IB MYP 2 Mathematics

1. Types of Numbers

Numbers are classified into different types based on their properties.

Natural Numbers (ℕ)

Whole Numbers

Integers (ℤ)

Rational Numbers (ℚ)

Irrational Numbers

2. Operations with Fractions (Criterion D)

Addition & Subtraction

Example:
2/3 + 1/6
LCM of 3 and 6 = 6
= 4/6 + 1/6 = 5/6

Multiplication

Example:
3/4 × 2/5 = 6/20 = 3/10

Division

Example:
4/5 ÷ 2/3 = 4/5 × 3/2 = 12/10 = 6/5
please please please fully simplify your answwer these examiners can't read too long

Patterns and Sequences

A pattern is a regular and repeated way in which numbers, shapes, or objects change. A sequence is an ordered list of numbers that follows a specific rule.

Sequences are important because they help us identify relationships, predict future values, and describe mathematical rules.

Arithmetic and Geometric Sequences

A sequence is an ordered list of numbers that follows a clear rule. Each number in the sequence is called a term. The position of a term is represented by n.

The two most important types of sequences studied in MYP are:

  • Arithmetic sequences (add or subtract)
  • Geometric sequences (multiply or divide)

1. Arithmetic Sequences

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. This constant value is called the common difference, written as d.

Identifying an Arithmetic Sequence

To check whether a sequence is arithmetic:

  1. Subtract consecutive terms
  2. If the difference is the same each time, the sequence is arithmetic

Example:

7, 12, 17, 22, 27, …

  • 12 − 7 = 5
  • 17 − 12 = 5
  • Common difference, d = 5

Arithmetic sequences can be:

  • Increasing if d > 0
  • Decreasing if d < 0

2. Finding the nth Term of an Arithmetic Sequence

The general formula for the nth term of an arithmetic sequence is:

Tₙ = a + (n − 1)d

  • a = first term
  • d = common difference
  • n = term number

Worked Example

Find the nth term rule for the sequence:

4, 9, 14, 19, …

Step 1: Identify a and d
First term, a = 4
Common difference, d = 5

Step 2: Substitute into the formula
Tₙ = 4 + (n − 1)5

Step 3: Simplify
Tₙ = 5n − 1

This rule allows us to find any term without listing the sequence.

3. Geometric Sequences

A geometric sequence is a sequence in which each term is found by multiplying or dividing the previous term by a constant value called the common ratio, written as r.

Identifying a Geometric Sequence

To check whether a sequence is geometric:

  1. Divide consecutive terms
  2. If the ratio is constant, the sequence is geometric

Example:

3, 6, 12, 24, 48, …

  • 6 ÷ 3 = 2
  • 12 ÷ 6 = 2
  • Common ratio, r = 2

4. Finding the nth Term of a Geometric Sequence

The general formula for the nth term of a geometric sequence is:

Tₙ = a × rⁿ⁻¹

  • a = first term
  • r = common ratio
  • n = term number

Worked Example

Find the nth term rule for the sequence:

5, 15, 45, 135, …

Step 1: Identify a and r
a = 5
r = 3

Step 2: Substitute into the formula
Tₙ = 5 × 3ⁿ⁻¹

5. Arithmetic vs Geometric Sequences

  • Arithmetic sequences change by adding or subtracting
  • Geometric sequences change by multiplying or dividing
  • Arithmetic growth is linear
  • Geometric growth is exponential

6. Applying Sequences in Real-Life Situations

a) Arithmetic Sequence Example (Salary Increase)

A worker earns ₹30,000 in the first year. Each year, their salary increases by ₹2,000.

This forms an arithmetic sequence:

30,000, 32,000, 34,000, 36,000, …

The nth-year salary can be found using:
Tₙ = 30000 + (n − 1)2000

b) Geometric Sequence Example (Population Growth)

A bacteria population doubles every hour. The initial population is 500.

This forms a geometric sequence:

500, 1000, 2000, 4000, …

The population after n hours is:
Tₙ = 500 × 2ⁿ⁻¹

This model is used in science, finance, and economics.

3. BODMAS (Fractions & Decimals)

BODMAS shows the correct order of operations. It tells us **which calculations to do first** to get the correct answer.
  • B – Brackets (solve inside brackets first)
  • O – Orders (powers and roots)
  • D – Division
  • M – Multiplication
  • A – Addition
  • S – Subtraction

⚠️ Division and multiplication are done from left to right. Addition and subtraction are also done from left to right.

BODMAS order of operations diagram
Worked Example:
(3/4 + 1/2) × 8

Step 1: Solve the bracket
(3/4 + 1/2) = (3/4 + 2/4) = 5/4

Step 2: Multiply
5/4 × 8 = 10

Common Mistakes

  • Ignoring brackets
  • Doing addition before multiplication
  • Forgetting to convert fractions to a common denominator
  • Not following left-to-right order for × and ÷

Practice Questions

  1. (1/2 + 3/4) × 8
  2. 6 + 3 × 1/2
  3. (5 − 2/5) ÷ 3
  4. 4 + 2² × 1/2
  5. (0.6 + 1.4) ÷ 2 × 5
Show all working steps clearly to get full marks in Criterion D.

4. LCM and HCF (Criterion D)

Highest Common Factor (HCF)

  • Largest number that divides two or more numbers
  • Used for simplifying fractions

Lowest Common Multiple (LCM)

  • Smallest number that is a multiple of two or more numbers
  • Used when adding or subtracting fractions
Example:
Find HCF and LCM of 12 and 18
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
HCF = 6
LCM = (12 × 18) ÷ 6 = 36

Tips & Tricks

  • HCF is useful to simplify fractions to lowest terms
  • LCM is important for adding/subtracting fractions with different denominators
  • Prime factorization is a reliable method for both HCF and LCM
  • Always check by multiplying HCF and LCM: HCF × LCM = product of the numbers

Practice Questions

  1. Find HCF and LCM of 15 and 20
  2. Find HCF and LCM of 8 and 12
  3. Find HCF and LCM of 9 and 18
  4. Find HCF and LCM of 14 and 35
  5. Find HCF and LCM of 24 and 36
Show your full working using lists or factor trees for more marks.

5. Terminating & Non-Terminating Decimals

Terminating Decimals

  • End after a finite number of digits
  • Can be written as fractions easily
Example:
0.25 = 25/100 = 1/4

Recurring (Non-Terminating) Decimals

  • Digits repeat forever
  • Use algebra to convert
Worked Example:
Let x = 0.333…
10x = 3.333…
10x − x = 3
9x = 3 → x = 1/3

Practice Questions

  1. Convert 0.75 to a fraction
  2. Convert 0.125 to a fraction
  3. Convert 0.666… to a fraction
  4. Convert 0.142857… to a fraction
  5. Convert 0.8 repeating (0.888…) to a fraction

6. Significant Figures

Significant figures show the accuracy of a number.
  • All non-zero digits are significant
  • Zeros between numbers are significant
  • Leading zeros are NOT significant
Example:
0.00456 → 3 significant figures
4050 → 3 significant figures

Practice Questions

  1. Find the significant figures of 0.00789
  2. Find the significant figures of 56007
  3. Find the significant figures of 0.009400
  4. Find the significant figures of 1200
  5. Find the significant figures of 0.03450

7. Squares and Square Roots

Square

  • A number multiplied by itself
  • Example: 7² = 49

Square Root

  • The number that multiplies by itself to give the original number
  • √49 = 7

Practice Questions

  1. Find the square of 12
  2. Find the square root of 144
  3. Find the square of 15
  4. Find the square root of 196
  5. Find the square of 8

8. Square Roots by Prime Factorization

Worked Example:
Find √72
72 = 2 × 2 × 2 × 3 × 3
Pair equal numbers:
√72 = 2 × 3 × √2 = 6√2

Practice Questions

  1. Find √50 using prime factorization
  2. Find √98 using prime factorization
  3. Find √180 using prime factorization
  4. Find √200 using prime factorization
  5. Find √288 using prime factorization

9. Laws of Exponents

  • am × an = am+n
  • am ÷ an = am−n
  • (am)n = amn
  • a0 = 1
  • a−n = 1 / an
Example:
2³ × 2⁴ = 2⁷ = 128

Practice Questions

  1. 3² × 3⁵
  2. 5⁷ ÷ 5³
  3. (2³)⁴
  4. 7⁰
  5. 4−2
Write all answers using the laws of exponents.

IB MYP 2 – Unit 2: Statistics and Probability

1. Terminology in Statistics

  • Data: Information collected for a purpose
  • Observation: Each individual value in a data set
  • Population: The complete group being studied
  • Sample: A part of the population
  • Discrete data: Countable data (e.g. number of students)
  • Continuous data: Measured data (e.g. height, weight)
  • Frequency: Number of times a value occurs

Practice Questions

  1. Identify the population and sample in a survey of student heights.
  2. Give an example of discrete data in your classroom.
  3. Give an example of continuous data.
  4. Define frequency for the number of pets owned by students.
  5. Classify each type of data (discrete/continuous) for number of books read per month.

2. Data Collection and Generation

Data can be collected in different ways depending on the situation.

Methods of Data Collection

  • Surveys and questionnaires
  • Interviews
  • Observation
  • Experiments

Good Surveys

  • Questions must be clear and unbiased
  • Sample size should be reasonable
  • Questions should be relevant to the aim

Practice Questions

  1. Design a small survey to find students’ favourite fruit.
  2. List two advantages of using interviews over surveys.
  3. Give one example of data you could collect by observation.
  4. Why is sample size important in a survey?
  5. Give an example of a biased question and explain why it is biased.

3. Tally Marks and Frequency Table

Tally marks are used to record data quickly.

Value Tally Frequency
Red |||| 4
Blue ||||| | 6

Practice Questions

  1. Record the following data using tally marks: {2, 3, 3, 4, 2, 3, 4, 4}
  2. Complete a frequency table for the number of pets owned by students: {1, 2, 2, 1, 3, 2, 1}
  3. Convert the tally marks ||| ||| |||| into a frequency table.
  4. Explain why tally marks are useful for quick data recording.
  5. Draw a tally chart for the colors: Red, Green, Blue with counts {3, 5, 2}

4. Graphical Representation of Data

Bar Graph

  • Used for discrete or categorical data
  • Bars have equal width and gaps

Double Bar Graph

  • Compares two related data sets
  • Bars are placed side by side

Pie Chart

  • Shows data as parts of a whole
  • Total angle = 360°
Angle = (Frequency ÷ Total) × 360°

Histogram

  • Used for continuous data
  • No gaps between bars
  • Area represents frequency

Practice Questions

  1. Draw a bar graph for the following data: {Apples: 5, Oranges: 8, Bananas: 3}
  2. Create a double bar graph comparing two classes’ test scores.
  3. Calculate the angles for a pie chart if frequencies are {10, 15, 25}.
  4. Draw a histogram for continuous data: {2–4: 3, 4–6: 5, 6–8: 2}
  5. Explain why a histogram has no gaps between bars.

5. Frequency Polygon

  • Constructed from a histogram
  • Plot midpoints of class intervals
  • Join points using straight lines

Practice Questions

  1. Draw a histogram and frequency polygon for the data: {1–2: 2, 2–3: 5, 3–4: 3}
  2. Identify the midpoints for class intervals: 5–10, 10–15, 15–20
  3. Explain why frequency polygons are useful compared to histograms
  4. Plot a frequency polygon for this histogram: {2–4: 4, 4–6: 6, 6–8: 2}
  5. Describe how to join the points for a frequency polygon correctly

6. Mean, Median, Mode, and Range

Mean

Average of the data

Mean = Sum of observations ÷ Number of observations

Median

  • Middle value when data is arranged in order
  • If even number of values → mean of two middle values

Mode

  • Most frequently occurring value

Range

Range = Highest value − Lowest value

Practice Questions

  1. Find the mean of {4, 7, 9, 2, 6}
  2. Find the median of {12, 5, 8, 3, 10}
  3. Find the mode of {3, 5, 3, 7, 8, 3}
  4. Calculate the range of {15, 22, 19, 10, 18}
  5. Explain why the median is less affected by extreme values than the mean

7. Finding Mean Using Variables

When data includes variables, use algebra.

Worked Example:
Mean of x, x+2, x+4 is 10
(x + x+2 + x+4) ÷ 3 = 10
3x + 6 = 30
x = 8

Practice Questions

  1. Find x if the mean of x, x+3, x+6 is 12
  2. Find x if the mean of 2x, x+4, 3x is 10
  3. Find the mean of 5, x, x+2, given mean = 8
  4. Find x if the mean of x, 2x, 3x, 4x = 10
  5. Check your answer: mean of x, x+1, x+2 = 7, find x

8. Stem and Leaf Diagram (Single)

example
  • Shows original data values
  • Stem = first digit(s)
  • Leaf = last digit
Example:
Data: 12, 15, 18, 21, 24

Stem | Leaf
1 | 2 5 8
2 | 1 4

Practice Questions

  1. Draw a stem-and-leaf diagram for {11, 14, 16, 19, 22, 24}
  2. Identify the stem and leaf for each data value in {13, 15, 17, 20}
  3. From the stem-and-leaf diagram, list all the original data values
  4. Explain why stem-and-leaf diagrams are useful for small datasets
  5. Create a stem-and-leaf diagram for {5, 7, 9, 12, 14, 16}

9. Sample Space

  • Set of all possible outcomes of an experiment
  • Written using brackets
Example:
Tossing a coin:
Sample Space = {H, T}

Practice Questions

  1. List the sample space for rolling a 6-sided die
  2. List the sample space for tossing two coins
  3. Write the sample space for picking a card from {A, B, C, D}
  4. List the sample space for spinning a spinner with numbers 1–4
  5. Identify the sample space for selecting a day of the week at random

10. Probability Scale

Probability values range from 0 to 1.

  • 0 → Impossible
  • 0.5 → Even chance
  • 1 → Certain
The probability of an event is always between 0 and 1.

Practice Questions

  1. State whether these events are impossible, even chance, or certain: Tossing a coin and getting heads, Rolling a die and getting 7
  2. Explain why probability cannot be less than 0 or greater than 1
  3. Classify the probability of raining tomorrow as 0, 0.5, or 1
  4. Give an example of an impossible event in your classroom
  5. Give an example of a certain event in daily life

11. Probability of Single Events

Formula:
Probability = Number of favourable outcomes ÷ Total outcomes
Example:
Rolling a die, P(rolling 3) = 1/6

Practice Questions

  1. Find the probability of rolling a 4 on a 6-sided die
  2. Find the probability of tossing a coin and getting tails
  3. Find the probability of picking A from {A, B, C, D}
  4. Find the probability of rolling an even number on a die
  5. Find the probability of tossing a coin twice and getting two heads

12. Probability of Pair of Events

Pair events are often represented using tables.

Red Blue
Circle 2 3
Square 1 4

Practice Questions

  1. Using the table above, find P(Circle and Red)
  2. Find P(Square and Blue)
  3. Calculate the probability of picking a red shape (any type)
  4. Calculate the probability of picking a circle (any color)
  5. Create your own table for {Green, Yellow} × {Triangle, Hexagon} and find P(Triangle and Green)

13. Dependent and Independent Events

Independent Events

  • One event does not affect the other
  • Example: Tossing two coins

Dependent Events

  • One event affects the outcome of the other
  • Example: Picking cards without replacement

Practice Questions

  1. Give one real-life example of independent events
  2. Give one real-life example of dependent events
  3. Explain why tossing two coins is independent
  4. Explain why picking cards without replacement is dependent
  5. Identify whether the following is independent or dependent: Rolling a die twice without changing the die

14. Theoretical vs Experimental Probability

Theoretical Probability

  • Based on all possible outcomes
  • Calculated before an experiment

Experimental Probability

  • Based on actual results of an experiment
  • May change as number of trials increases
Experimental Probability = Number of times event occurs ÷ Total trials

Practice Questions

  1. Calculate theoretical probability of rolling a 5 on a die
  2. Explain why experimental probability may differ from theoretical probability
  3. Conduct a small experiment tossing a coin 10 times and find experimental probability of heads
  4. Compare your experimental probability with theoretical probability for the coin toss
  5. Give an example of a situation where experimental probability is more useful than theoretical probability

IB MYP 2 – Unit 3: Ratio & Percentages

1. Ratio

A ratio compares two or more quantities of the same type.

  • Written using : or as a fraction
  • Order is important
Boys : Girls = 3 : 5
This means for every 3 boys, there are 5 girls.

Simplifying Ratios

  • Divide all parts by the highest common factor (HCF)
Simplify 12 : 18
HCF of 12 and 18 = 6
= 2 : 3
Always simplify ratios to their simplest form unless stated otherwise.

Practice Questions

  1. Simplify the ratio 15 : 25
  2. For every 4 apples, there are 9 oranges. Write the ratio as a fraction
  3. Divide 24 pencils between two students in the ratio 5:7. How many pencils does each get?
  4. Simplify the ratio 18:30:12
  5. Explain why the order of numbers matters in a ratio

2. Proportion

Two ratios are in proportion if they are equal.

Example:
2 : 3 = 4 : 6
2/3 = 4/6

Checking Proportion

  • Cross multiply
Check if 3 : 5 and 6 : 10 are in proportion
3 × 10 = 30
5 × 6 = 30
Therefore, they are in proportion.

Practice Questions

  1. Check if 4 : 7 and 12 : 21 are in proportion
  2. Determine if 5 : 8 and 10 : 16 are in proportion
  3. Write two ratios that are in proportion
  4. Explain why cross multiplication works to check proportion
  5. Check if 2 : 3 and 5 : 7 are in proportion

3. Problem Solving with Ratios

Sharing Quantities in a Given Ratio

  • Add all parts of the ratio
  • Find the value of one part
  • Multiply accordingly
Divide ₹600 in the ratio 2 : 3
Total parts = 2 + 3 = 5
Value of 1 part = 600 ÷ 5 = 120
Shares = 2 × 120 = ₹240, 3 × 120 = ₹360

Practice Questions

  1. Divide ₹900 in the ratio 3 : 2
  2. Share 84 candies in the ratio 5 : 7
  3. Divide 120 liters of juice in the ratio 4 : 6
  4. Explain the steps to solve a ratio problem like this
  5. Divide ₹750 in the ratio 3 : 5 : 7

4. Direct and Inverse Variation

Direct Variation

  • As one quantity increases, the other increases
  • As one decreases, the other decreases
Example:
Cost ∝ Number of items
5 pens cost ₹50 → 1 pen costs ₹10
8 pens cost = 8 × 10 = ₹80

Inverse Variation

  • As one quantity increases, the other decreases
Example:
Workers ∝ 1 / Time
6 workers take 10 days
12 workers take = (6 × 10) ÷ 12 = 5 days
Identify whether the situation is direct or inverse before solving.

Practice Questions

  1. Cost of 12 pens is ₹120. Find the cost of 1 pen (Direct Variation)
  2. 6 machines produce 120 units in 8 hours. How long will 4 machines take?
  3. 10 workers can complete a job in 15 days. Find time for 15 workers (Inverse Variation)
  4. Identify if the following is direct or inverse: Speed ∝ Time to travel 100 km
  5. Write an example of a direct variation in daily life

Graph Transformations

A transformation changes the position or orientation of a graph or shape on the coordinate plane without changing its size or shape. In MYP 2, we mainly study:

  • Translations
  • Reflections
  • Rotations

1. Translations on a Graph

A translation moves a graph from one place to another without flipping, turning, or resizing it. Every point moves the same distance in the same direction.

Key idea: Translations change position only.

Translation Rule

A translation is written as:

(x, y) → (x + a, y + b)

  • a controls movement left or right
  • b controls movement up or down

Direction and Signs

  • Right → add to x
  • Left → subtract from x
  • Up → add to y
  • Down → subtract from y

Example: Translating a Point

Translate point A(2, 3) 4 units right and 1 unit down.

(x, y) → (x + 4, y − 1)

New point: A′(6, 2)

Translating a Shape

To translate a shape:

  1. Write the coordinates of every vertex
  2. Apply the translation rule to each point
  3. Plot the new points
  4. Join them in the same order

2. Reflections on a Graph

A reflection flips a graph across a line called the mirror line. The most common mirror lines are the x-axis and y-axis.

Reflection in the x-axis

The x-coordinate stays the same, but the y-coordinate changes sign.

(x, y) → (x, −y)

Reflection in the y-axis

The y-coordinate stays the same, but the x-coordinate changes sign.

(x, y) → (−x, y)

Example

Reflect point B(−3, 4) in the y-axis.

New point: B′(3, 4)

Important: Reflections reverse orientation (the shape looks flipped).

3. Rotations on a Graph

A rotation turns a graph around a fixed point called the center of rotation. In MYP 2, rotations are usually about the origin (0,0).

Common Rotations About the Origin

  • 90° anticlockwise: (x, y) → (−y, x)
  • 180° rotation: (x, y) → (−x, −y)
  • 90° clockwise: (x, y) → (y, −x)

Example: 180° Rotation

Rotate point C(5, −2) by 180° about the origin.

New point: C′(−5, 2)

Rotations change orientation but keep size and shape the same.

Summary Table

  • Translation: slide (position changes)
  • Reflection: flip (orientation reverses)
  • Rotation: turn around a point

5. Rates and Unit Rates

A rate compares two quantities with different units.

  • Speed = distance ÷ time
  • Cost per item

Unit Rate

Rate for one unit.

Example:
300 km in 5 hours
Speed = 300 ÷ 5 = 60 km/h

Practice Questions

  1. A car travels 450 km in 9 hours. Find speed in km/h
  2. Cost of 15 pens is ₹75. Find cost per pen
  3. A person earns ₹1200 in 8 days. Find earnings per day
  4. Water tank fills 180 liters in 3 minutes. Find liters per minute
  5. Explain the difference between a rate and a unit rate

6. Percentages

Percentage means per hundred.

Example:
25% = 25/100 = 0.25

Percentage of a Quantity

20% of 150
= (20 ÷ 100) × 150
= 30

Practice Questions

  1. Find 15% of 200
  2. Convert 45% to a decimal
  3. Calculate 30% of 500
  4. Express 0.75 as a percentage
  5. If 60 is 20% of a number, find the number

7. Percentage Increase and Decrease

Percentage Increase

Original price = ₹500
Increase = 10%
Increase amount = 10% of 500 = ₹50
New price = ₹550

Percentage Decrease

Original price = ₹800
Decrease = 15%
Decrease amount = 120
New price = ₹680
Always find the percentage change using the original value.

Practice Questions

  1. Increase ₹400 by 12%
  2. Decrease ₹600 by 20%
  3. The price of an item increases from ₹250 to ₹300. Find the percentage increase
  4. The price decreases from ₹500 to ₹425. Find the percentage decrease
  5. Explain why percentage change is always calculated on the original value

8. Discount

A discount is a reduction in the marked price.

Example:
Marked price = ₹2000
Discount = 20%
Discount amount = 400
Selling price = ₹1600

Practice Questions

  1. Marked price = ₹1500, discount = 10%. Find selling price
  2. Marked price = ₹2500, discount = 25%. Find selling price
  3. If selling price is ₹800 after 20% discount, find the marked price
  4. Explain why discount is always calculated on marked price
  5. A shop offers 15% discount on ₹1200 item. Find the discounted price

9. Profit and Loss

Cost Price (CP)

Price at which an item is bought

Selling Price (SP)

Price at which an item is sold

Profit

Profit = SP − CP

Loss

Loss = CP − SP

Profit Percentage

Profit % = (Profit ÷ CP) × 100

Loss Percentage

Loss % = (Loss ÷ CP) × 100
CP = ₹500, SP = ₹600
Profit = ₹100
Profit % = (100 ÷ 500) × 100 = 20%

Practice Questions

  1. CP = ₹800, SP = ₹1000. Find profit and profit %
  2. CP = ₹1200, SP = ₹1000. Find loss and loss %
  3. CP = ₹450, SP = ₹500. Find profit %
  4. SP = ₹600, profit = ₹50. Find CP
  5. Explain why profit % is calculated on CP

IB MYP 2 – Unit 4: Algebra & Sequences

1. Sequences

A sequence is an ordered list of numbers that follows a rule or pattern.

2. Arithmetic Sequences

An arithmetic sequence is a sequence where the difference between consecutive terms is constant.

  • The constant difference is called the common difference (d)
Example: 2, 5, 8, 11, 14, …
Common difference = +3

3. Geometric Sequences (Introduction)

A geometric sequence is a sequence where each term is multiplied by the same number to get the next term.

  • The constant multiplier is called the common ratio (r)
Example: 3, 6, 12, 24, …
Common ratio = ×2
In MYP 2, focus is mainly on recognising geometric sequences, not complex formulas.

Practice Questions

  1. Find the next three terms of 5, 8, 11, …
  2. Find the common difference of 7, 12, 17, 22, …
  3. Identify if 2, 6, 18, 54 is arithmetic or geometric
  4. Find the next three terms of 4, 12, 36, …
  5. Explain the difference between arithmetic and geometric sequences

4. Term-to-Term Rule

The term-to-term rule describes how you move from one term to the next.

Example:
Sequence: 1, 4, 9, 16, …
Rule: Add consecutive odd numbers (+3, +5, +7, …)

Practice Questions

  1. Find the term-to-term rule for the sequence 2, 4, 6, 8, …
  2. Sequence: 5, 10, 20, 40, … Identify the term-to-term rule
  3. Explain the difference between term-to-term and nth term rules
  4. Sequence: 3, 6, 12, 24, … Find the term-to-term rule
  5. Write a sequence with a term-to-term rule of +7

5. Position-to-Term Rule (nth Term)

The nth term rule gives a formula to find any term directly.

nth Term of an Arithmetic Sequence

nth term = a + (n − 1)d
Sequence: 3, 7, 11, 15, …
a = 3, d = 4
nth term = 3 + (n − 1)4 = 4n − 1

Practice Questions

  1. Find the 10th term of 2, 5, 8, 11, …
  2. nth term of 7, 12, 17, 22, …
  3. Find the 15th term of 4, 9, 14, 19, …
  4. Write the nth term for 5, 10, 15, 20, …
  5. Explain how the nth term formula relates to the term-to-term rule

6. Algebraic Expressions

An algebraic expression contains numbers, variables, and operations.

Examples: 3x + 5, 2a − 4b, 5xy

Formation of Algebraic Expressions

Five more than twice a number x → 2x + 5
Perimeter of a rectangle with length l and breadth b → 2(l + b)

Linear Expressions with More Than One Variable

Linear expressions have variables raised to the power of 1 only.

3x + 2y − 5
−4a + 7b

Substitution into Algebraic Expressions

Substitute the given values of variables and evaluate.

Expression: 2x + 3y
x = 4, y = 2
= 2(4) + 3(2) = 8 + 6 = 14

Practice Questions

  1. Form an algebraic expression for: 7 more than 3 times a number x
  2. Form an expression for the perimeter of a rectangle with length l and width w
  3. Evaluate 4a − 3b for a = 2, b = 5
  4. Evaluate 5x + 2y − 3 for x = 1, y = 4
  5. Explain why 2x² + 3x is not a linear expression

10. Simplifying Algebraic Expressions

Like terms have the same variables raised to the same powers.

3x + 5x − 2y + y = 8x − y
Only like terms can be added or subtracted.

Practice Questions

  1. Simplify: 4a + 7a − 3b + 2b
  2. Simplify: 5x − 2x + 3y − y
  3. Explain why 2x + 3xy cannot be simplified with 4x + 5xy
  4. Simplify: 7m − 3m + 2n − n
  5. Simplify: 6p + 2q − 4p + 5q

11. Division of Polynomials

Divide each term in the polynomial by the divisor.

(6x² + 4x) ÷ 2x
= 3x + 2

Practice Questions

  1. (8x² + 12x) ÷ 4x
  2. (10a² + 5a) ÷ 5a
  3. (15y² − 5y) ÷ 5y
  4. (12m² + 8m − 4) ÷ 4
  5. (9x³ + 6x²) ÷ 3x²

12. Expanding Algebraic Expressions

Use the distributive law: a(b + c) = ab + ac

3(x + 4) = 3x + 12
(x + 5)(x + 2) = x² + 7x + 10

Practice Questions

  1. Expand: 5(a + 3)
  2. Expand: (x + 4)(x + 5)
  3. Expand: 2(3x + 7)
  4. Expand: (y + 2)(y + 3)
  5. Expand: (2a + 5)(a + 1)

13. Factorising Algebraic Expressions

Factorising is the reverse of expanding.

6x + 12 = 6(x + 2)
x² + 5x = x(x + 5)

Practice Questions

  1. Factorise: 8a + 12
  2. Factorise: x² + 7x
  3. Factorise: 10y + 15
  4. Factorise: 6m² + 9m
  5. Factorise: 4p² + 8p

14. Changing the Subject of a Formula

Rearrange the formula to make a different variable the subject.

y = 2x + 3
Make x the subject:
y − 3 = 2x
x = (y − 3)/2

Practice Questions

  1. Make y the subject: x = 3y + 4
  2. Make a the subject: P = 2a + b
  3. Make x the subject: 5x − 7 = 18
  4. Make t the subject: s = ut + ½at²
  5. Explain the steps to change the subject of a formula

15. Linear Equations

A linear equation has variables with highest power 1.

Practice Questions

  1. Identify whether 3x + 5 = 0 is a linear equation
  2. Is 2x² + 3x − 5 linear or non-linear?
  3. Write a linear equation for a line passing through (0, 3) with slope 2
  4. Check if y = 4x + 7 is linear
  5. Explain why x + y + z = 10 is considered linear

16. Solving Linear Equations

Transposition Method

3x + 5 = 20
3x = 15
x = 5

Elimination Method

x + y = 10
x − y = 2
Adding equations:
2x = 12 → x = 6, y = 4

Practice Questions

  1. Solve 5x − 7 = 18 using transposition
  2. Solve 2x + 3y = 12 and x − y = 2 using elimination
  3. Solve 4a + 5 = 21 using transposition
  4. Solve x + y = 8, x − y = 2 using elimination
  5. Explain the difference between transposition and elimination methods

17. Linear Graphs (Introduction to y = mx + c)

The equation y = mx + c represents a straight line.

  • m = slope (gradient)
  • c = y-intercept
y = 2x + 1
Slope = 2, y-intercept = 1

Practice Questions

  1. Find slope and y-intercept: y = 3x − 4
  2. Write equation for a line with slope 5 and y-intercept 2
  3. Graph y = −x + 3
  4. Explain what the slope tells you about the line
  5. Explain what the y-intercept represents on a graph

18. Distance-Time Graph (No Slope)

  • Horizontal line → object at rest
  • Straight rising line → constant speed

Practice Questions

  1. Draw a distance-time graph for an object at rest for 5 seconds
  2. Draw a graph for an object moving at 10 m/s for 6 seconds
  3. Explain the meaning of a horizontal line on a distance-time graph
  4. Explain the meaning of a steep slope on a distance-time graph
  5. Compare two objects moving at different constant speeds on the same graph

A distance–time graph shows how the distance travelled by an object changes with time.

Axes on a Distance–Time Graph

  • X-axis: Time (seconds, minutes, or hours)
  • Y-axis: Distance (metres or kilometres)
Always check units on both axes before interpreting the graph.

Key Types of Lines on a Distance–Time Graph

1. Horizontal Line (Object at Rest)

  • Distance does not change as time increases
  • The object is not moving
  • Speed = 0
Example interpretation:
From 2 minutes to 6 minutes, the graph is horizontal at 40 m.
The object stayed at 40 m and was stationary.

2. Straight Sloping Line (Constant Speed)

  • Distance increases uniformly with time
  • The object is moving at a constant speed
  • Steeper line → faster movement
If the distance increases by the same amount every minute,
the object is travelling at a constant speed.

3. Curve (Changing Speed – Awareness Only)

  • Distance does not increase evenly
  • Speed is changing
  • Not required to calculate in MYP 2
Focus only on interpretation. Slope calculations are NOT required.

Reading Information from the Graph

  • Find distance travelled at a given time by reading the y-value
  • Find time taken to reach a distance by reading the x-value
  • Compare speeds by comparing steepness of lines
At time = 5 minutes, distance = 100 m.
The object has travelled 100 m in 5 minutes.

Common Interpretation Questions (Criterion D)

  • When was the object at rest?
  • During which interval was the object moving fastest?
  • How long did the object remain stationary?
  • Describe the motion in words
Model answer:
“From 0 to 4 minutes, the object moved at a constant speed.
From 4 to 7 minutes, it remained stationary.”

Common Mistakes to Avoid

  • Confusing distance with speed
  • Assuming a higher graph means faster (steepness matters)
  • Trying to calculate slope when not required