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Mathematics sec 1

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APPRECIATE ~ GRADE 7 / SEC 1 MATH

APPRECIATE ~ GRADE 7 / SEC 1 MATH

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Mathematics sec 1 Mathematics sec 1 Presentation Transcript

  • MATHEMATICS secondary 1Nicco AlysshaParikh
  • PRIME NUMBERSPrime 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.Step by step finding Prime numbers .- Cross out the number 1- Circle the number 2 and cross out all the other multiples of1 .- Circle the number 3 and cross out all the other multiples of3 .- Circle the number 5 and cross out all the other multiples of5 .- Circle the number 7 and cross out all the other multiples of7 .- Continue the process unit all unit all the numbers are either
  • HIGHEST COMMONFACTOR(common)How to find the highest common factor?Find the highest commonmultiple of 15 and 75 ?15,755,251,5HCF= 3x5=1535Express 252 in PRIME FACTORS2522 x 1262 x 2 X 6322 3 X 213 3 x 7From the abovefactor tree,We have 252 =2x2x3x3x7
  • INDEX NOTATIONIndex notation is using the power of a certain number.e.g.) 252= (first, primefactorise the numbers)= 2x2x3x3x7= 22 x 32 x 712= 2x2x3 can bewritten as 12= 2 x 32
  • LOWEST COMMON MULTIPLE(max)Lowest common multiple of 65 , 175 , 13565,175,135Please note that we have toarrive to the answers to all be oneat the last ladder13, 35,271,35,271,1,271,1,15133527LCM(LOWEST COMMON MULTIPLE)= 5x13x35x27=61425
  • FINDING CUBE ROOT AND SQUAREROOTS1) FIRST PRIME FACTORISE THE NUMBEREG.) square root of 144= 24 x 322) Arrange them into 2 bracketsSquare root ( 2x2x3) (2x2x3)3) Solve what is in 1 bracket2x2x3=4x3=12Cube rootDo the same only atstep to , instead of 2brackets , it becomes3 brackets .
  • IntegersPositive and Negative integers. In the number line , the more left you go , the largerthe 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 fromleft to right UNLESS there is a bracket .
  • ADDITION OF INTERGERS3+2=53+(-3)=0-3+4=-1-4+(-2)=-6Owe someone 4 dollarsand another 2 dollars.
  • SUBTRACTION OF INTEGERS-7+(-11)+9=-7-11+9=-18+9= -934+(-18)+9=34-18+9=16+9=25
  • MULTIPLICATION OF INTEGERS+x+=+-x+=--x-=+-x0=? (0)-2x3x(-1)=-6x(-1)=6
  • DIVISION OF INTEGERS3x6 3-2= (18 3)-2=6-2=46 2x4 + (-3)= 3 x 4 +(-3)= 12 + (-3)= 9*ALWAYS DO FROM LEFT TORIGHTALWAYS DO THE “ POWERS “ FIRST(-4)2 (-8) + 3 x (-2)3= 16/(-8) + 3 x (-8)= -2 + 3 x (-8 )= -2 + (-24)=-26
  • RATIONAL NUMBERSa/b - b cannot be “0”e.g : mixed numbersimproper 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’TCHANGE – DURING DIVISION OF FRACTIONS ONLY !! DENOMINATORS MUST BE THE SAME.
  • ALGEBRA• Actually writing numbers in the form of letters• IF YOU ARE 40 YEARS OLD , I AM 20 YEARSYOUNGER 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 NEGATIVEINTEGERS . THE (-)MINUS SIGN IS ACTUALLYTHE “NEGATIVE” SIGN .
  • 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)
  • SUBTRACTION IN ALGEBRA1) (+)3a-2b+2a-3b= 3a+2a – 2b – 3b= 5a-5b2)[3a+3b(a-bc)] FROM 3a-3b=(2a-3b) – (3a2 -3b2c)=2a-3a2-3b-3b2c=-1a3-3b3cStep 1 : rearrangeStep 2 : evaluate
  • DIVISION IN ALGEBRA27a / 3a= 9 ( cancel the “a”)27/3a=27/3aDIVIDED AWAY
  • TERMS , VARIABLE , COEFFICIENTWhen x = 4 ,When y = 6When z = 10 ,x+y= 4+6= 10z-(x+y) = 10-10= 05x = 5x4= 20THE VALUE OF x IS CALLED A VARIABLE5 IS ATTACHED TO x , SO 5 is the coefficient OF x.E.G)10a _ a is a variable and 10 is the coefficient OF a6ab - ab is a variable and 6 is the coefficient of ab2B - 2 of B’sB2– 1b 2so , B is the variable and 1 is the coefficient of b2
  • ADDITION AND SUBTRACTION OFALGEBRAIC EXPRESSIONSRECALL : addition / subtraction of integerse.g.) sum of 4 and 2 = 4+2= 6Subtract 2 from 5 = 5-2= 3
  • Exponents often are used in the formula for area and volume. In fact, the wordsquared comes from the formula for the area of a square.ssArea of a square: A = s2The word cubed comes from the formula for the volume of a cube.sssVolume of Cube: V = s3SQUARE ROOTS AND CUBE ROOTS
  • FACTORISATION4p2 + 2pq= 2p(2p+q)Common factor1) Factorisation is the processof finding a term or analgebraic expression.2) The common factors ofseveral algebraic terms arenumbers or terms that are thefactors of all algebraic terms3) An algebraic expression with2 or more terms can befactorised by taking out all thecommon factors of theexpressions from the brackets.2xy + 6y + 3x +9= 2y(x+3)+3(x+3)=( 2y+3)(x+3)Same
  • FACTORISATIONFactorisation means taking out the commonfactors .Factorisation is NOT expansion .Factorisation vs expansion => oppositeOPERATION OPPOSITEADDITION SUBTRACTIONSQUARE SQUARE ROOTFACTORISTION EXPANSIONCUBE CUBE ROOTDIVISION MULTIPLICATION
  • EXPANSIONExpansion – final answer should not have fractions . (Using the“rainbow” method )e.g) 3(2+x)= 6+xe.g) -3(2h-2k)+4(k-3h)= -6 -6k +4k – 12h= -6h-12h+6k+4k= -18h+10kSTEP 1 : Remove thebracket by doingEXPANSION .STEP 2 : Rearrange toput the “like” termstogetherNOTE : 2 SETS OFBRACKETS , 2EXPANSIONS
  • ALGEBRASquare root is the opposite of squareE.G.) p(square) is opposite of p-DETAILS MUST BE STATED CLEARLY- Times (x) must be written in “bracket format” such as3x4= 3(4)2P= 2 x PP2= P x PP3= p x p x p3P= 3 X P
  • What algebraic expression can be used to find the perimeterof the triangle below?a bcPerimeter = a + b + cIn this algebraic expression, the letters a, b, and c are called ________.variablesIn algebra, variables are symbols used to represent unspecified numbers or values.NOTE: Any letter may be used as a variable.Variables and Expressions
  • It is often necessary to translate verbalexpressions into algebraic expressions.English word(s) Math Translationmore thanless thanproductadditionsubtractionmultiplicationof multiplicationquotient divisionWrite an algebraic expression for eachverbal expression:a) Eight more than a number n. 8 + ntranslates tob) Seven less the product of 4 and a number x. 4x-7translates toc) One third of the size of the area a. translates to ora313aVariables and Expressions
  • Find the perimeter of the triangle.If a is 8 , b is 15 and c is 17a bcPerimeter = a + b + c Write the expression.= 8 + 15 + 17 Substitute values.= 40 Simplify.= 8= 17= 15SUBSTITUTION
  • FINDING THE UNKNOWNe.g.) 3x – 2 = 43x= 4+23x=6x = 6/3X=2(+)11-2k=17-2=17-11-2k=(-2)17-11=66/-2 = -3(k)K=-32h +1.3=2.82h=2.8-1.32h=21.5-2=0.5h= 0.5*If “ –” , do “+”If “x” do “/”
  • FINDING THE UNKNOWN II3.14 => recurring numberFURTHER EXAMPLES ON EQUATIONS7 + 2x = 6x-52x=6x-5+72x=6x-122x-6x=-12-4x=-12X= -12/-4= +3.6hx + 12ky +9kx +8hy=6hx + 9kx + 12ky + 8hy= 3x (2h+3k) + 4y (3k+2h)=(3x+4y) (2h+3k)*REARRANGETHE ONE WITH THEMOST COMMONFACTOR
  • ESTIMATION1003 x 78~ 1000 x 80= 80,0001003 x 85~ 1000 x 90= 90,000~~*LESS THAN 5- ROUNDDOWN / ignore (“0”)*5 OR MORE – ROUNDUP1300 + 6~ 1000+10= 1010~
  • AREA AND PERIMETERAREA)Triangle = ½ x base x heightRectangle = Length x BreadthSquare = Length x LengthCircle= π x radius x radius (πr2)Parallelogram = Base x height (perpendicularheight)Trapezium = ½ x (a+b) x height (a&b 2 parallellines)
  • AREA AND PERIMETERPERIMETER)Triangle = plus (+) all outer sidesRectangle = plus(+) all outer sidesSquare = plus (+) all outer sidesCircle= (circumference) π x diameter (πD)Parallelogram = Plus(+) all outer sidesTrapezium = plus(+) all outer sides
  • FORMULAS FOR MEASURINGVOLUMECUBE = Length x Length x LengthCUBOID = Length x Breadth x HeightPRISM = Base area x Height= 1/2 x Length x Breadth x HeightPARALLELOGRAM = Base x HeightCONE = 1/3 x x radius2 x heightSPHERE= 4/3 X x radius3
  • NUMBER SEQUENCENUMBER SEQUENCE PATTERN2,4,6,8,10,12 2 times table1,3,5,7,9,11 Odd numbers/add 21,2,4,8,16,32 Power of 22,5,8,11,14,17,20 Add 30,10,20,30,40,50,60… Add 10 / 10 times table1,3,6,10,15 Add 1 to the top1,1,2,3,5,8,13,21 Add the 1st 2 numbersto get the 3rd number
  • FINDING SEQUENCES1st layer 1 = 12nd lay+0er 1+2= 33rd layer 1+2+3=64th layer 1+2+3+4= 1030th 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
  • FINDING SEQUENCESStep 1 : Find the patternStep 2 : See how the pattern flowsStep 3 : Continue the pattern
  • SOLVING INEQUALITIESSymbol 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
  • SOLVING INEQUALITIES12 < x + 5If we subtract 5 from both sides, we get:12 - 5 < x + 5 - 57 < xBut put an "x" on the left hand side ...so let us flip sides (and the inequality sign):x > 7Do you see how the inequality sign still "points at" thesmaller value (7) ?ANS: x > 7
  • VOLUMEVolume of cuboidLength x breadth x heightVolume of cubeLength x Length x LengthVolume of pyramid1/3 x Base x HeightVolume of CylinderBase x HeightVolume of cone1/3 x Base x HeightVolume of sphere 4/3 x π x r3
  • UNIT CONVERSIONUnits : mm,cm,m,km,ha (perimeter): mm2,cm2,m2, km2,ha2 (area)10mm= 1cm1mm=0.1cm100cm= 1m1cm=0.01 m1000mm= 1m1mm= 0.001 m1 ha = 10000 m21kg=1000g (1k-1000 , g – grams)
  • VOLUME AND TOTAL SURFACE AREA1. CUBEVolume : length3Area : 6xlength22. CUBOIDVolume : length x breadth x heightArea: 2(lb + bh + hl )3. PRISMArea : Base area x heightVolume: (Perimeter of base x h ) + 2base area4. CylinderVolume : πr2hArea: 2πr2 + 2πrh
  • UNIT CONVERSION185mm= 185 x 0.1 cm = 18.5 cm21cm = 21 x 10mm = 210 mm21cm = 21 x 0.01m = 0.21 cm1 hectare = ?x?
  • CONVERSION1m = 100cm (x100)1cm = 0.01m (/100)1m=0.001km(/1000)1000m =1km (x1000)1hour=60mins1minute=60 seconds
  • RATIO (REPEATED IDENTITY)If a:b = 3:5 and a:c = ½ : 3/5 , find the ration of a:b:c.a:c½:3/55:6a:b3:5LCM of 3 and 5 =15a:b:c = 15:25:18
  • ANGLESao84oA = 84O (vertically opposite angle)ao84OA= 840 (correspondingangles)
  • ANGLESao840A = 84o ( ALTERNATEANGLES)yoxo aoA = xo + yo ( interiorangles = exterior angle)
  • UNITS OF LENGTH• 1cm = 10mm• 1dm = 10cm• 1m= 100cm• 1km = 1000m
  • UNITS OF LENGTH• 1g = 1000mg• 1kg= 1000g• 1 ton = 1000kg• Capacity = volume1l = 1000ml1ml = 1cm2