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Introduction

Prologue 1     Elementary Addition

Sums less than 10.  Composing 10 itself.
Sums between 10 and 20.  Zero.

Prologue 2     The Multiplication Table

Lesson 1    Numeration of the Whole Numbers

Numerals versus numbers.
The names and numerals of the numbers
1 - 999.

The classes: Ones, thousands, millions, billions, and so on.
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.
How to write and how to read a decimal.
Trailing 0's. Leading 0's.
How to compare decimals.

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.

Regrouping to form a multiple of 10.
Adding by endings.
Adding from left to right by place value.
Rounding off.

Written calculations.

What number must we add?
Completing 100.  Rounding off.

Written calculation by compensating.
Written calculation by regrouping.

The multiplier, the multiplicand, the product.
Factors and the order property.
Multiplying whole numbers that end in 0's.
How to square a number mentally.
Decomposing the multiplier.
Decomposing the multiplicand:
  The "distributive" property of multiplication.
Rounding off.

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  

The divisor a decimal.

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  

Increase or decrease by a given part.
A mixed number of times.
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  1  

The definition of the ratio of two natural numbers.
Percents are ratios.
The definition of a proportion.
How to express the ratio of any smaller number to a larger.
Lowest terms.
The theorem of the common divisor.

Lesson 18    Ratio and Proportion  2  

The theorem of the same multiple.
How to find a percent by the method of proportions.
Mixed ratio.

Lesson 19    Proportionality  

"Directly" proportional.
"Inversely" proportional.

Lesson 20   Introduction to fractions

Fractions are numbers we need for measuring.
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.
The relative sizes of fractions.
"Out of."

Lesson 21    Unit Fractions  

Every fraction is a number 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  

How to find the LCM of two numbers.
How to compare fractions with different denominators.
The ratio of two fractions.
How to compare a fraction to œ.

Lesson 24    Fractions into Decimals  

What is a "decimal" fraction?
Frequent decimals and frequent percents:     Half, quarters, eighths, thirds, fifths.
Exact versus inexact decimals.

How to multiply a fraction by a whole number.
How to multiply a fraction by a fraction.
Dividing fractions compared to adding them.

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.

Lesson 28    Percents are Ratios  

The ratio to 100%.
Percent increase or decrease. Estimations.

Lesson 29    Percent of a Number  

How to find the Amount.
What œ% means.
How to find the Base.

How to find the Percent.

Lesson 31    Percent Increase or Decrease  

"Difference" problems versus "out of"
 problems.
Banking problems.

The definition of a prime number.
Is there a last prime?

How to find the GCD and the LCM of two numbers from their prime factorizations.

Appendix 1   Continuous versus Discrete