Number and Algebraic Patterns

Collaborative Learning: Learners will work in convenient groups (ability, mixedability, mixed gender, or pairs etc.) to solve problems related to identity and inverse elements and other properties of binary operations.
Talk for Learning Approaches: Learners will brainstorm through think-pairshare/square and debate to discuss and establish the identity, inverse elements, and other properties of a binary operation.
Experiential Learning: Learners will collaboratively engage in hands-on activities (learning by doing) to create binary operations, define them over sets and determine the inverse and identity elements and other properties of binary operations.
Activity 1: Identity and inverse elements of a binary operation. 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 relationship between inverse and identity elements of a binary operation.
Small group discussions/Think-pair-share: Learners in small convenient groups (for instance, mixed gender, mixed-ability, etc.) investigate the elements of binary operations to establish that the identity element 𝑒 of a set 𝑆 under an operation ∆ on 𝑆 exists if there is an element 𝑒 such that 𝑎∆𝑒 = 𝑒∆𝑎 = 𝑎
Example 1: A binary operation ∇ is defined on the set 𝑅 or real numbers by 𝑝 = 𝑝 + 𝑞 − 𝑝 where 𝑝 and 𝑞 ∈ 𝑅.Find the identity element under the operation ∇ .
Solution: Let 𝑒 be the identity element, where 𝑒 ∈ 𝑅, then for the right identity
𝑝 = 𝑝 + 𝑒 − 𝑝 = 𝑝. ⟹ 𝑒 = 0.
For the left identity e𝛻 = e+𝑝 − 𝑒 = 𝑝. ⟹ 𝑒 = 0. Since the left identity=right identity, the identity element 𝑒 of ℝ under the operation ∇ exists, and it is 0.
Example 2: Suppose the operation ∗ is defined on the set of real numbers R by 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 2𝑎. Find the identity element under the operation ∗.
Small group discussions/Think- pair-share: Learners in small convenient groups (for instance, mixed gender, mixed-ability, etc.) investigate the elements of binary operation to establish that the inverse of an element 𝑎 of a set 𝑆 under an operation ∆ on 𝑆 is an element 𝑎 *! ∈ 𝑆 such that 𝑎∆𝑎 *! = 𝑎 *! ∆𝑎 = 𝑒, where 𝑒 is the identity element of 𝑆 under the operation ∇.
Example: A binary operation ∇ is defined on the set 𝑅 or real numbers by 𝑝 = 𝑝 + 𝑞 − 𝑝 where 𝑝 and 𝑞 ∈ 𝑅.Find the inverse element under the operation ∇ .
Solution: Let 𝑝 *! ∈ 𝑅 be the inverse element, then
- + 𝑝 *! = 𝑝 + 𝑝 *! − 𝑝 *! = 𝑒. 𝐵 𝑒 = 0. So 𝑝 + 𝑝 *! − 𝑝 *! = 0 ⟹ 𝑝 *! = !*+
- + Similarly, 𝑝 *! 𝛻 = 𝑝 *! + 𝑝 − 𝑝 *! 𝑝 = 0 ⟹ 𝑝 *! = !*+ .
Since the left inverse = right inverse, then the inverse element 𝑝 of 𝑆 under the
- + operation ∇ exists, and it is !*+
Example: Suppose the operation ∗ is defined on the set of real numbers R by 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 2𝑎. Find the inverse element under the operation ∗.
Activity 2: Commutative, Associative Distributive and Closure properties of binary operations.
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 explore the commutative, associative, distributive and closure properties of binary operations.
Small group discussions/Think-pair-share: Learners in small convenient groups (e.g., mixed gender, mixed-ability, etc.) investigate the commutative property of a binary operation and recognise from Cayley's tables that a binary operation is commutative if the table is symmetric along its principal diagonal.
Example: A binary operation ⨀ is defined on the set 𝑃 = {𝑥, 𝑦, 𝑧} by the table below. Determine whether the operation ⨀ is commutative.
⨀ 𝑥 𝑦 𝑧
𝑥 𝑥 𝑦 𝑧
𝑦 𝑦 𝑧 𝑥
𝑧 𝑧 𝑥 𝑦 Solution: By inspection we identify that 𝑥⨀𝑦 = 𝑦⨀𝑥 = 𝑦; 𝑥⨀𝑧 = 𝑧⨀𝑥 = 𝑧 and 𝑦⨀𝑧 = 𝑧⨀𝑦 = 𝑥 ∴ ⨀ is commutative Alternatively, since the table is symmetric along the principal diagonal, we conclude that the operation ⨀ is commutative.
⨀ 𝑥 𝑦 𝑧
𝑥 𝑥 𝑦 𝑧
𝑦 𝑦 𝑧 𝑥
𝑧 𝑧 𝑥 𝑦
Example: 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 determine whether or not the operation ∇ is commutative.
Inter-group competition: Learners from one group create binary operations for the other group to investigate the associative property of the operation. Groups switch roles. Thus, learners recognise that a binary operation ⊘ on a set 𝑅 is associative if (𝑎 ⊘ 𝑏) ⊘ 𝑐 = 𝑎 ⊘ (𝑏 ⊘ 𝑐) for all 𝑎, 𝑏 and 𝑐 ∈ 𝑅.
Inter-group competition: Learners from one group create two binary operations for the other group to investigate the distributive property of the operations. Group switch roles. Thus, learners recognise that two binary operations ∅ and on a set 𝑅 are such that associative if. (𝑎 ⊘ 𝑏)∆𝑐 = (𝑎∆𝑐) ⊘ (𝑏∆𝑐) for all 𝑎, 𝑏 and 𝑐 ∈ 𝑅
Inter-group competition: Learners from one group create binary operations over a given set, and the other group investigates the set is closed under the given operation. Group switch roles. Thus, learners recognise that a set 𝑆 is closed under a binary operation ⋇ if for any 𝑎, 𝑏 ∈ 𝑆 𝑎 ⋇ 𝑏 ∈ 𝑆
Example: The table below is defined by the operation ⊚ on the set 𝐵 = {𝑟, 𝑠, 𝑡, 𝑢 } ⊚ 𝑟 𝑠 𝑡 𝑢
𝑟 𝑠 𝑢 𝑟 𝑡
𝑠 𝑢 𝑡 𝑠 𝑟
𝑡 𝑟 𝑠 𝑡 𝑢
𝑢 𝑡 𝑟 𝑢 𝑠
- Find, giving reasons, whether or not α) B is closed under β) ⊚ is commutative γ) there is an identity element
- Find, where possible, the inverse of the elements of set B.
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.2). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding.
A binary operation ∇ is defined on the set 𝑆 = { 2,3,4,5} by 𝑝 = 𝑝 + 𝑞 − 𝑝 where 𝑝 and 𝑞 ∈ 𝑆. a) Construct the table for the operation ∇ on the set S. b) Determine whether or not the operation ∇ is i. closed under 𝑆, ii. commutative, iii. associative and iv. distributive over ∗ if 𝑎 ∗ 𝑏 = 𝑎 − 3𝑎, 𝑎, 𝑏 ∈ 𝑆