,965 6

7 Multiplying Whole Numbers and Area Multiplication is repeated addition with a different notation. A factor is a whole number that divides evenly into another number. A product is the result when two numbers are multiplied. 7 6 = 42 factors Unit 1 Lesson 1b product Ways to represent multiplication: 7 x 6 = 42, 7 6 = 42, (7)(6) = 42, 7(6) = 42, and (7)6 = 42 Example 1: Identify the factors and the product. a) 59 x 26 = 1534 b) = 80 Multiplication Property of 0 the product of 0 and any number is 0 Example: 0 11 = 0 and 11 0 = 0 Multiplication Property of 1 the product of 1 and any number is that same number Example: 1 21 = 21 and 21 1 = 21 Example 2: Which property is being used? a) 7 4 = 4 7 c) 1 43 = 43 Commutative Property of Multiplication changing the order of two factors does not change their product Example: 9 4 = 4 9 Associative Property of Multiplication changing the grouping of factors does not change their product Example: (4 1) 9 = 4 (1 9) b) 12 0 = 0 d) (2 8) 3 = 2 (8 3) Distributive Property of Multiplication over Addition a(b + c) = ab + ac Example: 2(5 + 4) = The factor outside the parentheses is multiplied by each term in the sum Example 3: Apply the distributive property and simplify. a) 2(4 + 6) = b) 9(10 8) = 7

8 Multiplying Whole Numbers 1) Arrange the numbers vertically, lining up the rightmost digits. 2) Multiply each digit of the top number by each digit of the bottom number, keeping in mind the place value of each digit. 3) Add the results of the second step. Example 4: Multiply. a. 24 x 6 b. (725)(3) c d. 274(586) Area of a Rectangle Area measures the amount of surface of a region. Area (space enclosed by a plane figure) is measured in square units such as square inches, square feet, square miles, square centimeters, square meter, and so on. The formula: A = L W A = the area L = the length W = the width Example 5: Txtbk Go to page 49 and complete questions 46, 48, and 70. width length 8

9 Dividing Whole Numbers Ways to represent division Unit 1 Lesson 2 The process of separating a quantity into equal parts is called dividing. Dividend is the number being divided. Divisor is the number doing the dividing. Quotient is the result (answer) of division. Remainder is what s leftover = 8 long division symbol /9= 8 fraction bar division sign Example 1: Divide. a) 40 5 b) 24 6 c) 9 72 Division Properties of 1 The quotient of any number and that same number is 1 Example: = The quotient of any number and 1 is that same number 9 7 Example: = 4 Division Properties of 0 The quotient of 0 and any number (except 0) is 0 Example: 0 3 = The quotient of any number and 0 is not a number. We say that it s undefined. Example: 1 0 is undefined 5 is undefined 0 undefined 08 Example 2: Which property is used here? a) 0 5 c) b) 20 1 d)

10 Long Division Process Step 1: Divide Step 2: Multiply Step 3: Subtract Step 4: Bring down Then repeat these steps until you can t bring down or divide any longer. Example 3: Divide. a) b) 9 45,900 c) d) 164, The average of a list of numbers is the sum of the numbers divided by the number of numbers. sum of numbers average = number of numbers Example 4: Txtbk Go to page 63 and complete question

11 An exponential notation contains a base and exponent. Exponents and Order of Operations 5 3 Unit 1 Lesson 2 An exponent is a shorthand notation for repeated multiplication of the same factor = 125 Example 1: Write using exponential notation. a b c d Example 2: Evaluate. a. 4 2 b c d. 7 3 e Order of Operations A rule for the order in which operations should be done. The acronym for this order of operations is 4 step process PEMDAS. 1. Parentheses (or any grouping symbols i.e. brackets, braces, absolute value, fraction bar, square root) Parentheses Exponents Multiplication Division Addition Subtraction A popular expression for remembering this is Please Excuse My Dear Aunt Sally. 2. Exponents 3. From left to right comes first 4. From left to right comes first Multiply Divide whatever Add Subtract whatever 11

12 Example 3: Simplify. Circle each step. a P E M D A S b P E M D A S c (8 3) 2 P E M D A S d. (10 7) P E M D A S e. 36 [20 (4 2)] P E f (3 2) 3 P E M D M D A S A S Area of a Square Area measures the amount of surface of a region. The formula: Area (space enclosed by a plane figure) is measured in square units such as square inches, square feet, square miles, square A = side side = (side) 2 centimeters, square meter, and so on. A = the area s = side Example 4: Txtbk Go to page 74 and complete questions 88 and 90. side side 12

13 Introduction to Variables and Algebraic Expressions Example of an Algebraic Expression Unit 1 Lesson 3 A letter that represents a number is a variable. A collection of numbers, variables, operations, and grouping symbols is an algebraic expression. 3x + 6y 9w Replacing a variable in an expression by a number and then finding the value of the expression is called evaluating an expression. Example 1: Evaluate. a. x 2 for x = 7 b. y(x 3) for x = 8 and y = 4 c. y 6 x for x = 6 and y = 18 d. 25 z 3 + x for z = 2 and x = 1 13

14 Example 2: Evaluate. e. 9y 4w + 3z for y = 12, w = 4, & z = 25 f. 3yz for y = 5 and z = 3 g. 3x 2 + 2x 5 for x = 2 h for z = 3 and y = 5 2y z i. (4y + 3z) 2 for y = 5 and z = 3 j. xz(2y + x z) for x = 2, y = 5 and z = 3 14

15 Key words or phrases that represents Addition added to rise plus exceeds increased by grow more than larger than total longer than sum enlarge gain Translating subtract difference take away less decreased by subtracted from** less than** Subtraction diminish drop lose loss lower minus smaller than** **whatever comes first is written last Unit 1 Lesson 3 Example 1: Translate. a) The sum of 802 and y b) The total of 89, 45, 2, 19, and z c) 712 increased by 38 d) The difference of 16 and 5 e) 25 less d f) 86 subtracted from 90 Key words or phrases that represents Multiplication Division multiply divide multiply by quotient of per product divided by times divided or shared twice/double equally Example 2: Translate. a) Multiply z by 12 b) The product of 9 and 900 c) The quotient of 94 and m d) 640 divided by 32 15

16 Example 3: Write as an algebraic expression. Use x to represent a number. a) Twice a number b) 8 increased by a number c) 10 minus a number d) 10 subtracted from a number e) The quotient of 6 and a number f) The sum of three and a number g) A number less thirty h) A number less than thirty i) The product of a number and 7 j) Twenty decreased by twice a number k) A number divided by 11 l) The sum of a number and 7 16