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Basic Operations on whole and rational numbers
Middle school 12-15 years

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   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, x0 = 1 (x 0).
  •            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.