Applications of Algebra

Talk for Learning, Think-Pair-Share, building on what others say.
Learning Experience: Learners, in pairs, discuss statements and draw implications from those statements.
Activity 1: Implicative statements Learners in groups discuss the concept of implications.
Implication: Consider the following statements;
- If it is a dog, then it is an animal with four legs.
- If two triangles are congruent, then the two triangles are similar.
- If a = b, then a 2 = b 2 .
- If he has passed the examination, then he is promoted.
Each of these sentences consists of two clauses - the "if clause" and the "then clause". Each of the two clauses is a statement in itself. Let us consider the "if clause" as statement P and the "then clause" as statement Q. It is observed that in each case, if statement P is true, then it implies that statement Q is also true. This is expressed as P implies Q, symbolically P ⇒ Q Sentences of the type P ⇒ Q are referred to as Implication.
Statements 1, 2 and 3 may appear as follows:
- It is a dog ⇒ It is an animal with 4 legs
- Two triangles are congruent ⇒ Two triangles are similar
- a = b, ⇒a 2 = b 2
Note that the arrow "⇒" always points to the "then clause".
Example 1: Which implications are true or false;
- x 2 > 4, then x > 2
- If a number is a perfect square, then it is positive
Solution
- False, (-3) 2 > 4, (But -3 < 2)
- True
Converse of an Implication For all implications, the arrow "⇒" always points at the "then clause", as shown in the following implications:
- It is a dog⇒ It is an animal with 4 legs.
- Two triangles are congruent ⇒ Two triangles are similar.
- a = b, ⇒a 2 = b 2
If the arrow is reversed like this "⇐", the implications appear as follows:
It is a dog ⇐ It is an animal with four legs.
Two triangles are congruent ⇐ Two triangles are similar. a = b⇐a 2 = b 2 Implications a, b and c are the respective converses of the implications 1, 2 and 3.
For converse (1);
- It is a dog ⇐ It has four legs
This means that "if it has four legs, then it is a dog."
The implication is false because not only dogs have four legs. Since this implication is not true, it is written as:
It is a dog ⇐ It has four legs.
Activity 2: The Chain Rule of Implication
Learners in groups discuss the chain rule of implications.
Example 1 Consider the statements that P, Q and R are defined by: P: x > 7, Q : x > 5, R : x > 3 and the implications P ⇒Q : x > 7 ⇒x > 5 Q ⇒R : x > 5 ⇒ x > 3
This can be represented in a venn diagram as shown below: ∪ = {numbers} A = {numbers > 7} B = {numbers > 5} C = {numbers > 3}

It is observed that A ⊂ B ⊂ C ⇒ A ⊂ C. Thus, if a number is greater 7, then it is also greater than 3.
Generally, if P ⇒ Q and Q ⇒ R, the conclusively, P ⇒ R. This is called the chain rule of implication, written as P ⇒ Q ⇒ R.
Teaching and Learning Resources:
- SHS Curriculum, Graph boards, mathematical set, ICT tools
Assessment (3.1.2.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning.