MILESTONE 11: Decimal Numbers

The following lessons provide a basic understanding of decimal numbers.

MATH MILESTONE #B6: DECIMAL NUMBERS

A “division” or “ratio” notation is not the only notation possible to express fractions. We may use the place values to the right of ONE to account for fractions. Since the successive place values change by TENS, we call such numbers the DECIMAL NUMBERS.

The place values successively magnify by TENS as one moves to the left. And, the place values successively shrink by a TENS as one moves to the right. Thus, to the right of ONES we have, tenths, hundredths, thousandths, and so on. With these fractional place values we may express fractions to a desired accuracy.

The DECIMAL POINT is used in a decimal to separate the fractional portion. Thus, it appears  to the right of the place value of ONE in the number. It is this decimal notation that we use on calculators and computers.

Computation with decimals follows the same procedure as with the whole numbers. The only additional requirement is keeping the track of the position of the decimal point.

In short, the decimals are a natural extension of the existing whole number system to account for the fractions.

Here are some videos on decimal numbers from Khan Academy.

Adding Decimals (Old)

Subtracting decimals (old)

Dividing decimal

Converting fractions to decimals

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December 16, 2011 – 8:39 AM Categories: Mathematics | Comments (9)

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MILESTONE 10: Fractions (Part II)

Here are some simple lessons on operations with fractions.

MATH MILESTONE #B5: OPERATIONS WITH FRACTIONS

To summarize:

Like fractions are added by adding the numerators. Like fractions are subtracted by subtracting the numerators. The denominator remains the same. To add or subtract unlike fractions, one must convert them to like fractions first.

To convert unlike to like fractions, we first calculate the LCM (least common multiple) of all the unlike denominators.  Then we calculate the equivalent fractions for unlike fractions with the LCM as the new denominator.

To multiply fractions, we simply multiply the numerators together to get the numerator of the product, and multiply the denominators together to get the denominator of the product. To divide by a fraction, we simply multiply by its reciprocal.

In general practice, a fraction in the final answer is expressed in its lowest terms. The lowest terms are obtained by taking all the common factors out of the numerator and the denominator.

A “division” notation is not the only notation possible to express fractions. Another way is to extend the place value notation to account for fractions. That notation is covered under the milestone on DECIMAL NUMBERS.

Here are some videos from Khan Academy on Fractions.

Least Common Multiple

Adding and subtracting fractions

Multiplying fractions (old)

Dividing fractions

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December 16, 2011 – 8:23 AM Categories: Mathematics | Post a comment

MILESTONE 9: Fractions (Part I)

Here are some lessons that provide a basic understanding of fractions.

MATH MILESTONE #B4: PROPERTIES OF FRACTIONS

To summarize:

When the division is not exact, a remainder is left after division. The remainder is less than the divisor, and it may be looked upon as a portion of the divisor. Such portions are called fractions. A proper fraction, such as “half,” is always less than one.

In the absence of a proper notation for a quantity less than 1, a fraction is presented as a “dividend over divisor.” These two numbers are called numerator and denominator respectively to emphasize the fact that a fraction is a single quantity even when two numbers are used to represent it.

When a unit is divided into equal number of smaller parts, each part is called a unit fraction. The larger is the number of parts the smaller is each part or unit fraction. The numerator of a unit fraction is always 1. All other fractions are multiples of unit fractions.

In a proper fraction the numerator is less than the denominator making it less than 1. In an improper fraction, the numerator is equal to, or greater than the denominator making it equal to, or greater than 1. Improper fractions may be written as mixed numbers.

Equivalent fractions are those which are written with different numerator/denominator pair, but represent the same portion of a unit. For example, both 1/2 and 2/4 represent “half” of a unit. In such a case, the numerator/denominator pair of a fraction is “magnified” or “shrunk” by the same amount to become the numerator/denominator pair of the equivalent fraction.

Like fractions are multiples of the same unit fraction. Unlike fractions are multiples of different unit fractions. Like fractions may be compared simply by their numerators. To compare unlike fractions, one must convert them to like fractions first.

Here are some related videos from the Khan Academy.

Mixed numbers and improper fractions

Equivalent fractions

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December 16, 2011 – 8:09 AM Categories: Mathematics | Post a comment

MILESTONE 8: Factors & Primes

The following lessons provide some basic understanding of factors and prime numbers.

MATH MILESTONE #B3: FACTORS

The use of FACTORS and PRIME NUMBERS has declined in the world of calculators today. However, a conceptual understanding of these concepts leads to insights that calculators and computers cannot provide.                       

The factors are obtained from EXACT DIVISION. The divisor and the quotient are the factors of the dividend. When a number cannot be factored into a pair of smaller numbers then it is a prime number.

A composite number has a unique set of prime factors.

The following is a list of prime numbers to a thousand or so. You may find this list useful.

You may now attempt to find the next ten prime numbers after 1013.

Here are some videos from Khan Academy on this subject.

Prime Numbers

Greatest Common Divisor

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December 16, 2011 – 6:55 AM Categories: Mathematics | Comments (3)

MILESTONE 7: Integers

The following lessons provide some basic understanding of Integers.

MATH MILESTONE #B2: INTEGERS

Mathematics of integers appears to be quite troublesome to most students. But when we look at it as arithmetic with increase and decrease from zero, it becomes easy to grasp.

Confusion takes place because “plus” and “minus,” which represent operations between two numbers, are also used to show a number as “positive” or ‘”negative.” This becomes clear when integers are defined as being referenced from zero.

–1   =    0 – 1

+1   =    0 + 1

This allows us to convert “positive” or ‘”negative” signs into “plus” and “minus” operations, and vice versa.

Furthermore, confusion arises when “plus” and “minus” operate on “positive” and “negative” numbers, giving consecutive signs. However, once we understand that LIKE consecutive signs produce a positive number…

– (–1)      =     +1

+ (+1)      =     +1

…and UNLIKE consecutive signs produce a negative number, the operations are greatly simplified.

– (+1)      =     –1

+ (–1)      =     –1

Here are some videos from Khan Academy on the subject of Integers.

Negative Numbers Introduction

Adding/Subtracting negative numbers

Multiplying and dividing negative numbers

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December 14, 2011 – 1:43 PM Categories: Mathematics | Comments (5)