“It is better to solve one problem five different ways, than to solve five problems one way.” ~ George Polya

Many years ago I had an instructor who used to read the “magic” tricks from *The Mad Book Of Magic And Other Dirty Tricks*^{1} during class to break the monotony.

Once he got our attention away from the classwork and onto the trick, he would then say: “Let me tell you how the trick is done.” He would then read the next page that explained the trick.

After reading, he would then laugh and so would we.

The story above illustrates a perceived problem I see on people’s math tricks posts on *Instagram* and *YouTube*: The trick is presented but not how the trick is done, i.e., Why does the trick work? What is the mathematics behind the trick?

### The Butterfly Method In Fractions

From *TeachableMath*^{2}: “In our opinion, tricks like the butterfly method should be avoided when students are first introduced to fractions. There are several reasons, e.g.,

- There is no conceptual understanding in the instruction.
- It reinforces the belief that fractions is just a bunch of tricks.
- What happens if you add three or more fractions?”

If we teach students conceptually how to add and subtract functions with unlike denominators using both models, as shown below, and the least common denominator (LCD) method (on the *MathIsFun* webpage^{4}), then students will understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

### Vedic Math Tricks

Let’s start with an example from the *Vedicfeed*^{5} website:

**Sutra 3. Urdva – Triyagbhyam**

Vertically and crosswise: For multiplication of any two two-digit numbers, using **45 x 87** as an example:

**Step 1:** Multiply the last digits of the two numbers.

5 x 7 = 35

**Step 2:** Multiply numbers diagonally and add them.

(4 x 7) + (5 x 8) = 28 + 40 = 68

**Step 3:** Place Step 1 at the end and Step 2 at the beginning.

68 | 35

**Step 4:** Multiply the first digit both numbers and put it at the beginning.

4 x 8 = 32

32 | 68 | 35

**Step 5:** For the final result (i.e., the product), more than 2 or more digits, add the beginning digits to the beginning numbers.

45 x 87 = 32 | 68 | 35 = 32 | 68 + **3 | 5** = 32 | 71 | 5 = 32 + **7** | 15 = 3915

What is the bottom line?

- I agree with
*Cuemath*: “However fascinating it might be to calculate faster using Vedic mathematics tricks, it fails to make a student understand the concepts, applications, and real-life scenarios of those particular problems.” - According to the
*Vedantu*website Vedic Mathematics is needed to pass tests (e.g., JEE, ICSE and CBSE). - To enrich a student’s understanding of mathematics, Vedic Mathematics should be introduced once the student understands why these methods work and when they can be applied.
- Even if the person is not a genius and still can’t get it, those who post math tricks should be sure post the solution so the user can learn a technique that can be used to solve those types of problems.

“8 Vedic Maths Tricks: Calculate 10x Faster”. 2021. *Vedantu*. https://www.vedantu.com/blog/vedic-maths-tricks.

**What Is Meant By Vedic Mathematics?**

The term ‘Vedic’ came from a Sanskrit word ‘Veda’, that means ‘Knowledge’. And, Vedic Math is a super collection of sutras^{3} to solve math problems in a faster & easy way.**What Are The Benefits Of Learning Vedic Mathematics?**

You can solve any difficult/ time-consuming JEE problem or ICSE/CBSE Math immediately using Vedic Math Tricks. Moreover, just by using Vedic Math you can solve a problem mentally and that’s the beauty of Vedic Maths. While you encounter polynomial functions & quadratic sums in a higher class in CBSE or ICSE Board, knowledge of Vedic Math will lend a helping hand to beat the difficulty level of those sums.

“Vedic Maths| Tricks And Importance”. 2021. *Cuemath*. https://www.cuemath.com/learn/vedic-maths-tricks/.

Vedic Maths is a collection of techniques/sutras to solve mathematical problem sets in a fast and easy way. These tricks introduce wonderful applications of Arithmetical computation, theory of numbers, mathematical and algebraic operations, higher-level mathematics, calculus, and coordinate geometry, etc.

It is very important to make children learn some of the Vedic maths tricks and concepts at an early stage to build a strong foundation for the child. It is one of the most refined and efficient mathematical systems possible.

Vedic maths was discovered in the mid-1900s and has certain specific principles to perform various calculations in mathematics. But the question that arises is that is mathematics only about performing calculations?**However fascinating it might be to calculate faster using Vedic mathematics tricks, it fails to make a student understand the concepts, applications, and real-life scenarios of those particular problems.**

### Cube Root Trick

I have seen the **Special Cube Roots** trick on a number of different personal *Instagram* accounts and presented in different ways. Interesting trick but raises the following questions.

- How do you tell if a number is a perfect cube, e.g., 19683, so you can use the trick shown below? This is not mentioned in the posts.
- Where and when would you use this trick?
- Is it worth teaching?

“How To Calculate The Cube Root Of Any Number Easily Without A Calculator (Vedic Maths Trick) – Fully Electronics”. 2020. *Fully Electronics*. https://fullyelectronics.com/how-to-calculate-cube-root-of-any-number-easily-without-calculator-vedic-maths-trick/.

This article discusses the short trick to find out **the cube root of a perfect cube** in less than 5 seconds without the use of a calculator.

This math trick allows you to work out the cube root of any number – NOT JUST PERFECT CUBES – instantly. With decimals. With ease. Can you work faster than a calculator? With this tecmath trick you just might! The math shortcut magic is back!

### Only a Genius

Or 99% of the people fail to answer. Here is an example that is written on a white board on which the viewer is asked to solve.

1 + 4 = 5

2 + 5 = 12

3 + 6 = 21

8 + 11 = ?

However, the *Instagram* post does not tell you how the trick is done! One way to answer the question is shown below, which is basically using a defined function: f( x, y ) = ( x * y ) + x

1 + 4 = 5 = (1 * 4) + 1

2 + 5 = 12 = (2 * 5) + 2

3 + 6 = 21 = (3 * 6) + 3

8 + 11 = ? = **(8 * 11) + 8 = 96**

The above can also be solved in another way by seeing that the pattern is the sum of the two numbers plus the previous sum, i.e.,

1 + 4 = 5

2 + 5 + **5** = 12

3 + 6 + **12** = 21

4 + 7 + **21** = 32

5 + 8 + **32** = 45

6 + 9 + **45** = 60

7 + 10 + **60** = 77

8 + 11 + **77** = 96

Remember, as with other sequences/patterns, a minimum of three items are need to establish a pattern.

## Bottom Line

Teaching is more than filling your student’s brains with tricks and facts, or getting students to memorize content and pass tests. Teaching is about creating thinkers and helping them to understand the *why* along with the *how*.

“The mind is not a vessel to be filled, but a fire to be ignited.”

Plutarch

## References

^{1} “The Mad Book Of Magic And Other Dirty Tricks : Jaffee, Al : Free Download, Borrow, And Streaming : Internet Archive”. 2021. *Internet Archive*. https://archive.org/details/madbookofmagicot00jaff.

^{2} “The Butterfly Method In Fractions And The Danger Of Overemphasizing Tricks – TeachableMath”. 2016. *TeachableMath*. https://teachablemath.com/butterfly-method-fractions-danger-overemphasizing-tricks/.

^{3} “16 Sutras, Or Mathematical Formulas, Found In The Vedas”. 2021. *Learn Religions*. https://www.learnreligions.com/vedic-math-formulas-1770680.

Vedic Math essentially rests on the 16 Sutras, or mathematical formulas, as referred to in the Vedas. (The Vedas are most ancient Hindu scriptures, written in early Sanskrit and containing hymns, philosophy, and guidance on ritual for the priests of Vedic religion. Believed to have been directly revealed to seers among the early Aryans in India, and preserved by oral tradition, the four chief collections are the Rig Veda, Sama Veda, Yajur Veda, and Atharva Veda.)

^{4} “Least Common Denominator”. 2021. *mathsisfun.com*. https://www.mathsisfun.com/least-common-denominator.html.

Uses pizza slices to demonstrate how to calculate the smallest number (LCD) that can be used for all denominators of 2 or more fractions.

^{5} “25+ Vedic Maths Tricks In Simplified Version”. 2017. *Vedicfeed*. https://vedicfeed.com/vedic-maths-tricks/.

## Additional Reading

Muller, Gretchen. 2022. “George Polya”. *cmc-math.org*. https://www.cmc-math.org/george-polya.

Dr. Polya was a distinguished mathematician and professor at Stanford University. Polya (1887-1985) made important contributions to probability theory, number theory, the theory of functions, and the calculus of variations. He was the author of the classic works How to Solve It, Mathematics and Plausible Reasoning, and Mathematical Discovery, which encouraged students to become thoughtful and independent problem solvers. He was an honorary member of the Hungarian Academy, the London Mathematical Society, and the Swiss Mathematical Society, and a member of the (American) National Academy of Sciences, the American Academy of Arts and Sciences, and the California Mathematics Council, as well as a corresponding member of the Academie des Sciences in Paris.