Number and… · SHS1 Additional Mathematics

Lesson notes

Learning Objectives

Indicator: 1.1.1.LI.1 - Explain binary operations and apply that knowledge in solving related problems.

By the end of the lesson, learners can:

Define a binary operation as a rule that combines two elements from a non-empty set to produce another element of that set.
Evaluate expressions involving a given binary operation by substituting values correctly into the defining rule.
Construct an operation table for a binary operation defined on a finite set.
Use an operation table to evaluate compound expressions such as (a ∇ b) ∇ c.
Determine whether a given pair of elements can be combined under a specific binary operation, giving reasons.

Sign in with your phone number to read the full note and download the GES plan - free.

Sign in with phone number

Curriculum details

Strand: Modelling with Algebra (Strand 1)
Sub-strand: Number and Algebraic Patterns (1.1)
Content standard: 1.1.1.CS.1 - Demonstrate knowledge and understanding of binary operations, sets and binomial theorem and solve related problems in real life situations. 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.1 - Explain binary operations and apply that knowledge in solving related problems.
Suggested placement: Semester 1, Week 1 (Week 1 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:

exemplars - p.37: 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. 37

Exemplars (from the NaCCA curriculum)

Collaborative Learning: Learners will work in convenient groups (ability, mixedability, mixed gender, or pairs etc.) to identify and define binary operations over given sets and solve related problems.
Talk for Learning Approaches: Learners will brainstorm by way of think-pairshare/square and debate on sets defined by binary operations and establish the rules for such binary operations.
Experiential Learning: Learners will collaboratively engage in hands-on activity (learning by doing) to create binary operations, define them over sets and solve related problems.
Activity 1: Define and interpret Binary Operations as a rule 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 recall the four basic operations and use them to define a binary operation.
Small group discussions/Think-pair-share: Learners in small convenient groups (for instance, mixed gender, mixed-ability, etc.) examine binary operations and recognise that binary operations are rules that combine elements from a non-empty set 𝑅 to produce another element.
Example: Suppose the operation * is defined on the set of real numbers ℝ by 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 − 2𝑎.
- Find (α)4 ∗ 2 !
- (β) " ∗ −2
- If 𝑎 ∗ 4 = −2, find the value of 𝑎.
Inter-group competition: Learners from one group devise their own binary operations for another group to examine and interpret the rule. Group switch roles.
Work with a partner: Construct tables for a given binary operation.
Examples:
- A binary operation ∇ is defined on the set 𝑆 = { 2, 3, 4, 5} by 𝑝 = 𝑝 + 𝑞 − 𝑝 where 𝑝 and 𝑞 ∈ 𝑆.
- Construct the table for the operation ∇ on the set 𝑆.
- Use your table to evaluate (2𝛻3)𝛻3
- Which of these can be solved using the operation ∇ on the set 𝑆. Give reasons.
(α) 3 𝛻 − 2 (β) 4𝛻3 (γ) 3𝛻5 (δ) 5𝛻1
#$%
- Given that 𝑚 ⊝ 𝑛 = , evaluate (2 ⊖ 3) + (3 ⊝ 2) #
&'
- 𝑎 and 𝑏 are integers such that 𝑎 ⋄ 𝑏 = . Find the values of 𝑎 and 𝑏 if 𝑎 ⋄ 𝑏 = !( ! (6 ⋄ 𝑎) − )
Activity 2: Discuss the history and relevance of learning binary operations.
Teaching and Learning Resources:
- Textbooks
- Rectangular paper cut-out
- Addition pyramid
- Curriculum
- Cardboards
- Reading resources
- Colour pens
- Notebook
- Graph sheets
- Mathematical sets
- Technological tools.
Assessment (1.1.1.AS.1). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 4 Extended critical thinking and reasoning.
The operation ∗ is defined on the set of real numbers ℝ by 𝑝 ∗ 𝑞 = 2𝑝 + 𝑞 − 2𝑝. a) Find i. 3 ∗ −2 ii. 3∗5 b) If 𝑝 ∗ 4 = −2 find the value of 𝑝. You have six shirts, two trousers and two pairs of shoes. Explain how you will use the binary operation to determine how many ways a shirt, a trouser and a pair of shoes can be worn