Operations: The checkerboard
The checkerboard was designed to help children become aware of multiplication in different categories. For example, units times units makes units, units times tens makes tens, tens times tens makes hundreds, and so on. It also allows children to do very large multiplication problems without the necessity of having memorized all the multiplication facts. The checkerboard has many items that are already familiar to the child such as the hierarchical colors and the bead bars. The checkerboard is divided into colored squares, green, blue, and red, representing the category colors. This arrangement results in a diagonal display of the colors.
Operations: Racks and Tubes
Small test tubes, standing in racks of ten, contain ten color coded beads, each representing the categories through one million. Color coding on the racks represent the number families. Millions in black, thousands in gray and simple in white. A green bead from the white rack represents one unit. A green bead from the grey rack represents one thousand. A green bead from the black rack represents one million. A blue bead from the white rack represents one ten. A blue bead from the grey rack represents one ten thousand and so on. Skittles, three boards and color coded cups to hold the dividend complete this material for short and long division, using up to a seven digit dividend and a three digit divisor.
Small Metal Inset Fractions
Segments of ten metal circles and ten metal squares represent halves through tenths. Used for designing and comparing, these small metal insets establish a foundation for understanding the concept of fractions. Using the same small metal inset material, several sequenced lessons lead the child to experiment with equivalencies, laying the groundwork for the concept of lowest common denominator. Initial lesson introduce the vocabulary, integer, denominator and numerator.
Advanced Fraction Work
These additional fraction materials show ways to break one into parts. Children use them to add, subtract, multiply and divide fractions, and to understand them in relation to decimal fractions and percentages. Rather than beginning with a rule, the student does much manipulative work and arrives at rules for working fractions abstractly.
This material acquaints the child with patterns in our number system. With each tenth number, we can increase a hierarchy and we can also decrease in hierarchies with each ten beyond the whole number. This material gives practice in composing amounts and performing the four operations.
Study of Decimal Fractions
The decimal checkerboard material includes the checkerboard, loose squares, the bead bars and symbols for the multiplicand and the multiplier. This material allows the child to experience geometric representation of decimal fraction multiplication, with an emphasis on place value.
The extensive set of bead material is used for the exercises of linear and skip counting, the quantities of the squares and cubes of the numbers 1-10. It prepares the child for later activities in multiplication, squaring and cubing, as well as base number work.
This cube material for the powers of numbers is designed to bring an awareness of powers beyond squares and cubes for decimal numbers. The language of power, base and exponent are introduced. This is an indirect preparation for non-decimal basis and is a preparation for algebraic manipulation.
The algebra materials consists of skittles and dice on a fulcrum which are manipulated to solve linear equations. As with most of the Montessori math curriculum, children develop a concrete understanding of algebra using a hands on approach. Elementary aged children develop this basic understanding of algebra prior to entering middle school.
The Montessori math curriculum provides children with an extraordinary understanding of the meaning of numbers. Anyone who has learned mathematics in a Montessori classroom can easily conceive of the difference between one hundred and one thousand, or the difference between a squared number and a cubed number. The carefully designed movement from concrete to the abstract allows the child to deeply understand complex mathematical principles.
Comprehensive geometry studies begin with experiences with the line and its parts and continue through studies of angles, polygons, triangles, quadrilaterals, circles area and volume. The concepts of similarity, congruency and equivalency are also studied.
Children of this age love to reach back into history with their imaginations and reconstruct these needs and solutions and the creation of systems of learning. The Hindus introduced the use of “0.” Let the child try to do math without it! Where did algebra, calculus, trigonometry come from? They want to know!”
-Child of the World,
Essential Montessori for Age three to twelve