BES-143 Jan 2024

How knowledge in mathematics is validated and proved? Discuss with the help of suitable examples.

Answer:

Validation and Proof in Mathematics

1. Logical Reasoning

1. Given: A right-angled triangle with sides $a$$a$aa$a$, $b$$b$bb$b$, and hypotenuse $c$$c$cc$c$.
2. To Prove: $a^2+b^2=c^2$$a^2+b^2=c^2$a^(2)+b^(2)=c^(2)a^2 + b^2 = c^2$a^2+b^2=c^2$.
3. Construction: Construct a square with side length $a+b$$a+b$a+ba + b$a+b$. Within this square, place four right-angled triangles, leaving a smaller square in the center with side length $c$$c$cc$c$.
4. Area Calculation:
   • Total area of the large square: $(a+b{)}^2$$(a+b{)}^2$(a+b)^(2)(a + b)^2$(a+b{)}^2$.
   • Area of four triangles: $4\times \frac{1}{2}ab=2ab$$4\times \frac{1}{2}ab=2ab$4xx(1)/(2)ab=2ab4 \times \frac{1}{2}ab = 2ab$4\times \frac{1}{2}ab=2ab$.
   • Area of the smaller square: $c^2$$c^2$c^(2)c^2$c^2$.
5. Equating Areas:
   • $(a+b{)}^2=4\times \frac{1}{2}ab+c^2$$(a+b{)}^2=4\times \frac{1}{2}ab+c^2$(a+b)^(2)=4xx(1)/(2)ab+c^(2)(a + b)^2 = 4 \times \frac{1}{2}ab + c^2$(a+b{)}^2=4\times \frac{1}{2}ab+c^2$.
   • Expanding and simplifying: $a^2+2ab+b^2=2ab+c^2$$a^2+2ab+b^2=2ab+c^2$a^(2)+2ab+b^(2)=2ab+c^(2)a^2 + 2ab + b^2 = 2ab + c^2$a^2+2ab+b^2=2ab+c^2$.
   • Therefore, $a^2+b^2=c^2$$a^2+b^2=c^2$a^(2)+b^(2)=c^(2)a^2 + b^2 = c^2$a^2+b^2=c^2$.

2. Axioms and Postulates

3. Theorems and Proofs

• Fermat’s Last Theorem was famously conjectured by Pierre de Fermat in 1637 and remained unproven for 358 years.
• Andrew Wiles, in 1994, provided a proof using sophisticated techniques from algebraic geometry and modular forms, specifically the Taniyama-Shimura-Weil conjecture.
• The proof, although complex, follows a logical structure, verifying the theorem’s validity through meticulous argumentation and mathematical constructs.

4. Inductive and Deductive Reasoning

• Inductive Reasoning: Involves observing patterns and forming conjectures. For example, observing the sequence 2, 4, 8, 16, one might conjecture that the $n$$n$nn$n$-th term is $2^n$$2^n$2^(n)2^n$2^n$. While useful for hypothesis formation, induction alone does not constitute a proof.
• Deductive Reasoning: Involves proving statements by logically deducing consequences from axioms and theorems. For example, proving that the sum of the angles in a triangle is 180 degrees involves deductive reasoning based on the properties of parallel lines and transversals.

5. Proof by Contradiction

1. Assume the opposite: $\sqrt{2}$$\sqrt{2}$sqrt2\sqrt{2}$\sqrt{2}$ is rational.
2. Representation: Then, $\sqrt{2}=\frac{a}{b}$$\sqrt{2}=\frac{a}{b}$sqrt2=(a)/(b)\sqrt{2} = \frac{a}{b}$\sqrt{2}=\frac{a}{b}$ where $a$$a$aa$a$ and $b$$b$bb$b$ are coprime integers.
3. Squaring both sides: $2=\frac{a^2}{b^2}$$2=\frac{a^2}{b^2}$2=(a^(2))/(b^(2))2 = \frac{a^2}{b^2}$2=\frac{a^2}{b^2}$, leading to $a^2=2b^2$$a^2=2b^2$a^(2)=2b^(2)a^2 = 2b^2$a^2=2b^2$.
4. Implication: $a^2$$a^2$a^(2)a^2$a^2$ is even, so $a$$a$aa$a$ is even. Let $a=2k$$a=2k$a=2ka = 2k$a=2k$.
5. Substitute and simplify: $4k^2=2b^2$$4k^2=2b^2$4k^(2)=2b^(2)4k^2 = 2b^2$4k^2=2b^2$, thus $b^2=2k^2$$b^2=2k^2$b^(2)=2k^(2)b^2 = 2k^2$b^2=2k^2$, implying $b$$b$bb$b$ is even.
6. Contradiction: Both $a$$a$aa$a$ and $b$$b$bb$b$ are even, contradicting the assumption that they are coprime.
7. Conclusion: $\sqrt{2}$$\sqrt{2}$sqrt2\sqrt{2}$\sqrt{2}$ is irrational.

Conclusion

Discuss the differences between inductive and deductive approaches. Discuss with the help of an example how these approaches work in the mathematics classroom.

Answer:

Differences Between Inductive and Deductive Approaches

Inductive Approach

Deductive Approach

Key Differences

1. Starting Point:
   • Inductive: Begins with specific observations or examples.
   • Deductive: Begins with general principles or axioms.
2. Nature of Reasoning:
   • Inductive: Moves from specific to general.
   • Deductive: Moves from general to specific.
3. Outcome:
   • Inductive: Results in conjectures or hypotheses that need further verification.
   • Deductive: Results in definite conclusions that are logically certain if the premises are true.
4. Proof:
   • Inductive: Provides strong evidence but not absolute proof.
   • Deductive: Provides conclusive proof.

Example in the Mathematics Classroom

Inductive Approach

1. Observation and Pattern Recognition:
   • The teacher presents the sum of the first few natural numbers:
     • $1+2=3$$1+2=3$1+2=31 + 2 = 3$1+2=3$
     • $1+2+3=6$$1+2+3=6$1+2+3=61 + 2 + 3 = 6$1+2+3=6$
     • $1+2+3+4=10$$1+2+3+4=10$1+2+3+4=101 + 2 + 3 + 4 = 10$1+2+3+4=10$
   • Students observe these sums and look for a pattern.
2. Generalization:
   • Students might notice that the sums 3, 6, 10, and so on, can be arranged in a triangular pattern. They may conjecture that the sum of the first $n$$n$nn$n$ natural numbers is given by a specific formula.
   • The teacher guides them to write down the sums and look for a pattern:
     • $S_1=1$$S_1=1$S_(1)=1S_1 = 1$S_1=1$
     • $S_2=1+2=3$$S_2=1+2=3$S_(2)=1+2=3S_2 = 1 + 2 = 3$S_2=1+2=3$
     • $S_3=1+2+3=6$$S_3=1+2+3=6$S_(3)=1+2+3=6S_3 = 1 + 2 + 3 = 6$S_3=1+2+3=6$
     • $S_4=1+2+3+4=10$$S_4=1+2+3+4=10$S_(4)=1+2+3+4=10S_4 = 1 + 2 + 3 + 4 = 10$S_4=1+2+3+4=10$
3. Formulating a Conjecture:
   • After examining the pattern, students may conjecture that the sum of the first $n$$n$nn$n$ natural numbers is given by the formula $S_n=\frac{n(n+1)}{2}$$S_n=\frac{n(n+1)}{2}$S_(n)=(n(n+1))/(2)S_n = \frac{n(n+1)}{2}$S_n=\frac{n(n+1)}{2}$.

Deductive Approach

1. Stating the Formula:
   • The teacher starts with the formula $S_n=\frac{n(n+1)}{2}$$S_n=\frac{n(n+1)}{2}$S_(n)=(n(n+1))/(2)S_n = \frac{n(n+1)}{2}$S_n=\frac{n(n+1)}{2}$ and aims to prove it deductively.
2. Using Mathematical Induction (A Form of Deductive Proof):
   • Base Case:
     • For $n=1$$n=1$n=1n = 1$n=1$, $S_1=1$$S_1=1$S_(1)=1S_1 = 1$S_1=1$, and the formula gives $\frac{1(1+1)}{2}=1$$\frac{1(1+1)}{2}=1$(1(1+1))/(2)=1\frac{1(1+1)}{2} = 1$\frac{1(1+1)}{2}=1$, which is true.
   • Inductive Step:
     • Assume the formula holds for $n=k$$n=k$n=kn = k$n=k$, i.e., $S_k=\frac{k(k+1)}{2}$$S_k=\frac{k(k+1)}{2}$S_(k)=(k(k+1))/(2)S_k = \frac{k(k+1)}{2}$S_k=\frac{k(k+1)}{2}$.
     • Show that it holds for $n=k+1$$n=k+1$n=k+1n = k + 1$n=k+1$.
     • Consider $S_{k+1}=1+2+\dots +k+(k+1)$$S_{k+1}=1+2+\dots +k+(k+1)$S_(k+1)=1+2+dots+k+(k+1)S_{k+1} = 1 + 2 + \ldots + k + (k + 1)$S_{k+1}=1+2+\dots +k+(k+1)$.
     • By the inductive hypothesis, $S_k=\frac{k(k+1)}{2}$$S_k=\frac{k(k+1)}{2}$S_(k)=(k(k+1))/(2)S_k = \frac{k(k+1)}{2}$S_k=\frac{k(k+1)}{2}$.
     • Therefore, $S_{k+1}=S_k+(k+1)=\frac{k(k+1)}{2}+(k+1)$$S_{k+1}=S_k+(k+1)=\frac{k(k+1)}{2}+(k+1)$S_(k+1)=S_(k)+(k+1)=(k(k+1))/(2)+(k+1)S_{k+1} = S_k + (k + 1) = \frac{k(k+1)}{2} + (k + 1)$S_{k+1}=S_k+(k+1)=\frac{k(k+1)}{2}+(k+1)$.
     • Simplifying: $S_{k+1}=\frac{k(k+1)+2(k+1)}{2}=\frac{(k+1)(k+2)}{2}$$S_{k+1}=\frac{k(k+1)+2(k+1)}{2}=\frac{(k+1)(k+2)}{2}$S_(k+1)=(k(k+1)+2(k+1))/(2)=((k+1)(k+2))/(2)S_{k+1} = \frac{k(k+1) + 2(k+1)}{2} = \frac{(k+1)(k+2)}{2}$S_{k+1}=\frac{k(k+1)+2(k+1)}{2}=\frac{(k+1)(k+2)}{2}$.
     • Hence, the formula holds for $n=k+1$$n=k+1$n=k+1n = k + 1$n=k+1$.
3. Conclusion:
   • By mathematical induction, the formula $S_n=\frac{n(n+1)}{2}$$S_n=\frac{n(n+1)}{2}$S_(n)=(n(n+1))/(2)S_n = \frac{n(n+1)}{2}$S_n=\frac{n(n+1)}{2}$ is proven true for all natural numbers $n$$n$nn$n$.

Application in the Classroom

• Students could be given a series of numbers to sum and asked to observe and record patterns. They may use visual aids like dot diagrams to see the triangular numbers forming, encouraging exploration and conjecture formation.

• The teacher can introduce the formula and guide students through a deductive proof using mathematical induction, emphasizing the logical steps and reasoning. This approach teaches students how to derive specific results from general principles and how to construct rigorous mathematical proofs.

What do you understand about the use of concept mapping in unit planning? Select a unit from secondary level mathematics. Develop a unit plan through concept mapping on the selected unit.

Answer:

Understanding Concept Mapping in Unit Planning

Selected Unit: Quadratic Equations (Secondary Level Mathematics)

Developing a Unit Plan through Concept Mapping

Step 1: Identify Key Concepts

1. Introduction to Quadratic Equations
2. Standard Form of Quadratic Equations
3. Methods of Solving Quadratic Equations
   • Factoring
   • Completing the Square
   • Quadratic Formula
4. Graphing Quadratic Equations
5. Applications of Quadratic Equations

Step 2: Create the Concept Map

1. Introduction to Quadratic Equations
   • Definition
   • Examples
   • Differences from linear equations
2. Standard Form of Quadratic Equations
   • General form: $a{x}^2+bx+c=0$$a{x}^2+bx+c=0$ax^(2)+bx+c=0ax^2 + bx + c = 0$a{x}^2+bx+c=0$
   • Identifying coefficients (a, b, c)
3. Methods of Solving Quadratic Equations
   • Factoring
     • Finding factors
     • Zero-product property
   • Completing the Square
     • Steps to complete the square
     • Deriving the vertex form
   • Quadratic Formula
     • Derivation of the formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$x=(-b+-sqrt(b^(2)-4ac))/(2a)x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
     • Discriminant analysis ($\mathrm{\Delta }=b^2-4ac$$\mathrm{\Delta }=b^2-4ac$Delta=b^(2)-4ac\Delta = b^2 – 4ac$\mathrm{\Delta }=b^2-4ac$)
     • Types of roots based on the discriminant
4. Graphing Quadratic Equations
   • Parabolic shape
   • Vertex
   • Axis of symmetry
   • Direction of opening (upwards/downwards)
   • Finding the vertex and intercepts
5. Applications of Quadratic Equations
   • Real-world problems
   • Projectile motion
   • Area optimization problems

Step 3: Develop the Unit Plan

• Day 1: Introduction to quadratic equations. Discuss real-life situations where quadratic equations are used.
• Day 2: Standard form of quadratic equations. Identifying coefficients.
• Day 3: Practice problems on identifying the standard form and coefficients.

• Day 1: Introduction to factoring method. Step-by-step process with examples.
• Day 2: Practice problems on factoring.
• Day 3: Introduction to completing the square method. Steps involved.
• Day 4: Practice problems on completing the square.

• Day 1: Derivation of the quadratic formula.
• Day 2: Practice problems using the quadratic formula.
• Day 3: Introduction to the discriminant. Types of roots based on discriminant.
• Day 4: Practice problems on discriminant analysis.

• Day 1: Introduction to the parabolic shape and vertex form.
• Day 2: Finding the vertex, axis of symmetry, and intercepts.
• Day 3: Practice graphing quadratic equations.
• Day 4: Real-life applications of graphing quadratic equations.

• Day 1: Solving real-world problems using quadratic equations.
• Day 2: Problems on projectile motion.
• Day 3: Problems on area optimization.
• Day 4: Review and unit test.

Concept Map Visual Representation

                             Quadratic Equations
                                    /       |        \
          Introduction    Standard Form    Methods of Solving
            /             /         |        \
Definition  Examples     Form  Identifying  Solving Methods
                                Coefficients      /       |        \
                                        Factoring  Completing  Quadratic Formula
                                         |                       the Square                 |
                                     Steps                Steps                Derivation
                                      Practice           Practice           Discriminant
                                                                                           Analysis

                             |   Graphing Quadratic Equations   |
                             |  Parabolic Shape     |
                             |  Vertex, Axis of Symmetry   |
                             |  Intercepts     |
                             |  Practice Graphing   |
                
              |   Applications of Quadratic Equations  |
              |  Real-World Problems  |
              |  Projectile Motion  |
              |  Area Optimization  |