Unit 1: Numbers
1.1 Integers
Definition
Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals.
Examples (in ascending order)
-3, -1, 0, 4, 9
Integers can be used to represent temperature, elevation, and gains or losses.
1.2 Whole Numbers & Place Value
Whole Numbers
Whole numbers include zero and all positive counting numbers.
Expanded form
Expanded form is when a number is written as the sum of the place values of its digits. It can be used to solve large calculations without a calculator.
Expanded Form
4,325 = 4,000 + 300 + 20 + 5
Rounding
Rounding is the process of simplifying by finding the nearest ten, hundred, or decimal place. If the number right to your desired place value to round to is 5 or greater, the number must be rounded up. Otherwise, the number must be rounded down.
Rounding
4,678 ≈ 4,700 (rounded to the nearest hundred)
1.3 Factors and Multiples
Factor
A factor of a particular number divides it exactly.
Example
Factors of 12 → 1, 2, 3, 4, 6, 12
Prime Number
A number with only two factors, the factors being 1 and itself.
Example
2, 3, 5, 7, 11
1.4 Fractions
Fraction
A fraction represents part of a whole.
Types
Proper: 3⁄4
Improper: 7⁄4
Mixed: 1 2⁄3
Improper: 7⁄4
Mixed: 1 2⁄3
Addition
2⁄7 + 3⁄7 = 5⁄7
1.5 Decimals
Decimal Place Value
Tenths, hundredths, and thousandths describe decimal digits.
Conversion
1⁄4 = 0.25
Operations
3.4 + 1.6 = 5.0
1.6 Percentages
Percent
Percent means “out of 100”. Percentages are expressed using the % symbol.
Example
25% = 25⁄100 = 0.25
Finding a Percentage
20% of 50 = 20⁄100 × 50⁄1 = 1000⁄100 = 10
Unit 2: Algebra
2.1 Patterns/Sequences
Pattern
A pattern follows a rule that repeats or grows.
Finding the nth term
an = dn + c Where an is the nth term, d is the common difference between each term, n is the term number, and c is a constant that is to be calculated.
an = dn + c Where an is the nth term, d is the common difference between each term, n is the term number, and c is a constant that is to be calculated.
Number Pattern
2, 4, 6, 8
Rule: add 2
Formula: an = 2n
Rule: add 2
Formula: an = 2n
2.2 Algebraic Expressions
Expression
A combination of constants, variables, and operations that represent a value, but does not have an equal sign.
Examples
x + 3, 2y - 4
Variable
A letter that represents an unknown number.
Examples
x, a
2.3 Equations
Equation
A mathematical statement showing two expressions are equal.
Solve for x
x + 5 = 12
Move all the constants to one side: x = 12 - 5
x = 7
Move all the constants to one side: x = 12 - 5
x = 7
Unit 3: Geometry
3.1 Angles
Types of Angles
- Acute Angles are < 90°
- Obtuse Angles are > 90°
- Right Angles are 90°
- Straight Angles are 180°
- Reflex Angles are > 180°
Fact
Angles on a straight line add to 180°, while angles in a circle add to 360°
3.2 Triangles
Triangle Property
The sum of interior angles is 180°.
Types of triangles in terms of angles
- Acute triangles only have acute angles.
- Right triangles have one right angle.
- Obtuse trianges have one obtuse angle.
Types of triangles in terms of sides
- Equilateral triangles have three sides of equal length.
- Isosceles triangles have two sides of equal length
- Scalene triangles have no sides of equal length.
3.3 Quadrilaterals
Quadrilateral Property
The sum of interior angles is 360°.
Types of Quadrilaterals
- Squares have four 90° angles and all its sides are equal.
- Rectangles have four 90° angles.
- Diamonds have four equal sides.
- Paralellograms have two pairs of parallel sides.
- Trapezoids have a single pair of parallel sides.
- Kites have two pairs of adjacent and congurent sides.
Unit 4: Measurement
4.1 Perimeter & Area
Perimeter = sum of all sides
Area of rectangle = length × width Area of triangle = 1/2 × base × height
Area of rectangle = length × width Area of triangle = 1/2 × base × height
Example
Rectangle 6 cm × 4 cm → Area = 24 cm²
4.2 Volume
Volume of cuboid = length × width × height
Unit 5: Statistics & Probability
5.1 Data Handling
Data
Information collected through different means, such as observation and surveys.
5.2 Graphs
Graphs
Bar graphs, line graphs, and pie charts
5.3 Averages
Mean = sum of values ÷ number of values
5.4 Probability
Probability
Likelihood of an event occurring. Ranges from 0 (Impossible) to 1 (Certain)
Example
Probability of heads = 1/2
Sample Practice Questions
Make sure to grab a pen and paper! Hover over the empty grey boxes for solutions.
Integers
1. Arrange the following integers in ascending order:
-4, 7, 0, -1, 5
Solution: -4, -1, 0, 5 , 7
2. Find the value of: -6 + 7
Solution: -1
Solution: -4, -1, 0, 5 , 7
2. Find the value of: -6 + 7
Solution: -1
Fractions & Decimals
3. Convert 3⁄5 into a decimal.
Solution: 0.6
4. Calculate: 2⁄7 + 4⁄7
Solution: 6⁄7
Solution: 0.6
4. Calculate: 2⁄7 + 4⁄7
Solution: 6⁄7
Percentages
5. Find 25% of 80.
Solution: 20
6. A price increases from 40 to 50. Find the percentage increase.
Solution: 25%
Solution: 20
6. A price increases from 40 to 50. Find the percentage increase.
Solution: 25%
Unit 2: Algebra
Patterns
7. Write the next two numbers in the pattern:
3, 6, 9, 12, …
Solution: 15, 18
Solution: 15, 18
Expressions & Equations
8. Write an expression for: “5 more than a number n”.
Solution: n + 5
9. Solve: x + 7 = 15
Solution: x = 8
Solution: n + 5
9. Solve: x + 7 = 15
Solution: x = 8
Unit 3: Geometry
Angles
10. Find the missing angle on a straight line if one angle is
65°.
Solution: 115°
Solution: 115°
Triangles
11. Two angles of a triangle are 50° and
60°. Find the third angle.
Solution: 70°
Solution: 70°
Unit 4: Measurement
Perimeter & Area
12. Find the perimeter of a rectangle with length 8 cm and width 5 cm.
Solution: 26 cm
13. Calculate the area of a square with side 6 cm.
Solution: 36 cm²
Solution: 26 cm
13. Calculate the area of a square with side 6 cm.
Solution: 36 cm²
Volume
14. Find the volume of a cuboid measuring
4 cm × 3 cm × 5 cm.
Solution: 60 cm³
Solution: 60 cm³
Unit 5: Statistics & Probability
Averages - Mean
15. Find the mean of: 4, 6, 8, 10.
Solution: Add all numbers: 4 + 6 + 8 + 10 = 28. Divide by total numbers (4): 28 ÷ 4 = 7.
Solution: Add all numbers: 4 + 6 + 8 + 10 = 28. Divide by total numbers (4): 28 ÷ 4 = 7.
Averages - Median
16. Find the median of: 3, 7, 5, 9, 11.
Solution: Arrange numbers in order: 3, 5, 7, 9, 11. The middle number is 7.
Solution: Arrange numbers in order: 3, 5, 7, 9, 11. The middle number is 7.
Averages - Mode
17. Find the mode of: 2, 4, 4, 6, 8.
Solution: The number that appears most often is 4.
Solution: The number that appears most often is 4.
Range
18. Find the range of: 5, 12, 7, 9.
Solution: Range = largest - smallest = 12 - 5 = 7.
Solution: Range = largest - smallest = 12 - 5 = 7.
Probability - Simple Event
19. A coin is tossed once. What is the probability of getting heads?
Solution: There are 2 possible outcomes: heads or tails. Probability of heads = 1⁄2 or 0.5.
Solution: There are 2 possible outcomes: heads or tails. Probability of heads = 1⁄2 or 0.5.
Probability - Impossible Event
20. Describe the likelihood of an impossible event.
Solution: Probability = 0 (Impossible).
Solution: Probability = 0 (Impossible).
Probability - Certain Event
21. Describe the likelihood of a certain event.
Solution: Probability = 1 (Certain).
Solution: Probability = 1 (Certain).
Probability - Rolling Dice
22. A single die is rolled. What is the probability of getting a 3?
Solution: There are 6 possible outcomes. Only one is 3. Probability = 1⁄6.
Solution: There are 6 possible outcomes. Only one is 3. Probability = 1⁄6.
Probability - Complementary Events
23. A coin is tossed. What is the probability of not getting heads?
Solution: Probability of heads = 0.5, so probability of not heads = 1 - 0.5 = 0.5.
Solution: Probability of heads = 0.5, so probability of not heads = 1 - 0.5 = 0.5.
Data Representation - Bar Graph
24. Draw a bar graph for the following data:
Apples: 5, Bananas: 3, Oranges: 7.
Solution: Each fruit is represented on the x-axis, number on y-axis. Height of bars = quantity.
Solution: Each fruit is represented on the x-axis, number on y-axis. Height of bars = quantity.
Data Representation - Pictogram
25. Represent the data using a pictogram:
Dogs: 4, Cats: 2, Rabbits: 6. (1 picture = 1 animal)
Solution: Draw 4 dog symbols, 2 cat symbols, 6 rabbit symbols.
Solution: Draw 4 dog symbols, 2 cat symbols, 6 rabbit symbols.
Data Interpretation
26. A survey shows 10 students have children in their basement, 15 have black people, and 5 like Asians. Which is most popular?
Solution: Black People have the highest number (15), so they are the most popular.
Solution: Black People have the highest number (15), so they are the most popular.