The Math Lab for Young Minds: Logic, Geometry & Algebra
Winter 2024
Rs 5,500/- PKR
- Duration: 2 Weeks
- Starting: 26 December 2024
- Mode: Online
- Age: 11 and Above
- 5 Days a week
- Timing: 03:00 pm - 04:30 pm
About Course
This course introduces fundamental concepts in physics, mathematics, and computer science, using interactive simulations and hands-on activities. Students will explore motion, including horizontal and vertical movement, gravity, and projectile motion, with tools like GeoGebra. They'll deepen their understanding of classical mechanics by modeling falling objects, optimizing projectile trajectories, and discussing advanced topics such as Newton's cannonball and orbital motion. The course also introduces logical reasoning, proof techniques, and combinatorics, including graph theory, topological sorting, and pathfinding algorithms (DFS and BFS). By the end of the course, students will have developed a solid foundation in both mathematical proofs and practical applications, including probability simulations, to solve real-world problems.
What Will You Learn?
- Understanding basic principles of motion, including horizontal and vertical movement.
- Simulating simple motion scenarios using GeoGebra to visualize speed and trajectory.
- Exploring the effects of gravity on falling objects and dispelling common misconceptions.
- Analyzing free fall motion and graphing velocity and displacement over time.
- Mastering depth-first search (DFS) and breadth-first search (BFS) algorithms for pathfinding and problem solving.
- Simulating and optimizing projectile motion, exploring how velocity affects flight time and path.
- Studying Newton’s cannonball experiment and the concept of orbital motion around a spherical Earth.
- Learning logical reasoning and proof techniques, including direct proof and proof by contradiction.
- Understanding graph theory concepts, including vertices, edges, and directed acyclic graphs (DAGs).
- Combining horizontal and vertical motion to understand projectile trajectories.
Certifications
“The Math Lab for Young Minds: Logic, Geometry and Algebra”
Dec’2024-Jan’2025
Certified Students 🌟
Here you can view and download your certificates
With lifetime access to our lecture content,
Lecture 01: Reasonings - Propositional Logic and Visual proof
- Proposition
- Connectors and Truth Tables
- Not
- And/Conjunction
- Or/Disjunction
- XOR
- Quantifiers
- Universal Quntifiers
- Existential Quantifers
- Visual Proofs
- 1+2+3+4+…+97+98+99+100
- Geometric Proof
- Algebraic Proof
- 1+2+3+4+…+97+98+99+100
- Sum of Odd numbers:
- 1+3+5+7+…
- Geometric Proof
- 1+3+5+7+…
- (a+b)^2 = a^2 + 2ab + b^2
- (a-b)^2 = a^2 – 2ab + b^2
Lecture 02: Implications/Prove by Contropositive and Visual Proofs II
- Revision of Propositions and Quantifiers and Connectors
- Propositional Functions
- Making several propositions based on quantifiers
- Implications
- Truth Table
- English interpretation
- Given an Implication, drawing the conclusion
- Definitions (Evens, Odds, Primes, Composites)
- Proofs Examples
- Evens * Evens = Even
- Odd * Odd = Odd
- Implication and Contraposition
- How they are equivalent
- Examples Proofs: If an integer’s square is an odd number then the given number is an odd number.
- Visual Proofs
- Sum(1/2^i) = 1 where 1<=i<=n
- Sum(1/3^i) = 1/2
- Sum(1/4^i) = 1/3
Lecture 03: GeoGebra, Proof by Contradiction And Primes are Infinite
- Introduction to GeoGebra
- How to make shapes
- Points
- Line and perpendicular lines (1D)
- Polygons and regular polygons (2D)
- How to move shapes and play with them
- Concept of Equations, inputs and outputs
- Use of tools and other functionalities
- How to make shapes
- Visual Proofs using GeoGebra
- (a+b)^2=a^2+b^2+2ab
- (a-b)^2=a^2+b^2-2ab
- Revision of Implications
- Revision of Quantifiers
- Existential Quantifiers
- Universal Quantifiers
- Examples
- English Interpretation
- Proof By Contradiction
- Proof that Primes are infinite
Lecture 04: Rationals, Irrationals and De-Morgan's Law
- Discussion about Homework 1 & 2
- Necessary Conditions
- Sufficient Conditions
- Proofs
- If n is odd then n^2 is odd
- If n is even then n^2 is even
- If 3n+2 is odd then n is odd
- Closure Property
- Rational and Irrational Numbers
- Definitions
- Examples
- Proofs
- Rational+Rational=Rational
- Rational+Irrational=Irrational
- √2 is an Irrational number
- How to find Square Root
- Pythagoras Theorem
- De-Morgan’s Law
Lecture 05: Counting and Combinations, Biology of fingerprints and DNA
- Discussion of Homework 3
- Discussion about Konigsberg Bridge
- Counting Rules
- Product Rule
- Password Exmple
- Number Plate Example
- Sum Rule
- Gift Example
- Division Rule
- Handshakes Example
- Subtraction Rule
- Bit String Example
- Product Rule
- Biology of fingerprints
- Biology of DNA
Lecture 06: Pigeonhole Principle, Graphs and Adjacency Matrix
- Pigeonhole Principle
- Cake Slice Problem
- Euler’s idea of graphs
- Introduction to Graphs
- Edges
- Vertices
- Degree
- Reachability Problem
- Adjacency Graph Matrix
- Adjacency Graph List
- Facebook in terms of Graphs
- Unidirectional Graph
- Twitter in terms of Graphs
- Directional Graph
- Dependancy Graph
Lecture 07: GeoGebra, Graph Plotting and Geometric Shapes
- Introduction to Functions
- Making 2D shapes
- Plotting Graphs on GeoGebra
- Plotting Square Side
- Plotting Rectangle sides
- Moving Car using Functions
- Plotting graphs of
- x^2, x^3, x^4, x^5
- x^ 1/2 , x^ 1/3, x^ 1/4
- x^ 1/100, x^ 1/1000
- Making sliders in GeoGebra
- Using tools in GeoGebra
Lecture 08: GeoGebra; Curves and their Traslation
- Use of Functions
- 1D motion of a car
- Function for Speed of Car
- 2D motion of car using two funtions
- Curves in Graphs
- Translation of Curves
- Shifts in Graphs
- Horizontal Shift
- Vertical Shift
- y=f(x)+k
- Horizontal Shift
- Scaling of Graphs
Lecture 09: GeoGebra; Circular, Eliptical and Free Fall Motion
- Vectors
- Movement from a vector
- Movement on a changing velocity vector
- Movement from a velocity vector and acceleration vector
- Free Fall motion of cars
- Eliptical motion
- Circular motion
- Scripting Derivative
- Scripting Integration
Lecture 10: GeoGebra; Newton's Canon Ball Experiment
- Planetary motion
- How planets orbit around the Sun
- Newton’s CanonBall Experoment’
- Velocity and acceleration in GeoGebra
- Simulating Free fall Motion
- Centripetal Acceleration and Force