Discovering a product without doing a multiplication

Notable identity: sqare of a binomial

Introduction

The children that have played the games and challenged have understood through verification and manipulation of triangles. This as well favors the comprehension of algebra procedures. In this challenge children will understand that the square of a binomial is a notable identity.

In arithmetic multiplication and in algebraic multiplication an algorithm (a system in which we manipulate symbols) has to be followed, and it takes us to a product. Nevertheless, there are algebraic products that respond to a rule that simplifies the process of getting to the results. The goal is to get the children to discover this rule. A notable identity is a product that can be performed without doing the multiplication.

Preparation

Build a composite square as shown in the diagram.

Procedure

  1. Tell the children: Here I have a square because its four sides are equal.
  2. Introduce the rules pointing out the upper extern side of the red square and the yellow rectangle.
  3. Say: Let this line be a, and this other b.
  4. Ask: How long is the side of the square?
  5. Children should answer a+b.
  6. Reinforce the notion by saying: It is the square of a+b” or “a+b squared.
  7. Challenge the children: Tell me what is inside this square of a+b.
  8. The children answer.
  9. Insist that the measure must be included in the description.
  10. Reinforce the development of vocabulary by saying: Then a square of a+b equals a square of a plus a square of b plus two rectangles a•b.

Second level

On this level the challenge is that children test the Square binomial product with another kind of puzzle.

Preparation

Build a construction as shown on the diagram.

Procedure

  1. Show the construct to the child and point out that it is a square and each of its sides measures a + b.
  2. Challenge the child by saying: “If it is true that the square of a+b is equal to a square, plus b square, plus two rectangles that measure a•b, then we should find those same figures inside this square.
  3. Reinforce the child’s intuition by asking: Are the figures there?
  4. Get the children to verify it.
  5. The kid should construct the product.