MATHEMATICS STUDY GUIDE - E-monsite

MATHEMATICSSTUDY GUIDECurriculum Planning and Development Division

The booklet highlights some salient points for each topic in the CSEC Mathematics syllabus. Atleast one basic illustration/example accompanies each salient point. The booklet is meant to beused as a resource for “last minute” revision by students writing CSEC Mathematics.

Number TheoryBasic RulesPoints to RememberThe sum of any number added to zero gives thesame numberThe product of any number multiplied by 1 givesthe same numberAny number that is multiplied by zero gives aproduct of zeroThe sum (or difference) of 2 real numbers equals areal numberZero divided by any number equals zero.Any number that is divided by zero is undefined.The denominator of any fraction cannot have thevalue zero.The Associative Law states The "AssociativeLaws" say that it doesn't matter how we group thenumbers, the order in which numbers are added ormultiplied does not affect their sum or product.The Commutative Law states that in a set ofnumbers, multiplication must be applied beforeaddition.Illustration/ ExampleThe Additive Identitya 0 0 a a7 0 70 3.6 3.6Multiplicative Identitya 1 1 a a7 1 1 7 7a 0 0 a 07 0 0 7 04 5 94 (- 5) -14.3 5.2 9.50/5 00/x 0 x 05/0 is undefined0/0 is undefinedx/0 is undefined x 0(a b) c a (b c)(6 3) 4 6 (3 4) 13(a b) c a (b c)(6 3) 4 6 (3 4) 72a b c c b a b c a2 3 4 4 3 2 3 4 2 9a b c c b a b c a2 3 4 4 3 2 3 4 2 24BODMAS provides the key to solvingmathematical problemsB - Brackets firstO - Orders (ie Powers and Square Roots, etc.)DM- Division and Multiplication (left-to-right)When positive numbers are added together theresult is positiveWhen two or more negative numbers are to beadded, we simply add their values and get anothernegative number7 (6 52 3)Start inside Brackets, and then use"Orders" first 7 (6 25 3) Then Multiply 7 (150 3)Then Add 7 (153)Final operation is addition 160DONE!4 5 9-4–5 -9Curriculum Planning and Development Division - 1

Number TheoryBasic RulesPoints to RememberTo find the difference between two numbers whenone number is positive and one number is negativethe result will be “ ” if the larger value is positiveor “–“ negative if the larger number is negative.When multiplying, two positive numbersmultiplied together give a positive product; and anegative number multiplied by another negativenumber gives a positive product. Also, a negativenumber multiplied by a positive number gives anegative productNumber TheoryPositive and Negative NumbersPoints to RememberThe rules for division of directed numbers aresimilar to multiplication of directed numbers.Use manipulatives- counters (yellow and red)There are different type of numbers:Natural Numbers - The whole numbers from 1upwardsIntegers- The whole numbers, {1,2,3,.} negativewhole numbers {., -3,-2,-1} and zero {0}.Rational Numbers- The numbers you can makeby dividing one integer by another (but notdividing by zero). In other words, fractions.Irrational Number – Cannot be written as a ratioof two numbersReal Numbers - All Rational and Irrationalnumbers. They can also be positive, negative orzero.Number TheoryDecimals – RoundingPoints to RememberRounding up a decimal means increasing theterminating digit by a value of 1 and drop off thedigits to the right.Round down if the number to the right of ourterminating decimal place is four or less (4,3,2,1,0)Illustration/ Example20 – 10 10-20 10 -10( ) ( ) (-) (-) ( ) (-) (-) ( ) -e.g.8 5 40-8 -5 408 -5 -40-8 5 -40Illustration/ Example( ) ( ) e.g. 10 5 2(-) (-) -10 -5 2( ) (-) 10 -5 -2(-) ( ) -10 5 -2Natural Numbers (N) : {1,2,3,.}Integers (Z) : {., -3, -2, -1, 0, 1, 2, 3, .}Rational Numbers (Q) :. 3/2 ( 1.5), 8/4 ( 2), 136/100( 1.36), -1/1000 ( -0.001)Irrational Number : 𝜋, 3.142 (cannot be written as afraction)Real Numbers (R): 1.5, -12.3, 99, 2, πIllustration/ Example5.47 to the tenths place, it can be can be rounded up to5.56.734 to the hundredths place, it can be rounded down to6.73Curriculum Planning and Development Division - 2

Number TheoryOperations with DecimalsPoints to RememberFind the product of 3.77 x 2.8 ?1. Line up the numbers on the right,2. multiply each digit in the top number byeach digit in the bottom number (likewhole numbers),3. add the products,4. and mark off decimal places equal to thesum of the decimal places in the numbersbeing multiplied.Illustration/ ExampleFind the product of 3.77 2.83.77(2 decimal places)2.8 (1 decimal place)754301610.556 (3 decimal places)When dividing, if the divisor has a decimal in it,make it a whole number by moving the decimalpoint to the appropriate number of places to theright. If the decimal point is shifted to the right inthe divisor, also do this for the dividend.Fractions can always be written as decimals.For example:2514 0.4 0.25Curriculum Planning and Development Division - 31235 0.5 0.63434 0.75 0.75

Number TheorySignificant figuresPoints to RememberIllustration/ ExampleThe rules for significant figures:1) ALL non-zero numbersNumber(1,2,3,4,5,6,7,8,9) are ALWAYSsignificant.2) ALL zeroes between non-zero numbers48,923are ALWAYS significant.3.9673) ALL zeroes which areSIMULTANEOUSLY to the right of the900.06decimal point AND at the end of the0.0004 ( 4 E-4)number are ALWAYS significant.4) ALL zeroes which are to the left of a8.1000written decimal point and are in a number 10 are ALWAYS significant.501.040A helpful way to check rules 3 and 4 is to write the3,000,000 ( 3 E 6)number in scientific notation. If you can/must getrid of the zeroes, then they are NOT significant.10.0 ( 1.00 E 1)Number TheoryBinary NumbersPoints to RememberEach digit "1" in a binary number represents apower of two, and each "0" represents zero.Binary numbers can be addedNumber 3,41131,3,4Illustration/ Example0001 is 2 to the zero power, or 10010 is 2 to the 1st power, or 20100 is 2 to the 2nd power, or 41000 is 2 to the 3rd power, or 810001 11101101110Binary numbers can be subtractedCurriculum Planning and Development Division - 4

Number TheoryComputation – FractionsPoints to RememberWhen the numerator stays the same, and thedenominator increases, the value of the fractiondecreasesWhen the denominator stays the same, and thenumerator increases, the value of the fractionincreases.Equivalent fractions are fractions that may lookdifferent, but are equal to each other.Equivalent fractions can be generated bymultiplying or dividing both the numerator anddenominator by the same number.Fractions can be simplified when the numeratorand denominator have a common factor in themFractions with different denominators, can beconverted to a set of fractions that have the samedenominatorIllustration/ Example1 1 13 , 4 , 53 3 3 34 ,5 , 6 77 892 , 2 , 23 41 22 , 4 , 6 812 35 610 1x 222x2 43x265 x 2 103x2 35x2 53 2984 , 3 is the same as 12 , 12Addition and subtraction of fractions are similar toadding and subtracting whole numbers if thefractions being added or subtracted have the samedenominator98112 – 12 12When multiplying fractions, multiply thenumerators together and then multiply thedenominators together and simplify the results.5 21056 3 18 9Number TheoryPrime NumbersPoints to RememberA prime number is a number that has only twofactors: itself and 1e.g. 5 can only be dividedevenly by 1 or 5, so it is a prime number.Numbers that are not prime numbers are referredto as composite numbersIllustration/ ExampleCurriculum Planning and Development Division - 5

Number TheoryComputation of Decimals, Fractions and PercentagesPoints to RememberIllustration/ ExamplePercent means "per one hundred"20 % 20 per 100To convert from percent to decimal, divide thepercent by 10010 % 100 0.1To convert from decimal to percent, multiply thedecimal by 10067.5% 100 0.6750.10 as a percentage is 0.10 100 10%0.675 is 0.675 100 67.5%1067.5To convert from percentages to fractions, dividethe percent by 100 to get a fraction and thensimplify the fractionTo convert from fractions to percentages, convertthe fraction to a decimal by dividing the numeratorby the denominator and then convert the decimalto a percent by multiplying by 100.1212 4312% 100 100 4 25325 0.120.12 as a percentage is 0.12 100 12%Curriculum Planning and Development Division - 6

TrianglesClassification of TrianglesPoints to RememberTriangles can be classified according to lengths oftheir sides to fit into three categories:-Illustration/ ExampleScalene TriangleScalene: No equal sides ;No equal anglesIsosceles triangleIsosceles: Two equal sides ; Two equal anglesEquilateral Triangle: Three equal sides ; Threeequal 60 anglesTriangles can be classified according to angles:-Equilateral Triangleacute angle triangleAcute- All three angles are acute angles.Obtuse- An obtuse triangle is a triangle in whichone of the angles is an obtuse angle.obtuse angle triangleRight angle- A triangle that has a right angle (90 )right angle triangleCurriculum Planning and Development Division - 7

Angles formed by a Transversal Crossing two Parallel LinesVertical Angles are the angles opposite eachIllustration of all angles mentioned on a singleother when two lines cross.diagram. The transversal crosses two ParallelVertically opposite angles are equalLinesa df gb ce hThe angles in matching corners are calledCorresponding Angles.Corresponding Angles are equala ec gb fd hThe pairs of angles on opposite sides of thetransversal but inside the two lines are calledAlternate Angles.Alternate Angles are equald ec fThe pairs of angles on one side of thetransversal but inside the two lines are calledConsecutive Interior Angles.Consecutive Interior Angles are supplementary(add up to 180 )d fc eCurriculum Planning and Development Division - 8

TrianglesPythagoras' TheoremPoints to RememberPythagoras' Theorem states that the square of thehypotenuse is equal to the sum of the squares onthe other two sidesIllustration/ Examplec2 a2 b2Find cThe Hypotenuse is cc2 52 122 25 144 169c 169 13 unitsTrianglesSimilar Triangles & Congruent TrianglesPoints to RememberDefinition: Triangles are similar if they have thesame shape, but can be different sizes.Illustration/ ExampleShow that the two triangles given beside are similar andcalculate the lengths of sides PQ and PR.(They are still similar even if one is rotated, or oneis a mirror image of the other).There are three accepted methods of proving thattriangles are similar:If two angles of one triangle are equal to twoangles of another triangle, the triangles are similar.Solution: A P and B Q, C R(because C 180 A - B and R 180 - P - Q)If angle A angle D and angle B angle ETherefore, the two triangles ΔABC and ΔPQR aresimilar.Then ABC is similar to DEFCurriculum Planning and Development Division - 9

TrianglesSimilar Triangles & Congruent TrianglesPoints to Remember1) If the three sets of corresponding sides of twotriangles are in proportion, the triangles aresimilar.Illustration/ ExampleConsequently:𝐴𝐵𝑃𝑄 𝐵𝐶𝑄𝑅𝐴𝐶 𝑃𝑅implies𝐴𝐵𝑃𝑄 𝐵𝐶𝑄𝑅Substituting known lengths give:Therefore PQ Also,𝐵𝐶𝑄𝑅12 46𝐴𝐵𝐴𝐶𝐵𝐶 𝐷𝐸 𝐷𝐹 𝐸𝐹Then ABC is similar to DEF 612or 6PQ 4 12 8𝐴𝐶 𝑃𝑅Substituting known lengths give:If4𝑃𝑄Therefore PR 12 766127 𝑃𝑅 or 6PR 12 7 14Find the length AD (x)2) If an angle of one triangle is equal to thecorresponding angle of another triangle and thelengths of the sides including these angles arein proportion, the triangles are similar.The two triangles ΔABC and ΔCDE appear to be similarsince AB DE and they have the same apex angle C.It appears that one triangle is a scaled version of theother. However, we need to prove this mathematically.If angle A angle D and𝐴𝐵AB DE, CD AC and BC EC BAC EDC and ABC DEC𝐴𝐶 𝐷𝐸 𝐷𝐹Then ABC is similar to DEFConsidering the above and the common angle C, wemay conclude that the two triangles ΔABC and ΔCDEare similar.Curriculum Planning and Development Division - 10