The Algebraic Decanomial — (a+b+c+…+j)²

The elementary reasoning mind is ready to see that the familiar decanomial is (a+b+…+j)². This is algebra made sensorial — the child names relationships they can already see. Indirect aim: preparation for the binomial and trinomial squares at symbolic level, and later for algebraic expansion.

• Framework: Montessori
• Unit: Sensorial
• Learning area: Algebraic Sensorial
• Level: Ages 7-9

Objectives

• Relabel the decanomial's ten rows as a, b, c, …, j
• Identify squares (a², b², … j²) along the diagonal
• Identify rectangles (ab, ac, …) as the off-diagonal pairs
• Express the total as the square of the sum (a+b+c+…+j)²

Materials

• Long decanomial already laid (see id 211)
• Letter cards: a, b, c, d, e, f, g, h, i, j
• Symbol cards: a², b², …, j², ab, ac, …, ij
• Control chart of the algebraic identity

Instructions

1. With the decanomial laid, place letter card a at the first row, b at the second, through j. 2. Along the diagonal, lay the square cards: a², b², … j². 3. Off-diagonal, lay the rectangle cards on matching cells: ab, ac, ad. 4. Notice ab appears twice (row a col b, and row b col a) — write 2ab. 5. Express: the full square = a² + b² + c² + … + j² + 2ab + 2ac + 2ad + … 6. Compute: substitute a=1, b=2, c=3… j=10; verify against the bead count. 7. Reference the control chart and identity. 8. File notes in the child's mathematics journal.

Educator tip

Do not lecture algebra. Invite the child to see the letters on the decanomial they already built. The substitution step is where recognition arrives; give it time.

Assessment

Observe the child's recognition of squares and rectangles by type, ability to verbalize the identity, and willingness to substitute numerical values to check.

More from Sensorial

• The Power of Two Cube — 2³
• The Power of Three Cube — 3³
• Trinomial Cube at Algebraic Level — (a+b+c)³
• Constructive Triangles — Third Box (Mixed Polygons)
• Constructive Triangles — Large Blue Triangle Box
• Geometric Cabinet — Polygon Nomenclature & Angle Count
• Yellow Area Material — Decomposing Shapes to Find Area
• The Five Platonic Solids — Euler's Formula