Geometry and Measurement: Ages Nine to Twelve

Overview

The word geometry is derived from the Greek words for earth and measurement. The word geometry literally means ‘measuring the earth’. The Montessori approach to teaching geometry involves the use of concrete materials, which engage with the children’s sense of wonder and which encourage children to measure for themselves the world they live in.

In the Montessori curriculum the first knowledge about shapes and space is gained through activity with concrete materials in the Children’s House. This sensory work is extended in the geometry lessons in the environment prepared for children from six to nine years old. All new concepts continue to be introduced through activities in which children manipulate concrete materials, but from the age of six children independently complete a sequence of activities that build deeper understanding and move them towards abstraction. Manipulative work with concrete materials always precedes the presentation of an abstract concept or formula. Ideally, children, during their own independent work, begin to verbalise the concepts and formulae in conversation, although if, unusually, children do not make this step for themselves, an adult will model how to talk about the concept using the language of geometry.

Learning the names of geometric shapes and their parts is an important aspect of the study of geometry. As in the Children’s House children take part in spoken language lessons in which they learn the names of geometric shapes and their parts, but for children beyond the age of six, naming lessons also include:

• learning the origin, or etymology, of the name

• learning definitions and systems of classification

• reading and writing the names.

In addition to the names, children of this age want to know the reasons for things so they are given opportunities to explore questions such as:

• Why do we say this is an acute angle?

• How many right angles would make a whole angle?

Following most presentations, children take part in follow-up activities with card materials and booklets.

The lessons in geometry are keys to enable children to follow their own interests in this area of study. They are encouraged to explore the concrete materials to find new relationships, variations and extensions. As in all areas of the Montessori curriculum for children of this age, the initial lessons and the children’s use of the concrete materials provides them with the knowledge they need as a starting point for their own exploration and discovery.

The study of geometry for children from the age of six begins with an exploration of the foundation concepts: point – line – surface – solid. These concepts are then explored in more detail in a sequence of follow-up lessons. From the same starting point, the actual sequence may vary from child to child. For example, work on lines and angles might be presented parallel to one another, and some early polygon work can be presented parallel to later work with angles. Whatever sequence is followed, work in each area progresses logically, with each new lesson building on the one before. For example, children complete work that builds an understanding of equivalence, before they work with area, and the work with area precedes the work on volume. In the case of the study of circles, the work follows a progression such as the following:

• identification of linear, then surface parts

• relationships between a line and a circle

• relationships between two circles

• relationships between circles and polygons

• circumference, radius and their relationship

• area of a circle

As in all areas of the Montessori Cosmic Education curriculum, the study of geometry is designed to encourage activity, repetition, exploration, discovery, logical thinking and reasoning. The initial lessons are clear and succinct, providing children with enough knowledge to allow them to ask pertinent questions, and enough guidance to initiate independent activity with the materials. It is through this activity that children make the knowledge their own and learn to research and think for themselves.

The study of measurement in the Montessori Cosmic Education curriculum, begins with an account of the historical development of measuring, comparing and estimating. The children explore the reasons humans in earlier times might have wanted to, or needed to, measure, compare and estimate. They also experiment with the use of non-standard units and units used in earlier times, in particular, units based on parts of the body, including, for example, the hand, the palm and the cubit. They use these units to measure objects in the environment. During these activities children recognise the need for standardised units, as exemplified, for example, in the story of the royal cubit from Ancient Egypt. They are then introduced to the standard measurement systems in use today, beginning with the International Metric System. As students learn different elements of this system, they apply their knowledge, first, to measuring activities in the everyday environment, selecting the most appropriate standard unit and recording the measurement of, for example, the length of the school hall, the temperature of the classroom, the volume of water in a watering can or the mass of a can of pet food. Finally, students apply their knowledge during activities and research projects across all areas of the curriculum and in the solving of real-life problems, including, for example, calculating how much water the garden needs each day, or how many days the bird food will last before more needs to be purchased.

 Knowledge, Skills and Understanding

Typically, children will:

01. Explore and research development and application of geometry through history and across cultures

02. Study ancient and modern scholars in the field of geometry

 Knowledge, Skills and Understanding

Typically, children will:

01. Explore and research development and application of measurement through history and across cultures

 Knowledge, Skills and Understanding

Typically, children will:

01. Review, consolidate and extend knowledge of solid geometry

02. Apply knowledge of solid geometry in a range of contexts

.03 Connect objects to their nets and build objects from their nets using spatial and geometric reasoning

.04 Compare the parallel cross-sections of objects and recognise their relationships to right prisms

 Knowledge, Skills and Understanding

Typically, children will:

.01 Use wooden triangles of different types to construct a variety of compound and complex shapes

.02 Recognise and name constructed shapes and identify objects in the environment

.03 Draw constructed shapes

.04 Use knowledge of measurement of angles to explore the mathematics of shapes constructed with the triangles

 Knowledge, Skills and Understanding

Typically, children will:

01. Review, consolidate and extend the detailed study of plane, including:

• lines

• angles

• polygons

• tessellation

02. Extend vocabulary for talking about plane shapes, including bisect, median, perpendicular, points of concurrency

.03 Recognise and use combinations of transformations to create tessellations and other geometric patterns, using dynamic geometric software where appropriate

 Knowledge, Skills and Understanding

Typically, children will:

01. Review, consolidate and extend detailed study of angles

02. Recognise relations between:

• pairs of angles

• angles formed by two lines and transversal: interior-exterior, adjacent-opposite, complementary-supplementary

.03 Estimate, construct and measure angles in degrees, using appropriate tools including a protractor, and relate these measures to angle names

.04 Explore the sums of angles of polygons

.05 Explore angle bisectors

.06 Build definitions related to relations between angles

 Knowledge, Skills and Understanding

Typically, children will:

01. Identify and label the linear parts of a circle: centre, radius, diameter, chord, arc, circumference

02. Identify and label the parts of the surface of a circle: sector, segment

03. Explore relationships between a circumference and a straight line (external, tangent, secant)

04. Explore relationships between two circles (external, internal, externally tangent, internally tangent, intersecting, concentric)

 Knowledge, Skills and Understanding

Typically, children will:

.01 Review, consolidate and extend study of symmetry, similarity, congruence and equivalence

.02 recognise line and rotational symmetry of shapes and create symmetrical patterns and pictures, using dynamic geometric software where appropriate

.03 Describe and perform translations, reflections and rotations of shapes, using dynamic geometric software where appropriate; recognise what changes and what remains the same, and identify any symmetries

 Knowledge, Skills and Understanding

Typically, children will:

01. Use concrete material to explore ‘families’ of polygons for relations of equivalence:

• triangle

• rhombus

• common parallelogram

• trapezium

• regular polygons

02. Use concrete materials to build theorems of equivalence, including theorem of Pythagoras

 Knowledge, Skills and Understanding

Typically, children will:

01. Review and extend knowledge of surface area

02. Explore units of measurement for area (square metre)

03. Choose the most appropriate unit of measurement for area

04. Use concrete material to build formulae for calculating area of:

• rectangle

• common parallelogram

• triangle (including right-angled and obtuse-angled)

• square

• circle

05. Extend knowledge of area to build formulae for calculating area of other quadrilaterals: rhombus, kite, trapezium, common quadrilateral

 Knowledge, Skills and Understanding

Typically, children will:

01. Extend knowledge of volume to solids

02. Use unit of measurement for volume of solids (cubic metre)

03. Apply the concept of equivalence to volume of solids

04. Use concrete material to build formulae for calculating volume of:

• solids

• rectangular prism

• other prisms

• solids that are not prisms (pyramid, cylinder, cone, sphere)

 Knowledge, Skills and Understanding

Typically, children will:

01. Review, consolidate and extend knowledge of measurement (length, mass and temperature)

02. Use knowledge of and experience with measurement to extend and apply estimation and comparison skills

03. Apply knowledge of measurement to a range of projects and problems