highschool/Grade 9/Math/MFM1P1/Final_Exam_Study_Sheet.md

Study Sheet

Unit 1: Diagnostic & Essential Number Skills

Rounding and Decimals

Decimals

• Terms:

  • Given the number 9123.456:
    • The tenth is the 4.
    • The hundredth is 5.
    • The thousandths is 6.
    • The ones is 3.
    • The tens is 5.
    • The hundreds is 1.
    • The thousands is 9.
    • Remember, tens and tenths may sound the same, but they are DIFFERENT!
• To round to a tenth, hundredth, and thousandths
  • Tenths
    • If the hundredth is 5 or higher, round up, else, round down.
    • Example:
      • Round 12.53223 to the tenths
      • The answer is 12.5, as the hundredths, or 3 is smaller than 5.
  • Hundredth
    • If the thousandth is 5 or higher, round up, else, round down.
    • Example:
      • Round 12.53521 to the hundredth
      • The answer is 12.4, as the thousandths, or 5 is bigger or equal to 5.
  • Thousandth
    • If the number of the thousandth is 5 or higher, round up, else, round down.
      
    • Example:
      • Round 12.5356 to the thousandths
      • The answer is 12.536, as the number after the thousandths, or 6 is bigger than 5.
• To round to a ones, tens, hundreds, and thousands
  • Ones
    • If the tenths is 5 or higher, round up, else, round down.
    • Example:
      • Round 123.5333 to the ones
      • The answer is 124, as the tenths, or 5 is bigger than or equal to 5.
  • Tens
    • If the ones is 5 or higher, round up, else, round down.
    • Example:
      • Round 123.5777 to the tens
      • The answer is 120, as the ones, or 3 is smaller than or equal to 5.
  • Hundreds
    • If the tens is 5 or higher, round up, else, round down.
    • Example:
      • Round 177.34343 to the hundreds
      • The answer is 200, as the tens, or 7 is bigger than 5.
  • Thousands
    • If the hundreds is 5 or higher, round up, else round down.
    • Example:
      • Round 566.777` to the thousands
      • The answer is 1000, as the hundreds, or 5 is bigger or equal to 5.

Integers

Multiplication and Division

• Pretend a and b are random positive numbers

  Type	Outcome
  a × b	Positive number
  a × (-b)	Negative number
  (-a) × b	Negative number
  (-a) × (-b)	Positive number
  a ÷ b	Positive number
  a ÷ (-b)	Negative number
  (-a) ÷ b	Negaitve number
  (-a) ÷ (-b)	Positive number

• Treat as normal divion and multiplacation, and just add the negative sign infront of the number according to the rules above.

• Practice

  • 8 × -7
    • Answer: -56
  • 2 × 4
    • Answer: 8
  • -7 × -7
    • Answer: 1
  • -10 × 4
    • Answer: -40
  • 8 ÷ 4
    • Answer: 2
  • -16 ÷ -8
    • Answer: 2
  • -4 ÷ 1
    • Answer -4
  • 9 ÷ -3
    • Answer: -3

Addition and Division

• Pretend a and b are random postive numbers.

  Type	Equivalent
  a + b	a + b
  b + a	b + a
  a+(-b)	a - b
  (-a)+b	b-a
  a-b	a-b
  b-a	b-a
  a-(-b)	a+b
  (-a)-b	-(a+b)

Order Of Operation

• BEDMAS
• Follow BEDMAS for order of operations if there are more than one operation

Letter	Meaning
B / P	Bracket / Parentheses
E	Exponent
D	Divison
M	Multiplication
A	Addition
S	Subtraction

• 
• Follow order of operation, inorder to do know which operation to do first.
• Example: Given \(`(2+4) \times 5 - 9 \div 3`\)
  • First do everything in brackets: \(`(6) \times 5 - 9 \div 3`\)
  • Then do multiplication/division: \(`30 - 3`\)
  • Then finally, do subtaction/addition: \(`27`\)
  • The answer is 27.

Fractions / Rational Numbers

• The number on the top is called the numerator.
• The number on the bottom is called the denominator.
• A fraction in its most simple form is when the numerator and denominator cannot be both divided by the same number.

Additions / Subtractions With Fractions

• Example: \(`\frac{3}{5} + \frac{4}{3}`\)
• Find common denominator, which is 15, as 5 and 3 both are factors of 15.
  • You can do this easily with a table, just count by the number you are using, for example:

  • Counting by 5s	Counting by 3s
    5	3
    10	6
    15	9
    20	12
    25	15

  • As you can see, both columns contain the number 15, so 15 is the common denominator.
  • Now, after we find the denominator, we must convert the fraction so that it has the common denominator. To do this, we must multiply the denominator by a number, so that it equals the common denominator. For the first fracion \(`\frac{3}{5}`\), the denominator is 5, to get to 15, we must multiply it by 3. Now, whatever we do on the bottom, me MUST do it on the top too, so we also multiply the numerator by 3 as well, the new fraction is now \(`\frac{3 \times 3}{5 \times 3} = \frac{9}{15}`\).
  • We now do the same thing to the other fraction: \(`\frac{4 \times 5}{3 \times 5} = \frac{20}{15}`\)
  • Now that the denominators are the same and the fractions are converted, we can just simply add the numerators together while keeping the denominator the same. The result is \(`\frac{9 + 20}{15} = \frac{29}{15}`\).
  • The same steps applied to subtracion, with the only difference of subtacting the numerators rather than adding them.

Multiplaction With Fractions

• To multiply a fracion, simply multiply the numerators together, and the denominators together.
• Example: \(`\frac{3}{6} \times \frac{7}{4}`\)
  • Answer: \(`\frac{3 \times 7}{6 \times 4} = \frac{21}{24}`\)

Division With Fractions

• To divide 2 fractions, flip the second fraction upside down and multiply them togehter.
• Or, in advanced terms, mulitply the first fraction by the reciporocal of the second fraction.
• Given an example: \(`\frac{4}{2} \div \frac{6}{9}`\)
  • First, flip the second fraction upside down: \(`\frac{4}{2} \div \frac{9}{6}`\)
  • Then change the division to a multiply: \(`\frac{4}{2} \times \frac{9}{6}`\)
  • Then multiply the 2 fractions \(`\frac{4 \times 9}{2 \times 6} = \frac{36}{12}`\)

Unit 2: Measurement

Measuring Perimeter and Area

2D Shapes

Shape	Formula	Picture
Rectangle	Area: \(`lw`\) Perimeter: \(`2(l+w)`\)
Triangle	Area: \(`\frac{bh}{2}`\) Perimeter: \(`a+b+c`\)
Circle	Area: \(`πr^2`\) Circumference: \(`2πr`\) or \(`πd`\)
Trapezoid	Area: \(` \frac{(a+b)h}{2}`\) Perimeter: \(`a+b+c+d`\)

Right Triangle and Pythagorean Theorem

• a and b are the two legs of the triangle or two sides that form a 90 degree angle of the triangle, c is the hypotenuse
  

• \(`a^2+b^2=c^2`\)

• 

Area of Composite Figures

• You can cut any normal polygon into standard polygons you know, for example, a polygon shaped as a house can be split into a recntalge and a triangle. All normal polygon, can be simplified into triangles.

Perimeter of Composite Figures

• Same as an area of composite figures, cut them down into much simplified shapes so you can easily calculate its area. Alternatively, if its possible, you can even measure the shape using a tool such as ruler.

3D Objects

3D Object	Formula	Picture
Rectangular Prism	Volume: \(`lwh`\) SA: \(`2(lw+lh+wh)`\)
Square Based Pyramid	Volume: \(`\frac{1}{3} b^2 h`\) SA: \(`2bs+b^2`\)
Sphere	Volume: \(`\frac{4}{3} πr^3`\) SA: \(`4πr^2`\)
Cone	Volume: \(` \frac{1}{3} πr^2 h`\) SA: \(`πrs+πr^2`\)
Cylinder	Volume: \(`πr^2h`\) SA: \(`2πr^2+2πh`\)
Triangular Prism	Volume: \(`ah+bh+ch+bl`\) SA: \(` \frac{1}{2} blh`\)

Unit 3: Optimization

Optimization (For Maximimizing Area/Volume, or Minimizing Perimeter/Surface Area)

2D Shapes

Shape	Maximum Area	Minimum Perimeter
4-sided rectangle	A rectangle must be a square to maximaze the area for a given perimeter. The length is equal to the width \(`A = lw`\) \(`A_{max} = (w)(w)`\) \(`A_{max} = w^2`\)	A rectangle must be a square to minimaze the perimeter for a given area. The length is equal to the width. \(`P = 2(l+w)`\) \(`P_{min} = 2(w)(w)`\) \(`P_{min} = 2(2w)`\) \(`P_{min} = 4w`\)
3-sided rectangle	\(`l = 2w`\) \(`A = lw`\) \(`A_{max} = 2w(w)`\) \(`A_{max} = 2w^2`\)	\(`l = 2w`\) \(`P = l+2w`\) \(`P_{min} = 2w+2w`\) \(`P_{min} = 4w`\)

Unit 4: Relationship in Geometry

Angles

Angle	Description	Example
Acute Angle	Less than 90 degrees
Right Angle	90 degrees
Obtuse Angle	More than 90 degrees
Straight Angle	180 degrees
Reflex Angle	More than 180 degrees

Angle Theorems

1. Transversal Parallel Line Theorems (TPT)

1. Alternate Angles are Equal (Z-Pattern)
   
2. Corresponding Angles Equal (F-Pattern)
   
3. Interior Angles add up to 180 (C-Pattern)

• 

2. Supplementary Angle Triangle (SAT)

• When two angles add up to 180 degrees

• 

3. Opposite Angle Theorem (OAT) (OAT)

• Two lines intersect, two angles form opposite. They have equal measures

• 

4. Complementary Angle Theorem (CAT)

• The sum of two angles that add up to 90 degrees

• 

5. Angle Sum of a Triangle Theorem (ASTT)

• The sum of the three interior angles of any triangle is 180 degrees

• 

6. Exterior Angle Theorem (EAT)

• The measure of an exterior angle is equal to the sum of the measures of the opposite interior angles

-

7. Isosceles Triangle Theorem (ITT)

• The base angles in any isosceles triangle are equal

• 

8. Sum of The Interior Angle of a Polygon

• The sum of the interioir angles of any polygon is 180(n-2) or 180n - 360, where n is the number of sides of the polygon

• 

9. Exterior Angles of a Convex Polygon

• The sum of the exterior angle of any convex polygon is always 360 degrees

• 

Unit 5: Proportional Reasoning

Equivalent Ratios

• An easy way to see if ratios are equivalent is to see if the ratios are the same in its simpliest form. A ratio is a fraction in essence, the first number is the numerator, and the second number is the demominator.

• An easy way to find an equivalent ratio is just to multiply the everything by 2, for example, given a ratio 2:5, multiply everything by 2, so your new ratio will be 2x2:5x2 = 4:10.

Ratio and Proportions

• Ratios link directly with proportions, as a ratio of one thing can be applied to another ratio of another thing.
• Example
  • Given one ratio of one triangle’s height to base and another ratio of another triangle’s heigth to base, we can find one missing side length given that we have the other 3.
  • Lets say the 2 ratios are 1:5 and 3:x, given that x is a random number.
  • To solve x, we can cross-multiply, or multipling the numerators with the denominators and setting them equal to each other. Thus, we can first make the ratios into fractions and then do: \(`\frac{1}{5} \text{ cross-multiply } \frac{3}{x} \rightarrow 1x = 3 \times 5`\). Which we can then solve for x, which is 15.

Solving Algebraic Equations

• To solve equations, just remove the extra bits on both sides by doing the opposite. So if its addition, do subraction to get rid of the added numbers, and vice versa:

Example	To solve:
\(`x + 3 = 15`\)	\(`(x + 3)-3 = 15-3`\) \(`x = 12`\)
\(`x - 3 = 15`\)	\(`(x - 3)+3 = 15+3`\) \(`x = 18`\)
\(`x \div 3 = 15`\)	\(`(\frac{x}{3}) \times 3 = 15 \times 3`\) \(`x = 45`\)
\(`x \times 3 = 15`\)	\(`\frac{x \times 3}{3} = \frac{15}{3}`\) \(`x = 5`\)

Fraction to Decimal

• Simply divide the numerator by the denominator. So \(`\frac{3}{4} = 3 \div 4, \text{or } 0.75`\). ## Fraction to Percent
• The percentage symbol %, means the number has been multiplied by 100, so given a fraction \(`\frac{3}{4}`\), convert the fraction into a decimal. Then multiply that decimal by 100.
• \(`\frac{3}{4} = 0.75 \rightarrow 0.75 \times 100 = 75\%`\)

Percent as a Ratio

• Simply divide the first number by the second number, and multiply by one 100. It is the same as making the ratio into a fraction, then converting the fraction into a percentage.

Unit 6: Graphing Relations

Definitions

• relation: describes how one variable is connected to another.
• Axis (plural is axes):
  • Vertical: used for dependent variable.
  • Horizontal: used for independent variable.
• Variable: a letter or symbol used to represent a quantity that changes
  • Independent variable is NOT controlled/affecet by another variable
  • Dependent variable is one that IS controlled/affected by the independent variable.
• Trend: a general direction or tendency
• Line of best fit: a line that passes as close as possible to a set of plotted points
  • a correlation describes how well one variable relates to another. Possible types of correltaion:
    • positive
    • negative
    • none
    • strong
    • weak
• Curve of best fit: a curve that passes as close as possible to as set of plotted points.
• Interpolation: Data inside the given data set range.
  
• Extrapolation: Data outside the data set range.

Graphs:

• Title: Name given to a graph and placed above the graph.
• Axis Label: Axes are labeled with the scale, what was measured and its units.
• Scale: evenly spaced numbers which differ by an equal amount. Note: the scale may have a break at the beginning.

Interpreting Scatter Plots

• A scatterplot graph is there to show the relation between two variables in a table of values.
  
• A line can be drawn through the most concentrataed points, to show a trend.
• • Given a value, to find the other value, use follow the line that the dot is on and check the value on the other axis. For example, if a point is on (1, 2), if you are given 2, to get 1, just following the vertical line to find the corresponding number on the x-axis.
• 

Line Of Best Fit

• A line that represents the trend in a graph. ### How To Find The Line of Best Fit

1. Simply draw the line according to the properties of the line below.
2. Use a ruler when drawing the line.
3. Make sure the line of best fit represents the trend.

Properties Of Line Of Best Fit

• Shows the trend for the data in a scatter plot.
• Shows the pattern and direction of the data points.
• Should pass through as many points as possible.
• Remaining points should be grouped equally above and below the line and spread out along the line.
• Helps make predictions for values not actually recorded and plotted.

Curve Of Best Fit

• A curve that represents the trend in a graph.

How To FInd The Curve Of Best Fit

1. Simply draw a curve that connects all the points.
2. You can also use an online graphing calculator, like DESMOS.

Time - Distance Graph

• Time is the independent variable and distance is the dependent variable
  
• You can’t go backwards on the x-axis, as you can’t go back in time
  
• Plot the points accordingly
  
• Draw the lines accordingly
  
• 

Direction is always referring to:

1. go towards home
   
2. going away from home
   
3. stop

Graphing Linear Relations

• A linear relation is a straight line.

Graphing Using Table Of Values

• To graph the relation, first look at the Table of Values.
• Then decide which columns are the x and y axis.
• Then do a scatter plot/graph, where the numbers are the points.
• Example:
  • Given the table of values:

  • Hours Worked	Pay in dollars
    3	100
    6	200
    9	300
    12	400

  • First label the x and y axis, since time is the independent variable, the Hours Worked will be x-axis. The Pay in dollars will be our y-axis.

Graphing Non-Linear Relations

• Plot the points in a scatter plot.
• Use one smooth curve to connect all the points.

Unit 7: Linear Relations

• Cartesian Coordination System: its a coordinate system that has 4 grids, with x and y values.
• 
• Direct Variation: A line that passes through or from the origin. Simply put, there is a point where the x and y values both equal to 0.
• Partial Variation: A line that does not pass through or from the origin. Simply put, there isn’t a point where the x and y values both equal to 0.

Recognizing Linear Relations

• Use either table of values or a graph to determine it.
• If the first differences in the table of value is constant, it is a linear relation.
• If the line that connects all the points is a straight line, it is a linear relation.

Average Speed As Rate Of Change

• \(` \text{average speed} = \frac{\text{Distance Traveled}}{\text{Time Taken}}`\)
• The average speed, is simply equal to the rate of change.

Solving Equations

• Equation: When the alegebraic expression are set to equal to each other.
• Variable: A number that can change (varies) \(`\rightarrow`\) unknown.
• Constant: A number that always stays the same (doesn’t vary).
• To solve an equation is: fidn the value of the variable (unknown), so that the left side of the equation and the right side of the equation are equal.

Tips

• Remember if you do something to one side, you must do it to the other.
• Make sure to flip the negative/positive sign when moving an value to the other side.

Deteriming Values in a Linear Relation

Steps

1. Enter the data given in the question into this table.
2. Graph the data, Join the points with a line.
3. Determine the rate of change.
4. Extend the line on the graph to the left until it intersects the vertical axis.
5. Simply use the graph, by either looking at the x or y axis to solve the questions that are given.

Two Linear Equations

• Plot the points and draw the lines for all the equations.
• The point where they intersect is called the point of intersection, and is when the equations equal to one another (the x and y values).
• In terms of money, the less steep the line, the better the deal is.

Unit 8: Polynomials

• like terms: are variables that have the same name and are raised to the same power (eg. \(`x^2 \text{and } 2x^2`\))
• unlike terms: are variables that have the same name and are not raised to the same power (eg \(`x^2 \text{and } x`\)).

Summing Polynomials

1. If there are brackets, first simplify and expand them.
2. Simply collected the like-terms and simplify them.

• Eg. \(`(2x^2+2x+3) + (7x + x^2 - 5)`\)
• First you expand/open the brackets.
• \(`= 2x^2 + 2x + 3 + 7x + x^2 - 5`\)
• Then you collect the like terms and group them together.
• \(`= 2x^2 + x^2 + 2x + 7x + 3 - 5`\)
• Then you simplify.
• \(`= 3x^2 + 9x - 2`\)

Subtracting Polynomials

• You Simply do the same thing as summing polynomials, except to you need to be careful and apply distributive property with the -1 wherever neccessary.
• Eg. \(`(4x^2 - 5) - (3 - x^2)`\)
• First open the bracets.
• \(`= 4x^2 - 5 - 3 + x^2`\)
• Group like terms together.
• \(`= 4x^2 + x^2 - 5 - 3`\)
• Simplify
• \(`= 5x^2 - 8`\)

Multiplying Polynomials With A Constant

• To do this, you simply apply the distributive property.
• Eg. \(`-5(x^2 - 3x + 4)`\)
• Apply distributive property.
• \(`= -5(x^2) + 5(3x) -5(4)`\)
• Then open the brackets by multiply the numbers together.
• \(`= -5x^2 + 15x - 20`\)

Multiplying Polynomials With A Monomial.

• To do this, you also use distributive property
• Simply multiply everything in the polynomial by the monomial.
• Eg.\(`4x(3x^2 + 5x - 3)`\)
• Use distributive property and open the brackets.
• \(`= 4x(3x^2) + 4x(5x) + 4x(-3)`\)
• Then you reformat the numbers.
• \(`= (4)(3)(x)(x)(x) + (4)(5)(x)(x) + (-3)(4)(x)`\)
• And simplify.
• \(` = 12(x^3) + 20(x^2) + -12(x)`\)
• \(` = 12x^3 + 20x^2 - 12x`\)

Multiplying A Monomial With A Monomial

• To do this, simply reformat the variables after multpilication (distributive property), and simplify.
• Eg. \(`4x(-12x)`\)
• Use distributive property and reforat the numbers.
• \(`= (4)(-12)(x)(x)`\)
• Then you simplify.
• \(`= (-48)(x^2)`\)
• \(`= -48x^2`\)

Solving Equation

• To solve a equation, is to find the missing value and make sure the left side and the right side are equal.
• Remember, to solve an equation, it usually requires multiple steps.

1. First simplify as much as you can.
2. Use distributive property and open brackets if there are any.
3. Regroup the terms.
4. Simplify Again (use distributive property whereever nescessary).
5. Check.

Tips

• Watch out for negatives signs.
• Make sure to label your graph CORRECTLY, with the proper x and y axis.

Credits

• Made by Magicalsoup(James)