Math Reviewer
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Fundamentals of Mathematics
Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Arithmetic • • • •
Sum = addend + addend (or terms) Difference = minuend – subtrahend Product = multiplicand multiplier (or factors) Quotient = dividend divisor
a • Take note! is undefined 0 2 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Arithmetic • Addition – Same sign: add the numbers, copy the sign – Opposite signs: subtract the numbers, copy sign of larger number
• Subtraction – Change the operation into addition, then change the sign of the subtrahend
• Multiplication/division – Count the number of negative signs; when even = 3 positive; when odd = negative Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Properties of arithmetic • • • • •
Commutative (,) a + b = b + a Associative (,) Distributive ( over and over +) Identity element (0 for and 1 for ) Inverse elements – Additive inverses have the sum 0 – Multiplicative inverses have the product 1
• Multiplicative property of zero 4 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Properties of equality • Reflexive • Symmetric
aa
a b; then b a
• Transitive
a b and b c, then a c
• Substitution
If a b, then any expression involving a retains the same value if a is replaced by b. 5
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Philippines.
Order of operations: PEMDAS • Parenthesis or grouping symbols • Exponents • Multiplication and division (whichever comes first) • Addition and subtraction (whichever comes first)
http://www.mahalo.com/pemdas/
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6 Philippines.
Laws of exponents • A power is an expression involving repeated multiplication. • base: b; power or exponent: m or n • Take note! 0
a 1 1 a a
m n
m n
b b b m n mn (b ) b m b mn 0 b ; hence b 1 n b m m m (ab) a b m m a a m b b 0 1 a 1 n a n ; since n n a a a
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Philippines.
Scientific notation • Write numbers as the product of a number less than 10 and a power of 10. • The sign of the exponent is positive when the original figure is less than 1 and negative when more than 1.
2
245 2.45 10 3 0.0056 5.6 10 8 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Prime, Composite or Denominate? • Prime – Numbers whose only factors are 1 and itself
• Composite – Numbers with factors other than itself and 1
• Denominate – Numbers with a unit of measurement
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Philippines.
GCF and LCM • Greatest common factor – Useful in reducing fractions to lowest or simplest terms
• Least common multiple – Useful in operations involving dissimilar fractions
10 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Fractions • Improper fractions to mixed numbers
a remainder quotient (a b) b b
• Mixed numbers to improper fractions
a (b * c) a c b b 11 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Fractions • Equivalent fractions – Have the same value – Result from the multiplication and division of the numerators and denominators of a fraction by the same number
• Reducing to simple fractions – Determine the GCF of the numerator and denominator; divide them by the GCF 12 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Ratio and proportion • Ratios may be written as fractions or by using colons
2 is the same as 2 : 3 3 • Proportions are the equality between two ratios
2 4 is the same as 2 : 3 :: 4 : 6 3 6 13 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Sharing/dividing in a ratio When sharing/dividing any amount/quantity (e.g., n) in a ratio (e.g., a : b : c): • Add
a b c number of parts
n share per part • Divide number of parts • Multiply share per part * a share of a
or b or c
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or b or c
Philippines.
Rules for ratio and proportion a c If , then ad bc. b d a c a b If , then b d c d. a c b d If , then . b d a c Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
15
Variation • Direct:
y kx
– Variable y varies at the same rate as variable x
• Inverse:
k y x
– As variable x increases, y decreases and vice-versa
• Joint – More than one variable is involved in a direct variation Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
16
Algebra
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Philippines.
Mathematical sentences • is – was – will be – has: • plus – sum – more than – greater than – farther than – years from now – increased by – exceeds by: • decreased by – minus – less than – diminished by – younger than – years ago – subtracted from: • divided by – quotient – per: 18 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Factoring polynomials • Factor out the common monomial factor • Difference of two squares 2
2
a b (a b)(a b) • Perfect square trinomial 2
2
a 2ab b (a b)
2
• FOIL method and quadratic polynomials
(a b)(c d ) ac (ad bc) bd 2 standard form : ax bx c 19 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Notes in solving word problems • Consecutive numbers: x , x 1, x 2 • Consecutive even/odd numbers: x , x 2, x 4 • Mixture (e.g., of coins, vehicles, chemicals): total amt = number of coins * denomination number of feet/wheels = animals/vehicle * # of feet/wheels of each 20 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Notes in solving word problems • Motion:
• Work:
distance speed time part of a job done by one person (in 1 day/hour) 1 number of days/hours it takes the person to complete
• Age: add when it says “# of years from now” and subtract when it says “# of years ago” 21 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Notes in solving word problems • Interest
I Prt
• Compound interest
r nt A P(1 ) n
or future value FV PV (1 r ) t • Continuous compound interest
A Pe
rt
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Philippines.
Notes in solving word problems • Percent change
new original % change original • Discount
Discount rate original price Discounted price original price discount • Commission
Commission rate total earnings 23 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Functions • A function is a set of ordered pairs such that for each first component, there is at most one value of the second component • Three things that specify a function: – Domain – Range – A rule which assigns to each member of the domain exactly one member of the range 24 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Functions • Domain – set of all first components in each element of the set
• Range – set of all the second components in each element of the set
• When y is a function of x, the set of values of x is the domain of the function and the set of values of y is the range of the function. 25 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Geometry
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Philippines.
Lines • Parallel lines do not meet. • Intersecting lines meet at one point. The sum of any 2 adjacent angles formed is 180°. • Perpendicular lines meet at one point, at 90°.
27 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Kinds of Angles • • • • •
Acute: less than 90° Right: 90° Obtuse: more than 90°, but less than 180° Straight: 180° Reflex: more than 180 degrees, less than 360°
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Philippines.
Angle relationships • Supplementary angles – Sum of measures is 180°
• Complementary angles – Sum of measures is 90°
29 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Pythagorean theorem • Applies to right triangles • Legs: a and b • Hypotenuse: c
2
2
c a b
2
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Philippines.
Polygons • Polygons are flat, closed figures consisting of line segments. • Exterior angles of a polygon add up to 360°. • The sum of the interior angles of a polygon, with the number of sides n: 180° (n 2) • The sum of the exterior angles of a simple, closed polygon is 360°. 31 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Measurements • Perimeter: sum of the length of all sides
Prectangle 2l 2w
Circumference 2r
Ptriangle a b c Psquare 4s
• Area: size covered by a surface
Asquare s
2
Acircle r
2
Arectangle lw 1 Atriangle bh 2
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Philippines.
32
Measurements • Volume: the space occupied or contained by a substance or shape
Vcube s V pyramid
3
1 bh 3
4 3 Vsphere r 3 1 2 Vcone r h 3
Vrectangular prism abc or lwh 33 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Statistics, Probability and Number Series
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Philippines.
Measures of central tendency • Mean – Add the numbers and divide the sum by the number of values (which were added).
• Median – The middle number when numbers are written in order. (When the number of values is even, the median is the mean of the two middle values.)
• Mode – The number which occurs most often 35 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Counting techniques • If two or more actions are performed in a particular order, such that the first action produces a results, the second action produces b results, the third action produces c results, and so on, then:
a * b * c * ... total number of possible results
36 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
Counting techniques: Permutation • Objects are arranged in a definite order: – n different things taken r at a time – n things taken n at a time in which q are alike, r are alike and so on – n distinct objects are arranges along a fixed circle
n! n Pr (n r )! n! n Pr (q! r!...) Pn (n 1)!
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Philippines.
37
Counting techniques: Combination • Objects are not arranged in a particular order – n different things taken r at a time
n! n Cr (n r )! r!
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Philippines.
Probability Probability of an outcome number of ways an outcome can happen the total number of possible outcomes
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Philippines.
Arithmetic sequences and series • Arithmetic sequence/progression: a sequence of numbers where successive members of the sequence have a constant difference (d). • Formula for n-th term an a1 (n 1)d • Formula for sum of first n values of a sequence n(n 1)
Sn
2 40
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Philippines.
Geometric sequences and series • Geometric sequence/progression: a sequence of numbers where successive members of the sequence are produced by multiplying them by a constant multiplier (r, or common ratio)
a n a1 * r • Formula for n-th term • Formula for sum of first n values of a n sequence a1 (1 r )
Sn
n 1
1 r
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41 Philippines.
References and further reading British Broadcasting Corporation. (2012). GCSE Bitesize: Maths. Retrieved 27 December 2012, from http://www.bbc.co.uk/schools/ gcsebitesize/ maths/ Fisico, M., L. Sia, J. Ho, Z. Maralit, J. Tan, . . . J. Rosello, R. Franco. (1998). 21st Century Mathematics Series. Quezon City: Phoenix Publishing House, Inc. Pierce, Rod. (2012). Math Is Fun. Retrieved 28 December 2012, from http://www.mathsisfun.com/index.htm Stapel, E. (2012). The Purplemath Forums. Retrieved 27 December 2012, from http://www.purplemath.com/ index.htm
42 Exclusively for My Review Coach online LET review. Not for redistribution. © 2011 My Review Coach of
Philippines.
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