Fraction Practice
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Slicing the structure and forms of fractions
"Fraction" comes from the Latin word fractus, meaning broken. In mathematics, it is used to express parts of a whole. A common example to picture the concept is a pizza. The pizza contains eight slices. If you take four slices, they represent four parts of the whole, or 4/8. This can be reduced to 1/2. The concept of fraction is used widely in different computational fields.
Fractions are also called non-integer rational numbers. They are written in the form of a/b. A and b are whole numbers, and b cannot be zero. A fraction has two parts. The numerator is the top number. It indicates the number of parts taken. The denominator is the bottom number, which is the total number of parts.
A fraction is actually a quotient of numbers. You can divide the numerator by the denominator. In the pizza example, if you divide 4 by 8 or 1 by 2, you can get 0.5 as an answer. This decimal is equivalent to 50 percent. So the portion of the pizza that you ate is 50%.
There are five kinds of fractions. The first form is vulgar fractions. A vulgar fraction, or common fraction, has an integer as the numerator and a non-zero number for its denominator. This can still be classified into two: proper and improper. Proper fraction is a fraction with a smaller numerator (in absolute value) compared to its denominator (in absolute value). An example is 2/5 or 4/7. On the other hand, an improper fraction results when the numerator is higher than the denominator in absolute values. 8/3 is an example.
The second form are mixed numbers. It has a whole number and a proper fraction in its form. The fractions 2 ¾ and 5 6/7 are examples of mixed numbers. They can be converted to improper fractions.
Equivalent fractions are fractions that have the same value. If you multiply the numerator and denominator of a fraction by the same non-zero number, this results to a fraction which has the same value. For example, in the fraction 2/3, if you multiply numerator 2 and denominator 3 both by the same number 3, you get 6/9. The new fraction has the value as 2/3. The same is also true if you use division. This process is called simplifying or reducing fractions.
Another form is the reciprocal. It is a fraction with the numerator and denominator interchanged. For 4/5, its reciprocal is 5/4. In the case of the number 5, 1 is called the "silent denominator". It can be written as 5/1, and its reciprocal is 1/5.
The last form of fraction is the complex fraction. Another name is compound fraction. Its numerator and denominator are both in the form of fractions. An example is 1/2/1/5. You can simplify it by dividing the numerator by the denominator. The example's simple form is 5/2 or 2 ½.
The Maya Civilization - La Civilización Maya
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Mayan Numeric System
Ancient Maya discovered two fundamental ideas in mathematics: positional value and the concept of zero. This feat was accomplished by only one other great culture of antiquity, the Hindu. But they did it 300 years or so after the Maya.
These two elements, positional value and zero, might be considered simple and basic concepts nowadays. In fact, they are, and that is precisely what set them apart as a distinct stroke of genius. Greek and Romans, with all the force of their spirit and all the strength of their institutions, did not manage to find these principles. Just try to write down a large number using the Roman notation to see how important are the notions of positional value and zero.
The Maya system is based on the number 20, not on the number 10 as our own. This means that the Maya counted from zero to nineteen before they had to move to the next order, instead of using 10 digits, from zero to nine, as we do. Perhaps they employed fingers and toes to keep the count.
In a decimal system the positional value is met as soon as we reach beyond number nine. A one followed by a zero is a ten. In the Maya system, a one followed by a zero equals twenty.
Our numeric system employs ten symbols to represent each one of the digits. Maya numerals were written with only three symbols: a dot for one; a line, which is a five, and the glyph of a sea shell to represent zero.
In that way, 
And the first twenty numerals would be:
Concepts in Number Theory
Objectives
Students will
- Understand Maya achievements in mathematics.
- Understand the Maya calendar.
- Learn how to convert Maya numbers to decimal numbers, and vice versa.
- Learn basic Maya arithmetic: addition, subtraction, multiplication, and division.
Materials
- Computer with Internet access
- Print resources about the history of Maya mathematics
Procedures
- Research the Maya culture and create a time line of Maya civilization using print and Web resources. The following Web sites are a good starting point: You may create a bulletin board from the time lines.
- Research the Maya calendar using print and Web resources. The following Web sites are a good starting point:
- Summarize your research in a one-page report.
- You may share your report with the class and answer any questions.
- Review the decimal number system, place value, and expanded notation. Show the class how to convert base 20 numbers with decimal digits to base 10 numbers. Have the class convert several base 20 numbers to base 10 numbers on their own.
- To familiarize the class with Maya numbers, have the class write the Maya numbers from 1 to 26. Show the class examples of converting Maya numbers to decimal numbers and converting decimal numbers to Maya numbers.
Remember the three rules of addition: a dot equals one, five dots equals one bar, and four bars equals one dot in the next place. Show the class examples of adding Maya numbers and allow them time for practice. - Show the class examples of subtracting Maya numbers and provide students time for practice.
- Review the properties of multiplication as they apply to Maya numbers; that is, the identity property, the zero property, and the commutative, associative, and distributive properties. Show the class examples of multiplying Maya numbers and allow them time for practice.
- Show the class examples of dividing Maya numbers and allow them time for practice.
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Evaluation
The following three-point rubric will be used to evaluate your work during this lesson. - Three points: You were highly engaged in class discussions; produced complete reports, including all of the requested information; clearly demonstrated the ability to convert between Maya and decimal numbers, and showed a complete understanding of Maya arithmetic.
- Two points: You participated in class discussions; produced an adequate report, including most of the requested information; satisfactorily demonstrated the ability to convert between Maya and decimal numbers, and showed a satisfactory understanding of Maya arithmetic.
- One point: You participated minimally in class discussions; created an incomplete report with little or none of the requested information; were not able to convert between Maya and decimal numbers or adequately perform Maya arithmetic.
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Vocabulary
decimal number system
Definition: A positional system of numeration that uses decimal digits and a base of 10
Context: The decimal number system is the most common numeral system used around the world. expanded notation
Definition: A numeral expressed as a sum of the products of each digit and its place value
Context: Expanded notation is used when converting between the Maya number system and
glyph
Definition: A symbolic figure or a character usually incised or carved in relief
Context: The Maya used glyphs to represent days and months in their calendar.
place value
Definition: The value of a digit as determined by its position in a number
Context: In the Maya number , the top dot is in the 202 (four hundreds) place, the shell is in the 201 (twenties) place, and the bottom dot is in the 200 (ones).
Maya number system
Definition: A positional system of numeration that uses a base of 20 and three symbols: zero(a shell-shaped glyph), one (a dot), and five (a bar)
Context: Advanced features of the Maya number system are the zero, represented by a shell, and the place value system.
vigesimal number system
Definition: Any positional system of numeration that uses a base of 20
Context: The Maya number system is a vigesimal number system.
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Academic Standards
National Council of Teachers of Mathematics (NCTM)
The National Council of Teachers of Mathematics provides guidelines for teaching mathematics in grades K-12 to promote mathematical literacy. To view the standards, visit this Web site: http://standards.nctm.org/document/chapter3/index.htm
This lesson plan addresses the following national standards:
- Understand numbers, ways of representing numbers, relationships among numbers, and number systems
Mid-continent Research for Education and Learning (McREL)
McREL's Content Knowledge: A Compendium of Standards and Benchmarks for K-12 Education addresses 14 content areas. To view the standards and benchmarks, visit http://www.mcrel.org/compendium/browse.asp.
This lesson plan addresses the following national standards:
- Mathematics: Understands and applies basic and advanced properties of the concepts of numbers; Understands and applies basic and advanced properties of functions and algebra; Understands the general nature and uses of mathematics
- Science: Physical Science: Understands the structure and properties of matter; Understands the sources and properties of energy
- World History: Understands Maya achievements in mathematics
- Historical Understanding: Understands the historical perspective
Number Sense and Operations 
Standards:
6.N.1: Demonstrate an understanding of positive integer exponents, in particular, when used in powers of ten, e.g., 10 to 2nd power, 10 to the 5th power.
6.N.2: Demonstrate an understanding of place value to billions and thousandths.
6.N.3: Represent and compare very large (billions) and very small (thousandths) positive numbers in various forms such as expanded notation without exponents, e.g., 9724 = 9 x 1000 + 7 x 100 + 2 x 10 + 4.
6.N.4: Demonstrate an understanding of fractions as a ratio of whole numbers, as parts of unit wholes, as parts of a collection, and as locations on the number line.
6.N.5: Identify and determine common equivalent fractions, mixed numbers, decimals, and percents.
6.N.6: Find and position integers, fractions, mixed numbers, and decimals (both positive and negative) on the number line.
6.N.7: Compare and order integers (including negative integers), and positive fractions, mixed numbers, decimals, and percents.
6.N.8: Apply number theory concepts-including prime and composite numbers, prime factorization, greatest common factor, least common multiple, and divisibility rules for 2, 3, 4, 5, 6, 9, and 10-to the solution of problems.
6.N.9: Select and use appropriate operations to solve problems involving addition, subtraction, multiplication, division, and positive integer exponents with whole numbers, and with positive fractions, mixed numbers, decimals, and percents.
6.N.10: Use the number line to model addition and subtraction of integers, with the exception of subtracting negative integers.
6.N.11: Apply the Order of Operations for expressions involving addition, subtraction, multiplication, and division with grouping symbols (+, -, x, ÷).
6.N.12: Demonstrate an understanding of the inverse relationship of addition and subtraction, and use that understanding to simplify computation and solve problems.
6.N.13: Accurately and efficiently add, subtract, multiply, and divide (with double-digit divisors) whole numbers and positive decimals.
6.N.14: Accurately and efficiently add, subtract, multiply, and divide positive fractions and mixed numbers. Simplify fractions.
6.N.15: Add and subtract integers, with the exception of subtracting negative integers.
6.N.16: Estimate results of computations with whole numbers, and with positive fractions, mixed numbers, decimals, and percents. Describe reasonableness of estimates.
Strand Overview 3 + 3 + 3 +3 = 3 x 4
The study of numbers and operations is the cornerstone of the mathematics curriculum. Learning what numbers mean, how they may be represented, relationships among them, and computations with them is central to developing number sense.
Research in developmental psychology and in mathematics education has shown that young children have a great deal of informal knowledge of mathematics. As early as age three, children begin counting and quantifying, and demonstrate an eagerness to do so. Capitalizing on this informal knowledge and interest, education in the early years focuses on developing children’s facility with oral counting and recognition of numerals and word names for numbers. Experience with counting naturally extends to quantification. Children count objects and learn that the sizes, shapes, positions, or purposes of objects do not affect the total number of objects in a group. One-to-one correspondence, with its matching of elements between two sets, provides the foundation for the comparison of groups. Combining and partitioning groups of objects set the stage for operations with whole numbers and the identification of equal parts of groups.
In the early elementary grades, students count and compute with whole numbers, learn different meanings of the operations and relationships among them, and apply the operations to the solutions of problems. “Knowing basic number combinations—the single-digit addition and multiplication pairs and their counterparts for subtraction and division—is essential. Equally essential is computational fluency—having and using efficient and accurate methods for computing.”8 Once teachers are confident that students understand the underlying structure of a particular operation, they should teach students the conventional algorithm for the operation. While students will not be asked to demonstrate use of standard algorithms on the grade four MCAS mathematics tests, they are expected to be introduced to them as theoretically and practically significant methods of computing.9 After students have learned how to use the conventional algorithm for an operation, whatever they then choose to use on a routine basis should be judged on the basis of efficiency and accuracy. No matter what method students use, they should be able to explain their method, understand that many methods exist, and see the usefulness of methods that are efficient, accurate, and general.10
As they progress through the elementary grades, students compute with multi-digit numbers, estimate in order to verify results of computations with larger numbers, and use concrete objects to model operations with fractions, mixed numbers, and decimals. By the end of their elementary school years, students choose operations appropriately, estimate to solve problems mentally, and compute with whole numbers.
Mathematics in the middle school centers on understanding and computing with rational numbers, and on the study of ratio and proportion (what they are and how they are used to solve problems). Students achieve competence with rational number computations and the application of the order of operations rule to prepare for high school.
At the high school level, understanding systems of numbers is enhanced through exploration of real numbers and computations with them. Thereafter, students investigate complex numbers and relationships between the real and complex numbers. Students expand their knowledge of counting techniques, permutations, and combinations, and apply those techniques to the solution of problems.
As students develop competence with numbers and computation, they construct the scaffolding necessary to build an understanding of number systems. Students not only compute and solve problems with different types of numbers, but also explore the properties of operations on these numbers. Through investigation of relationships among whole numbers, integers, rational numbers, real numbers, and complex numbers, students gain a robust understanding of the structure of our number system.
Technology in the Number Sense and Operations strand is used to facilitate investigation of mathematical concepts, skills, and strategies. Calculators and computers enhance students’ abilities to explore relationships among different sets of numbers (e.g., the relationship between fractions and decimals, fractions and percents, and decimals and percents), investigate alternative computational methods (e.g., generating the product of a pair of multi-digit numbers on a calculator when the multiplication key cannot be used), verify results of computations done with other tools, compute with very large and very small numbers using numbers in scientific notation form, and learn the rule for the order of operations.
