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# Mathematics sec 1

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APPRECIATE ~ GRADE 7 / SEC 1 MATH

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### Mathematics sec 1

1. 1. MATHEMATICSsecondary 1 Nicco Alyssha Parikh
2. 2. PRIME NUMBERS Prime numbers : Numbers that can only be divided by 1 and itself . Composite numbers : Not prime numbers . Prime numbers from 1-20 = 2,3,5,7,11,13,17,19 * 1 is not a prime number – because it can only be divided by itself . Q.) How many prime numbers are there from 1-100? A.) 25. Stepby stepfindingPrimenumbers . - Cross outthe number1 - Circlethe number2andcross out allthe othermultiples of 1. - Circlethe number3andcross out allthe othermultiples of 3. - Circlethe number5andcross out allthe othermultiples of 5. - Circlethe number7andcross out allthe othermultiples of 7. - Continuetheprocess unit allunitall thenumbers areeither circled or crossed.
3. 3. HIGHEST COMMON FACTOR(common) How to find the highest common factor? Find the highest common multiple of 15 and 75 ? 15,75 5,25 1,5 HCF = 3x5=15 3 5 Express 252 in PRIME FACTORS 252 2 x 126 2 x 2 X 63 2 2 3 X 21 3 3 x 7 From the above factor tree, We have 252 = 2x2x3x3x7
4. 4. INDEX NOTATION Index notation is using the power of a certain number. e.g.) 252= (first, prime factorise the numbers) = 2x2x3x3x7 = 22 x 32 x 7 12= 2x2x3 can be written as 12= 2 x 3 2
5. 5. LOWEST COMMON MULTIPLE(max) Lowest common multiple of 65 , 175 , 135 65,175,135 Please note that we have to arrive to the answers to all be one at the last ladder 13, 35,27 1,35,27 1,1,27 1,1,1 5 13 35 27 LCM(LOWEST COMMON MULTIPLE) = 5x13x35x27 =61425
6. 6. FINDING CUBE ROOT AND SQUARE ROOTS 1) FIRST PRIME FACTORISE THE NUMBER EG.) square root of 144 = 24 x 32 2) Arrange them into 2 brackets Square root ( 2x2x3) (2x2x3) 3) Solve what is in 1 bracket 2x2x3 =4x3 =12 Cube root Do the same only at step to , instead of 2 brackets , it becomes 3 brackets .
7. 7. Integers Positive and Negative integers . In the number line , the more left you go , the larger the number gets(smaller value) . . Zero is an integer by itself- not positive or negative. *note that there is no such thing as +0 or -0 . - BODMAS rule stated that everything should be from left to right UNLESS there is a bracket .
8. 8. ADDITION OF INTERGERS 3+2=5 3+(-3)=0 -3+4=-1 -4+(-2)=-6 Owe someone 4 dollars and another 2 dollars.
9. 9. SUBTRACTION OF INTEGERS -7+(-11)+9 =-7-11+9 =-18+9 = -9 34+(-18)+9 =34-18+9 =16+9 =25
10. 10. MULTIPLICATION OF INTEGERS +x+=+ -x+=- -x-=+ -x0=? (0) -2x3x(-1) =-6x(-1) =6
11. 11. DIVISION OF INTEGERS 3x6 3-2 = (18 3)-2 =6-2 =4 6 2x4 + (-3) = 3 x 4 +(-3) = 12 + (-3) = 9 *ALWAYS DO FROM LEFT TO RIGHT ALWAYS DO THE “ POWERS “ FIRST (-4)2 (-8) + 3 x (-2)3 = 16/(-8) + 3 x (-8) = -2 + 3 x (-8 ) = -2 + (-24) =-26
12. 12. RATIONAL NUMBERS a/b - b cannot be “0” e.g : mixed numbers improper fraction - Using the cancellation method …. Such as: - 21/17 X 19 /7 = - ? THE INTEGERS IN RATIONAL NUMBERS CAN BE BOTH POSITIVE AND NEGATIVE. THE CHANGING OF SIGNS MUST BE INCLUDED!!!!!  REMEMBER WHEN DIVIDING A FRACTION OR FRACTIONS , SAME-CHANGE-INVERT !!  ALSO REMEMBER THAT EVEN IF THERE ARE 3 OR MORE FRACTIONS ONLY ONE DOESN’T CHANGE – DURING DIVISION OF FRACTIONS ONLY !!  DENOMINATORS MUST BE THE SAME.
13. 13. ALGEBRA • Actually writing numbers in the form of letters • IF YOU ARE 40 YEARS OLD , I AM 20 YEARS YOUNGER THAN YOU , MY AGE WILL BE (40- 20) . • BUT IF I AM x YEARS OLD , YOU ARE (x- 20)years old • OF BOTH , POSITIVE AND NEGATIVE INTEGERS . THE (-)MINUS SIGN IS ACTUALLY THE “NEGATIVE” SIGN .
14. 14. ALGEBRA Only like terms can combine into a single term ( BY ADDITION OR SUBTRACTION ONLY ) Like terms : 1) ab , 2 ab ( yes) 2) x , 2x2 (no) 3) 3p,7p (yes) 4) xy , 2x2y (no)
15. 15. SUBTRACTION IN ALGEBRA 1) (+)3a-2b+2a-3b = 3a+2a – 2b – 3b = 5a-5b 2)[3a+3b(a-bc)] FROM 3a-3b =(2a-3b) – (3a2 -3b2c) =2a-3a2-3b-3b2c =-1a3-3b3c Step 1 : rearrange Step 2 : evaluate
16. 16. DIVISION IN ALGEBRA 27a / 3a = 9 ( cancel the “a”) 27/3a =27/3a DIVIDED AWAY
17. 17. TERMS , VARIABLE , COEFFICIENT When x = 4 , When y = 6 When z = 10 , x+y= 4+6 = 10 z-(x+y) = 10-10 = 0 5x = 5x4 = 20 THE VALUE OF x IS CALLED A VARIABLE 5 IS ATTACHED TO x , SO 5 is the coefficient OF x. E.G) 10a _ a is a variable and 10 is the coefficient OF a 6ab - ab is a variable and 6 is the coefficient of ab 2B - 2 of B’s B2– 1b 2so , B is the variable and 1 is the coefficient of b2
18. 18. ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS RECALL : addition / subtraction of integers e.g.) sum of 4 and 2 = 4+2 = 6 Subtract 2 from 5 = 5-2 = 3
19. 19. Exponents often are used in the formula for area and volume. In fact, the word squared comes from the formula for the area of a square. s s Area of a square: A = s2 The word cubed comes from the formula for the volume of a cube. s s s Volume of Cube: V = s3 SQUARE ROOTS AND CUBE ROOTS
20. 20. FACTORISATION 4p2 + 2pq = 2p(2p+q) Common factor 1) Factorisation is the process of finding a term or an algebraic expression. 2) The common factors of several algebraic terms are numbers or terms that are the factors of all algebraic terms 3) An algebraic expression with 2 or more terms can be factorised by taking out all the common factors of the expressions from the brackets. 2xy + 6y + 3x +9 = 2y(x+3)+3(x+3) =( 2y+3)(x+3) Same
21. 21. FACTORISATION Factorisation means taking out the common factors . Factorisation is NOT expansion . Factorisation vs expansion => opposite OPERATION OPPOSITE ADDITION SUBTRACTION SQUARE SQUARE ROOT FACTORISTION EXPANSION CUBE CUBE ROOT DIVISION MULTIPLICATION
22. 22. EXPANSION Expansion – final answer should not have fractions . (Using the “rainbow” method ) e.g) 3(2+x) = 6+x e.g) -3(2h-2k)+4(k-3h) = -6 -6k +4k – 12h = -6h-12h+6k+4k = -18h+10k STEP 1 : Remove the bracket by doing EXPANSION . STEP 2 : Rearrange to put the “like” terms together NOTE : 2 SETS OF BRACKETS , 2 EXPANSIONS
23. 23. ALGEBRA Square root is the opposite of square E.G.) p(square) is opposite of p -DETAILS MUST BE STATED CLEARLY - Times (x) must be written in “bracket format” such as 3x4= 3(4) 2P= 2 x P P2= P x P P3= p x p x p 3P= 3 X P
24. 24. What algebraic expression can be used to find the perimeter of the triangle below? a b c Perimeter = a + b + c In this algebraic expression, the letters a, b, and c are called ________.variables In algebra, variables are symbols used to represent unspecified numbers or values. NOTE: Any letter may be used as a variable. Variables and Expressions
25. 25. It is often necessary to translate verbal expressions into algebraic expressions. English word(s) Math Translation more than less than product addition subtraction multiplication of multiplication quotient division Write an algebraic expression for each verbal expression: a) Eight more than a number n. 8 + ntranslates to b) Seven less the product of 4 and a number x. 4x-7translates to c) One third of the size of the area a. translates to ora 3 1 3 a Variables and Expressions
26. 26. Find the perimeter of the triangle. If a is 8 , b is 15 and c is 17 a b c Perimeter = a + b + c Write the expression. = 8 + 15 + 17 Substitute values. = 40 Simplify. = 8 = 17 = 15 SUBSTITUTION
27. 27. FINDING THE UNKNOWN e.g.) 3x – 2 = 4 3x= 4+2 3x=6 x = 6/3 X=2 (+)11-2k=17 -2=17-11 -2k=(-2) 17-11=6 6/-2 = -3(k) K=-3 2h +1.3=2.8 2h=2.8-1.3 2h=2 1.5-2 =0.5 h= 0.5 *If “ –” , do “+” If “x” do “/”
28. 28. FINDING THE UNKNOWN II 3.14 => recurring number FURTHER EXAMPLES ON EQUATIONS 7 + 2x = 6x-5 2x=6x-5+7 2x=6x-12 2x-6x=-12 -4x=-12 X= -12/-4 = +3 . 6hx + 12ky +9kx +8hy =6hx + 9kx + 12ky + 8hy = 3x (2h+3k) + 4y (3k+2h) =(3x+4y) (2h+3k) *REARRANGE THE ONE WITH THE MOST COMMON FACTOR
29. 29. ESTIMATION 1003 x 78 ~ 1000 x 80 = 80,000 1003 x 85 ~ 1000 x 90 = 90,000 ~ ~ *LESS THAN 5- ROUND DOWN / ignore (“0”) *5 OR MORE – ROUND UP 1300 + 6 ~ 1000+10 = 1010 ~
30. 30. AREA AND PERIMETER AREA) Triangle = ½ x base x height Rectangle = Length x Breadth Square = Length x Length Circle= π x radius x radius (πr2) Parallelogram = Base x height (perpendicular height) Trapezium = ½ x (a+b) x height (a&b 2 parallel lines)
31. 31. AREA AND PERIMETER PERIMETER) Triangle = plus (+) all outer sides Rectangle = plus(+) all outer sides Square = plus (+) all outer sides Circle= (circumference) π x diameter (πD) Parallelogram = Plus(+) all outer sides Trapezium = plus(+) all outer sides
32. 32. FORMULAS FOR MEASURING VOLUME CUBE = Length x Length x Length CUBOID = Length x Breadth x Height PRISM = Base area x Height = 1/2 x Length x Breadth x Height PARALLELOGRAM = Base x Height CONE = 1/3 x x radius2 x height SPHERE= 4/3 X x radius3
33. 33. NUMBER SEQUENCE NUMBER SEQUENCE PATTERN 2,4,6,8,10,12 2 times table 1,3,5,7,9,11 Odd numbers/add 2 1,2,4,8,16,32 Power of 2 2,5,8,11,14,17,20 Add 3 0,10,20,30,40,50,60… Add 10 / 10 times table 1,3,6,10,15 Add 1 to the top 1,1,2,3,5,8,13,21 Add the 1st 2 numbers to get the 3rd number
34. 34. FINDING SEQUENCES 1st layer 1 = 1 2nd lay+0er 1+2= 3 3rd layer 1+2+3=6 4th layer 1+2+3+4= 10 30th layer ? 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17 +18+19+20+21+22+23+24+25+26+27+28+29+30 = 465
35. 35. FINDING SEQUENCES Step 1 : Find the pattern Step 2 : See how the pattern flows Step 3 : Continue the pattern
36. 36. SOLVING INEQUALITIES Symbol Words Example > greater than x + 3 > 2 < less than 7x < 28 ≥ greater than or equal to 5 ≥ x - 1 ≤ less than or equal to 2y + 1 ≤ 7
37. 37. SOLVING INEQUALITIES 12 < x + 5 If we subtract 5 from both sides, we get: 12 - 5 < x + 5 - 5 7 < x But put an "x" on the left hand side ... so let us flip sides (and the inequality sign): x > 7 Do you see how the inequality sign still "points at" the smaller value (7) ? ANS: x > 7
38. 38. VOLUME Volume of cuboid Length x breadth x height Volume of cube Length x Length x Length Volume of pyramid 1/3 x Base x Height Volume of Cylinder Base x Height Volume of cone 1/3 x Base x Height Volume of sphere  4/3 x π x r3
39. 39. UNIT CONVERSION Units : mm,cm,m,km,ha (perimeter) : mm2,cm2,m2, km2,ha2 (area) 10mm= 1cm 1mm=0.1cm 100cm= 1m 1cm=0.01 m 1000mm= 1m 1mm= 0.001 m 1 ha = 10000 m2 1kg=1000g (1k-1000 , g – grams)
40. 40. VOLUME AND TOTAL SURFACE AREA 1. CUBE Volume : length3 Area : 6xlength2 2. CUBOID Volume : length x breadth x height Area: 2(lb + bh + hl ) 3. PRISM Area : Base area x height Volume: (Perimeter of base x h ) + 2base area 4. Cylinder Volume : πr2h Area: 2πr2 + 2πrh
41. 41. UNIT CONVERSION 185mm= 185 x 0.1 cm = 18.5 cm 21cm = 21 x 10mm = 210 mm 21cm = 21 x 0.01m = 0.21 cm 1 hectare = ?x?
42. 42. CONVERSION 1m = 100cm (x100) 1cm = 0.01m (/100) 1m=0.001km(/1000) 1000m =1km (x1000) 1hour=60mins 1minute=60 seconds
43. 43. RATIO (REPEATED IDENTITY) If a:b = 3:5 and a:c = ½ : 3/5 , find the ration of a:b:c. a:c ½:3/5 5:6 a:b 3:5 LCM of 3 and 5 =15 a:b:c = 15:25:18
44. 44. ANGLES ao 84o A = 84O (vertically opposite angle) ao 84O A= 840 ( corresponding angles)
45. 45. ANGLES ao 840 A = 84o ( ALTERNATE ANGLES) yo xo ao A = xo + yo ( interior angles = exterior angle)
46. 46. UNITS OF LENGTH • 1cm = 10mm • 1dm = 10cm • 1m= 100cm • 1km = 1000m
47. 47. UNITS OF LENGTH • 1g = 1000mg • 1kg= 1000g • 1 ton = 1000kg • Capacity = volume 1l = 1000ml 1ml = 1cm2