Math 10 ProvincialTheory:

Cross Multiplication:

Process used to solve equations with one fraction on each side:

ab = cd ad = bc

Direct Variation: Happens when the graph of a relation between x and y is a straight line passing through

the origin. Represented by the equation y = mx The variable y is said to vary directly as x, when the equation is in the form y = mx The variable m is called the constant of probability

Equation of a Line: y – intercept form:

o The equation y = mx + b is called the y-intercept form of the equation of a lineo The graph of the equation y = mx + b has:

Slope m y-intercept b.

Standard form:o The equation Ax + By + C = 0 is called the standard form of the equation of a lineo The graph of the equation Ax + By + C = 0 has:

Slope −AB

x-intercept CA

y-intercept CB

Finding the interceptso To find the x-intercept of a graph, take the equation, put the y value as 0, and

solve for x

o To find the y=intercept of a graph, take the equation, put the x value as 0, and solve for y

When given the graph:o The equation of a line can be found by taking any two points, calculating the

slope between them, and finding the y-intercept of the line.

When given two point on the line1. Calculate the slope of the line using the two points2. Let (x,y) be another point on the line3. Calculate the slope using any one given point and(x,y)4. Create an equation with the given slope on one side, and the calculated slope

from step 3 on the other side5. Simplify, combine like terms, and put into the slope-intercept form or standard

form

When given the slope and a point (a,b) on the line:1. Let (x,y) be another point on the line2. Calculate the slope using (a,b) and(x,y)3. Create an equation with the given slope on one side, and the calculated slope

from step 2 on the other side4. Simplify, combine like terms, and put into the slope-intercept form or standard

form

Factoring: Difference of Squares:

In a binomial with the form x2 – y2

The factored equation reads (x+y)(x-y)

Trinomials:

In a trinomial with the form: x2 + bx + c

1. Find two integers with a sum of b, and a product of c2. Let the two integers found be m and n3. The factored equation reads (x+m)(x+n)

Graphs (Statistics): Smoothing Graphs

1. Take the table of values for a graph2. The first and last values of the graph stay the same3. For each value (other than the first and last), take the median of the value to the

right, left, and itself.4. Take the smoothed values and graph them

Types of Graphs: Bar Graph

o Bars represent certain quantities of data (not ranges) Histogram

o Bar graph that represents a range of data (not quantities, but a range of data) Line Graph

o Broken-line graph (aka frequency polygon) Made of data plotted on a grid, and then the dots joined with line Only end points of lines represent data

o Continuous-line graph All points on the graph represent actual values of variables (data)

Circle Graph / Rectangle Grapho Used to show a whole quantity divided into partso Drawing a Circle Graph

1. Divide each value of data, by the total amount of data to express it a decimal

2. Multiply each decimal in Step 1 by 360o to determine the degrees of the region in the circle. Round to nearest degree

3. Draw the circle graph with the corresponding degrees Scatterplot

o See Glossary

Graphing: Linear Relations:

o With an equation Make a table of values using the equation, plot any two points (do more

to be safe) and join the dots with a line

o With a table of values Simply plot any two or three points and join them with a line

Linear Inequalitieso Begin by treating the inequality as an equation, and graphing that equationo Notice the graphed line divides the grid into two parts. o Take the point (0,0) and substitute the x and y value (0) of that point into the

inequalityo If the resulting simplified inequality is true, shade in the part of the grid (divided

by the line) with (0,0) in it.o If the resulting simplified inequality is false, shade in the part of the grid (divided

by the line) without (0,0) in it.o This will work with any point on the grid, as long as the substitutions were done

correctly, so do not always use (0,0)

Non-Linear Equations:o Plot all points in the table of values, and draw a smooth curve connecting them

Partial Variation: Happens when the graph of a relation between x and y is a straight line and does not

pass through the origin. Represented by the equation y = mx + b, b ≠ 0 The variable y is said to vary partially as x, when the equation is in the form y = mx + b,

b ≠ 0 The variables m and b are constants

Prime Numbers: To find out if a number is prime, divide that number by all the prime numbers below its square root. If that number divides any one of the primes evenly, that number is not prime. Otherwise, that number is prime.

Randomizing: Ways to Take Random Samples

o Random Draw Write data on slips of paper, and draw from a box or hat

o Pair of Dice Arrange data into a 6 by 6 array (or closest to a 6x6), and number the

sides. Roll dice and record the corresponding itemo Random Numbers

Number the values in the data, and use a random number chart or phone book to find random numbers

Rational Expressions:

The rules that govern the addition, subtraction, multiplication, division, of rational numbers also apply to rational expressions

Relations:

Can be represented in three ways A table of values A graph An equation

Repeating Decimal into Fraction:

1. Isolate the repeating digits2. Multiply by 10x where x is the number of repeating digits3. Subtract equation 1 from equation 24. Convert into a fraction

Change 0.444… into a fractionLet x =0.444….

10x = 4.444… x = 0.444… 9x = 4.000… x = 4/9

0.444… = 4/9

Situations:

Tournament, each team plays all the other teams once, or every person in a room handshakes with another person:

o Number of games or handshakes = n(n−1)2

Solving Equations:

With one variable:o Simply the equation, and break all brackets and parentheseso Combine like termso Isolate the variable on one side of the equation, performing operations on both

sides of the equation

Solving Inequalities:

The rules that govern the solving of regular equations also apply to the solving of inequalities, except that:

If an operation is done that causes both sides of the equation to be multiplied or divided by a negative number, the inequality sign must be reversed

Solving Linear Systems: By graphing:

o Graph both lines, and the solution is the coordinate at which the lines intersect If the lines are on top of each other, there are infinite solutions If the lines are parallel, and never intersect, there are no solutions

By addition or subtraction:1. Choose whether to eliminate the x value or the y value of the system2. Determine the LCM of the coefficients of the values in both equations3. Multiply both equations by different constants to create two identical

equations, but with the coefficients of the chosen value (x or y) being the same

4. Add or subtract the equation to eliminate the value5. Solve for the other value6. Substitute the other value in the equation to solve for the eliminated variable

By substitution:1. Choose one equation in the system and express one variable in the terms of

the other. In other words, isolate one variable.2. Substitute the variable into the other equation3. Solve for the other variable in the equation4. Substitute the solved value into either equation and solve for the final

variable

Solving Quadratic Equations:

The rules that govern the solving of regular equations also apply to the solving of quadratic equations

After isolating the variable, taking the square root of both sides will produce two value for the variable

Remember the square root of a number can be positive or negative. This produces the two solutions mentioned above

Standard Form to Scientific Notation:

1. Change the number into a decimal, with the point after the first non-zero digit.a. If the decimal place was moved to the right, multiply the decimal by 10n

where n is the number of places moved to the rightb. If the decimal place was moved to the left, multiply the decimal by 10-n

where n is the number of places moved to the left

Change 8 200 000 000 000 into scientific notation

8 200 000 000 000 = 8.2 x 1012

Change 0.000 000 000 000 000 3 into scientific notation0.000 000 000 000 000 3 = 3.0 x 10-16

Transformations: Language of Transformations:

o The transformation: (x,y) (x + 5, y – 2) means that for each coordinate point of the shape being transformed, its x value is increased by 5, and y value decreased by 2.

Translations:o Referred to as “sliding” or “moving”o Represented by the following transformation: (x , y) (x + h, y + k), for all real

values of h and k.o Properties of translations:

The arrows from all points to their corresponding image points (the arrow refers to the imaginary line between each point of the original image and the translated image) have the same length and same direction

Preserves length and slope Rotations:

o About the origin 90o rotation is represented by (x , y) (-y , x) 180o rotation is represented by (x , y) (-x , -y) 270o rotation is represented by (x , y) (y , -x)

o Rules depend on position of the point of rotation, or rotation center, and the angle of rotation

o All rotations are considered to be counter-clockwiseo Properties of rotations:

Preserves length Preserves location of rotation center Does not always preserve slope

Reflections:o Mapping Rules:

About the x-axis: (x , y) (x , -y) About the y-axis: (x , y) (-x , y) About the line y = x: (x , y) (y , x) About the line y = -x: (x , y) (-y , -x)

o Properties Preserves length Preserves location of points on the reflection line Does not preserve slope or orientation

Dilatations:o Mapping Rules:

A dilatation with scale factor k and dilatation center (0 , 0), assuming k > 0 can be represented by: (x , y) (kx , ky)

If 0 < k < 1, the dilatation is a deduction If k > 1, the dilatation is an enlargement

o Properties: Preserves slope Preserves the location of the dilatation center Does not preserve length or area (unless |k| = 1)

Images of Lines:

o Lines can be translated to form image lineso Finding the equation of an image line

1. Find the coordinates of any two points on the original line2. Find the coordinates of the images of these two points3. Find the equation of the line passing through these two points

Words into Symbols:

1. Assign a variable in the word equation 2. Use relations and operations to write an expression for the word equation. Write

out exactly what the word equation says

Five less than four times a number

1. Let the number be x x2. Five less….. x – 53. …Than four times a number 4x - 5

Rules:

Area: Triangle:

o Regular A = ½bh

o On a grid, with vertices (0,0), (a,b), (c,d) A = ½|ad-bc|

Congruency: Axioms for congruent triangles

o SSS: Side, Side, Side; All three sides of a triangle are equal to all three sides of another triangle

o SAS: Side, Angle, Side; Two sides, and the angle between them are equal to two sides, and the angle between them of another triangle

o ASA: Angle, Side, Side; Two angles, and the sides between them are equal to two angles, and the side between them of another triangle

Isosceles Triangle Theorem

o In any isosceles triangle, the angles opposite the equal sides are equal. In other words, with the isosceles triangle pointing up, and the two equal sides forming the diagonals, the bottom two angles are equal

Properties and Characteristics of a Parallelogram:o A parallelogram is a quadrilateral in which both pairs of opposite sides are

parallelo Properties:

Opposite sides are equal Opposite angles are equal Angles next to each other are supplementary

Exponent Law: Multiplying: am a∙ n = am + n

Dividing: am ÷ an = am - n

Exponent: (am)n = amn

Bracket: (xy)n = xnyn

Bracket Division: (x ÷y)n = xn ÷ yn, (y ≠ 0) Fractional: am/n = n√am

2D Geometry Theorems: Opposite Angle Theorem

o When two lines intersect, the opposite angles are equal Supplementary Angle Theorem

o If two angles are equal, their supplements are equal Parallel Line Axiom

o In the diagram to the right, angles 3 and 6 add up to 180o, as do angles 4 and 5

o Angles satisfying this axiom form a C pattern (follow the pattern to see the C: t, 3, 6, s)

Parallel Line Theoremo If a transversal intersects two parallel lines, then the:

Alternate angles are equal (Angles 3 and 5 in above diagram) Corresponding angles are equal (Angles 4 and 8 in above diagram)

o Alternate angles form a Z pattern with their lineso Corresponding angles form a F pattern with their lines

Angle Sum Theoremo The angles in any triangle add up to 180o

3D Geometry Formulas and Theorems: Cylinders:

o Net is formed by two circles on either sides of a rectangle. The circles should be near opposite angles of the rectangle

o Surface Area: SA = 2πrh, where r is the radius of the base circle, and h is the height of

the cylindero Volume:

V = πr2h Cones:

o Net is formed by using a pac-man like part-of-a-circle and a full circle. The two circles are different sizes

The radius of the pac-man part circle should be equal to the side slant height of the finished cone

o Calculating the slant height of a cone: Pythagorean Theorem s = √r2+h2, where r is the radius of the base circle, and h is the distance

from the tip of the cone to the center of the base circleo Surface Area:

SA = πrs + πr2, where r is the radius of the base circle, and s is the length of the side slant of the cone

o Volume:

V = 13πr2h, where r is the radius of the base circle

Spheres:o Surface Area:

SA = 4𝜋r2, where r is the radius of the sphere, also the straight line distance from any point on the surface of the sphere to its center

o Volume:

V = 43πr3, where r is the radius of the sphere, also the straight line

distance from any point on the surface of the sphere to its center

Integral Exponents:

Positive Integral Exponent: an = a a a a … (n times)∙ ∙ ∙ Negative Integral Exponent: a-n= 1 / a –n

Zero Exponent: a0 = 1, a ≠ 0

Irrational Number: Any number in the form √ x where x > 0 and x is not the square of a rational number, is irrational

Line Segments / Lines: Equations:

o The equation Ax + By = 0 represents a line through the origino The equation y = k represents a horizontal lineo The equation x = k represents a vertical line

Length:o The straight line distance between any two points on a grid P1 (x1, y1) and P2 (x2,

y2) can be determined by the formula: P1P2 = √(x¿¿2−x1)2+( y2− y1 )

2¿

Midpoint:o The midpoint (M) of any line segment on a grid with endpoints P1 (x1, y1) and P2

(x2, y2) can be determined by the formula: M ( x1+x22,y1+ y22 )

Slope:o The slope of a line segment joining two points on a grid P1 (x1, y1) and P2 (x2, y2)

can be determined by the formula: slope of P1P2 = y2− y1x2−x1

and (x2 ≠ x1)

Line segments rising to the right have a positive slope Line segments falling to the right have a negative slope The slope of any horizontal line segment is zero The slope of any vertical line segment is undefined

o If two line segments are parallel, they have equal slope. If two line segments have equal slope, they are parallel

o If two line segments are perpendicular, their slopes are negative reciprocals. If two line segments have slopes that are negative reciprocals, they are perpendicular

In other words, negative reciprocal refers to: Slope1 x Slope2 = -1

Linear Systems: Multiplying both sides of either equation of a linear system by a constant does not

change the solution

Adding or subtracting the equations of a linear system does not change the solution

Polynomials: Squaring:

o Monomials: (am)2= a2m2

o Binomials: (a+b)2 = a2 + 2ab + b2

(a-b)2 = a2 - 2ab + b2

Prime Factorization Theory: Every composite number can be expressed as a product of primes on only one way

Probability: If experiment has n equally likely outcomes of which r are favourable to event A, then

the probability of event A is: P(A) = rn

The probability of two or more events is the product of the probability of each event

Pythagorean Theorem: In a right triangle with legs a and b, and hypotenuse c, a2+ b2 = c2

Radicals: Multiplying:√a x √b = √ab, (a ≥ 0, b ≥ 0) Dividing:√a ÷ √b = √a÷b, (a ≥ 0, b > 0) Adding: a√ x + b√ x = (a + b)√ x Subtracting: a√ x - b√ x = (a - b)√ x

Relative Frequency:

If an outcome, A, occurs r times in n repetitions of an experiment, then the relative frequency

of A is rn

Similar Triangles: A pair of triangles are similar if:

o Two angles of one triangle are equal to two angles of the other triangleo Ratios of corresponding sides are equal

Transformations: See Theory section above

Trigonometry: Right Triangles

o Sine: sin θ=oppositehypotenuse

o Cosine: cosθ= adjacenthypotenuse

o Tangent: tanθ= oppositeadjacent

o SOHCAHTOA

Volume: Cone:

o V = 13 πr2h

Cylindero V = πr2h

Sphere

o V = 43 πr3

Rectangular Prismo V = lwh

Glossary:

Absolute value: Distance from zero on a number line of any number. If x is negative, |x| = -x. If x is positive, |x| = x

Axiom: Something we know is true, but cannot be explained or defined

Bar graph: Graph in which the lengths of bars represent certain quantities in a set of data

Binomial: Polynomial with two terms

Coefficient: The constant number factor in a term

Composite number: A number that is not prime, and is not one.

Constant of probability: Represented by m in the formula of direct variation: y = mx

Deductive reasoning: Reasoning that happens after making a conclusion based on statements we accept as true. This form of reasoning always produces a true conclusion

Dilatation: A transformation that enlarges or reduces all dimensions of a figure by a factor k, called a scale factor

Direct variation: An equation in the form y = mx; happens when the graph of a relation between x and y is a straight line passing through the origin

Factor: Numbers that can be evenly divided into a larger number. If a is a multiple of b, b is a factor of a

Frequency table: Table that shows how often a certain piece of data appears in the set of data

Greatest common factor (GCF): Largest factor that is shared by both numbers

Histogram: Bar graph that represents a range of values

Hypotenuse: The side of a right triangle that the right angle faces

Inductive reasoning: Reasoning that happens after observing the same result over and over again, and concluding that it will always occur. This never completely proves that the conclusion is always true

Integers ( I ) : Any whole number

Irrational number: A number that cannot be written in the form mn , where m and n are integers,

and n ≠ 0. The decimal representation of an irrational number never terminates, nor repeats

Least common multiple (LCM): Smallest whole number divisible by both numbers

Leg: The two sides in a right triangle that forms the right angle

Linear equation: An equation that can be expressed in the form Ax + By + C = 0

Linear system: Two linear equations, compared together on the same graph

Literal coefficients: Coefficients that are represented by letters, not numbers

Mean: Mathematical average; found by dividing the a sum of all the values in a set of data, by the number of values in the same set

Median: The middle value numerically of all the values in a set of data; found by arranging the values in order, and then selecting the middle value. If there are two middle numbers, take the mean of the two medians

Mode: Most frequent value in a set of data; there may be more than one mode, or no mode at all

Monomial: The product of a coefficient and one or more variables

Natural number: Non-positive integers; {1, 2, 3 …}. Disputed to include zero or not

Outcome: A desired result in experimenting

Partial variation: An equation in the form y = mx + b, b ≠ 0; happens when the graph of a relation between x and y is a straight line, but does not pass through the origin

Polynomial: Expressions formed by adding or subtracting monomials

Population: See “Sample” in the Glossary

Power: A term in the format an, where a is the base, and n is the exponent

Prime number: A natural number which only has two factors; itself, and 1. The number 1 is not prime

Probability: The chance a favourable outcome will appear in an experiment. Calculated by:

P (A )= rn , where the equation has n equally likely outcomes of which r are favourable to event

A.

Quadratic equation: An equation that can be put in the form Ax2 + Bx + C = 0

Radical: An expression in the format n√ x. If n is even, the expression represents the principal root, or the positive root

Rational expression: Any algebraic expression that can be written as the quotient of two polynomials. Rational expressions are not defined when the denominator is equal to zero

Rational number: A number that can be written in the formmn , where m and n are integers, and

n ≠ 0

Relation: A set of ordered pairs; usually used to represent data

Relative frequency: Number used to describe the number of times an outcome appears in repetitions of an experiment

Rotation center: Also known as point of rotation; point on the grid in which the image rotates about

Sample: A smaller group of items or data chosen to represent a larger group, called a population

Scale factor: Amount that an image is enlarged or reduced in a dilatation

Scatterplot: Data graphed with points or dots on a grid; may form clusters or trends

Scientific notation: A way of writing very large, or very small numbers, using powers of 10

Slope: The ratio of rise to the run between two points on a grid

Slope-intercept form: The equation y = mx + b, representing a line

Standard form (of a line): The equation Ax + By + C = 0, representing a line

Statistics: The collection, interpretation, and analysis of data for the purpose of drawing inferences and making predictions

Stem and leaf diagram: A way of representing data by showing multiples of powers of 10, and showing lesser digits to the right of them

Supplementary angles: Two angles that add up to 180o

Tree diagram: Diagram that shows the probability of an outcome for two or more events

Trinomial: Polynomial with three terms

Variable: Letter representing a number that when changed, still proves the equation true