Title price = 99 $
Chapter price = 10 $
Basic Operations on whole and rational numbers
Middle school 12-15 years
Click arrows to display the contents of chapters
Click an icon signaling courses and exercises which can be downloaded
Click on a box or boxes corresponding to the chapter (s) you want to buy
1. Addition in the Set of Natural Numbers. The Set N. Increasing and Decreasing Order.
1. Review of addition in the set of natural numbers. The set N.
2. Addition is associative.
3. Demonstrating associativity using colors on a graduated axis.
4. Addition is commutative.
5. Demonstrating commutativity on a graduated axis.
6. Using associativity in mental operations.
7. To order the set N: comparison of two natural numbers.
8. Comparison of one natural number to another and to a sum.
9. To order a set of natural numbers in increasing and decreasing order.
10. Free Drawing.
2. Addition and Subtraction of Integers. The Set Z. Comparison of Two Integers.
1. Arithmetic subtraction.
2. An impossible subtraction in the set N.
3. Extending the set N to the set of integers, Z.
4. Addition and subtraction of two natural numbers on a graduated axis.
5. Rules of the addition of integers. Representation on a graduated axis.
6. Calculating the sum of two integers, using their representation on a graduated axis.
7. Basic addition properties of numbers in the set Z.
8. What ever happened to subtraction ?
9. Ordering integers.
10. To arrange a set of integers in increasing and decreasing order.
3. Subtraction Allows an Algebraic Sum to Be Simplified.
1. Definition of algebraic subtraction. Link to the addition.
2. Transforming a subtraction operation into an addition.
3. Transforming an addition operation into a subtraction.
4. Removing brackets in algebraic expressions.
5. Simplifying and calculating algebraic expressions.
6. Calculating an algebraic expression given in literal form.
7. The opposite of an algebraic sum is the sum of the opposite terms which make up the sum.
8. Calculations involving additions/subtractions and brackets.
4. Multiplication of Integers. The Product Signs Rule.
1. Review of the multiplication of integers. Multiplication is associative.
2. Rule of writing a product of factors. To omit the multiplication sign.
3. Multiplication is commutative.
4. Extending multiplication to a single negative factor.
5. Extending multiplication to the whole set of integers Z.
6. To carry out the product of numerical factors. Symmetry properties of a product of factors.
7. Product of factors made up by sums/differences of integers.
5. The Distributive Rule and Factoring. Why " - " Times " - " Equals + ?
1. Distributivity of multiplication over addition and subtraction.
2. Applying the distributive rule twice.
3. Illustrate distributivity using a rectangle's area.
4. The basis of factoring: using distributivity.
5. Factoring simple expressions.
6. Why is the product of two negative numbers positive?
7. Showing that the product of two negative numbers is positive using a rectangle's area.
8. Distinguishing between terms and factors. The sign rules.
9. Using distributivity and factoring in numerical calculations.
6. Powers of Positive Whole Numbers in the Set Z.
1. Definition and notation of exponents. Values of some powers of natural integers.
2. Calculating powers of natural numbers by iteration.
3. Values of some powers of negative integers.
4. Calculating powers of negative integers.
5. Multiplying positive whole number powers of integers.
6. Calculating the product of two powers.
7. Assigning names to positive powers of 10.
8. Naming positive powers of 10.
7. Additions and Subtractions of Integers. Multiplications and Powers.
1. Regrouping addition and subtraction operations involving integers.
2. Multiplying powers of integers.
3. Simplifying single variable, first degree expressions without brackets.
4. Simplifying single variable, first degree expressions with brackets.
5. Simplifying 2-3 variable, first degree expressions without brackets.
6. Simplifying 2-3 variable, first degree expressions with brackets.
7. Simplifying single variable, second degree expressions in one variable.
8. Multiplying powers involving one, two or three variables.
8. Factoring Numbers, Prime Numbers, LCM's and GCD's. Euclid's algorithm.
1. Factoring natural numbers. Prime numbers. Introducing the tool "Premier".
2. Determining whether a natural number is prime.
3. List of all prime numbers less than 100.
4. Factoring natural numbers into powers of prime numbers using the tool "Premier".
5. Lowest common multiples (LCM's).
6. Geometrical representation of the LCM of two natural integers.
7. Calculating the LCM of whole numbers factored into prime numbers.
8. Greatest common divisor (GCD). Its determination from factoring into prime numbers.
9. Determination of GCD by Euclid's algorithm.
10. Finding GCD's by the factoring method and by Euclid's algorithm.
11. Exam exercises using Euclid's algorithm.
9. Expanding the Set of Integers Z to Include Fractions. The Set Q. Decimal Numbers.
1. An example of a division possible in the set Z.
2. An example of an impossible division in the set Z.
3. Equivalent forms of a fraction. Irreducible form.
4. Finding equivalent fractions to a given fraction. Irreducible form of a fraction.
5. Irreducible form of a fraction using the GCD.
6. Irreducible form of a fraction using Euclid's algorithm.
7. Positive and negative fractions. Correspondence with a point on a graduated axis.
8. Identifying a fraction from the equation it satisfies.
9. The decimal numbers are practical fractions. Decimal point notation.
10. Recognising whether a fraction is a decimal number, and converting it to one if possible.
11. Converting a decimal number to its fraction form. Replacing the power of 10 with a decimal point.
10. Addition / Subtraction and Comparison of Numerical Fractions.
1. Logical basis of the rule for adding fractions.
2. Reduce to the same denominator to add/subtract fractions.
3. Adding two positive fractions and simplifying if possible.
4. How many of us are the same age ?
5. Route of a sales representative.
6. The smallest common denominator of two or many fractions is their LCM p.
7. Sum of fractions with arbitrary signs.
8. A solved exercise: use of LCM.
9. Adding two fractions using their lowest common denominator.
10. Adding three fractions using their lowest common denominator.
11. Comparison of two fractions.
11. Multiplication of Fractions. Fraction to the Power n. Percentages and Interests.
1. Multiplication and division of a fraction by a positive whole number.
2. Multiplication of two fractions. Signs rule.
3. Review sheet: multiplication of fractions and the n-th power of a fraction.
4. Multiplying and simplifying two simple fractions.
5. Application exercises on the product of fractions.
6. Using decomposition into prime factors to simplify the product of fractions.
7. To draw a figure inside a square and to determine the fraction of area it represents.
8. Percentages and their applications: simple interests, taxes, an alloy's composition ...
9. Exercises involving percentages.
10. Compound interests.
11. Comparisons of simple and compound interests.
12. Inverse of a Fraction. Division of Two Fractions. The Algebra of the Set of Fractions Q.
1. The set Z is a subset of the set Q. Neutral element and symmetrical element for the addition.
2. The multiplicative neutral element. The inverse of a non-zero fraction.
3. Learning to calculate the opposite and multiplicative inverse of fractions.
4. Basic properties of addition and multiplication on the set Q.
5. Whatever happened to division ?
6. Using distributivity to calculate fractional numbers.
7. Ratio of two decimal numbers. In which case is the quotient decimal ?
8. Quotient of the sum of two fractions.
9. Using prime factorization to simplify the quotient of two fractions.
10. To extract the integer part of a fraction and to determine the complementary fraction.
11. To simplify the writing of a fraction by extracting the integer part.
13. Exponents in Set Q. Multiplication of Powers. Scientific Notation.
1. Positive whole number exponents of a fraction.
2. Calculating powers of fractions.
3. Multiplying and dividing powers, x
= 1 (x
4. Assigning names to negative powers of 10.
5. Scientific notation of decimal numbers.
6. Converting a decimal number less than 1 or greater than 1 to scientific notation.
7. Product and quotient of powers of a variable.
8. Raising the power of a variable to a power.
9. Operations on powers of a variable.
10. Raising the product of powers of many variables to a power.
11. Summary of the different rules on powers.
12. Powers involving many variables.
14. Equalities and Equations in the Set Q. Transposition of Terms and Factors.
1. Equalities, identities, and equations.
2. Basic properties of equalities. Adding the same number to both sides.
3. To group together similar terms of an equation using transposition.
4. Writing equivalent equalities using opposites.
5. The remaining distance to cover.
6. To transform a rectangle into a square.
7. To organize the terms of a linear equation and to solve it.
8. Equivalence between equalities resulting from the uniqueness of a number's inverse.
9. Multiplying or dividing both sides of an equality by the same number.
10. Solutions of linear equations in the form ax=b where a and b are fractions.
11. To translate problems into equations and to solve them.
15. Addition/Subtraction Side By Side of Two Equalities. Multiplication/Division Side By Side.
1. Addition/subtraction side by side of two equalities. Application.
2. Chasle's relation. Signed measure of a segment. The x-coordinate of its middle.
3. To find two numbers x and y with sum s and difference d. Geometrical meaning.
4. Allowed operations on equalities resulting from the uniqueness of the product. Product and quotient.
5. Finding the dimensions of a rectangular field, given its area and the quotient of its two sides.
16. Solving Linear Equations.
1. Linear equations without brackets with coefficients in Z.
2. Linear equations involving brackets with coefficients in Z.
3. Linear equations without brackets with coefficients in Q.
4. Linear equations involving brackets with coefficients in Q.
5. Equations with coefficients in Z, where the second degree terms cancel out.
6. Equations with coefficients in Q where the second degree terms cancel out.
17. Setting Down and Solving Linear Equations Containing One Unknown Variable.
1. Shopping at the grocery store.
2. Going to a concert.
3. How old are they.
4. Dining out on the town.
5. In the country.
6. Visiting the stock market.
7. Minute and hour hands playing hide-and-go-seek.