SHS1 Additional Mathematics · Semester 1 Week 6: Number and Algebraic Patterns

Lesson notes

Learning Objectives

Indicator: 1.1.1.LI.4 - Recognise the relationship between surds and indices and apply laws of indices to simplify expressions.

By the end of the lesson, learners can:

Express surds in index form and rewrite fractional indices as surds, showing the inverse relationship between the two notations.
Apply the laws of indices (multiplication, division, power of a power, zero index, negative index, and fractional indices) to simplify algebraic expressions correctly.
Use the relationship between surds and indices to evaluate numerical expressions involving surds, such as (25)^(1/2) = 5 and 8^(2/3) = 4.
Solve problems that combine squaring of surds with indicial equations to demonstrate the equivalence of the two forms.
Create and present original problems to peers that apply the laws of surds and indices to practical situations, communicating reasoning clearly.

This lesson builds on Week 4 and Week 5 work on surd operations and rationalisation, and prepares learners for Week 7 and beyond where indicial and logarithmic equations will be solved in full.

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Curriculum details

Strand: Modelling with Algebra (Strand 1)
Sub-strand: Number and Algebraic Patterns (1.1)
Content standard: 1.1.1.CS.2 - Demonstrate knowledge and understanding of numbers in relation to Surds, Indices and Logarithms. 1.1.1.LO.1 Solve problems involving properties of binary operations. 1.1.1.LO.2 Model and solve real life problems on sets. 1.1.1.LO.3 Expand binomials with positive integral indices and simplify coefficients of the terms. 1.1.1.LO.4 Perform basic operations on surds as well as solve simple indicial and logarithmic equations.
Indicator: 1.1.1.LI.4 - Recognise the relationship between surds and indices and apply laws of indices to simplify expressions.
Suggested placement: Semester 1, Week 6 (Week 6 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

content standard text - p.53: the source prints the content-standard code as 1.1.1.CS2 without the separator before 2; the CSV key uses 1.1.1.CS.2 so the otherwise unambiguous second standard can be imported
exemplars - p.59: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
Curriculum reference: NaCCA curriculum document, p. 59

Exemplars (from the NaCCA curriculum)

Research, Collaborative Learning, and Talk for Learning.
Learning Experience: Learners in groups conduct research on the relationship between surds and indices.
Activity 1: Group leads present the findings of the group on the relationship between indices and surds to the whole class.
Activity 2: Learners apply the laws of surds and the laws of indices to solve problems.
Example: If 𝑎 = 3 − √3, show that 𝑎 ) + ! = 24. "/ &
Solution 𝑎 = 3 − √3 (Squaring both sides) 𝑎 ) = 9 − 6√3 + 3
𝑎 ) = 12 − 6√3 (Squaring both sides) 𝑎 , − 24𝑎 ) + 144 = 36 × 3 𝑎 , − 24𝑎 ) + 36 = 0 (divide through by a 2 )
"/ 𝑎 ) − 24 + & ! = 0
"/ So, 𝑎 ) + & ! = 24
Activity 2: Learners create similar problems involving the application of the laws of surds and the laws of indices to solve real life problems.
Teaching and Learning Resources:
- Textbooks
- Curriculum
- Cardboards
- Reading resource
- Colour pens
- Notebook
- Technological tools
Assessment (1.1.1.AS.4). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 3 Strategic reasoning.