algebra file

Algebra________________________________________________________________________ Section 1: IntroductionDescription of Algebra and its place in the Montessori CurriculumIt has been said that the language of science is mathematical and the grammar of mathematics is algebra.The word algebra comes from the Arabic word “al-jabr” which means the reuniting of broken parts.Algebra is the science of equations using symbols that help to solve complicated problems.In Algebra, we use numbers and letters, and at all times the letters represent numbers. If we know the quantity we generally use a, b, c; if the quantity is unknown we use x, y, z. Algebra helps us to find the value of unknown quantities.Dr. Montessori developed a great deal of material for teaching mathematics. It was her belief that if the child absorbed a concept of mathematics at the sensorial level, then s/he would understand the abstract idea later on.With algebra we introduce the binomial cube and the trinomial cube. Binomial means consisting of or pertaining to two terms. Trinomial means consisting of or pertaining to three terms.Principles of Algebra Material•Isolation of stimulus•Control of error•Exercises are aesthetic in nature•Exercises are activity provoking and encourage repetition•Exercises progress from simple to complex•All exercises lay a foundation for future and lifelong learningSkills developed / Purposes of the Algebra Exercises•To develop concentration•To develop co-ordinated movement•To promote verbal skills and extend vocabulary•To promote development of cognitive skills: recognition, association, anticipation, sequencing, matching, classifying, gradingAlgebra________________________________________________________________________ Section 2: Theoretical foundationThe following concepts/topics underpin the Algebra Exercises.•Absorbent mind•Sensitive Periods•Sensorial EducationSection 3: Syllabus/ Programme Content•The Binomial Cube•The Trinomial Cube•Exploring and using patterns•Sequencing of patternsSection 4: Examples of exercises / illustrationsEach strand is introduced and developed as the child is ready.The Binomial CubeA wooden box containing eight blocks painted red, blue and black, together representing the cube of a binomial equation, (a + b)3.Presentation:The presentation is made on the child’s table. Place the box in the upper left hand comer of the table. The hinged sides of the box should face frontward and to the right.Remove the lid, open the sides of the box and place the lid in the angle made by opening the sides.Starting with the red cube “a3”, remove it from the box and place it to the left of the work space. Then remove the 2 blocks which were to the left and right of “a3” in the box, placing them under each other on the table to the right of the cube. Lastly remove the block in the far comer of the box and place it to the right again of the other blocks.Starting a second row, place the second layer of blocks on the table in order. The last block out of the box will be “b3”.Algebra________________________________________________________________________ Beginning with the red cube (a3), place the blocks from the first layer onto the lid of the box, matching colour to colour (i.e. red to red)Place your hand flat on this layer to indicate that they are all the same height. Now place the blocks from the second layer on top of the previous layer which is on the lid ending with the blue “b3”, matching colour to colour. Indicate that these are all the same height by placing your hand flat across them.Starting with the blue cube (b3), place the first layer of blocks back in the box. Again match colour to colour.Now place the second layer of blocks remaining on the lid into the box in the same way. When the blocks are assembled in the box, the large red cube will have been the last block returned.Fold up the sides, replace the lid and invite the child to do the exercise.Purpose:To develop the child’s perception of differences in proportion of three dimensionsTo encourage the perception of a mathematical patternTo indirectly prepare the child for algebraControl of Error:Visual, also the box will not close if the cube is incorrectly assembled.Age: 4 yearsNote:The blocks are set out precisely because they correspond to the algebraic formula. We do not teach this equation to the child at this stage.Algebra________________________________________________________________________ The Binomial CubeThe Binomial CubeAlgebra________________________________________________________________________Algebra________________________________________________________________________ The Trinomial CubeA wooden cube divided to represent a trinomial cube (a+b+c)3. The faces of the cubes are painted red, blue or yellow and the rectangles are black. Control Charts.Presentation:A large table is necessary for this exercise. Place the box at the top left hand corner of the table.The hinged side of the box should face front wards and to the right. Remove the lid and place it in the angle made by opening the sides of the box.Starting with the red cube “a3”, remove it from the box and place it to the left of the work space. Then remove the 2 blocks which were to the left and right of “a3” in the box, placing them under each other on the table to the right of the cube. Continue in this way until all blocks from this layer have been laid out.Remove the second and third layers in the same manner, placing the matching blocks directly beneath those of the previous layer.Beginning with the red cube (a3), place the blocks of the first layer from the table on to the lid. This red cube is placed at the back left hand corner of the lid. Match colour to colour.Place your hand flat on this layer to indicate that they are all the same height. Continue in this way placing the “b” layer on top of the “a” layer and finally the “c” layer on top of the “b” layer. Always indicate that each layer is the same height by placing the hand flat on the layer.Now return the cube to the box starting with the yellow cube. Each layer is returned separately. The teacher points now and then to the colours and in this way the child understands that alike colours are placed together.When the cubes are assembled in the box, the large red cube (a3) will have been the last block returned.Finally, fold up the sides, replace the lid and invite the child to use the material.Purpose:To develop the child’s perception of differences in proportion in three dimensionsAlgebra________________________________________________________________________To encourage the perception of a mathematical patternTo indirectly prepare the child for algebraControl of Error:Visual, also the box will not close if the cube is incorrectly assembled.Age: 4 ½ yearsAlgebra________________________________________________________________________ The Trinomial Cube(a+b+c)3 = a3 + 3a2b + 3a2c + 3ab2 + 6abc + 3ac2 + b3 + 3b2c + 3bc2 + c3 Trinomial cube laid out to correspond with Trinomial EquationAlgebra________________________________________________________________________ Exploring patternExercises in identifying and copying patterns in colour, shape and size:These exercises may be done using everyday objects (e.g. fruit), coloured blocks of different qualities and manipulative bears. Later coloured beads or pegs can be used.•Set out a pattern and ask child to identify and copy. Start with 3 elements. The number of elements in a pattern will increase as thechild becomes more capable.•Discover different arrays of the same number: Using the number rods, compose the different arrangements of the number 10, 9, 8 etc.Sequencing patternsExercises in recognising and continuing a sequence:•Recognise patterns and predict subsequent elements. Ask child to identify a pattern and to continue it.•Using patterns: Odd and even, using cards and counters•Use frame to show presence of an unknown number。