Grade 9 math exam
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GRADE 9 MATH EXAM
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PROJECTIONS AND VIEWS
AREA OF COMMON BASES
LATERAL AND TOTAL SURFACE AREA OF SOLIDS.
PYTHAGOREAN THEOREM
b2 = c2 - a2
a2 = c2 - b2
“c” must be the hypotenuse. In a right triangle that has 30o and 60o
angles, the longest side ( the hypotenuse) is always twice the length of the shortest side.
REAL NUMBERS Natural number: positive integers and no zero.
Example: 1,2,3,4,....89,.....756,.....1000000 Whole number: natural + zero.
Example: 0,1,2,3.....76....3456.....282763.... Integer: whole numbers and their opposites (no decimal)
Example: -45, -39, -8, 0, 123, 29874, 30000000 Rational: number can be written as a ratio (fraction) of
two integers. (in decimal form are terminating or repeating. Example: ½ , 5.2222..., 0.19, -11/3, 2, -4.5, √25 Terminating decimal numbers: 5/2 = 2.5, 5/8 = 0.625 Repeating decimal numbers: 1/9 = 0.1111111...... or 0.1
Irrational: number that cannot be written as a fraction of integers and whose decimal numbers are infinite and non-periodic (does not repeat). Example: √2, √5, ∏
INVERSE VARIATION FUNCTION Reverse x and y to get an inverse
function If x increases, y decreases and vice
versa When the product of each variables’
values is a constant you get an inverse variation function.
FUNCTIONAL RELATION a relation is a function when each
value of the x-axis (abscissa) has one y-axis (ordinate) associated with it.
x-axis (abscissa) = independent variable y-axis (ordinate) = dependent variable
INTERVALS [included] ]excluded[
Intervals with infinity: infinity is never included.
[-4, +∞[ = from -4 to positive infinity.]- ∞, -1[ = negative infinity up to but
excluding -1.
FUNCTION PROPERTIES Domain (X): all x values from left to right. Range (Y): all y values from down to up Variation (X): it can increase, decrease or
remain constant. Extrema (Y): The minimum: smallest value of y.
The maximum: largest value of y. Sign (X): above x-axis is positive and below is
negative. X-intercept (zero) & y-intercept (initial value).
FUNCTION PROPERTY EXAMPLE Domain: ]-∞,+ ∞[ Range: ]- ∞,8] Variation
Increasing: ]- ∞,-4] U [-1,3] Decreasing: [-4,-1] U [3, + ∞[ Constant: none
Extrema Min: - ∞ Max: 8
Sign Positive: [-6,-2] U [1,5] Negative: ]- ∞,-6] U [-2,1] U [5,+ ∞[
Zero: -6, -2, 1, 5 Initial value: -2
VARIABLES Variables are qualitative (words) or
quantitative (numbers).
Discrete quantitative (counting numbers) E.g. Dolls on a shelf
Continuous quantitative (all values included within an interval – can be decimal points) E.g. Height
REPRESENTATIVE SAMPLING 1. simple random: by chance (from a
hat) 2. systematic: regular intervals from a list
of the whole population (every 10th member)
3. cluster: A random selection of clusters is chosen to represent the whole. Every individual within a selected cluster is selected.
4. stratified: taking representative samples from each group.
CLUSTER AND STRATIFIED
Percentage: 10% of 254 = 10/100 x 254 = 25.4
SOURCES OF BIAS Sources of bias are different reasons
that could lead researchers or survey people to draw the wrong conclusion from a survey or census.
There are 6 different sources of bias:A non-representative sample of the populationA poorly formulated questionThe attitude of the person doing the survey Inadequate representation of the resultsLarge part of the sample is rejectedA processing error that occurs when compiling
the data.
MEASURES OF CENTRAL TENDENCY – CONDENSED (REGULAR) DATA TABLE Median: is the number in the middle
when values are placed in order. Mode: the number that occurs most
often in a distribution (list of numbers). Mean: average of all numbers (sum of
all values divided by the number of values).
Range: highest value – lowest value
2 DIFFERENT TYPES OF DATA TABLES:
Table of condensed data: mostly used when data values are repeated.
Table with data grouped into classes: data is grouped into intervals [a,b[ (included, excluded) – very few repeated values.
Need to determine the number of groups and how much data each one can carry (amplitude).Amplitude = range/number of classes.Amplitude of each interval must be the same!
MEASURES OF CENTRAL TENDENCY IN GROUPED DATA A) mode: class with highest frequency is
called the modal class. Middle of modal class ≈ mode
B) median: the class that includes the median is called the median class. Middle of median class ≈ median
C) mean: sum of midpoints of each class multiplied by its frequency divided by the number of data values.
D) range is a measure of dispersion In condensed data: Highest value – lowest value In grouped data: upper bound of highest group
or class – lower bound of smallest group or data.
RELATIVE FREQUENCY
Relative frequency
Relative frequency is a percentage of a group within the total (how many red pens in a box full of colored pens)
RATE OF CHANGE OR SLOPE
X AND Y Independent = x values Dependent = y values
______y______ depends on ____x________.
Before starting a slope type word problem, figure out which variable is x and which is y.
DETERMINE THE RULE FROM 2 ORDERED PAIRS (TABLE OF VALUES OR GRAPH) 1. locate two ordered pairs (table or
graph) 2. find the rate of change (y2-y1)/(x2-
x1) 3. using the a you just found, substitute
the variables of an ordered pair from your graph or table of values.
4. solve for b. 5. put a and b in the generic rule. 6. y=ax+b
DETERMINE THE RULE FROM 1 ORDERED PAIR AND “A” (TABLE OF VALUES OR GRAPH) 1. using the a you are given, substitute
the variables of an ordered pair from your graph, table of values or description.
2. solve for b. 3. put a and b in the generic rule.
VOLUME = AREA OF BASE X HEIGHT
CUBE ROOT
SIMILAR SOLIDS In 2 similar solids: corresponding
angles are congruent and the measures of corresponding edges (sides) are proportional.
Ratio of similarity = measure of one edge of the mirror-image solid ÷ measure of corresponding edge of the initial solid
RATIO OF SIMILARITY AREA AND VOLUME Ratio of areas = area of mirror image
solid/area of initial solid Ratio of volumes = volume of mirror
image solid/volume of initial solid
In 2 similar solids: Ratio of areas is equal to the square of the
ratio of similarity If ratio of area is 16, ratio of similarity is √16 = 4
Ratio of volumes is equal to the cube of the ratio of similarity If ratio of similarity is 4, ratio of volumes is 43 = 64
am = a x a x a x ... x a (m times)
a1 = a
a0 = 1
a-m =
a½ = √a
a1/3 = ∛a
am x an = am + n
am ÷ an = am - n
(ab)m = ambm
(am)n = amn
a m = am
b bm
NEGATIVE EXPONENTS With negative exponents we invert the
number to the denominator. If the denominator has a negative
exponent, we send it to the numerator position.
= x3
FOIL AND RAINBOW
SCIENTIFIC NOTATION
FACTORIZATION
INEQUALITIES
Inequality Sign Meaning Example
< less than x < 5
> greater than, more than 200 > 6
≤ no more than, at most less than or equal to h ≤ 1.8
≥ no less than, at least greater than or equal to n ≥ 180
EXAMPLE If dividing or multiplying both sides by a
negative number you must switch the direction of the inequality sign.
= -14a > 3 – 4 =-14a > -1 =-14a > -1 -14 -14 = a < 1/14
4 - 14a > 3
SYSTEM OF EQUATIONS By comparison (Exam type)
Both equations are equal to each otherSolve for xThen solve for y
SYSTEM OF EQUATIONS GRAPH Isolate y in equation so y = ........ Give x random values and solve for y Find two points for each equation Plot points on graph and draw straight
lines Intersection = solution
BOX AND WHISKER PLOTS
BOX AND WHISKER PLOTS Order values in increasing fashion Find the median (n+1)/2 = position of
medianQ2
Find median of left and rightQ1 and Q3
Draw number line with every number Put lines at Q1, Q2 & Q3 and draw box Whiskers go to min and max values Interquartile range = Q3 – Q1
PROBABILITIES Theoretical probability =
and = multiply probabilities
or = add probabilities.
PERMUTATION, ARRANGEMENT AND COMBINATION Permutation = all values of set used,
order important, formula = n! Arrangement = subset of values of the
set used, order important, formula is:
Combination = subset of values of the set used, order is not important, formula is:
n = total number of values in the setr = number of ways to arrange them
GEOMETRIC PROBABILITY One dimension = length Two dimensions = area Three dimensions = volume
Probability = favorable outcome/total outcomes
Example: probability that a point falls in circle is Area of circle Area of square