A Packet of Problems & Waffles

There is no doubt I love Waffles. There is no doubt I love Problems.

Yes, I had faced Problems while having Waffles. Waffles made me call my girlfriend’s name by a different name, which sounds similar. The waffles just twisted my tongue. Nevertheless, I still love waffles.

What happened after that? She just waffled me. XD.

But, this time I decided to give a try to solve problems with waffles together. This idea of a Packet of Problems just emerged from using the structure of waffles to develop a pedagogical method. I myself get excited to learn by this specific pedagogical method. Cheenta Ganit Kendra is currently adhering to this method to create Forerunner Problems, Quizzes and deliver our Problem-Solving classes.

Let’s start discussing this.

The New Pedagogy

The idea is similar to use problems to teach a Mathematical Theory. This is infamous as Moore’s Method. But for Moore’s Method is restricted to advanced mathematics. We have decided to generalize to teach it to everyone, by step by step problems, leading to the theory or derivation of a result. This is absolutely rewarding when a student solves in this method. We have used a similar method to provide solutions to a problem called Sequential Hints, rather a pet name for dividing a solution into important checkpoints.

Here, the Packet of Problems is to use problems step by step with basic knowledge to arrive at a higher mathematical result or proof, just like biting a small box of waffles one at a time.

Let’s give examples.

Example 1

I used the following packet of problems, to help a student to start counting and develop their prowess in Combinatorics from an early age. I got an overwhelming response from the students in terms of the quick slope of their learning curve. So, I am excited about this.

  • Draw a 2×2 square. Place two rooks on the square such that they don’t attack each other. Determine all such positions. [ Basic Counting ]
  • Can you place 3 rooks in a 2×2 square, so that they don’t attack each other? [ Pigeon Hole Principle ]
  • Let’s draw a 3×3 square. How many maximum rooks can you place on this square, so that they don’t attack each other? [ Pigeon Hole Principle ]
    ( This is the crux because there are lots of positions to explore. Therefore, they have to find an elegant way instead of computing all the cases. )
  • What about the same question for 100×100 square. Let’s draw a 100×100 square. How many maximum rooks can you place on this square, so that they don’t attack each other?
  • Can you draw all possible positions of three rooks on the 3×3 square board? [ Basic Counting ]
  • Can you draw all possible positions of three rooks on the 100×100 square board? [ Multiplication Principle ]

I used the first four problems, with a child of class 3, who just loves to play chess and he answered them all perfectly with logic. Yet, he doesn’t know algebra. He just loves to sit and think. What do you think, we could have done better to improve the problems? Reply in the comments section.

Example 2

The following is about Quadratic Equations. This is aimed at students, who are familiar with what is a quadratic equation and the definition of a root and don’t hate to sit, think, play and calculate with stuff.

  • Show that \(x^2 = x.x\) is never negative for any number x.
  • Show that \(x^2 + 1 = 0\) has no real roots.
  • Show that the roots of \(x^2 – 4 = 0\) are 2, -2.
  • Show that the roots of \(x^2 – 5 = 0\) are \(\sqrt{5}, – \sqrt{5} \).
  • Find the roots of \((x+1)^2 – 5 = 0\).
  • Find the roots of \(a(x+b)^2 – b = 0\).
  • Show that \( px^2 + qx + r = p(x+\frac{q}{2p})^2 -( \frac{q^2}{4p} – r ) = 0\).
  • Hence, find that roots of \( px^2 + qx + r = 0 \).

Thus, a student can easily use these problems to discover her own formula of quadratic equations easily, with thrill and pleasure. We used this problem packet in our Level 1A classes. The students were super excited to learn.

This is one of the best things, we can offer to a student to help them grow with better problem solving and love mathematics, who just loves to sit, think and play mathematics.

Just Waffle and Bewaffle Mathematics! Stay Tuned!

#waffle

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