grade 9 math exam

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