Number and… · SHS1 Additional Mathematics

Lesson notes

Learning Objectives

Indicator: 1.1.1.LI.3 - Recollect the initial laws of indices and establish other laws for negative powers and roots.

By the end of the lesson, learners can:

Recall and state the initial laws of indices for multiplication, division, powers of powers, and powers of products and quotients.
Establish the negative exponent rule (a to the power -n = 1/(a to the power n)) and use it to rewrite expressions with positive indices.
Establish the zero power rule (a to the power 0 = 1, where a is not 0) and apply it in simplification.
Simplify algebraic expressions involving indices by applying the laws of indices correctly.
Write final answers with positive indices only, as required by the assessment standard.

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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.3 - Recollect the initial laws of indices and establish other laws for negative powers and roots.
Suggested placement: Semester 1, Week 5 (Week 5 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.57: 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. 57

Exemplars (from the NaCCA curriculum)

Collaborative Learning, Talk for Learning, Building on what others say.
Learning Experience: Learners in groups explore the laws of indices.
Activity 1: Learners in pairs share ideas about indices and explain indices to their colleagues.
Expected Responses
- Index (indices) is the power or exponent raised to a number or a variable.
- The plural of index is indices.
Activity 1: Learners, in their groups, solve examples and use the examples to explore the derivation of the laws of indices.
Activity 2: Learners in their groups create and solve similar examples to establish the laws of indices.
Activity 3: Learners in their groups establish that:
- 𝑎 # × 𝑎 % = 𝑎 #$% , and by extension, once the base is equal 𝑎 # × 𝑎 % × 𝑎 + × ... = 𝑎 #$%$+$⋯ 𝑎 # ÷ 𝑎 % = 𝑎 #*%
-
- (𝑎 # ) % = 𝑎 #% , and by extension
- (𝑎 # × 𝑏 # × ... ) % = 𝑎 #% × 𝑏 #% × ... & # & %
- i ' j = ' # &×'×... # & % ×' % ×...
- i ;×<×... j = ; % ×< # ×...
- (𝑎) # = 𝑎 # 𝑏 % & % % & %#
- i % j = ' ' %# & % ×' % ×... % & %# ×' %# ×...
- i ; % ×< % ×... j = ; %# ×< %# ×...
Other rules are:
- Negative exponent rule: & *% ' % 𝑎 *# = & % by extension i ' j ! = i & j
- Zero power rule: 𝑎 ( = 1, where 𝑎 ≠ 0
Activity 4: Learners in their groups create and solve a variety of problems involving the laws of indices.
!) Example: Simplify 8 "
!) Expected solution = 8 " = 4 0
Example: Simplify 2 ) × 4 *, ÷ 16 *"
Expected Solution= 64
Teaching and Learning Resources:
- Textbooks
- Curriculum
- Cardboards
- Reading resource
- Colour pens
- Notebook
- Technological tools
Assessment (1.1.1.AS.3). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall.
Simplify and write =& ! ' ' > with positive (& ( ' " ) indices Show that ! ! i. 32 * ( = , . ! ") ii. i2𝑥 - ( j = 0 !