MPM1D – Grade 9 Academic Math Exam Notes

Math

 

Definitions:

BEDMAS: brackets, exponents, division and multiplication, addition and subtraction

circle graphs: find % of total, convert to decimal and multiply by 360

degree: sum of exponent of the variables in a polynomial

extrapolation: outside of range

independent variable: x, where y depends on x

interpolation: within given range on graph

origin: (0,0) on coordinate grid

qualitative variables: by catergory, cannot be measured—mainly on bar graphs

quantitative variables: discrete is whole numbers, continuous can be decimal value

 

Integers:
– adding:

+ and + = +

– and – = +

bigger + and – = +

bigger – and + = –

– subtraction:
subtraction sign becomes addition, and change sign of second number—follow addition rules

– multiplication/division:

– and – = +

+ and + = +

– and + = –

 

Fractions:
– adding/subtracting: find common denominator, what you do to top, do to bottom

– multiplying: multiply top and bottom

– dividing: flip second fraction and multiply

 

Ratio/Rate:

– ratio is comparison—5:6

– rate is a ratio per unit of something—5km/hour

 

Exponents:
(3)(3)(3) = 33 = 27

– expanded = exponent = standard
(2x)3 = (2x)(2x)(2x) = 8x3

 

Percents:
– 3 types of problems:
– find 20% of 65, so 20/100 X 65 = 13

– what % in 38 of 95, so 38/95 X 100 = 40%

– 60% of a number in 15, so number is (0.6)(x) = 15, x = 25

 

Exponent Laws:

– law #1: xn X xm = xn+m

– law #2: (xm)n = xm X n

– law #3: x/ym = xm/ym

– law #5: x-m = 1/xm

– law #6: x0 = 1 (where x cannot equal 0)

 

Scientific Notation:

– adding and subtracting:

2.0 X 102 + 3.0 X 103 – exponents must be equal

move decimal to left to increase exponent, or right to decrease exponent

0.0 X 103 + 3.0 X 103 = 3.2 X 103

– multiplying:

(4.0 X 104)(3.0 X 10-1) – collect like terms, then follow exponent laws

= (4.0)(3.0)(104)(10-1) – (104-1)

= 12.0 X 103

– dividing:

6.0 X 108

3.0 X 105 – subtract exponents, divide first terms

= 2.0 X 103

 

Algebra:

2x = a term, where 2 is coefficient and x is a variable—since it’s one term it’s a monomial

– degree of polynomials:

> add up exponents, only of variables and if there are two expressions take the degree that is bigger

 

Distributive Law:

2(3 + 4) = 2(7) = 14

expanded – (2)(3) + (2)(4) = 6 + 8 = 14

 

Common Factoring:

– take GCF (greatest common factor) and pull it out of terms

4x3 + 16x

4x(x2 + 4)

 

Geometry:

– complementary angles add up to 90

– supplementary add up to 180

– vertically opposite in an X formation – opposite sides are equal

– right angle is 90°, acute is less than 90°, obtuse in more than 90°, straight is exactly 180°

– isosceles, bottom angles equal

– equilateral all angles equal 60°

– scalene, all are different

– sum of interior angles on any triangle in 180°

> equation is (n – 2)(180) where n is the number of sides

– sum of exterior of any shape is 360°

 

Linear Correlations:

– can be positive or negative: perfect, strong, moderate, weak

– can also be nonlinear or no relationship

– mean fit line: find averages of x and y coordinates

 

Slopes:

– horizontal line: m = 0, vertical, m = undefined

– speed = distance/time

– slope = m = rise/run = rate of change

m = y2 – y1

x2 – x1

– standard form: y = mx + b

> m = slope, b = y-intercept

– parallel lines: slopes are the same

– perpendicular lines: slopes are negative reciprocals

> m = 1 is perpendicular to m = -1/1

 

Methods of Graphing:

– tables of values method:

> sub in values into equation to find coordinated to graph

– x and y intercept method:

> x-intercept when y = 0, y-intercept when x = 0

– slope y intercept method:

> plot the y-intercept, then use the slope to find other points—m=1, would mean one up, one across

 

First Difference:

– differences in the y values—subtract first y value from the second and so on

– if all are equal, then the relationship in linear, if the differences of the first differences (second difference) are the same, then the relationship is quadratic

– conditions: x values must be increasing/decreasing equally

 

Point of Intersection:

– graph and find an approximate coordinate

– algebraic method:

> put both equations in to y = mx + b form, then write both and equal to each other
example.

1/2x + 1 = -x + 4

> solve for x by collecting like terms

> sub back into either equation to find y coordinate

 

Packaging:

– minimum surface area is when height equals diameter

– maximum volume is when length = width = height