Textbooks & Curriculum

Visible & Tangible Math Textbook 1
Qualitative Arithmetic: Study of Numbers from 1 to 10

2 Qualitative work with the rods
Students discover equivalence, trains, patterns, inequalities, staircases, complementarities, trains of one color, transformations, and odd and even lengths.

3 Literal work
Students are introduced to:

• color name notation for rods
• signs for addition, difference and equivalence
• equations (such as r + g = y)
• brackets and parentheses
• multiples
• names of fractions and equivalent expressions

4 Number measure study of numbers up to 10
Children use a unit of measure to generate number names for rods and are introduced to the four operations with numbers up to 5, and later to 10.

5 Applications
Children work on word problems involving operations on numbers up to 10.

6 Study of numbers up to 20
Students work on problems involving all four operations, odd and even numbers, prime and composite numbers, series, ordinal numbers, zero, and short division.


Visible & Tangible Math Textbook 2
Study of Numbers up to 1000. The Four Operations

1 Study of numbers from 1 to 100
Problems involve multiples, first milestones, tables of products, factors, primes, composites, squares, common factors, and highest common factor.

2 Vertical notation
Addition and subtraction, equivalent subtractions, multiplication, and division are written in vertical form.

3 Clock
Children work on challenges involving reading the clock, finding the remaining minutes within the hour, counting the time from one hour to another, travel, the 24 hour day, and other time related questions.

4 Numbers up to 1000
Problems involve:

• equivalent crosses (rods)
• towers of three
• milestones: multiples of 100, transformations using towers, other multiples of 100, doubles providing milestones
• HTU: hundreds, tens, units
• completing naming numerals up to 1000
• sequences through doubling, equivalent products, distribution and multiplication
• towers of four, preparing for quick mental calculations multiplying by 25, vertical notation, quick calculations, multiples of 11, 3, 6, 9,
• criteria of divisibility
• more about squares, square of a sum or difference, product of a sum and a difference
• curiosities from Pythagoras¹ school

5 Days and weeks
Children learn about such things as the calendar year, seasons, making a model for the calendar, and leap years.

6 Reading and writing of numbers
Students learn about another way of naming numbers — “the number array,” as well as place value and zero as a place holder, and jumping from hundreds to millions and beyond.

7 Procedures, algorithms for the four operations
Topics include:

• addition and equivalent additions
• families of equivalent subtractions and best equivalent subtractions
• speeding up the operations
• multiplication and casting out 9’s
• milestones again: doubling, quicker and safer divisions, application to long division
• a very simple calculating machine

8 Grouping, cost price, selling prices, profit
Units such as dozens, grosses, and scores are covered as well as time, buying and selling, profit, and price.

9 Perimeters, area, volume
Students learn about rectangles, cubes and cuboids, and their perimeters, area, total area, volume, and units of volume.

Visible & Tangible Math Worksheets 1-6

Visible & Tangible Math Textbook 3
Applied Arithmetic

1 About money
Coins, bank- notes, uses of money, operations on money, applications, and division are explored.

2 Sharing, proportion, unitary method

3 Length
Topics include units of lengths, tables of lengths, conversion, length and money, and length and time.

4 Area and volume
Units including acres and square miles are looked at, as well as converting to metric units, measurable areas, different shapes of area, areas and money, volumes, conversion, and various applications.

5 Capacity and weight
Students discover units of capacity, units of weight, and conversion of these units. They also explore various applications.


Visible & Tangible Math Textbook 4
Fractions, Decimals, Percentages

1 Fractions as operators
Operators, inverse operators, composition of operators, and equivalent operators are explored.

2 Study of fractions
Students become knowledgeable about: ordered pairs, families of equivalent ordered pairs, first element of a family of equivalence, equivalent expressions for pairs, equivalent fractions, reciprocals, addition and subtraction of pairs, addition and subtraction of fractions, cross multiplication and equivalences, inequalities, transformation of pairs, ratios and proportions, fractions of fractions, division of fractions, mixed numbers, and operations on mixed numbers.

3 Study of decimal fractions
Students work on defining a decimal fraction, translating one notation for another, operating on decimals, operating on hundredths and thousandths, operating on powers of ten, and applying this knowledge in various ways.

4 Percentages
The new notation (%) and various forms of writing percentages are explored in conjunction with fractions, decimals, and applications.


Visible & Tangible Math Textbook 5
Study of Numbers

1 The formation of numbers
Students explore place value and powers of ten, largest and smallest number of n figures, index notation, horizontal and vertical notation, equivalent subtractions, growth of numbers, multiplication of numbers.

2 Different bases of numeration
Polynomials, the binary scale, scales of three, seven and twelve, and the four operations in any scale are covered.

3 Divisibility and prime numbers
Students use the Eratosthenes sieve, find criteria of divisibility, find (all) prime factors, do exercises and further explore prime numbers.

4 HCF and LCM
HCF - highest common factor, decomposition into prime factors, and LCM - lowest common multiple are worked on.

5 Squares, cubes, and cube roots
Students learn about the square of a sum, the formation of a table of squares, the square of a difference, the formation of a table of cubes, the cube of a difference, square roots, approximations, calculating square roots, and square roots to a certain approximation.

6 The set of integers
In this area, students learn about infinite sets, equivalent sets, partition of the set of integers, equivalence of (1) and one of its parts, other partitions of (1), how to think about infinite sets, the infinite nature of the set of prime numbers.


Visible & Tangible Math Textbook 6
Applied Mathematics

1 Length, area and volume
Topics include: the centimeter and its multiples, conversions, multiples and submultiples, errors and approximations, absolute and relative errors, the cubic centimeter, volumes of various bodies, areas and approximation of areas, figures reducible to rectangles, tables of units and applications.

2 Capacity and volume of round bodies
The liter and its multiples and submultiples, round bodies, and solids of revolution are explored.

3 Weight, mass and density
Students learn about weights and balances, how to weigh an object, finding a fraction of a weight, units of weight, making a box of weights, mass and weight, density, and applications.

4 Measures, their formalization
Dimensional formulae, measuring a length, isometric measures, measuring an area, transformation of a figure, measuring a volume, properties of measure, other magnitudes, and scalar and vector quantities are covered.

5 The C.G.S., M.K.S., and M.T.S. Systems of units historical notes, fundamental and derived units, the systems

6 Proportions and mixtures
Students explore areas including sharing, films and tapes, the post office, conversions of units, discounts, and the compound unitary method.

7 Problems depending on measurements & Problems in economics
Problems may include building a road, building a site, dividend and interest, budgets, consumer goods, and travel.

8 Problems on speed
Topics include: speed, paths/routes, speeds in various units, comparing actual speeds, relative measurements, graphical representation, and acceleration.

9 The perpetual calendar
Students explore the use of remainders in division by 7, days in the year, the number code for the months, days in different years, some historical dates worked out, and historical notes.


Visible & Tangible Math Textbook 7
Algebra and Geometry

1 Simultaneous equations about directed numbers
Topics include: finding a pair through knowledge of another pair, formalization of the solution, a word about algebra, a class of equivalence, and generalizations.

2 Permutations and combinations
Students are introduced to permutations, transpositions and substitutions, combinations, computation of permutations and combinations.

3 Sets and subsets algebra of sets
Sets and elements, the null set, relationships between sets, inclusion, equivalence of sets, operations on sets, and analogy between operations on sets and on products are explored.

4 Arithmetic progressions and geometric progressions
Students learn about arithmetic progressions, powers and their algebra, geometric progressions, sum of the terms of a G.P., and applications to decimal numbers.

5 The geometry of the Geoboards
Challenges on the nine-pin rectangular lattice Geoboard the regular polygon boards are worked on.