Mathematics for the International Student - MYP 1 Comprehensive Study Notes
CHAPTER 1: NUMBER SYSTEMS
History of Counting and Tallying
Archaeologists and anthropologists suggest early counting used tallies (scratches on cave walls, knots on ropes, pebbles, or branch notches).
Transition to efficiency: Replacement of every 10 strokes with a symbol. For example, 23 in ancient Egypt was written as two symbols for 10 and three single strokes.
The Egyptian System (Base Ten)
Evidence dating back to 3600 BC.
Hieroglyphics: Sacred picture writings.
Symbols:
Staff: $1$
Hock: $10$
Scroll: $100$
Lotus flower: $1,000$
Bent stick: $10,000$
Burbay fish: $100,000$
Astonished man: $1,000,000$
Religious symbol: $10,000,000$
Additive system with no place value; order of symbols does not affect value.
Ancient Greek (Attic) System
Included symbols for 5 ($\Gamma$), 10 ($\Delta$), 100 ($H$), 1000 ($X$).
Combinations used for 50, 500, 5000 (e.g., combining 5 and 10 to make 50).
Roman Numerals
Symbols: $I=1$, $V=5$, $X=10$, $L=50$, $C=100$, $D=500$, $M=1,000$.
Rules:
Subtractive principle: $IV$ is 4 ($1$ before $5$), $VI$ is 6 ($1$ after $5$).
Restrictions: $I$ only before $V$ or $X$; $X$ only before $L$ or $C$; $C$ only before $D$ or $M$.
Large numbers: A stroke above a symbol makes it $1,000$ times larger (e.g., $\overline{V} = 5,000$).
The Mayan System
Used Base 20.
Vertically arranged symbols; place value above another.
Recognized a symbol for zero (looks like a mussel shell).
Symbols based on dots ($1$) and strokes ($5$).
Hindu-Arabic System
Developed in India 2000 years ago; introduced to Europe by Arab traders 1000 years ago.
Digits: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$.
Natural Numbers: Counting numbers (infinite set).
Whole Numbers: Natural numbers including zero.
Place Value: Units, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, etc.
Commas/Spaces: Used to separate groups of three digits (thousands, millions).
Large Definitions:
Million: $1,000,000$ (a $10 \text{ cm} \times 10 \text{ cm} \times 10 \text{ cm}$ cube contains $1,000,000 \text{ mm}^3$).
Billion: $1,000$ million ($1,000,000,000$).
Trillion: $1,000$ billion ($1,000,000,000,000$).
CHAPTER 2: OPERATIONS WITH WHOLE NUMBERS
Addition and Subtraction
Column method: Line up place values vertically.
Word Problems: Translate text into a mathematical sentence.
Multiplication and Division by Powers of 10
Powers of 10: $10, 100, 1000, \dots$
Multiplying: Add $n$ zeros or move point $n$ places right.
Dividing: Remove $n$ zeros or move point $n$ places left.
Multi-Step Problem Solving
Identify intermediate steps (e.g., find total cost before calculating change).
Number Lines
Represent order and relative positions. Arrowheads show infinite continuation.
Operations: Jump right for addition, left for subtraction.
Rounding and Estimation
Rounding Rule: If the digit after the rounding place is $\geq 5$, round up; otherwise, round down.
Significant Figures: The number of digits from the left that are important ($37,621 \approx 38,000$ to 2 sig figs).
One Figure Approximation: Round numbers to one significant figure to estimate products (e.g., $3,789 \times 6 \approx 4,000 \times 6 = 24,000$).
Estimation of Objects
Method: Divide area into equal parts, count one part, and multiply by total parts.
CHAPTER 3: POINTS, LINES AND ANGLES
Geometry Basics
Point: Marks a position; no size. Named with a capital letter.
Vertex: Corner point of a figure (plural: vertices).
Line: Continuous infinite collection of points $(AB)$.
Line Segment: Part of a line with two endpoints $[AB]$.
Ray: Starts at a point and continues infinitely $[AB)$.
Plane: A flat surface.
Parallel Lines ($k$): Fixed distance apart, never meet.
Intersecting Lines: Meet at a point of intersection.
Angles
Formed by two arms meeting at a vertex.
Units: Degrees ($^{\circ}$). Full turn = $360^{\circ}$.
Classification:
Acute: $0^{\circ} < x < 90^{\circ}$
Right: $90^{\circ}$
Obtuse: $90^{\circ} < x < 180^{\circ}$
Straight: $180^{\circ}$
Reflex: $180^{\circ} < x < 360^{\circ}$
Revolution: $360^{\circ}$
Angle Properties
Angles at a point: Sum to $360^{\circ}$.
Angles on a line: Sum to $180^{\circ}$.
Angles of a triangle: Sum to $180^{\circ}$.
Angles of a quadrilateral: Sum to $360^{\circ}$.
Bisecting Angles
Dividing an angle into two equal parts using a compass and ruler (construction).
CHAPTER 4: LOCATION
Map and Grid References
Used in directories/atlases to locate regions (e.g., D4, E5).
The Cartesian Plane
Origin: Intersection of axes at $(0, 0)$.
x-axis: Horizontal.
y-axis: Vertical.
Ordered Pairs: Coordinates $(x, y)$. x-coordinate first, then y-coordinate.
Interpreting Points
Axes are used for variables (e.g., age vs. height).
Variable: A quantity whose value can change.
Compass and Bearings
Cardinal Directions: North ($N$), South ($S$), East ($E$), West ($W$).
Ordinal Directions: $NE, SE, SW, NW$.
Compass Bearings: Starts with $N$ or $S$ and an acute angle (e.g., $N35^{\circ}E$).
True Bearings: Degrees clockwise from True North, always 3 digits (e.g., $072^{\circ}$, $320^{\circ}$).
CHAPTER 5: NUMBER PROPERTIES
Operations
Sum: Result of addition.
Difference: Result of subtraction.
Product: Result of multiplication.
Quotient: Result of division (Dividend $\div$ Divisor).
Zero and One
$n + 0 = n$; $n \times 0 = 0$; $n \div 0$ is undefined.
$n \times 1 = n$; $n \div 1 = n$.
Index Notation (Exponents)
$2^5 = 2 \times 2 \times 2 \times 2 \times 2$. Base ($2$), Index/Exponent ($5$).
Order of Operations (BEDMAS)
Brackets, Exponents, Division/Multiplication (left to right), Addition/Subtraction (left to right).
Square and Cube Numbers
Square: $n^2$ (e.g., $4^2 = 16$). Square root: $\sqrt{16} = 4$.
Cube: $n^3$ (e.g., $2^3 = 8$). Cube root: $\sqrt[3]{8} = 2$.
Factors and Multiples
Factors: Divide exactly into a number.
Prime: Exactly two factors ($1$ and itself).
Composite: More than two factors.
HCF: Highest Common Factor.
LCM: Lowest Common Multiple.
Divisibility Tests:
By 3: Sum of digits is divisible by 3.
By 4: Last two digits divisible by 4.
By 6: Divisible by both 2 and 3.
CHAPTER 6: FRACTIONS
Representation
Fraction $\frac{a}{b}$: $a$ is numerator, $b$ is denominator.
Equivalent (Equal) Fractions
Found by multiplying/dividing numerator and denominator by the same non-zero number.
Lowest Terms: Simplify by dividing by HCF.
Types of Fractions
Proper: Numerator $<$ Denominator.
Improper: Numerator $\geq$ Denominator (e.g., $\frac{5}{4}$).
Mixed Number: Whole number and fraction (e.g., $2\frac{2}{3}$). To convert: $2\frac{4}{5} = \frac{2 \times 5 + 4}{5} = \frac{14}{5}$.
Fractions of Quantities
"of" means multiply (e.g., $\frac{2}{3}$ of 24 = $\frac{2}{3} \times 24 = 16$).
Comparing Sizes
Convert to Lowest Common Denominator (LCD) to compare.
CHAPTER 7: POLYGONS
Classification
Closed straight-sided plane figures.
$n$-gon: Polygon with $n$ sides (e.g., Decagon = 10 sides).
Regular Polygon: All sides and angles are equal.
Triangles
Scalene: No equal sides.
Isosceles: Two equal sides; base angles opposite equal sides are equal.
Equilateral: Three equal sides; all angles are $60^{\circ}$.
Quadrilaterals
Parallelogram: Opposite sides parallel and equal.
Rectangle: Parallelogram with $90^{\circ}$ angles.
Rhombus: Quadrilateral with all sides equal.
Square: Rhombus with $90^{\circ}$ angles.
Trapezium: One pair of opposite sides parallel.
Kite: Two pairs of adjacent sides equal.
Euler’s Rule for Plane Figures
Connection between Vertices ($V$), Regions ($R$), and Edges ($E$):
E = V + R - 2
CHAPTER 8: FRACTION OPERATIONS
Addition and Subtraction
Change to common denominator, then add/subtract numerators.
Mixed numbers: Either convert to improper fractions first or add whole parts/fraction parts separately.
Multiplication
$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$.
Cancellation: Cancel common factors before multiplying.
Reciprocals
Two numbers whose product is 1. Reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$.
Division
Multiply by the reciprocal of the divisor: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.
CHAPTER 9: DECIMALS
Definition
Represent numbers between whole numbers. Tenths ($0.1$), hundredths ($0.01$), thousandths ($0.001$).
Operations
Add/Subtract: Align decimal points vertically.
Multiplication: Count total decimal places in question; answer should have the same total.
Division: Align point in answer directly above point in question.
Conversions
Decimal to Fraction: Value over its place value power of 10 ($0.72 = \frac{72}{100} = \frac{18}{25}$).
Terminating Decimals: Fractions whose denominators have only 2 or 5 as prime factors.
Recurring Decimals: Decimals that repeat a sequence (e.g., $\frac{1}{3} = 0.\overline{3}$).
Rounding Decimals: Similar to whole numbers, look at the next digit to decide whether to round up.
CHAPTER 10: PROBLEM SOLVING TECHNIQUES
Trial and Error: Guessing a solution and checking it, then adjusting.
Making a List: Systematically recording all possibilities.
Modelling/Drawing a Picture: Visualizing the physical scenario.
Making a Table/Looking for Pattern: Solving simpler versions and extending the rule.
Working Backwards: Using the final result and inverse operations to find the starting point.
CHAPTER 11 & 12: MEASUREMENT
Metric Length Units: $mm, cm, m, km$.
$1 \text{ cm} = 10 \text{ mm}$
$1 \text{ m} = 100 \text{ cm}$
$1 \text{ km} = 1000 \text{ m}$
Perimeter: Distance around a boundary.
Rectangle: $P = 2(length + width)$.
Scale Diagrams: Proportional drawings.
Real Length = Scale Length $\times$ Scale Factor.
Mass: Amount of matter (constant).
$1 \text{ g} = 1000 \text{ mg}$
$1 \text{ kg} = 1000 \text{ g}$
$1 \text{ t} = 1000 \text{ kg}$
Area: Amount of surface inside a boundary.
Rectangle: $A = l \times w$.
Triangle: $A = \frac{1}{2} \times base \times height$.
Units: $mm^2, cm^2, m^2, ha, km^2$. ($1 \text{ ha} = 10,000 \text{ m}^2$).
Volume: Amount of space occupied.
Prism: $Area of Base \times height$.
Units: $mm^3, cm^3, m^3$.
Capacity: Space inside a container (fluids).
$1 \text{ L} = 1000 \text{ mL} \approx 1000 \text{ cm}^3$.
$1 \text{ mL} \approx 1 \text{ cm}^3$.
CHAPTER 13: DIRECTED NUMBERS
Positives and Negatives
Used for opposites (Profit/Loss, Above/Below sea level).
Zero is neither positive nor negative.
Operations with Directed Numbers
Addition: Move right on number line.
Subtraction: Move left on number line.
Rules for Multiplication and Division:
$(+) \times (+) = (+)$
$(-) \times (-) = (+)$
$(+) \times (-) = (-)$
Same signs = Positive, Different signs = Negative.
Average: Sum of values $\div$ number of values (can be negative).
CHAPTER 14: PERCENTAGES
Definition: "Out of every hundred." $100% = 1$.
Conversions
To Percentage: Multiply by $100%$.
To Fraction/Decimal: Divide by $100%$.
Applications
Profit/Loss as %: $\frac{Profit/Loss}{Cost Price} \times 100%$.
Discount: Reduction from marked price.
Simple Interest: $Interest = \text{Percentage p.a.} \times \text{Principal} \times \text{Time}$.
CHAPTER 15: TIME AND TEMPERATURE
Conversions
$1 \text{ year} = 52 \text{ weeks} \approx 365 \text{ days}$.
$1 \text{ millennium} = 1000 \text{ years}$.
Time Formats
12-hour: AM/PM.
24-hour: Four digits ($2015$ hours = $8:15 \text{ pm}$).
Average Speed: $Average Speed = \frac{Distance}{Time}$.
Temperature:
Celsius ($^{\circ}C$): Water freezes at $0$, boils at $100$.
Fahrenheit ($^{\circ}F$): Water freezes at $32$, boils at $212$.
Formula: $F = 1.8 C + 32$.
CHAPTER 16-24: ADVANCED TOPICS
Statistics (Ch 17)
Population vs Sample.
Mode: Most frequent score.
Mean (Average): Total sum $\div$ count.
Graphs: Dot plots, Column graphs, Pie charts, Stem-and-leaf plots.
Algebra and Patterns (Ch 18 & 20)
Variables: Symbols representing numbers (e.g., $x$).
Equation: Two expressions equal to each other.
Inverse Operations: Used to solve equations (add/sub, mult/div).
Coordinates and Lines (Ch 21)
Cartesian plane with four quadrants.
Vertical Lines: $x = k$; Horizontal Lines: $y = k$.
Transformations (Ch 22)
Translation: Slide.
Reflection: Mirror image.
Rotation: Spin around a centre point.
Enlargement/Reduction: Change size by a scale factor.
Tessellation: Tiling with no gaps.
Sets (Ch 23)
Element ($\in$): Member of a set.
Intersection ($\cap$): Elements in both A and B.
Union ($\cup$): Elements in A or B.
Venn Diagram: Visual display of sets.
Solids (Ch 24)
Polyhedron: Flat-faced solids (Prisms, Pyramids).
Curved Solids: Sphere, Cylinder, Cone.
Net: 2D shape foldable into a 3D solid.