Teach-in with Arik

Multiplication as visible area

A product can be understood by dividing a rectangle into simpler regions. For a square, this makes the identity (x + y)² = x² + 2xy + y² directly visible rather than merely symbolic.

• 11²: Split 11 into 10 + 1: 100 + 10 + 10 + 1 = 121.
• 21²: Split 21 into 20 + 1: 400 + 20 + 20 + 1 = 441.
• General form: The four regions of an (x + y)-by-(x + y) square have areas x², xy, yx, and y².

Mental multiplication by decomposition

The group practiced seeing each product as four manageable regions, building familiarity without relying on the standard carrying algorithm.

• 21 × 22: (20 + 1)(20 + 2) = 400 + 40 + 20 + 2 = 462.
• 51²: 50² + 50 + 50 + 1 = 2,601.
• 1,001²: 1,000² + 1,000 + 1,000 + 1 = 1,002,001.
• 1,024²: Using 1,024 = 1,000 + 24 gives 1,000,000 + 24,000 + 24,000 + 576 = 1,048,576.

Cubes and square roots

The area model was extended into three dimensions. A cube with side 11 contains one 1,000-block, three 100-faces, three 10-edges, and one unit corner, while square roots reverse the process of squaring.

• 11³: 1,000 + 300 + 30 + 1 = 1,331.
• 12³: Seen through a cube’s structure: 1,000 + six 100-faces + twelve 10-edges + eight corners = 1,728.
• Cubic measurement: A cubic foot contains 12³ = 1,728 cubic inches.
• Square roots: √N is the side length of a square whose area is N; for example, √576 = 24.

Powers, compounding, and musical pitch

Because 2¹⁰ = 1,024, squaring 1,024 reveals that 2²⁰ = 1,048,576. The same exponential structure describes both monthly compound growth and the equal-tempered musical scale.

• Monthly doubling: Multiplying by 2^(1/12) each month doubles an amount after 12 months; the monthly increase is approximately 5.95%.
• Equal temperament: Starting from A = 440 Hz, each successive note multiplies the frequency by 2^(1/12), reaching A = 880 Hz after twelve steps.
• Frequency formula: The frequency n semitones above A440 is 440 × 2^(n/12).

Further demonstrations

Adam invited interested students to follow “The Bhakti Math Guru” on Instagram for additional visual mathematics examples.

Multiplication as area and regrouping

A product can be divided into four visible regions. For example, 61² becomes (60 + 1)² = 60² + 60 + 60 + 1 = 3,721. This makes “squared” literal: multiplying a length by itself produces the area of a square.

• Mental examples: The group worked through 21² = 441, 31² = 961, 41² = 1,681, 51² = 2,601, 61² = 3,721, and 91² = 8,281.
• Unequal factors: The same four-box model gave 11 × 12 = 132 and 13 × 12 = 156.

From arithmetic to algebra

Replacing the side lengths with variables turns the picture into an algebraic identity. The four regions show (x + y)² = x² + 2xy + y²; dividing each side into three parts shows (x + y + z)² = x² + y² + z² + 2xy + 2xz + 2yz.

• Why the picture matters: The area model explains why every term appears, instead of treating expansion as a mechanical FOIL procedure.
• Powers-of-ten pattern: Dividing a square into lengths 1, 10, 100, and beyond reveals diagonal count patterns such as 111² = 12,321 and 11,111² = 123,454,321.

Changing the base

Place value always counts groups of the chosen base; base ten is only one possibility. In base 13, the available single-digit values run from 0 through 12, and 11₁₃ means one group of 13 plus one; therefore 11₁₃ × 11₁₃ = 121₁₃ by the same area pattern.

• General form: For any base b, (b + 1)² contains one b²-group, two b-groups, and one unit, so it is written 121 in that base.
• Representation versus quantity: A quantity stays the same while its written representation changes with the base—for example, binary uses only 0 and 1.

Suggested follow-up

Spend about 20 minutes reconstructing the area grid for yourself rather than trying to memorize the finished pattern.

• Practice: Draw side lengths 1, 10, 100, 1,000, and 10,000; multiply the intersecting labels and add along diagonals to see why the coefficients rise and then fall.
• Short videos: More visual-mathematics examples are available on Instagram at @thebhaktimathguru.

Seeing multiplication and algebra

Area and volume models turn arithmetic into something that can be decomposed and seen rather than merely memorized.

• Area models: Products such as 11 × 11 were split into hundreds, tens, and units: 100 + 10 + 10 + 1 = 121.
• Algebra beneath arithmetic: The same picture expresses (x + 1)² = x² + 2x + 1. Extending it into three dimensions gives (x + 1)³ = x³ + 3x² + 3x + 1; for x = 10, this shows 11³ = 1,331.
• Different bases: Changing the base changes what each place represents, but the underlying algebraic structure remains. Base 10 was the clearest starting point for visualization.

Exponential growth, music, and e

Exponential growth is recursive: each result becomes the input for the next step.

• Compound growth: Multiplying repeatedly by the same ratio models reinvested interest. Because each percentage increase acts on a larger amount, the graph curves upward rather than following linear growth.
• Musical octaves: An octave doubles frequency, from 440 Hz to 880 Hz. Dividing that doubling into 12 equal-ratio steps uses the factor 2^(1/12), approximately 1.0595—the same mathematical structure as repeated compounding.
• Understanding e: For a normalized 100% annual rate, increasingly frequent compounding approaches (1 + 1/n)^n → e ≈ 2.718. Without compounding, the corresponding linear endpoint is 2; the extra growth comes from repeatedly reinvesting each gain.

Mental multiplication practice

Draw or visualize products as rectangles divided into hundreds, tens, and units. The suggested exercise was to work through five multiplications mentally, using decompositions such as 21 × 21 = 400 + 20 + 20 + 1 = 441.

Correction from the demonstration

The animation incorrectly displayed 14 × 22 as 288 because of a code bug. The class checked the area decomposition and confirmed the correct result: 14 × 22 = 308.

Optional follow-up

Similar mathematical animations are available through the Instagram handle “thebhaktimathguru.” A summer course exploring mathematics in this visual style was also mentioned at “SFI.School”; no complete course URL was provided.