You are familiar with the terms “Binomial Cube” and “Trinomial Cube,” which are introduced as sensorial materials at the primary level of Montessori education. The Cubes are later used as a geometrical representation of algebraic formulas. The square of a binomial and a trinomial have several applications in real-life problems and are frequently used in various fields such as physics, engineering, finance, and others.
The square of a binomial is given by the formula (a + b)^2 = a^2 + 2ab + b^2. This formula is extensively used in algebra and is applied in finding the areas of squares and rectangles, factoring quadratic equations, and calculating probability distributions. The square of a trinomial is given by the formula (a +b +c)^2 =a^2 + b2 + c^2 + 2ab + 2ac + 2bc. It is used in algebra to factorize polynomials and perform calculations related to geometry and physics.
In Montessori, children progressively manipulate these formulas using the Bead Cabinet material. A bead square is divided using rubberbands, representing each term of the equation. This post presents the layout of our new supplemental material, Square of Binomial and Trinomial Task Cards, designed for children ages six to nine. The set contains 56 cards divided into six incrementally challenging sections.
Squaring of Binomial and Trinomial Task Cards
Section 1
This section reviews squares 1 through 10. It provides an opportunity to revisit the equation for each square and solve the equation to find the surface of a square.
Section 2
Section 2 helps learners transition from a concrete representation of a square to an abstract version on paper. Learners draw the equation on graph paper in their notebooks.
Section 3
This section provides preparatory work with a binomial’s square containing multiplication. For example, learners are given the equation: (6 + 3) x 4 = _____. They solve this equation by using four 6-bead bars and four 3-bead bars. They visualize 24 beads + 12 beads, which equals 36 beads.
Section 4
Section 4 is the same as section 3 but contains three terms to solve: (4 x 5) + (3 x 5) + (2 x 5) (as shown on picture)
Section 5
In section 5, learners already know how to solve a square and are ready to solve binomial equations. They are given a 100 square, so the product is predictable. They can focus on observing how a square can be divided into different terms.
Section 6
In section 6, learners make the equation for given numbers on a trinomial square. As for section 5, the product is predictable (100), so learners can focus on the process. They will find on the back of each task card a detailed answer.
The Task Cards have for the purpose of providing learners with independent work. They are not meant to replace presentational instruction. They can be used to illustrate examples and reactivate prior knowledge, as repetition is essential to permanent mastery! You may want to watch our video for more information. Also, visit our website at www.alisonsmontessori.com.
