Renaissance in Ancient Egypt: the 25th and 30th Dynasties - Grade ...
17 pages
English
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Renaissance in Ancient Egypt: the 25th and 30th Dynasties - Grade ...

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17 pages
English

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Renaissance in Ancient Egypt: the 25th and 30th Dynasties – Grade Seven 1 Ohio Standards Connection: History Benchmark B Describe the political and social characteristics of early civilizations and their enduring impact on later civilizations. Indicator 2 Describe the enduring impact of early civilizations in India, China, Egypt, Greece and Rome after 1000 B.C. including: a. The development of concepts of government and citizenship; b. Scientific and cultural advancements; c.
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Grade 6


Mathematics

Frameworks


Unit 2
Fun and Games
Student Edition
M A T H E M A T I C S
Georgia Performance Standards Framework
ndGrade 6 Mathematics Unit 2 2 Edition

Unit 2
FUN AND GAMES
TABLE OF CONTENTS

Overview ................................................................................................................................3

Enduring Understandings.......3

Essential Questions ................................................................................................................4

Key Standards and Related Standards ...................4

Selected Terms and Symbols .................................................................................................6

Tasks ......................................7
Back to School! ..........................................................................................................8
Arrays, Factors, and Number Theory ...... 11
You are the Teacher! Give „em homework! ............................ 14
Culminating Task: Three Number Theory Challenges ............ 15






Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 6  UNIT 2: FUN AND GAMES
September 1, 2009  Page 2 of 17
Copyright 2009 © All Rights Reserved

Georgia Performance Standards Framework
ndGrade 6 Mathematics Unit 2 2 Edition

Grade 6 Mathematics
Fun and Games: Extending and Applying Number Theory

OVERVIEW:
In this unit, students gain a deeper understanding of concepts and applications of number theory.
In Grade 5 Mathematics, students studied classification of counting numbers into subsets with
distinguishing characteristics such as odd and even numbers and prime and composite numbers.
Students also developed a strong foundation for understanding and applying multiples and
factors. They will extend this concept to include greatest common factors and least common
multiples which in turn will build a deep understanding of the Fundamental Theorem of
Arithmetic. This unit is a building block to students‟ deeper understanding of rational numbers
thwhich students will study extensively in unit 3 of the 6 grade mathematics framework.

Instruction should include the representation of these numbers and their relationships to other
concepts, such as multiplication and division using diagrams, charts, tables, multiple number
lines, and explanations. Number theory is a topic that begs for students to reason, discuss, make
sense of and justify their thinking. This can be accomplished by students playing games that are
based on number theory, working and debating with their peers and in sharing ideas through a
teacher-facilitated whole class discussion. Students also should be provided with opportunities
for revisions. In order to demonstrate mastery of the learning in this unit, students will explain
the Fundamental Theorem of Arithmetic to a friend who has been absent for the unit and solve a
puzzle involving factors, multiples and prime numbers.

By the conclusion of this unit, students should be able to demonstrate the following
competencies:
Students should be able to find all the factors of a number by constructing or drawing
arrays, thus “proving” the presence of 1 as a factor of all numbers.
Students should be able to list all the factors of any given number and discuss how they
know that the number is prime, composite or neither (the number 1 is neither prime nor
composite).
Students should be able to determine the greatest common factor of two or more numbers
and offer situations in which it would be useful to know common and greatest common
factors of two or more numbers.

ENDURING UNDERSTANDINGS:
Factors and multiples are related in ways that are similar to the way that multiplication
and division are related.
All natural numbers greater than one are either prime or can be written as a unique
product of prime factors.
The number 1 (one) is always a factor of any number.



Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 6  UNIT 2: FUN AND GAMES
September 1, 2009  Page 3 of 17
Copyright 2009 © All Rights Reserved

Georgia Performance Standards Framework
ndGrade 6 Mathematics Unit 2 2 Edition

ESSENTIAL QUESTIONS:
When or why would it be useful to know the factors of a number?
Whehy be useful t multiples of a?
What features does a number have if the number is prime?
What role does the number 1 have when you are finding factors of any number?
How can I use the array model to represent factors of a number?
How applicable are multiples in everyday life?
How applicable are factors in everyday life?
How can I use models to represent multiples of a number?




STANDARDS ADDRESSED IN THIS UNIT
Mathematics standards are interwoven and should be addressed throughout the year in as
many different units and activities as possible in order to emphasize the natural
connections that exist among mathematical ideas.

KEY STANDARDS:
M6N1. Students will understand the meaning of the four arithmetic operations as related to
positive rational numbers and will use these concepts to solve problems.
a. Apply factors and multiples.
b. Decompose numbers into their prime factorization (Fundamental Theorem of
Arithmetic).
c. Determine the greatest common factor (GCF) and the least common multiple (LCM)
for a set of numbers.


RELATED STANDARDS:
M6P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.

M6P2. Students will reason and evaluate mathematical arguments.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.

M6P3. Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers,
and others.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 6  UNIT 2: FUN AND GAMES
September 1, 2009  Page 4 of 17
Copyright 2009 © All Rights Reserved

Georgia Performance Standards Framework
ndGrade 6 Mathematics Unit 2 2 Edition

c. Use the language of mathematics to express mathematical ideas precisely.

M6P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce
a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.

M6P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical
ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical
phenomena.



Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 6  UNIT 2: FUN AND GAMES
September 1, 2009  Page 5 of 17
Copyright 2009 © All Rights Reserved

Georgia Performance Standards Framework
ndGrade 6 Mathematics Unit 2 2 Edition

SELECTED TERMS AND SYMBOLS:
The following terms and symbols are often misunderstood. These concepts are not an
inclusive list and should not be taught in isolation. However, due to evidence of frequent
difficulty and misunderstanding associated with these concepts, instructors should pay
particular attention to them and how their students are able to explain and apply them.

 Arrays: rectangular arrangements that have equal numbers in the rows and columns.

 Decompose: The process of factoring terms and numbers in an expression.

 Exponent: The number of times a number or expression (called base) is used as a factor of
repeated multiplication. Also called the power.

 Factor: When two or more integers are multiplied, each number is a factor of the product.
"To factor" means to write the number or term as a product of its factors.

 Fundamental Theorem of Arithmetic: Every integer, N > 1, is either prime or can be
uniquely written as a product of primes.

 GCF: Greatest Common Factor: The largest factor that two or more numbers have in
common.

 Identity property of multiplication: A number that can be multiplied by any second
number without changing the second number. The Identity for multiplication is “1”.

 LCM: Least Common Multiple: The smallest multiple (other than zero) that two or more
numbers have in common.

 Multiple: A number that is a product of a given whole number and another whole number.

 Prime factorization: The expression of a composite number as a product of prime numbers.

 Prime number: A positive number that is divisible only by itself and the number one.

 Square number: A number that is the product of a whole number and itself. This is also
known as a “Perfect Square”.

 Composite number: A composite number is a number that has factors in addition to one and
itself.

You may visit http://intermath.coe.uga.edu or http://mathworld.wolfram.com to see definitions
and specific examples of many terms and symbols used in the seventh-grade GPS.

Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 6  UNIT 2: FUN AND GAMES
September 1, 2009  Page 6 of 17
Copyright 2009 © All Rights Reserved

Georgia Performance Standards Framework
ndGrade 6 Mathematics Unit 2 2 Edition

TASKS:
The collection of the following tasks represents the level of depth, rigor and complexity
expected of all sixth grade students to demonstrate evidence of learning. These tasks or
tasks of similar depth and rigor should be used to demonstrate evidence of learning.

Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 6  UNIT 2: FUN AND GAMES
September 1, 2009  Page 7 of 17
Copyright 2009 © All Rights Reserved

Georgia Performance Standards Framework
ndGrade 6 Mathematics Unit 2 2 Edition

Back to School!

Part 1: Music

You and your friends have tickets to attend a music concert. While standing in line, the
promotion states he will give a free album download to each person that is a multiple of 2. He
will also give a backstage pass to each fourth person and floor seats to each fifth person.

Which person will receive the free album download, backstage pass, and floor seats? Explain
the process you used to determine your answer.



Backstage MUSIC
P A S S Gift Card

VIP












Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 6  UNIT 2: FUN AND GAMES
September 1, 2009  Page 8 of 17
Copyright 2009 © All Rights Reserved

Georgia Performance Standards Framework
ndGrade 6 Mathematics Unit 2 2 Edition

Part 2: School Supplies

The Parents Teachers Association (PTA) at your school donated school supplies to help increase
student creativity and student success in the classroom. Your teacher would like you to create
kits that include one package of colored pencils, one glue stick, and one ruler. When you receive
the supplies, you notice the colored pencils are packaged 12 boxes to a case, the rulers are
packaged 30 to a box, and glue sticks are packaged 4 to a box.




Rulers
30/box

Colored Pencils

12/box



Glue
4/box


1. What is the smallest number of each supply you will need in order to make the kits and not
have supplies left over? Explain your thought process.





2. How many packaged rulers, colored pencils, and glue sticks will you need in order to make
the kits? Explain the process you used to determine how many packages are needed for each
supply.











Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 6  UNIT 2: FUN AND GAMES
September 1, 2009  Page 9 of 17
Copyright 2009 © All Rights Reserved

Georgia Performance Standards Framework
ndGrade 6 Mathematics Unit 2 2 Edition

Extension
School Lunch

The Yearbook club at your school is sponsoring a fall festival to kick off the annual yearbook
drive. The club sponsor has asked for your help in determining what food items to sell and how
much of each item he needs to buy. Write a budget report supporting your decision.


Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 6  UNIT 2: FUN AND GAMES
September 1, 2009  Page 10 of 17
Copyright 2009 © All Rights Reserved