Cubic Identities (a+b)3 for School Maths Lab

Quick Answer: Cubic Identities (a+b)3 for School Maths Lab is a three-dimensional model of the identity (a + b)³ = a³ + 3a²b + 3ab² + b³. A cube of edge a + b is divided into smaller solids whose volumes match the terms on the right-hand side, so students can build the identity rather than just memorise it.

Eight Solids, Four Terms

Cutting a cube of edge a + b at distance a along each of its three edges divides it into eight solids. One is a cube of edge a, one a cube of edge b, three are cuboids measuring a by a by b, and three are cuboids measuring a by b by b. Grouping them by shape gives exactly the four terms of the expansion, with the coefficients 1, 3, 3, 1 appearing as the number of pieces of each shape.

The coefficients are the point students most often get wrong, writing (a + b)³ as a³ + b³ or dropping a middle term. Holding three identical a²b cuboids makes the 3 in 3a²b concrete, and the symmetry of the model, with the two groups of three cuboids mirroring each other, shows why the coefficients read the same forwards and backwards. The same 1, 3, 3, 1 pattern appears in Pascal’s triangle, which gives older students a link to the binomial theorem.

In a typical lesson, groups first build the full cube, then take it apart, sort the pieces, write the volume of each group, and finally add them to state the identity. Giving a and b numerical values, for example 4 cm and 1 cm, lets them check that 125 cm³ equals 64 + 48 + 12 + 1.

Specifications

Applications

• Building and dismantling (a + b)³ to identify each term
• Explaining the coefficients 1, 3, 3, 1 by counting pieces
• Numerical checks with chosen values of a and b
• Linking to Pascal’s triangle and the binomial expansion
• Rewriting the identity as a³ + b³ + 3ab(a + b) for factorisation work

Care & Handling

• Keep the pieces together in their box; losing one cuboid breaks the identity.
• Handle pieces over a table so corners are not chipped if dropped.
• Clean with a soft dry cloth.

Why Choose LabEquip

Maths teachers preparing the Class 9 identities unit buy this model to show the cube of a sum, often after using an Algebra Identity kit for (a + b)². LabEquip lists both in the Mathematics School Lab Products range; enquire through the contact page.

Frequently Asked Questions

How many pieces does a model of (a + b)³ need?

A complete model needs eight solids: one cube of edge a, one cube of edge b, three cuboids a by a by b and three cuboids a by b by b. They correspond to a³, b³, 3a²b and 3ab².

Why is (a + b)³ not equal to a³ + b³?

Because the cube of edge a + b also contains six cuboids between the two small cubes. Their volume, 3a²b + 3ab², is exactly what is missing from a³ + b³, and the model shows them physically.

How is the identity written as a³ + b³ + 3ab(a + b)?

Take out the common factor 3ab from the middle terms: 3a²b + 3ab² = 3ab(a + b). This form is the one used to find a³ + b³ quickly when a + b and ab are known.

How does the model connect to Pascal’s triangle?

The coefficients 1, 3, 3, 1 are the numbers of pieces of each shape, and they are also the row of Pascal’s triangle for power three. The same pattern extends to higher powers in the binomial theorem.

Can the model be used for (a − b)³?

It is built for the cube of a sum. The identity (a − b)³ = a³ − 3a²b + 3ab² − b³ follows algebraically by replacing b with −b, and students can compare its terms with the pieces of the model.

How does this differ from the Cubes of Algebra set?

This model is dedicated to (a + b)³, the cube of a sum. The Cubes of Algebra listing is more general in name, and suits schools that want the volume idea across more than one cube identity, depending on what that set includes.

Last Updated: September 2026