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MATHEMATICS GRADE 7 UNIT - INTRODUCTION 1 VOCABULARY FOR FLASHCARDS A.) Absolute Value: The absolute value of a number represents the positive distance the number is away from the origin (0). The symbol: x is used when the absolute value of a number needs to be found. Example 2: 2 = 2 Example 1: 2 = 2 B.) Associative Property: Addition and Multiplication PLEASE NOTE: CHANGE IN GROUPING Example 1: a + (b + c) = (a + b) + c Example 2: 4 + (6 + 7) = (4 + 6) + 7 Example 3: a • (b • c) = (a • b) • c Example 4: 2 x (3 x 4) = (2 x 3) x 4 C.) Commutative Property: Addition and Multiplication PLEASE NOTE: CHANGE IN ORDER Example 1: x + y = y + x Example 2: 3 + 8 = 8 + 3 Example 3: x • y = y • x Example 4: 3 1 1 3 4 3 3 4 2 D.) Distributive Property of Multiplication Over Addition(“sum of”): a(b + c) = ab + ac OR ab + ac = a(b + c) Distributive Property or Multiplication Over Subtraction(“difference of”): a(b – c) = ab – ac OR ab – ac = a(b - c) E.) Irrational Numbers: A number is irrational if it can not be written as a fraction with an integer numerator and a nonzero integer denominator. *Three Major Categories We Study Are: 1) pi: 2) Non-Perfect Square Roots: 3, 5, 6, 7,... 3) Decimals (Non-Terminating and Non-Repeating): 7.123145…, .03003000300003000003… F.) Order of Operations PROCESS: 1. Use PEMDAS (PARENTHESIS, EXPONENTS, MULTIPLICATION, DIVISION, ADDITION, SUBTRACTION) 2. PLEASE REMEMBER THAT DIVISION IS DONE FIRST IF IT COMES BEFORE MULTIPLICATION 3. PLEASE REMEMBER THAT SUBTRACTION IS DONE FIRST IF IT COMES BEFORE ADDITION 4. Underline the pair of numbers being worked on 5. Work down in a V-formation until expression is simplified 3 G.) Rational Numbers: A number is rational if it can be written as a fraction with an integer numerator and a nonzero integer denominator. 3 Examples: 0.4, 2. 9̅, 16 , , 5 4 4 Subsets of Real Numbers A.) Counting Numbers or Natural Numbers: 1, 2, 3, 4, 5… B.) Whole Numbers: 0, 1, 2, 3, 4, 5… C.) Integers: (negative infinity) - ,…..,-3, -2, -1, 0, 1, 2, 3, ….., (infinity) ADDITIVE INVERSE (OPPOSITE), PROPERTIES AND IDENTITY ELEMENTS A.) Additive Inverse (Opposite): A number and its opposite have a sum of zero. Example 1: g + (-g) = 0 Example 2: 5 + (-5) = 0 Example 3: 1 1 0 2 2 5 B.) Addition Property of Zero: A number added to zero leaves the number unchanged. Example 1: g + 0 = g Example 2: 5 + 0 = 5 Example 3: 1 1 0 2 2 C.) Associative Property: Addition and Multiplication PLEASE NOTE: CHANGE IN GROUPING Example 1: a + (b + c) = (a + b) + c Example 2: 4 + (6 + 7) = (4 + 6) + 7 Example 3: a • (b • c) = (a • b) • c Example 4: 2 x (3 x 4) = (2 x 3) x 4 D.) Commutative Property: Addition and Multiplication PLEASE NOTE: CHANGE IN ORDER Example 1: x + y = y + x Example 2: 3 + 8 = 8 + 3 Example 3: x • y = y • x Example 4: 3 1 1 3 4 3 3 4 6 E.) Multiplication Property of One: For any whole number a, a • 1 = a and 1 • a = a Example 1: 1 x 5 = 5 Example 2: 6 x 1 = 6 F.) Multiplication Property of Zero: For any whole number a, a • 0 = 0 and 0 • a = 0 Example 1: 0 x 4 = 0 Example 2: 4 x 0 = 0 G.) Additive Identity Element of 0: 0 is the additive identity element because anything added to 0 will be that number. H.) Multiplicative Identity Element of 1: 1 is the multiplicative identity because anything multiplied by 1 will be that number. 7 I.) Distributive Property of Multiplication Over Addition(“sum of”): a(b + c) = ab + ac OR ab + ac = a(b + c) Distributive Property or Multiplication Over Subtraction(“difference of”): a(b – c) = ab – ac OR ab – ac = a(b - c) HOW THE DISTRIBUTIVE PROPERTY CAN BE USED FINDING PRODUCTS Example 1: 4(3 + 2) = (4 x 3) + (4 x 2) = 12 + 8 = 20 Example 2: 2(8 – 3) = (2 x 8) – (2 x 3) = 16 – 6 = 10 Example 3: (9 x 20) + (9 x 7) = 9(20 + 7) = 9(27) = 243 Example 4: (5 x 7) – (7 x 2) = 7(5 – 2) = 7 x 3 = 21 CHANGING THE FORM OF AN ALGEGRAIC EXPRESSION Example 1: 8(x + y) = 8x + 8y Example 2: 3.4(a – b) = 3.4a – 3.4b Example 3: 1 1 1 (9 x 6 x) ( 9 x 6 x) 4.5 x 3 x 1.5 x 2 2 2 8 ABSOLUTE VALUE A.) Absolute Value: The absolute value of a number represents the positive distance the number is away from the origin (0). The symbol: x is used when the absolute value of a number needs to be found. Example 1: 2 = 2 Example 2: 2 = 2 Example 3: 0 = 0 Example 4: Example 5: 2 = 2 Example 6: 2 = 2 62 6-2 4 Example 7: 7 3 7–3 -10 Example 8: Example 9: 0 2 Example 10: 8 9 8+9 17 2 2 0–2 2 4 9 2 2 IRRATIONAL AND RATIONAL NUMBERS A. ) Rational Numbers: A number is rational if it can be written as a fraction with an integer numerator and a nonzero integer denominator. 3 Examples: 0.4, 2. 9̅, 16 , , 5 4 B. ) Irrational Numbers: A number is irrational if it can not be written as a fraction with an integer numerator and a nonzero integer denominator. *Three Major Categories We Study Are: 1) pi: 2) Non-Perfect Square Roots: 3, 5, 6, 7,... 3) Decimals (Non-Terminating and Non-Repeating): 7.123145…, .03003000300003000003… *AN EASY WAY TO REMEMBER IRRATIONAL NUMBERS* An irrational number is a nonrepeating/nonterminating decimal (never ends!!!!!) 10 EXPONENTS The example below has two parts: a base and an exponent(power). The base is 5 and the exponent(power) is 3. 5 3 Example 1: 5 2 , means 5 x 5, read: five to the second power, or five squared or 25 Example 2: 10 3 , means 10 x 10 x 10, read: ten to the third power, or ten cubed or 1000 Example 3: 6 0 , read: six to the zero power or 1 ANY NUMBER, OTHER THAN ZERO, RAISED TO THE ZERO POWER IS 1 SQUARE ROOTS Some counting numbers can be written as the product of two equal factors. For example: 1 = 1 x 1 16 = 4 x 4 4 = 2 x 2 25 = 5 x 5 9 = 3 x 3 36 = 6 x 6 Counting numbers such as 1, 4, 9, 16, 25 and 36 are called square numbers or perfect squares. These numbers can be written using the exponent of 2. 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 One of the two equal factors of a square number is called the square root of the number. To show the square root of a number, a radical sign (√ ) is used. Therefore the following is true: √𝟏 = 𝟏 √𝟒 = 𝟐 √𝟗 = 𝟑 √𝟏𝟔 = 𝟒 √𝟐𝟓 = 𝟓 √𝟑𝟔 = 𝟔 11 RATIONAL NUMBER VOCABULARY A.) Fraction: A number that has a numerator and denominator. 𝟐 Examples: 𝟕 , 𝟗 𝟏 , , 𝟓 𝟏𝟏 𝟖 𝟑 B.) Proper Fraction: When the numerator is less than the denominator. 𝟏 𝟐 Examples: , , 𝟒 𝟐 𝟑 𝟏𝟏 C.) Improper Fraction: When the denominator is greater than the numerator. 𝟗 𝟏𝟏 𝟏𝟎 Examples: 𝟖 , 𝟑 , 𝟒 D.) Mixed Number: Is a number that has a fraction attached to an integer. 𝟐 𝟒 𝟕 𝟏𝟏 Examples: 𝟏 , 𝟗 ,𝟕 𝟑 𝟖 E.) Reciprocal: When the numerator and the denominator of a fraction is flipped. Example: 𝟐 𝟑 , reciprocal is 𝟑 𝟐 F.) Simplifying(Vertically and/or Diagonally): Is a method of simplifying two or more fractions by simplifying a numerator and denominator. This may be repeated more than once if able. Example: 𝟏𝟎 Example: 𝟐 𝟏𝟐 𝟑 𝟑 𝟐 𝟏 𝟏 𝟏 𝟏 𝟓 𝟒 𝟏 𝟐 𝟏 𝟐 𝟏 𝟑 𝟑 𝟏 𝟓 𝟓 × = × = × = × 𝟗 𝟏𝟎 = × = 12 ADDING, SUBTRACTING, BORROWING WITH FRACTIONS AND MIXED NUMBERS METHOD (DENOMINATORS ARE THE SAME): 1. Perform the operation(s) asked 2. Simplify if asked METHOD (DENOMINATORS ARE NOT THE SAME): 1. Rewrite the fractions with a common denominator (be sure to adjust the numerator) 2. Perform the operation(s) asked 3. Simplify EXAMPLES 5 8 1. + 7 8 - ___ 1 7 = 1 1 2 2 1 72 14 10 9 +2 3 14 = +2 3 3 2 14 14 ___ ____ __________ 3 1 9 3 8 8 3 24 = = 9 9 3 27 3 3 3 + = + =+ 27 27 27 ____ 3. 1 ____ 12 4 1 1 1 8 8 2 4. 7 9 2. ___ 5. 3 5 14 1 1 4 4 40 4 4 3 [BORROWING] 9 9 4 36 36 2 29 18 18 2 2 - 2 2 4 49 36 36 4 ____ _____ ____ ____ 27 1 27 1 13 22 2 11 1 36 2 18 MULTIPLYING FRACTIONS AND/OR MIXED NUMBERS METHOD: 1. Convert all mixed numbers to improper first 2. Simplify if able 3. Multiply the numerators together 4. Multiply the denominators together EXAMPLES 1. 3 6 4 5 1 2 2. 2 3 5 10 2 3 3 3 2 5 5 5 1 3 9 10 3. 1 3 10 3 20 15 5 21 25 1 8 3 3 3 8 4 9 4. 3 1 2 1 4 3 1 7 1 2 15 8 3 4 9 2 5 2 3 1 3 2 8 21 5 1 1 5 = 5 1 1 1 1 5. 5 9 20 5 9 1 9 9 1 2 1 20 1 4 4 4 14 DIVIDING FRACTIONS AND/OR MIXED NUMBERS METHOD: 1. Convert all mixed numbers to fractions 2. Change all division signs to multiplication signs and take the reciprocal of all fractions following the changed operation sign(s) 3. Simplify if able 4. multiply fractions (numerator times numerator and denominator times denominator) EXAMPLES 1. 3 9 7 14 1 2 2. 2 5 8 2 5 3 14 7 9 1 4 = 4 1 1 2 1 2 = 3 1 3 2 5 27 1 27 2 1 5 5 25 25 3. 5 5 5 8 4. 8 12 16 9 21 11 8 21 11 9 12 16 1 7 11 77 5 1 3 72 12 2 72 5. 8 6 1 8 8 49 1 8 8 8 64 15 1 1 49 49 49 15 Changing Fractions and Mixed Numbers To Decimals Method: 1. Divide the numerator by the denominator .8 5 4.0 4 1. 5 0.8 -40 0 2. 2 .75 4 3.00 3 4 2.75 -28 20 -20 0 Changing Decimals To Fractions and Mixed Numbers Method: 1. Determine the place holder where the decimal terminates (ends) 2. Write the numeral of the decimal part over where the place holder where the decimal terminated (ended) 3. Simplify if asked Please Note: If the decimal has a whole number attached to it, write this in front of the fraction. The decimal will convert to a mixed number. 1. 0.7 = 7 10 2. 2.94 = 2 94 2 47 2 100 2 50 16 Changing Mixed Numbers To Improper Fractions Method: 1. Multiply the denominator by the whole number and then add it to the numerator (this becomes the new numerator) 2. Rewrite this new number over the original denominator Please Note: If it is a negative mixed number, ignore the negative sign and attach at the end. 1 4 1. 3 (4 3) 1 13 4 4 9 (17 1) 9 26 2. 1 17 17 17 Changing Improper Fractions To Mixed Numbers Method: 1. Divide the numerator by the denominator (this determines the whole number) 2. Write the remainder over the denominator 3. Write the whole number next to the fraction Please Note: If the improper fraction is negative, ignore the negative sign and attach at the end. 1. 17 3 5 3 17 -15 2 2. 2 5 3 9 8 1 89 -1 -8 1 17 1 8 Math Rap For Operations With Fractions by Mrs. Pirillo Top Times Top Bottom Times Bottom Cancel If You Can Reduce At The End And When You Divide You Flip The Other Side!!!!! Adding And Subtracting Is A Different Game The Bottoms Need To Be The Same And If They’re Not, Don’t Bother Me Go And Find The L.C.D!!!!! 18 ADDING AND SUBTRACTING DECIMALS METHOD: 1. Write the given numbers one above the other with the decimal points in line. 2. Add zeros to get the same number of decimal places and then add or subtract as if the numbers were whole numbers. 3. Place a decimal point in the number for the sum or difference in position under the decimal points in the given numbers. Example 1: 6.47 + 340.8 + 73.523 6.470 340.800 + 73.523 420.793 Example 2: 13.94 – 7.693 13.940 -7.693 6.247 19 ROUNDING METHOD: 1. Find the place to which you wish to round and mark it with a line under it. 2. Draw a vertical line segment after the place holder to rounded to. 3. If the digit to the right of the vertical line is 5 or greater, increase the place holder being rounded to by 1. If the digit to the right of the vertical line is less than 5, leave the marked digit unchanged. 4. If rounding a whole number, replace each digit to the right of the marked place with 0. 5. If a mixed decimal, do not add 0 to the right of the marked place to be rounded. Complete the table below by rounding to the appropriate place holder. NUMBER 456 ROUNDING PLACE TENS ANSWER 460 456 HUNDREDS 0.87872 TEN-THOUSANDTHS 0.87872 HUNDREDTHS 0.88 12.111 TENTHS 12.1 12.111 ONES 7465 HUNDREDS 7500 7465 TENS 7470 500 0.8787 12 20 MULTIPLYING DECIMALS METHOD: 1. Right justify the two numbers being multiplied 2. Write down to the right side of the numbers the amount of decimal places needed to be moved in order to make it a whole number 3. Find the product of the two factors 4. Place the decimal point in the product so that the number of places to the right of the decimal point in the product is the sum of the number of places to the right of the decimal point in the factors EXAMPLES 1. 14.92 x 7.2 14.92 x7.2 2984 +10444 107.424 2. 820.3 x 3.2 2 places 1 place 820.3 x3.2 16406 +24609 2624.96 3 places 21 1 place 1 place 2 places DIVIDING DECIMALS METHOD: 1. If needed, multiply the dividend and divisor by a power of ten that makes the divisor a counting number. 2. Divide the new dividend by the new divisor. 3. Check by multiplying the quotient and divisor. EXAMPLE 1.25 2.5232 3.14500 -250 654 -625 290 -250 400 -375 250 -250 0 22 COMPARING TWO MORE FRACTIONS METHODS: 1. Cross multiplying technique 2. Change to fractions with same denominator 3. Change to decimals-stack them on top of each other and compare by looking at columns EXAMPLE: Which is bigger? 𝟓 𝟔 𝐨𝐫 𝟕 𝟖 CROSS MULTIPLYING FRACTION DECIMAL 𝟓 𝟕 𝐨𝐫 𝟔 𝟖 𝟓 𝟒𝟎 = 𝟔 𝟒𝟖 𝟓 = 𝟎. 𝟖𝟑𝟑 … 𝟔 40 < 42 𝟕 𝟒𝟐 = 𝟖 𝟒𝟖 𝟕 = 𝟎. 𝟖𝟕𝟓 𝟖 𝟓 𝟕 < 𝟔 𝟖 0.833… 0.875 𝟓 𝟕 < 𝟔 𝟖 𝟓 𝟕 < 𝟔 𝟖 23 ORDER OF OPERATIONS PROCESS: 1. Use PEMDAS (PARENTHESIS, EXPONENTS, MULTIPLICATION, DIVISION, ADDITION, SUBTRACTION) 2. PLEASE REMEMBER THAT DIVISION IS DONE FIRST IF IT COMES BEFORE MULTIPLICATION 3. PLEASE REMEMBER THAT SUBTRACTION IS DONE FIRST IF IT COMES BEFORE ADDITION 4. Underline the pair of numbers being worked on 5. Work down in a V-formation until expression is simplified For Examples 1 – 6, use Order of Operations to simplify the expressions. 19 + 84 ÷ 4 x 8 – 11 (5.4 ÷ 0.6) x 5.5 – (3.6 + 6.7) 19 + 21 x 8 – 11 9 x 5.5 - 10.3 19 + 168 - 11 49.5 – 10.3 187 – 11 39.2 176 𝟖 2 x 𝟐 12 - ( + 𝟑 ÷ ) 𝟓 𝟑 𝟖 𝟐 𝟑 𝟗 12 - ( + ) 𝟓 𝟐 12 𝟓 𝟏 𝟑 ÷ 2 x ÷ 2 x 𝟏 𝟏 × 𝟑 𝟓 𝟔𝟏 𝟏𝟎 𝟏 𝟏𝟓 𝟗 𝟏𝟎 24 𝟏 𝟓 𝟏 𝟓