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Follow Five Year-Old Math Wizard

Let me start from an equation of five year-old prodigy:

1 + 3 = 2 * 2,

1 + 3 + 5 = 3 * 3,

…

After 15-20 years he may already wrote the same equation in a following form:

Σ j = m², where j, m: j = 1, 3, 5,..., 2*m-1; m = 1, 2, 3,... : (1)

The same genuine diamond in an expanded expression:

1 + 3 = 2², as we have 2 odd numbers in a sum

1 + 3 + 5 = 3², as we have 3 odd numbers in a sum

. . . . .

1 + 3 + ... + (2 * m - 1) = m², for m = 1, 2, ...

Students start to learn exponents in grades 5, 6. We may simplify the formula to make it accessible to students of lower grades:

1 = 1 * 1

1 + 3 = 2 * 2

1 + 3 + 5 = 3 * 3

1 + 3 + 5 + 7 = 4 * 4

1 + 3 + 5 + 7 + 9 = 5 * 5

. . . . .

Give your students a problem set to write down all the equations for

m = 1, 2,...,10. They should catch the fire.

Before going any further, I would like to invite you to the popular forum Math is Fun, where I placed few posts and a game to play and learn the above equality.

Here is an interactive game for students in grades 2 - 4:

- step 1: produce random number by Fingerplay method* (see at the bottom):

e.g., n = 5

- step 2: first student sets up the problem,

e.g., for random n = 5, we have 1 + 3 + 5 + 7 + 9,

other student has to come with a solution 5 * 5 = 25.

Continue in a similar creative spirit.

Following the above interactive games, ask students to create their own number relations on a graph paper. After 10 - 15 minutes suggest to exchange the output with fellow student to solve it. Repeat the process for couple weeks or even months and you will gather few new formulas for our world to enjoy.

To incite creativity, students should start from finding as many solutions for target number as possible. E.g., in how many ways student may present math solutions for number 6?

2 + 4 = 6, 1 + 5 = 6, 3 + 3 = 6, 2.5 + 3.5 = 6, (-1) + 7 = 6, (-2) + 7 = 6,

8 - 2 = 6, 7 - 1 = 6, 9 - 3 = 6, 8.4 - 2.4 = 6,

2 * 3 = 6, 4 * 1.5 = 6, 5 * 1.2 = 6, 1 * 6 = 6

12 / 2 = 6, 18 / 3 = 6, 15 / 2.5 = 6

[ ∫ x^5 dx ] ^(-1) = 6, integrate in a range of [0,1]:

1 -1

[ ∫ x^5 dx ] = 6

It would be a great honor to challenge your students to produce similar number plays. Best wishes in this creative endeavor.

This page is dedicated to similar number plays or playful math.

Just recently just came to me very simple formula:

2 + 1 = 2² - 1²

3 + 2 = 3² - 2²

4 + 3 = 4² - 3²

. . .

It took a while for me to comprehend that it comes from well know algebraic formula

a² - b² = (a + b) * (a - b), where (a - b) = 1, because we take consecutive numbers.

Factoring the difference of two squares comes in algebra curriculum, but playing with numbers in a simplified version is fun opportunity for earlier grades:

2 + 1 = 2* 2 - 1 *1

3 + 2 = 3 * 3 - 2 * 2

4 + 3 = 4 * 4 - 3 *3

. . .

* * * * *

Let's come back to five year old formula.

The day I read about the prodigy's formula to my wife, she enjoyed a brilliancy of the numbers' play. Next morning my darling suggested that the same result you may get not only by taking m, as amount of numbers in a sum, but also as a median number of the set: e.g., median for {1, 3, 5} is 3 and we already know that 1 + 3 + 5 = 3².

I checked how does it work for even amount set such as {1, 3, 5, 7} with 4 numbers: a median for this set is (3 + 5) / 2 = 4, then 1 + 3 + 5 + 7 = 4 * 4. Indeed it worked. And, if you remember, it came from Suzuki violin teacher. It challenged me at once, as I have a degree in math.

I thought of 3rd way to define number amount m:

add two border numbers of the set and divided by two. It works for even and odd number sets:

odd #: for {1, 3, 5, 7, 9} we have (1 + 9) / 2 = 5, thus 1 + 3 + 5 + 7 + 9 = 5 * 5

even #: for {1, 3, 5, 7} we have (1 + 7) / 2 = 4, thus 1 + 3 + 5 + 7 = 4 * 4

To find similar working number relations was the matter of playing mind games with numbers and sometimes kids are good at this. The beauty of the process is in its purity. It cleans you out by bringing forward your creative energies. It also provides enormous satisfaction and everyone may experience the bliss. Just give yourself time and play with numbers.

Many parents, who are searching for engaging form of activities with numbers, may consider proposed path of invoking a creative output. Start from simple number plays and let the path evolve naturally. Important: play 10-15 minutes daily. Good place to start and nourish your love for number plays is a Sudoku game which may be found here or there.

Free publication “Playful Math. Blank Grid to Incite Your Creativity” contains a section “A Word to Teacher Inside You” which discusses the playfulness in detail. You may find it at www.lulu.com/skymus

And lastly, a Prodigy question:

What was a name of the five year-old prodigy child, who discovered the equation: 1 + 3 + 5 = 3 * 3 ?

Fingerplay method* - on a count of four, two students toss out one hand fingers from 0 to 3 and then add them. The sum gives random number in a range 0 - 6. If 0, repeat the toss. Students may rhyme the count of four with their favorite rhyme or as follows: Ready - Set and - Ready - Go!

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