Number and Algebraic Patterns

Collaborative Learning: Learners will work in convenient groups (e.g., ability, mixed-ability, mixed gender, or pairs etc.) to explore the basic properties and operations on surds in different contexts.
Talk for Learning Approaches: Learners will brainstorm using participatory activities such as think-pair-share/square and debate on various properties of surds.
Experiential Learning: Learners will work with others in groups and pairs to create surds problems involving basic operations, properties and types of surds.
Activity 1: Definition and Types of Surds Use Talk for Learning Approaches (building on what others say, managing Talk for Learning, structuring Talk for Learning), collaborative learning approaches and experiential learning approaches to investigate the types of surds.
Learners in small and convenient groups (i.e., mixed gender, mixed-ability, etc.) identify and work to establish types of surds.
Example: Learners recognise that:
- Surds is used to refer to a number that does not have a root. E.g., √2, √3, √5 etc. "
- Surds represent numbers in the form of square roots since these numbers cannot be whole or rational numbers.
Types of surds Learners investigate and discover the following as types of surds:
- Pure Surds: A surd having only a single irrational number is called a pure surd. E.g., √7
- Mixed Surds: A surd having a mix of a rational number and an irrational number is called a mixed surd. E.g., 5√3
- Compound Surds: A surd composed of two surds or a surd, and a rational number is called a compound surd. E.g., √3 + √10, 3+ √7,
- Binomial Surd: When two surds give rise to one single surd, the resultant surd is known as a binomial surd.
Activity 2: Properties of surds Use Talk for Learning Approaches (building on what others say, managing Talk for Learning, structuring Talk for Learning), collaborative learning approaches and experiential learning approaches to investigate the rules and properties of surds, including conjugates and rationalisation.
- Learners in small convenient groups (i.e., mixed gender, mixed-ability, etc.) work together to establish the properties and rules of surds.
Example: Learners, in groups, establish the rules of surds, including conjugating surds. That is Rule 1: w(𝑎 × 𝑏) = √𝑎 × √𝑏 𝑎, 𝑏 ≥ 0
& √& Rule 2: y ' = 𝑏 > 0 √'
& & √' &√' Rule 3: = × = 𝑏 > 0 √' √' √' '
$ # Rule 4: √𝑎 = 𝑎 #

 Rule 5:
& & &*√' Rule 6: = &$√' × &*√' &$√'
Rule 7: &*√' = &*√' × &$√' & & &$√'
NB:
- √𝑎 + √𝑏 ≠ w(𝑎 + 𝑏)
- √𝑎 − √𝑏 ≠ w(𝑎 − 𝑏)
Activity 3: Simplification of Surds Inter-group competition: Learners from one group create surd problems whilst the other group applies properties of surds to solve the problem. Group switch roles.
Example: Simplify √108.
Solution √108 = √36 × 3 = √36 × √3 = 6√3
Example: Simplify the following surd expressions where possible. For those that cannot be solved, state the reasons why. a) √5 + √7 b) 3√2 + 5√2 c) √7 − √5 d) 3√2 − 5√2 e) √5 × √7 f) 3√2 × 5√2 g) √15 ÷ √5
NB: Learners in their groups create and solve examples and discover that surds
- cannot be added/subtracted, but similar surds can be added/subtracted.
- can be multiplied.
- can be divided.
- can be written in exponential form.
Teaching and Learning Resources:
- Textbooks
- Curriculum
- Cardboards
- Reading resource
- Colour pens
- Notebook
- Technological tools
Assessment (1.1.1.AS.1). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 3 Strategic reasoning; Level 4 Extended critical thinking and reasoning.
Given that √𝑎 + √𝑏 = 7 and √𝑎 − √𝑏 = 1, find the value of 𝑎 and 𝑏 Solve for the value of 𝑥 in 2√3𝑥 + 5 = 4√𝑥 + 1 What is the square root of (10 + √25)(12 − √49)?