The Whole Child Curriculum: Math K-8

1st Grade
In Mathematics, the first graders first encounter numbers through stories, clapping, musical rhythms, and other artistic activities. In this way, they are guided from their sensory experiences to the beginnings of abstract reasoning.

Students begin with the Roman numerals, which is less abstract than the Arabic. Whole numbers are introduced with emphasis on their archetypal character- 1 means unity, 2 is a duality, and so on, using pictures familiar to the child's world (the sun, parts of the body, petals of flowers, etc.).

Then students learn the four basic arithmetical operations and their different qualities. Students begin the actual figuring with something concrete and visible, stones, shells or other natural objects, always proceeding from wholes to parts. (20 is 10+10). Only after considerable practical experience in adding, subtracting, multiplying and dividing are the written symbols for these operations introduced in a pictorial way. Rhythmical counting, recitation of the times tables, number riddles, number bonds up to twenty, and mental arithmetic are all practiced intensively in the early years. This gives the children an experience of movement in mental activity, which compliments the way the letters of the alphabet are introduced.

FORM DRAWING
Forms are experienced initially though movement. At first the forms are simple, but become more complex as the child progresses. In the first 4 to 6 weeks of school the first graders learn the form elements of the straight and curved line, elements that are encountered again in the writing block as the Latin printed letters. They later go onto angles, triangles, rectangles and star forms and then semicircles, circles, spirals and ellipses, setting the stage for the study of Geometry in later years.

By the end of First Grade the students will be able to:

• Understand Roman numerals 1-V and Arabic numerals 1-110
• Count from 1-110
• Have working knowledge of the 4 processes and their symbols.

2nd Grade
The children carry out more complicated operations with the four processes. Imaginative stories still form the basis of these problems. Through rhythmic counting accompanied by accented clapping and movement of the whole body, they learn to count by two's, three's, four's and five's, and begin learning the multiplication tables. Tossing a beanbag to each other, chanting 4 is 2x2, 6 is 2x3 etc., the children are engaged and enthusiastic. They then learn to process through mental arithmetic and are taught carrying and borrowing. Years, months, days of the week, and time of day are also introduced at this time.

FORM DRAWING
Symmetry exercises with emphasis on an axis, mirroring exercises with emphasis on the above-below relationship, and four sided symmetries with rounded forms and their metamorphosis into angular forms.

By the end of second grade the children should be able to:

• Practice of the 4 rules using numbers up to 100
• Practice in combined calculation
• Up to 12 times tables by heart
• Representation of tables in drawing
• Further practice in mental math

3rd Grade
Mathematics in grade three should remain connected to the practical things of life. The main lesson themes of house building and gardening in grade three are an excellent source for arithmetic problems. It immediately becomes clear that such problems require measurements. In grade three measurement moves from the oral realm, which is comparative, qualitative and contextual (this is bigger, there are more here, etc.) to the use of formal units. The measurement of length is made an even greater experience for the children by beginning with the old measurements which were based on human body proportions (yard, foot, span, and hand and finger width). Moving on from the traditional measures, children are then introduced to the standard measurements of length, liquid, weight, time, money, and music in use today. The children themselves should measure and weigh many things.

By the end of third grade, most children of normal ability range will be proficient in:

• Long multiplication and long division
• Remainders of division
• Checking answers by doing reverse process
• Estimating answers to the nearest hundred or thousand
• Regular practice in oral and written arithmetic to firm up knowledge in basic addition and subtraction facts, and the times tables
• The recognition of number patterns in various multiplication tables, up to the 12 times tables.
• Telling the time on clocks (both analog and digital)
• Measurement of time, liquid capacity, length, rectangular area, weights, money, and all measurements connected with practical manual work (gardening, cooking, and building)
• Problems about measures written in words and sentences
• Shopping lists and money calculations; obtaining the correct change
• Freehand drawing of line symmetries and rotational symmetries. 19 Patrick McMahon Third Grade Mathematics Goals (2002-2003)
• Mirror picture exercises.
• Awareness of perpendicularity
• Experience of directions; north, south, east, west
• Prime numbers
• Place Value

FORM DRAWING

• Rather than painting it is usually form drawing that begins things. Form drawing becomes the focus throughout a series of main lesson block.
• A fitting preparation is made for writing by working with lines that do not illustrate an object but which meet the impulse for movement in the child, that train his feeling for form and develop his manual dexterity
• Exercises with asymmetric symmetries are now added to the preceding one. We are concerned here with lines that develop from a point in the center out towards threes side. The child is to find supplemental forms that lead back inside in order to restore equilibrium and harmony. This requires a great deal of independence and mobility in being able to imagine. Such exercises are a significant preparation for the geometry of later grades in which construction with the compass and ruler begins.

4th Grade
In 4th Grade the students begin to work with fractions. Fractions will be introduced graphically. The teacher will introduce interesting and significant teaching ideas by drawing from the historical development of fraction calculations in Egypt. In order to do general justice to the subject of fractions it is recommended to use the following three methods as an introduction: To proceed from the whole to the parts, from the parts to the whole, and to establish the principle of equivalence. After this the four rules are practiced with fractions, the same with simplifying, expansion and division of the denominator into prime factors. After this, decimal fractions follow as a practical application.

Form drawing leads into elementary geometry. In order that the pupils get as intensive an image as possible of these forms, it is recommended that they do not initially use compasses and ruler, but draw free-hand. The Pythagorean rope is presented as a first introduction to Pythagoras' Theorem. By the end of fourth grade, most children will be able to:

NUMBERS

• Carry out all four number processes confidently
• Read and understand numbers up to six figures
• Know the multiplication tables up to 12 out of sequence
• Do long multiplication with numbers up to 122 as multiplier
• Find factors of a given number
• Identify prime numbers less than 100
• Answer more complex mental arithmetic questions involving a mix of processes (e.g. I doubled a number and added 8 and got 32, what was the number?)
• Do long division including making use of remainder and estimating approximate answers.
• Find Lowest Common Multiple or Highest Common Factors
• Create a 'book of rules' introduced in the course of the fraction work.

MEASUREMENT

Record information such as height, weight, volume, etc.

5th Grade
Constant practice in mental arithmetic.

• Combinations of the four rules
• Calculations with fractions and mixed numbers: expansion and reduction of equivalents (division into prime factors)
• Illustration and comparison of fractions. Introduce calculation with decimals.
• Work with table of place values, rhythmically, through movement, and qualitatively introduced
• Introduce of the relationship of decimals to place values
• Measurements using decimals
• Recognition of connections between decimal numbers and decimal fractions

ASSESSMENT

• Answer more complex mental arithmetic questions involving a mix of processes (e.g. The 12:38 train to Santa Barbara takes 118 minutes but left 29 minutes late. When did it arrive?)
• Do long division including making use of remainder and estimating approximate answers
• Find lowest common multiple or highest common factors
• Use all four processes with fractions including mixed numbers and improper fractions
• Understand how to use decimal notation, decimal fractions and interchange of decimal with common fractions
• Carry out four processes with decimals
• Use long division and multiplication using the decimal point
• Work with aspects of time including 24 hour clock
• Calculate average speeds

GEOMETRY

• Starting with the circle, discovery of the main geometrical figures
• Construction of different triangles; equilateral, isosceles, scalene, right angled
• The Various angles; acute, obtuse, reflex.
• Circles touching a triangle; inside and Pythagorean Theorem; visually using knotted string (did Egyptians used this to construct their pyramids?)
• Introduce and work with metric measurement including estimation

ASSESSMENT

• Draw freehand archetypal geometric shapes: different kinds of triangle, rectangle, quadrilaterals, polygons and circles
• Divide circles into 2,3, 4, 6 and multiples of these, deriving regular figures like square, triangle and hexagon

6th Grade
In 6th Grade students can increasingly create order out of what has been gained with the strength of their ability to experience internal logic. As they become confident and secure with mathematical laws, they learn self-confidence.

• Continue mental arithmetic exercises
• Calculation with natural numbers, fractions and decimals
• Introduce ratio and proportion, with direct and inverse proportion
• Percentages
• Convert percentages and decimal numbers to fractions and vice versa
• Estimate results by rounding off number prior to accurate calculation
• Application of percentages to business: simple interest and discount
• Block graphs, pie charts, bar charts, linear graphs
• Make time and speed calculations
• Tessellation (tiling) involving accurate construction of parallel lines
• Exact construction of pentagon/pentagram

GEOMETRY
Geometry is taught in a separate main lesson.

• Geometrical proof of sums of angles of triangle: using cut outs, protractors
• Proof of above using calculations
• Accurate construct of angles using compasses, bisecting angles
• Construction of triangles from description
• Congruent triangles; the four principle cases for congruency
• Movement properties of triangles and quadrilaterals, triangles in the same segment of a circle

7th Grade
Beginning in 7th Grade and continuing into 8th Grade, pupils create order with the strength of their new ability to experience internal logic. This is exemplified in algebra.

• Continuing practice with mental arithmetic
• Revision; the four rules in natural and positive numbers
• Basic bookkeeping
• Intro to negative integers through debt calculation
• The four rules with negative numbers
• Extension to cover all radicals
• The four rules with rationals and their connections
• Intro of brackets
• Recurring decimals, value of !
• Compound interest
• Statistical data
• Graphing, business math

ALGEBRA

• Simple equations
• Formulas
• Powers and roots of numbers
• Ratio and proportion
• Areas
• Simple set theory

GEOMETRY

• Areas - construction, calculation
• Circle
• Pythagorean theorem
• Tangents
• Perspective drawing - linked to modern history lessons.

ASSESSMENT

• Know power of numbers
• Work out ratio and scale
• Use algebra as a general solution to specific problems
• Use negative and positive intergers

8th Grade
8th Graders continue with the above and will know:

• Know how to work with square roots
• Calculate compound interest, mortgage rates, income tax
• Make time and speed calculations
• Calculate mechanical advantage in simple machines like pulleys, levers
• Present information graphically - pie charts, bar graphs
• Foreign currency exchange
• Algebraic graphs
• Precise use of a compass, ruler, set squares to draw constructions of major geometric figures
• Make use of freehand perspective
• Use a protractor
• Draw translations, reflections, rotations
• Know Pythagorean theorem and its applications
• Use instruments to draw linear perspective
• Know properties of triangles, parallel lines, and intersecting lines
• Know and apply formulas for area of regular geometric forms
• Calculate areas of irregular forms

(See also "John Morse Waldorf Methods Magnet School" - Sacramento Unified School District APPENDIX G )

For more information about the curriculum, please email: curriculum@oceancharterschool.org