Number and Algebraic Patterns

Think-pair-share, Talk for Learning, Project-Based Learning.
Learning Experience: Learners in pairs investigate and determine the identity element and use it to find the inverse of a given element
Activity 1: Learners in pairs investigate the identity element of addition and multiplication and establish the definition of identity element.
- Learners identify the identity element of addition and multiplication by solving the following questions: 7 − 𝑥 = 7, 5 × 𝑦 = 5, 12 + 𝑧 = 12
- Learners in pairs discuss and conclude that the identity element of addition is zero and the identity element of multiplication is one.
- Learners conclude that an identity element or neutral element leaves a combination unchanged or unaffected.
Through discussion, learners establish that, for a binary operation *, if there exist just one element e such that 𝑎 ∗ 𝑒 = 𝑒 ∗ 𝑎 = 𝑎 where 𝑎, 𝑒 ∈ ℝ a, then e is called an identity element.
Activity 2: Learners in groups investigate identity elements in a given table. Example 1: The combination table for the set 𝑄 = {𝑎, 𝑏, 𝑐, 𝑑} under the operation ∗ is given below
∗ a b c d 𝑎 𝑐 𝑎 𝑏 𝑑 𝑏 𝑎 𝑏 𝑐 𝑑 𝑐 𝑑 𝑐 𝑏 𝑎 𝑑 𝑏 𝑑 𝑎 𝑐
- State the identity element. Through brainstorming and discussion, learners find out that: 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎 = 𝑎 𝑏 ∗ 𝑏 = 𝑏 𝑐 ∗ 𝑏 = 𝑏 ∗ 𝑐 = 𝑐 𝑑 ∗ 𝑏 = 𝑏 ∗ 𝑑 = 𝑑 ∴ 𝑏 𝑖 𝑡ℎ𝑒 𝑖 𝑒
Learners in groups create and solve similar examples. Learners establish that the clue for identifying the identity element in a table is to look out for where the elements match the outer elements.
Activity 3: Learners in groups investigate how to find the identity element when given a binary definition.
Example 1: If 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 5, where 𝑎, 𝑏 ∈ 𝑅, find the identify element.
Solution: Definition of identity element: 𝑎 ∗ 𝑒 = 𝑒 ∗ 𝑎 = 𝑎 ∴ 𝑎 ∗ 𝑒 = 𝑎 + 𝑒 + 5 = 𝑎 𝑒 = −5 And 𝑒 ∗ 𝑎 = 𝑒 + 𝑎 + 5 = 𝑎 𝑒 = −5
Therefore, the identity element of 𝑎 ∗ 𝑒 = 𝑎 + 𝑏 + 5 is −5
- Using example 2, i.e. 𝑎 ∗ 𝑏 = 𝑎 − 𝑏, learners in groups discover that the binary operation does not have a unique identity element.
- Learners establish that it is important to check that both the left and right sides give a unique solution because for a binary operation to have an identity 𝑎 ∗ 𝑒 = 𝑒 ∗ 𝑎 = 𝑎
Activity 4: Through Talk for Learning, discuss the inverse of a binary operation and solve examples.
- Learners establish that: An element 𝑎 ∈ 𝐴 is called invertible if there exists an element 𝑏 ∈ 𝐴 such that 𝑎 ∗ 𝑏 = 𝑒 = 𝑏 ∗ 𝑎, which means 𝑎 ∗ 𝑎 *! = 𝑒 = 𝑎 *! ∗ 𝑎
- Learners in groups sole examples to determine the condition for finding the inverse of a binary operation
Examples: Find the inverse of 1. 𝑎 ∗ 𝑏 = 𝑎 − 𝑏 2. 𝑎 ∗ 𝑏 = 𝑎 ) + 𝑏 ) , &' 3. 𝑎 ∗ 𝑏 = , , if possible.
- Learners establish that it is not possible to find the inverse of examples 1 and 2 since the identity element is not unique.
- In Example 3, Learners discover that the binary operation has a unique identity element; therefore, the inverse can be found.
&' Solution: The Identity element for 𝑎 ∗ 𝑏 = is 4. If 𝑒 = 4, then let b be the inverse , such that 𝑎 ∗ 𝑏 = 𝑒 = 𝑏 ∗ 𝑎 ∴ 𝑎 ∗ 𝑏 = 𝑒 𝑎 ∗ 𝑎 *! = 𝑒 𝑎 *! =4 4
(𝑎) *! = 16 16 ∴ 𝑎 *! = 𝑎 And for 𝑏 ∗ 𝑎 = 𝑒 𝑎 *! ∗ 𝑎 = 𝑒 𝑎 *! 𝑎 =4 4 𝑎 𝑎 = 16
- ! 16 ∴ 𝑎 *! = 𝑎
Learners in groups research how to find the inverse element in a given table and present findings in class.
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.3). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 3 Strategic reasoning.
If 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 2, where 𝑎, 𝑏 ∈ ℝ, find the identity element Given that 𝑚 ∗ 𝑛 = 3𝑚 + 2𝑛 - 𝑚, investigate whether the operation * has a unique identity