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Geometry and Fractions: Overview

Children have learned to associate shapes with the word geometry. Shapes of all different kinds are part of their daily lives. Most children are familiar with the most common shapes. You can introduce children to the vocabulary they need to work with the different shapes.

A plane shape is a flat, closed, two-dimensional figure. Different plane shapes have different numbers of sides and vertices. A side is a straight line that makes part of the shape, and a vertex is a corner, or a place where two sides of a shape meet.

A triangle is a shape with 3 sides and 3 vertices. A rectangle is a shape with 4 sides and 4 vertices in which opposite pairs of sides are the same length. A square is a rectangle in which all four sides are of equal length. A circle is a round, closed shape that has 0 sides and 0 vertices.

Triangle, rectangle, square, circle

Many of the everyday objects with which children are familiar are solid shapes. For example, building blocks are often cubes or rectangular prisms. They have six faces, or flat surfaces, and 12 edges. An edge is where two faces meet. Some other solid shapes are spheres, which children might recognize as being shaped like playground balls; cones, like ice-cream cones or traffic cones; and cylinders, which are shaped like cans. One shape that children might not immediately recognize is a pyramid, which has one square face and three triangular faces.

Cube, rectangular prism, sphere, cone, cylinder, pyramid

Most children are familiar with shape-matching. Shapes that match exactly (are the same size and shape) are congruent shapes. Children may also be familiar with the concept of symmetry. Many things in our daily life are symmetrical, or have at least one line of symmetry—a line that separates a shape into two matching parts. For example, children can look at a picture of a butterfly and see that the two sides of the butterfly are the same. Children may have practiced using symmetry by folding paper and cutting a shape, such as part of a snowflake, then unfolding the paper. The fold is a line of symmetry.

congruent shapes and symmetrical shape

Children must understand shapes and their attributes before they can really understand fractions, which at this level involve equal parts of a whole. Fractions are found in many common activities. For example, everyday tasks such as measuring, telling time, and cooking depend on understanding fractions. It is easy to illustrate fractions by using simple shapes divided into equal parts. Children may recognize things such as pieces of pie or pizza as fractions. They begin to work with unit fractions by coloring in one part of a whole shape. Once they learn to write the total number of parts of the whole under the line in the fraction and the number of parts colored over the line, they can begin to practice identifying and writing fractions.

Coloring in parts of whole shapes prepares children to compare fractions. For example, children can look at a circle divided into five parts with one part shaded. They can also look at a circle divided into four parts with one part shaded. The shaded part of the second circle is larger than the shaded part of the first circle. Children can use this visual comparison to see that one-fourth is larger than one-fifth.

When children have had sufficient experience identifying and writing fractional parts of a whole, they are ready to work with parts of a group, or set. Help children grasp the idea that a group of objects can represent a whole by giving them practice partitioning groups into equal parts. Guide children to see the relationship between a group of objects and the fraction that describes it. For example, give each child 4 counters. Have children divide the group of counters into 4 equal parts. Explain that 1 counter of 4 is one fourth of the group. Elicit that 2 counters are two fourths of the group, and so on. Give them practice partitioning groups of different numbers of counters into equal parts and identifying the fractions that name various parts of the groups. This will reinforce the idea that a group of objects can be considered a unit and lay the groundwork for work with ratios in later grades.

The ability to identify and compare fractions is necessary for understanding probability—at this level, deciding if an event is more likely or less likely. Eventually, children will learn to express probability using fractions: a one-in-four chance can be expressed using the fraction one-fourth.


Houghton Mifflin Math Grade 2