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Introduction
Prologue 1
Elementary Addition
Sums less than 10. Sums equal to 10.
Sums between 10 and 20. Zero.
Prologue 2
The Multiplication Table
Lesson 1
Numeration of the Whole Numbers
Units.
Numerals versus numbers.
The names and numerals of the numbers
1  999.
Lesson 2 
Reading and Writing Whole Numbers 

The Powers of 10

How to read a whole number, however large.
Place value. Positional numeration.
Expanded form.
Relationship between units of adjacent place value.
How to round off a whole number.
How to multiply a whole number by a power of 10.
Lesson 3
The Meaning of Decimals
Cardinal and ordinal numbers.
Division into equal parts.
The decimal units.
The function of the decimal point.
How to write and how to read a decimal.
Trailing 0's. Leading 0's.
How to compare decimals.
Lesson 4 
Multiplying and Dividing 

by Powers of 10



The Meaning of Percent 
How to multiply or divide a whole number
by a power of 10.
How to multiply or divide a decimal
by a power of 10.
Per centum: For each 100.
How to take 1% of a number.
How to take 10%.
Scientific notation.
The metric system.
Lesson 5 
More Elementary Addition


Mental Calculation

Composing a multiple of 10.
Adding by endings.
Adding from left to right by place value.
Rounding off.
Lesson 6 
Adding Whole Numbers and Decimals


Addition with Regrouping

Written calculations.
Lesson 7 
The Meaning of Subtraction



Mental Calculation

What number must we add?
Completing 100. Rounding off.
Lesson 8 
Subtracting Whole Numbers and Decimals

Written calculation by adding to both numbers.
Written calculation by regrouping, or borrowing.
Lesson 9 
The Meaning of Multiplication


Mental Calculation

The multiplier, the multiplicand, the product.
Factors and the order property.
How to multiply whole numbers that end
in 0's.
How to square a number mentally.
Decomposing the multiplier.
Decomposing the multiplicand:
The "distributive" property.
Multiply mentally by rounding off.
Lesson 10 
Multiplying Whole Numbers and Decimals

Written calculation.
How to find a percent of a number.
The area of a rectangle.
Lesson 11
The Meaning of Division
The divisor, the dividend, and the quotient.
How to divide a number into equal parts.
Exact versus inexact division.
The division bar.
Mental calculation:
Decomposing the dividend.
The divisor a decimal.
Three properties of division.
Lesson 12
Short Division
What is short division?
The dividend a decimal.
How to round off a decimal.
How to express the quotient as a decimal.
The dividend less than the divisor.
Calculator problems.
Lesson 13
Dividing Decimals
Lesson 14
Percent with a Calculator
Lesson 15
Parts of Natural Numbers 1
What is a natural number?
The definition of a part of a number.
Percent  parts of 100%.
Lesson 16
Parts of Natural Numbers 2
Finding a part by decomposing.
How to divide by taking a part.
A mixed number of times.
Increase or decrease by a given part.
The percent that means a third.
Even and odd numbers.
Half of any number.
A fourth or 25% of a number.
5% of a
number  15%.
Lesson 17
Ratio and Proportion
The definition of the ratio of two natural numbers.
Percents are ratios.
What is a proportion?
How to express the ratio of any smaller number to a larger.
Lowest terms.
The theorem of the common divisor.
Lesson 18
Solving Proportions
The theorem of the same multiple.
The Rule of Three.
How to find a percent by the method of proportions.
Mixed ratio.
Lesson 19
Proportionality
Directly proportional.
Inversely proportional.
Lesson 20 What is a Fraction?
Common fractions, proper fractions, improper fractions, mixed numbers.
How to place a fraction on the number line.
How to change an improper fraction to a mixed number.
How to change a mixed number to an improper fraction.
How to compare fractions that have equal numerators
How to compare fractions that have equal denominators
"Out of."
Lesson 21 
Unit fractions: Fractions with Numerator 1

Every fraction is a sum of unit fractions.
How to express a whole number as a fraction.
The complement of a proper fraction.
How to subtract a proper fraction from a whole number.
Lesson 22
Equivalent Fractions
How to recognize equivalent fractions.
How to change denominators.
How to simplify or reduce a fraction.
How to clear a fraction of decimals.
Lesson 23 
Lowest common multiple Comparing fractions 
How to compare fractions by crossmultiplying.
How to compare a fraction to ½.
Lesson 24
Fractions into Decimals
What is a decimal fraction?
Frequent decimals and frequent percents: Half, quarters, eighths, thirds, fifths.
Exact versus inexact decimals.
Lesson 25

Adding and Subtracting Fractions and Mixed Numbers

Lesson 26 
Multiplying Fractions Dividing Fractions

How to multiply a fraction by a whole number.
How to multiply a fraction by a fraction.
Dividing fractions compared to adding them.
Lesson 27 
The Meaning of Multiplying Fractions Parts of Fractions

The meaning of multiplying a number by
a
fraction.
How to express a fraction as a percent.
How to take a part of a fraction.
The general definition of multiplication.
Lesson 28
Percents are Ratios
The ratio to 100%.
Percent increase or decrease. Estimations.
Lesson 29
Percent of a Number
How to find a percent of a number.
What does ½% mean?
A number is a given percent of what other number?
Lesson 30 
What Percent is One Number 

of Another? 
The method of proportions.
Lesson 31
Percent Increase or Decrease
"Difference" problems versus "out of"
problems.
Banking problems.
Lesson 32 
Prime Numbers and 

Prime Factorization

The definition of a prime number.
Is there a last prime?
Lesson 33 
Greatest Common Divisor 

Lowest Common Multiple

How to find the GCD and the LCM of two numbers from their prime factorizations.
Appendix 1
Continuous versus Discrete
Appendix 2 
Triangular and Square Numbers 

The Sum of Consecutive Cubes

Appendix 3 
Two Theorems 


The distributive property of multiplication 


The order property 
Copyright © 2016 Lawrence Spector
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