CLASS 8 Page 1
SYNA INTERNATIONAL SCHOOL LEARNING PAPERS CLASS 8 SUBJECT MATHEMATICS 1 x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2 x 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 3 x 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 4 x 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 5 x 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 6 x 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 7 x 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 8 x 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 9 x 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 10 x 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 Squares And Cubes Square Cube Square Cube 1 1 1 16 256 4096 2 4 8 17 289 4913 3 9 27 18 324 5832 4 16 64 19 361 6859 5 25 125 20 400 8000 6 36 216 21 441 9261 7 49 343 22 484 10648 8 64 512 23 529 12167 9 81 729 24 576 13824 10 100 1000 25 625 15625 11 121 1331 26 676 17576 12 144 1728 27 729 19683 13 169 2197 28 784 21952 14 196 2744 29 841 24389 15 225 3375 30 900 27000 CLASS 8 Page 2
RATIONAL NUMBER Natural Number [N] 1, 2,3,4,5,. Whole Number [W] 0,1,2,3,4,5. Integer [Z] Rational Number [Q] The Number of the form are called the Rational number.where q Ex CLOSURE PROPERTY ADDITION SUBTRACTION MULTIPLICATION DIVISION RATIONAL YES YES YES NO NUMBERS INTEGERS YES YES YES NO WHOLE YES NO YES NO NUMBERS NATURAL NUMBERS YES NO YES NO COMMUTATIVE PROPERTY ADDITION SUBTRACTION MULTIPLICATION DIVISION RATIONAL YES NO YES NO NUMBERS INTEGERS YES NO YES NO WHOLE YES NO YES NO NUMBERS NATURAL NUMBERS YES NO YES NO ASSOCIATIVE PROPERTY ADDITION SUBTRACTION MULTIPLICATION DIVISION RATIONAL YES NO YES NO NUMBERS INTEGERS YES NO YES NO WHOLE YES NO YES NO NUMBERS NATURAL NUMBERS YES NO YES NO CLASS 8 Page 3
Additive And Multiplicative Identities For Rational Numbers 0 It is Additive Identity 1 It is Multiplicative Identity DISTRIBUTIVITY OF MULTIPLICATION OVER ADDITION FOR RATIONAL NUMBERS If x, y and z are any three rational numbers, then x (y + z) = (x y) + (x z). DISTRIBUTIVITY OF MULTIPLICATION OVER SUBTRACTION FOR RATIONAL NUMBERS If x, y, and z are any three rational numbers, then x (y z) = (x y) (x z). ADDITIVE AND MULTIPLICATIVE INVERSE OF RATIONAL NUMBERS If the sum of two rational numbers is 0, then the two rational numbers are said to be additive inverse or negative of each other. Example If the multiplication of two numbers gives the result as 1, then the two numbers are called reciprocal or multiplicative inverse of each other. CLASS 8 Page 4
Definition of Algebraic Expression ALGEBRAIC EXPRESSION An algebraic expression is an expression that contains one or more numbers, one or more variables, and one or more arithmetic operations. Examples of Algebraic Expression Related Terms for Algebraic Expression Variable Expression Operation MONOMIAL Definition of Monomial A Monomial is an algebraic expression containing only one term. For example:3xy 2 More about Monomial A monomial can be a constant number or a variable expression. A monomial should not have negative and fractional exponents. Example: a- 2 and a 1/2 (are not monomials.) A monomial multiplied by a monomial is also a monomial. A monomial multiplied by a constant is also a monomial. Examples of Monomial 5, xy, 36x2y, a, 14b are the examples of monomial. 8x 1, 2x2 + 3x + 6 are not monomials. BINOMIAL Definition of Binomial Binomial is an algebraic expression (or a polynomial) containing two terms that are not like terms. For example 6x 3 and 2t 5 CLASS 8 Page 5
TRINOMIAL Definition of Trinomial A Trinomial is an algebraic expression containing three terms. For example:3xy 2 +xy COEFFICIENT Definition of Coefficient The Coefficient of a term in an expression is the number which is multiplied by one or more variables or powers of variables in the term. More about Coefficient Coefficients can be positive or negative or zero. While adding or subtracting polynomials, you just have to add or subtract the coefficients in the like terms. Examples of Coefficient 5x 3: Here 5 is the coefficient of the linear term 5x. 3xy 2-7z: Here 3 and - 7 are the coefficients of the first and second terms respectively. like terms Definition of Like Terms Like terms are monomials that contain the same variables raised to the same powers. They can be combined to form a single term. Examples of Like Terms 7a 2 b and 31a 2 b are like terms. In the expression 6x 2 + 7xy - 9y 3-3xy +7, the like terms are 7xy and - 3xy. More about Like Terms While adding or subtracting, we add or subtract the like terms. Like Radical Expressions: Like radical expressions are the radical expressions that have the same indices and radicands. Rules and formulas Rules and formulas in mathematics are written in a concise and general form using algebraic expressions: Thus, the area of rectangle = lb, where l is the length and b is the breadth of the rectangle. CLASS 8 Page 6
Exponents and powers For any non-zero integers a and b and whole numbers m and n, (a) a m a n = a m + n (b) a m a n = a m n, m > n (c) (a m ) n = a mn (d) a m b m = (ab) m (e) a m b m =( a/b) m (f) a 0 = 1 (g) ( 1) even number = 1 ( 1) odd number = 1 Math Magic DOUBLE AND HALF METHOD Ex [1] 35 x 24 =. a) 35 x 2 = 70. b) 24 2 = 12. c) 70 x 12 = 840. d) The answer is 840. CLASS 8 Page 7
CONSTANTS AND VARIABLES- LINEAR EQUATION IN ONE VARIABLE A CONSTANT is a quantity whose value remains the same throughout a particular problem. A VARIABLE is a quantity whose value is free to vary. Linear Equation 5x + 8 = 38 Constant Variable Equality DEGREE OF AN EQUATION The degree of an equation that has not more than one variable in each term is the exponent of the highest power to which that variable is raised in the equation. The equation 3x - 17=0 is a FIRST-DEGREE equation, since x is raised only to the first power. The Equation of Degree 1 is called the Linear Equation. It represent a number Line on the graph paper. An example of a SECOND-DEGREE equation is 5x 2-2x+1=0. 1. ADDITION How to Solve Linear Equation Find the value of x in the equation x-3=12 Perform the following steps: 1. Add 3 to both members of the equation, as follows: x - 3 + 3 = 12 + 3 CLASS 8 Page 8
2. Combining terms, we have x = 15 2. SUBTRACTION Find the value of x in the equation x + 14 = 24 1. Subtract 14 from each member. In effect, this undoes the addition indicated in the expression x + 14. x + 14-14 = 24-14 2. Combining terms, we have x = 10 3. MULTIPLICATION Find the value of y in the equation y/5 = 10 1. The only way to remove the 5 so that the y can be isolated is to undo the indicated division. Thus we use the inverse of division, which is multiplication. Multiplying both members by 5, we have the following: 5(y/5) = 5(10) 2. Performing the indicated multiplications, we have y = 50 4. DIVISION Find the value of x in the equation 3x = 15. 1. The multiplier 3 may be removed from the x by dividing the left member by 3. This must be balanced by dividing the right member by 3 also, as follows: 2. Performing the indicated divisions, we have x=5 CLASS 8 Page 9
DATA HANDLING WAYS OF REPRESENTING INFORMATION 1. A Pictograph: Pictorial representation of data using symbols. 2. A bar graph: A display of information using bars of uniform width, their heights being proportional to the respective values. (i) What is the information given by the bar graph? (ii) In which year is the increase in the number of students maximum? (iii) In which year is the number of students maximum? (iv) State whether true or false: The number of students during 2005-06 is twice that of 2003-04. 3. Double Bar Graph: A bar graph showing two sets of data simultaneously. It is useful for the comparison of the data. (i) What is the information given by the double bar graph? (ii) In which subject has the performance improved the most? (iii) In which subject has the performance deteriorated? (iv) In which subject is the performance at par? CLASS 8 Page 10
CIRCLE GRAPH OR PIE CHART Drawing pie charts PROBABILITY- There are certain experiments whose outcomes have an equal chance of occurring. 1. A random experiment is one whose outcome cannot be predicted exactly in advance. 2. Outcomes of an experiment are equally likely if each has the same chance of occurring. 3. Probability of an event = CLASS 8 Page 11
SQUARE AND SQUARE ROOTS 1. If a natural number m can be expressed as n2, where n is also a natural number, then m is a square number. 2. All square numbers end with 0, 1, 4, 5, 6 or 9 at unit s place. 3. Square numbers can only have even number of zeros at the end. 4. Square root is the inverse operation of square. 5. There are two integral square roots of a perfect square number. For example, 3 2 = 9 gives = 3 CLASS 8 Page 12
PLAYING WITH NUMBERS 1. Numbers can be written in general form. Thus, a two digit number ab will be written as ab = 10a + b. 2. The general form of numbers are helpful in solving puzzles or number games. 3. The reasons for the divisibility of numbers by 10, 5, 2, 9 or 3 can be given when numbers are written in general form. Math Magic Squaring A Number In The Range (90 99): Ex [1] 97 2 =. a) 100 97 = 3. b) 3 2 = 9. Write 09 to take up 2 place values. c) 97 3 = 94. Write 94. d) The answer is 9409. Ex [2] 92 2 =. a) 100 92 = 8. b) 8 2 = 64. Write 64. c) 92 8 = 84. Write 84. d) The answer is 8464. CLASS 8 Page 13
FACTORISATIONS CLASS 8 Page 14
CHAPTER 8 COMPARING QUANTITIES CLASS 8 Page 15
Review Ratio It means comparing two quantities. 5: 20 Equivalent to fraction OR 1 : 4 Percentage 20% equivalent to Remember 1. % to fraction Divide by 100 2. Fraction to Percentage Multiply by 100 3. Finding Percentage of a number Find 5% of 200 CHECK YOURSELF 1. Michael spends 30% of his income on rent. If he is left with $ 800. Find his income? 2. If 72% of 25 students are good in Physics. How many are not good in Physics? 3. Convert the following ratios to percentage. a. 5 : 8 b. 12 : 25 Finding the Increase and Decrease Percentage [Discount ] Ex The strength of the school is 800 in 2010.Find the number of students in academic year 2011,If the strength increase by 20% Step1 Add the increase percentage to 100 100 + 20 = 120 Step 2 Multiply the increase percentage to the original strength Discount or Decrease in percentage [strength of school in 2011 ] Ex The list price of a book is Rs 2000/. A discount of 10% is given on its sale price.find the amount of discount and its sale price. CLASS 8 Page 16
Finding Discount Sale price = List price Discount = 2000 200 = Rs 1800 OR Direct Formula Sale Price = FORMULA Discount = List price [ Marked Price ] Selling Price Discount %= CHECK YOURSELF 1. A chair marked at Rs 4000 is sold for 3600.Find the discount and Discount percentage. 2. A Book is sold at Rs 5000 after giving a discount of 10%. Find its Marked Price. HOTS A person get consecutive discount of 10%, 20%,5% and 15% on a Computer book. If the Marked price of the book is $1600. Find the price for which he buy the Book. [NOTE Can you frame a Formula for the Consecutive Discount] CLASS 8 Page 17
CHECK YOURSELF Chapter 10 - Visualising Solid Shapes Recognizing Solids CLASS 8 Page 18
Identifying Faces Edges and Vertices CLASS 8 Page 19
Polyhedron Each of these solids is made up of polygonal regions which are called its faces; these faces meet at edges which are line segments; and the edges meet at vertices which are points. Such solids are called polyhedrons Analysis of the Above Table CLASS 8 Page 20
CHAPTER 11 MENSURATION DIRECT PROPORTION Chapter 13 Direct and Inverse Proportions INVERSELY PROPORTIONAL CLASS 8 Page 21
Chapter 14 Factorisation Chapter 15.Introduction to Graphs 1. Graphical presentation of data is easier to understand. 2. (i) Abar graph is used to show comparison among categories. (ii) A pie graph is used to compare parts of a whole. (iii) AHistogram is a bar graph that shows data in intervals. 3. Aline graph displays data that changes continuously over periods of time. 4. A line graph which is a whole unbroken line is called a linear graph. 5. For fixing a point on the graph sheet we need, x coordinate and y coordinate. 6. The relation between dependent variable and independent variable is shown through a graph. Chapter 16. CLASS 8 Page 22
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