Math-Standard

MYP 4 Notes

Year 4 Syllabus-- summarised

Rational & Irrational Numbers

Rational numbers illustration

Irrational numbers: A real number that cannot be expressed as a simple fraction (ratio) of two integers. Their decimal expansions are non-terminating and non-repeating. Example: π

Rational numbers: Numbers that can be expressed as a fraction p/q with integers p and q. Their decimals are either terminating or repeating. Example: 69

--> The only task that they would ask you on this is classifying number types in a venn diagram

Practice Task

Write whether the following numbers are rational or irrational:

Number System Recap

Name Symbol Properties
Integer ℤ All positive and negative whole numbers
Natural ℕ Positive integers, counting numbers
Real ℝ All numbers on the number line
Irrational — Real numbers that aren’t rational
Rational ℚ Real numbers that can be expressed in p/q form with a terminating/repeating decimal

Note: Complex and imaginary numbers exist but we don't need to think about that for now.

In this topic they'll probably also ask about simplification, but that's a topic best learned through practise

  1. Combine like terms: Terms with the same variable and power can be added or subtracted.
    Example: 3x + 5x = 8x
  2. Remove brackets: Multiply everything inside the bracket.
    Example: 2(x + 3) = 2x + 6
  3. Cancel common factors: Divide numerator and denominator by the same value.
    Example: 6x / 3 = 2x

    4. 2️⃣ Simplification with Exponents

    • Multiplying powers (same base): Add the exponents.
      Rule: am × an = am+n
      Example: x2 × x3 = x5
    • Dividing powers (same base): Subtract the exponents.
      Rule: am ÷ an = am−n
      Example: x5 ÷ x2 = x3
    • Power of a power: Multiply the exponents.
      Rule: (am)n = amn
      Example: (x2)3 = x6
    • Zero exponent: Any non-zero base raised to the power 0 equals 1.
      Example: x0 = 1

➗ Simplifying Polynomial Fractions

1️⃣ Factor Everything

Always factor the numerator and denominator completely before simplifying.

Example:

(x² + 5x) / x = (x(x + 5)) / x

2️⃣ Cancel Common Factors

Cancel factors that appear in both the numerator and denominator.

Example:

(x(x + 5)) / x = x + 5   (x ≠ 0)

3️⃣ State Restrictions

The denominator cannot equal zero. Always state any restrictions.

Example:

x ≠ 0

4️⃣ Final Answer

Write the simplified expression and include restrictions if needed.

Note: only cancel factors, not just random terms please

√Radicals

A radical represents the root of a number or expression. General form:

√[n]{a} = x means xⁿ = a

Where:

Rules / Properties of Radicals

Worked Examples

Radicals illustration

Sets & Venn Diagrams

What are Sets?

A set is a collection of objects or data grouped together because they share a common property. The objects in a set are called elements.

Sets are often written using curly brackets { } and can contain numbers, letters, or other objects.

What are Venn Diagrams?

A Venn diagram is a visual way to represent sets using circles. It shows how sets are related by displaying:

Venn diagrams make it easier to organise data, compare groups, and solve word problems involving sets.

Example:

Disjoint (Mutually Exclusive) Sets

Disjoint sets (also called mutually exclusive sets) are sets that have no elements in common.

In a Venn diagram, disjoint sets do not overlap at all — they are completely separate.

Interpreting Venn Diagrams (3 Sets)

You will often be asked to interpret survey data using a three-set Venn diagram. The key is to fill in the diagram from the centre outward.

Example Problem

50 students were surveyed about their favourite sport.

Find:
a) How many play only basketball?
b) How many play exactly one sport?
c) How many play both tennis and soccer?
d) How many play exactly two sports?

How to Solve These Problems

--> Always check that the total of all regions matches the universal set before finalising your answers.

Patterns

There are three main types of patterns: Arithmetic, Geometric, and Quadratic.

1️⃣ Arithmetic Patterns

If the pattern follows addition or subtraction, it is arithmetic.

Example: 6, 8, 10, 12, 14, 16

The general term (Tn) is given by:

Tn = a + (n - 1)d

Example calculation:

Tn = 6 + (n-1)·2 = 2n + 4

2️⃣ Geometric Patterns

If the pattern follows multiplication or division, it is geometric.

Example: 2, 4, 8, 16, 32

The general term (Tn) is:

Tn = a · rn-1

Example calculation: Tn = 2 · 2n-1

3️⃣ Quadratic Patterns

If a sequence has a constant second difference but inconsistent first difference, it is quadratic.

Example: 1, 4, 9, 16, 25

Second differences:

Formula for quadratic sequences:

Tn = a·n² + b·n + c

a = second difference ÷ 2

Finding b and c

Use the first two terms to create equations:

Shortcut to find b:

b = (T2 - T1) - 3a

Then find c:

c = T1 - a - b

Practice – Sequences

Find the general rule for each sequence:

Quadratic sequences illustration

More practice problems

Command Terms – Criteria B

Describe

Use describe when you need to explain how values change in a sequence or pattern:

Test

Use test to check if a general rule produces the correct value:

Method: Substituting a given value of n into the general rule.

Example statement: “The value calculated using the general rule is equal to the value given in the table for column __ when n = __.”

Verify

Use verify when you predict a value and check it against the general rule:

Method: Substituting a predicted value of n into the general rule.

Example statement: “The value calculated using the general rule when n = __ is ___.”

Prove / Justify

Use prove/justify when you apply the context of the problem to use an alternative method and explain why the rule works.

Proportionalities

Direct Proportionality

In direct proportionality, as one quantity increases, the other increases proportionally.

The constant of proportionality (k) is given by:

k = y / x

Example:

If 7 kids cost Rs. 140, how much will 12 kids cost?

Indirect Proportionality

In indirect proportionality, as one quantity increases, the other decreases. (e.g., pressure and volume in Boyle’s law)

The constant of proportionality (k) is given by:

k = y × x

Example:

If 4 workers take 6 hours to paint a house, how many hours will 8 workers take?

Exponential Equations

This is solving equations with exponents by using rules of simplification. It is best taught with an example:

16y-1 = 8

Given that 24 is 16 and 23 is 8
=> 24y-4 = 23
This implies that 4y-4 = 3
Therefore 4y = 7
and y = 7/4

25m+1 / 5 = 125

This can be taken as 52m+2 / 5 = 53
Therefore 2m+2 = 3
And 2m = 1
This implies m = ½

Task

Solve the following exponential equations for the unknown variable

49p / 7 = 343

128y-8 / 4 = 256

27h+1 = 9

42k = 32

5r+1 = 252r-3

Answers:

p = 21/2
y = 88/7
h = -1/3
k = 5/4
r = 7/3

Quadratic Equations

They are expressed in the standard form ax2 + bx + c = 0 where a ≠ 0

They can have 2,1 or 0 solutions . There are three methods to solve for these solutions:

Quadratic formula

-b ± √(b² - 4ac)
x = -------------------
2a

√b2-4ac is known as the discriminant. When this is equal to 0, there is one solution, when this > 0 there are 2 solutions and when the discriminant is negative there will be no real solution.

Example Question:

Solve the quadratic equation using the quadratic formula:

2x² + 3x - 2 = 0

Solution:

Step 1: Identify a, b, and c

a = 2
b = 3
c = -2

Step 2: Write the quadratic formula

x = (-b ± √(b² - 4ac)) / (2a)

Step 3: Substitute values

x = (-3 ± √(3² - 4(2)(-2))) / (2(2))

Step 4: Simplify

x = (-3 ± √(9 + 16)) / 4

x = (-3 ± √25) / 4

Step 5: Solve

x = (-3 + 5) / 4 = 2/4 = 1/2

x = (-3 - 5) / 4 = -8/4 = -2

Final Answer:

x = 1/2 or x = -2

Task : Solve for all real solutions of ‘x’ by usage of the quadratic formula

1. x² + 5x + 6 = 0

2. x² - 7x + 10 = 0

3. 2x² + x - 3 = 0

4. 3x² - 2x - 1 = 0

5. x² + 4x = 12

Answers:

1. x = -2 or x = -3

2. x = 5 or x = 2

3. x = 1 or x = -3/2

4. x = 1 or x = -1/3

5. x = 2 or x = -6

Null factor law

The null factor law states that if ab = 0, then a = 0 or b = 0.

Example Question:

Solve the quadratic equation using the null factor law:

(x + 4)(x - 3) = 0

Solution:

Step 1: Apply the null factor law

If ab = 0, then a = 0 or b = 0

Step 2: Set each factor equal to zero

x + 4 = 0     or     x - 3 = 0

Step 3: Solve each equation

x = -4

x = 3

Final Answer:

x = -4 or x = 3

Task: Solve for ‘y’ using the null factor law

Hint: Factorise the ones given in standard form.

  1. (y + 2)(y - 5) = 0
  2. y² - y - 12 = 0
  3. 2y² + 7y + 3 = 0
  4. (y - 6)(y + 1) = 0
  5. 3y² - 10y - 8 = 0

Answers:

  1. y = -2 or y = 5
  2. y = 4 or y = -3
  3. y = -3 or y = -1/2
  4. y = 6 or y = -1
  5. y = 4 or y = -2/3

Completing the Square Method

Completing the square is a method used to solve quadratic equations or rewrite them into a perfect square form.

The goal is to turn part of the expression into this form:

(x + a)²

Example Question:

Solve x² + 6x + 5 = 0 by completing the square

Step 1: Move the constant to the other side

x² + 6x = -5

Step 2: Take half of the coefficient of x, then square it

Coefficient of x = 6

Half of 6 = 3

Square it: 3² = 9

Step 3: Add this number to BOTH sides

x² + 6x + 9 = -5 + 9

x² + 6x + 9 = 4

Step 4: Rewrite the left side as a perfect square

(x + 3)² = 4

Step 5: Square root both sides

x + 3 = ±2

Step 6: Solve for x

x + 3 = 2 or x + 3 = -2

x = -1 or x = -5

Final Answer:

x = -1 or x = -5

Task: Solve for x by completing the square

  1. x² + 4x + 3 = 0
  2. x² + 8x + 7 = 0
  3. x² - 6x + 5 = 0
  4. x² + 10x + 16 = 0
  5. x² - 4x - 5 = 0

Answers:

  1. x = -1 or x = -3
  2. x = -1 or x = -7
  3. x = 1 or x = 5
  4. x = -2 or x = -8
  5. x = 5 or x = -1

Example Question (word problem)

In a flight of 600 km an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/h and the time increased by 30 minutes. Find the flight's duration.

Step 1: Define the variable

Let the original speed of the aircraft be v km/h.

Distance = 600 km
Time = Distance ÷ Speed

Step 2: Write expressions for time

Original time = 600 / v

New speed = v - 200

New time = 600 / (v - 200)

The time increased by 30 minutes.
30 minutes = 0.5 hours

Step 3: Form the equation

New time = Original time + 0.5

600 / (v - 200) = 600 / v + 0.5

Step 4: Multiply everything by v(v - 200) to remove fractions

600v = 600(v - 200) + 0.5v(v - 200)

Step 5: Expand

600v = 600v - 120000 + 0.5v² - 100v

Step 6: Simplify

Move everything to one side:

0 = 0.5v² - 100v - 120000

Multiply everything by 2 to remove decimals:

0 = v² - 200v - 240000

Step 7: Solve the quadratic

v² - 200v - 240000 = 0

Factorise:

(v - 600)(v + 400) = 0

Step 8: Solve

v = 600 or v = -400

Speed cannot be negative, so:

v = 600 km/h

Step 9: Find the original duration

Original time = 600 / 600

Original time = 1 hour

Final Answer:

The flight's original duration was 1 hour.
The new duration is 1.5 hours.

Question:

A train travels 480 km between two cities. Due to track maintenance, its average speed was reduced by 20 km/h, which increased the travel time by 1 hour.

Find the train's original speed.

→ You may solve this using any method of choosing

Linear Inequalities

Example: Solve for ‘x’

2x - 5 ≥ 5x - 7

⇒ 2x - 5x ≥ -7 + 5

⇒ -3x ≥ -2

⇒ x ≤ -2 / -3 (flip the inequality when the coefficient of ‘x’ is negative)

⇒ x ≤ ⅔

This can also be represented in set builder form:

{x | x ≤ ⅔ , x ∈ ℝ }

Questions:

  1. -2(x - 3) < 5(x + 1) - 12
  2. (-3x + 1) / 2 ≤ 11
  3. 3x + 8 < -2(x - 1) + 5

Answers:

  1. x > 13/7
  2. x ≥ -7
  3. x < -1/5

→ Note that this won’t give an exact value for ‘x’ never say x = whatever in an inequality

Word Problem (Linear Inequalities)

Lucy has $17 to spend on a taxi ride. The taxi charges $5 per mile.

(a) Write and solve a linear inequality to find the maximum number of miles Lucy can travel.

(b) Lucy had to spend $5 on baby oil, and her friends took $6 and ran. How many miles can she travel now?

(c) Lucy’s home is 10 miles away, will she be able to get home through the taxi?

Compound/Double Inequalities

You solve this the same way, except there are two inequalities instead of one. When ‘or’ is used to describe the inequalities’ relationship, it indicates a union and when ‘and’ is used to describe their relationship it indicates an intersection.

Task: Solve each double inequality.

  1. x + 3 ≥ 7 and x + 3 ≤ 15
  2. 2x - 4 > 6 and 2x - 4 < 16
  3. x/2 - 1 ≥ 3 and x/2 - 1 ≤ 9
  4. 3x + 5 ≤ 2 or 3x + 5 ≥ 14
  5. x - 8 < -2 or x - 8 > 10

Answers:

1. x + 3 ≥ 7 and x + 3 ≤ 15
x ≥ 4 and x ≤ 12

2. 2x - 4 > 6 and 2x - 4 < 16
2x > 10 and 2x < 20
x > 5 and x < 10

3. x/2 - 1 ≥ 3 and x/2 - 1 ≤ 9
x/2 ≥ 4 and x/2 ≤ 10
x ≥ 8 and x ≤ 20

4. 3x + 5 ≤ 2 or 3x + 5 ≥ 14
3x ≤ -3 or 3x ≥ 9
x ≤ -1 or x ≥ 3

5. x - 8 < -2 or x - 8 > 10
x < 6 or x > 18

Rationalizing the Denominator of a Fraction with a Surd

Single-term denominator: Multiply both numerator and denominator by the same surd. The denominator becomes a whole number since a surd squared is rational.

Multiple-term denominator: Multiply numerator and denominator by the conjugate of the denominator (swap the ± sign). This removes the surd because multiplying by itself would still leave a surd term.

Linear Programming

Linear programming is used to find the optimal solution to linear functions or inequalities. It helps determine maximum or minimum values in problems like profit optimization.

Key concept: Identify all inequalities, graph the feasible region, and check the corner (extreme) points for the optimum value.

Simultaneous Equations

3x + 2y = 4
4x + 5y = 17

There are two methods to solve these equations, elimination method and substitution method. Elimination method is preferred but it's lowkey ur choice (or is it?)

Elimination method

Given:

3x + 2y = 4
4x + 5y = 17

Multiply the first equation by 4 and the second equation by -3:

12x + 8y = 16
-12x - 15y = -51

Add the equations:

-7y = -35

Solve for y:

y = -35 / -7
y = 5

We can input this into the first equation:

3x + 2(5) = 4
3x + 10 = 4
3x = -6
x = -2

Final Answer:

x = -2
y = 5

Substitution method

Given:

3x + 2y = 4 ...(1)
4x + 5y = 17 ...(2)

Step 1: Rearrange the first equation to make x the subject

3x + 2y = 4

3x = 4 - 2y

x = (4 - 2y) / 3

Step 2: Substitute this into the second equation

4x + 5y = 17

4((4 - 2y) / 3) + 5y = 17

Step 3: Simplify

(16 - 8y) / 3 + 5y = 17

Multiply everything by 3:

16 - 8y + 15y = 51
16 + 7y = 51

Step 4: Solve for y

7y = 35

y = 5

Step 5: Substitute y = 5 into x = (4 - 2y) / 3

x = (4 - 10) / 3

x = -6 / 3

x = -2

Final Answer:

x = -2
y = 5

Word Problem:

You're stalking someone to try stealing from them. You notice they withdraw ₹9000 from an ATM using only ₹500 notes and ₹200 notes.

Let x be the number of ₹500 notes and y be the number of ₹200 notes.

(a) Write a simultaneous equation representing this situation.

(b) If the total number of notes withdrawn was 24, find how many ₹500 notes and how many ₹200 notes were withdrawn.

Here’s some practise on simultaneous equations:

Simultaneous Equations Practice

Application of Simultaneous Equations Practice

Mixed Questions (Past Summative Paper)

Survey Question

In a survey, 100 students were asked if they like Basketball (B), Football (F) and Swimming (S). The results were as follows:

a) Represent the given information in the Venn diagram and find the value of x.

b) Find the number of students who liked exactly one sport.

c) Find the number of students who liked Football or Basketball.

Answer:

x = 3

40 students liked exactly one sport

89 students like either football or basketball

Find the value of x.

Answer: x = ⅙

Inverse Proportionality Question

Mili has decided to encourage her class to participate in ‘Adopt a Sumatran Tiger’ which funds research and conservation of these critically endangered animals. The cost of adoption per person depends on the number of people participating.

a) Verify that the number of people and the cost per person are inversely proportional.

b) Find the constant of proportionality and write down the equation connecting the number of people and the cost per person.

c) Calculate the cost per person when 25 people participate.

Answers:

a) Yes, it is inversely proportional as the number of people increases the cost decreases at a constant rate.

b) k = 100

c) C = 12

Splitting The Middle Term

This uses a technique known as the “Master Product” method to “split” the middle term into two terms, so that the trinomial can be factorized into two binomials.

ax² + bx + c

Changing The Subject

Simply shift the terms in an equation around till the desired term is expressed as a multiple of other terms.

Functions

A function is a relationship between inputs and outputs where one input can only be assigned one output.

Quadratic Functions

Quadratic functions are similar to quadratic equations. When plotted, they form a parabola shape on a graph.

Forms of Quadratic Functions

Standard Form:

f(x) = ax² + bx + c

Vertex Form

The vertex form of a quadratic function is:

f(x) = a(x-h)² + k

Intercept Form

The intercept form of a quadratic function is:

f(x) = a(x-p)(x-q)

Quadratic Equations

Quadratic equations are polynomial expressions with at least one squared term. They do not include terms raised to powers higher than two.

They can be written as:

ax² + bx + c = 0

Quadratic equations are set equal to 0 to simplify solving, allowing us to focus on values that make the product of the binomial terms equal to 0. Any value on the right side can be moved to the left, keeping the equation equal to 0.

To solve for x, the quadratic formula is used:

x = [-b ± √(b² - 4ac)] / 2a

Note: If a problem asks for “infinite solutions,” it means that, when simplified, the first equation equals the second. The lines overlap entirely, so any solution satisfies both.

Completing the Square

Completing the square is a method used to factorize quadratic equations by rewriting them as the sum of a squared binomial plus a constant.

Reference: mathsisfun.com

Gradients

The gradient of a line measures its slope, showing both steepness and direction.

It is calculated using the formula:

Gradient = rise / run = (y₂ - y₁) / (x₂ - x₁)

An equation of a line expresses the relationship between all y-coordinates and their corresponding x-coordinates.

Exponential Growth and Decay

Exponential growth and decay describe quantities that increase or decrease at extremely rapid rates, such as populations or radioactive substances.

The formula for both is:

f(t) = a(1 ± r)t

Depending on whether the quantity is growing or decaying, r is added to or subtracted from 1: