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The following instructional plan is part of a GaDOE collection of Unit Frameworks, Performance Tasks, examples of Student Work, and Teacher Commentary.
Many more GaDOE approved instructional plans are available by using the Search Standards feature located on GeorgiaStandards.Org.
Georgia Performance Standards Framework for Mathematics – Grade 6
Unit Two Organizer: “FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY”
(3 weeks)
OVERVIEW:
This unit addresses concepts and applications of number theory. Topics include prime numbers, composite numbers, factors,
multiples, least common multiple and greatest common factor. Instruction should include the representation of these numbers and
their relationships to other concepts, such as multiplication and division using diagrams, charts, tables 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 teacherfacilitated whole class discussion. 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.
To assure that this unit is taught with the appropriate emphasis, depth and rigor, it is important that the tasks listed under “Evidence of
Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely
replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The
tasks in these units illustrate the type of learning activities that should be utilized from a variety of sources.
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.
ESSENTIAL QUESTIONS:
•
•
How are multiplication and division related?
How are factors and multiples related to multiplication and division?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 1 of 19
Copyright 2006 © All Rights Reserved
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Georgia Performance Standards Framework for Mathematics – Grade 6
•
•
•
•
When or why would it be useful to know the factors of a number?
When or why would it be useful to know the multiples of a number?
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?
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.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 2 of 19
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Georgia Performance Standards Framework for Mathematics – Grade 6
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
d. 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.
The following standards may not necessarily be used in this unit on the number theory, but are related in the following ways: Arrays, rectangular
arrangements that have equal numbers in the rows and columns, are good conceptual models for multiplication and are an outstanding way to
convey the concept of factors. All of the possible arrays for a number, i.e., 1x20, 2 x 10, 4x5, (5x4, 10x2, 20x1) are the facts and factors of the
number. Arrays that “cover” an area with equal-sized squares are also called area models. When students are working with scale drawings they
will enlarge or reduce a drawing or model from its original size by a “scale factor”. That is, they are making the entire drawing or model larger by
multiplying by a factor or smaller by dividing by a factor. All conversions, measurement in this case, involve a “factor” for the conversion.
M6M2. Students will use appropriate units of measure for finding length, perimeter, area and volume and will express each
quantity using the appropriate unit.
a. Measure length to the nearest half, fourth, eighth and sixteenth of an inch.
b. Select and use units of appropriate size and type to measure length, perimeter, area and volume.
c. Compare and contrast units of measure for perimeter, area, and volume.
M6G1. Students will further develop their understanding of plane figures.
c. Use the concepts of ratio, proportion and scale factor to demonstrate the relationships between similar plane figures.
d. Interpret and sketch simple scale drawings.
e. Solve problems involving scale drawings.
M6A2. Students will consider relationships between varying quantities.
a. Analyze and describe patterns arising from mathematical rules, tables, and graphs.
b. Use manipulatives or draw pictures to solve problems involving proportional relationships.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 3 of 19
Copyright 2006 © All Rights Reserved
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Georgia Performance Standards Framework for Mathematics – Grade 6
CONCEPTS/SKILLS TO MAINTAIN:
It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be
necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a
deeper understanding of these ideas.
•
•
Multiples and factors
Perimeter, capacity and area of geometric figures
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.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 4 of 19
Copyright 2006 © All Rights Reserved
One Stop Shop For Educators
Georgia Performance Standards Framework for Mathematics – Grade 6
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”.
You may visit www.intermath-uga.gatech.edu and click on dictionary to see definitions and specific examples of terms and symbols
used in the sixth grade GPS.
EVIDENCE OF LEARNING:
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.
• Students should be able to use their understanding of number theory in this unit to solve puzzles and problems. (Number
theory, especially as pertains to prime numbers, was often used in creating “codes” before sophisticated computers took over
the job. Some of the toughest codes in the world are made using mega prime numbers. Search for “prime numbers in codes” in
your Internet search engine or see http://www.murderousmaths.co.uk/games/primcal.htm)
The following task represents the level of depth, rigor, and complexity expected of all 6th grade students. This task or a task of similar
depth and rigor should be used to demonstrate evidence of learning.
Culminating Activity: “Three Number Theory Challenges”
This activity will have three parts:
• “Hi Mike! Let me tell you about the Fundamental Theorem of Arithmetic!”
• “Secret Number”
• “Slammin’ Lockers”
They are activities where students will visit practical applications of the number theory concepts taught in this unit.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 5 of 19
Copyright 2006 © All Rights Reserved
One Stop Shop For Educators
Georgia Performance Standards Framework for Mathematics – Grade 6
STRATEGIES FOR TEACHING AND LEARNING:
•
•
•
•
Students should be actively engaged in developing their own understanding.
Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols and words.
Appropriate manipulatives and technology should be used to enhance student learning.
Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition
which includes self-assessment and reflection.
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.
• Codes
Codes
Number theory, especially as pertains to prime numbers, was often used in creating “codes” before sophisticated computers
took over the job. Some of the toughest codes in the world are made using mega prime numbers. To learn more about this
search for “prime numbers in codes” in your Internet search engine or see
http://www.murderousmaths.co.uk/games/primcal.htm
*********************************************************************************
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 6 of 19
Copyright 2006 © All Rights Reserved
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Georgia Performance Standards Framework for Mathematics – Grade 6
• Multiple/Divisible – What’s the Difference?
Multiple/Divisible – What’s the Difference?
All multiples of two are divisible by two, all multiples of three are divisible by 3, all multiples of 7 are divisible by 7.
This relationship for multiples and divisibility is true in all cases.
Name some other examples and explain why this is always true?
What are some common multiples for 2, 3 and 7?
What is the least common multiple for 2, 3, and 7?
*********************************************************************************
Multiple/Divisible – What’s the Difference?
Discussion, Suggestions, Possible solutions:
When we divide a number by another number and there is no remainder, we say that the first number (let’s say 20) is
divisible by the second number (let’s say 5). There are many ways that we can express this. We can say that 5 divides 20, 5
is a divisor of 20, 5 is a factor of 20, 20 divided by 5 is 4 and there is no remainder, 20 is divisible by 5, 20 is a multiple of 5.
If we count by 5’s, we name the multiples of 5: 5, 10, 15, 20, 25, and so on. Notice that any of the multiples we just named
(5, 10, 15, 20, 25, ...) can all be divided by 5 with no remainder. A multiple of a number is always divisible by that number
with no remainder.
Other examples may vary: All multiples of 8 are divisible by 8. 8, 16, 24, 32, 40, …can all be divided by 8 without a
remainder.
Common multiples for 2, 3, and 7 are 42, 84, 126, 168, etc. The least common multiple for 2, 3 and 7 is 42.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 7 of 19
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Georgia Performance Standards Framework for Mathematics – Grade 6
One way to have students physically produce (and then sketch on paper) a solution is to make “trains” of twos and threes
with Cuisenaire rods. This same effect can be produced with a “double number line” with 2’s being shaded above and
threes being shaded below. In either case, the 2/s and 3/s trains are equi-distant from zero (or the starting point) at 6, 12,
18, 24, 36, 42, etc. When you create another train or a triple number line that includes 7, students realize that the first time
a number that they have in common arises is when they reach 42.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23 24
• Arrays, Factors, and Number Theory
Arrays, Factors, and Number Theory
Create, draw or shade all possible arrays for the numbers 1-20.
a) Label all of the dimensions of the arrays, which are the factors of each number.
b) Look for patterns in the arrangements, factors, or drawings.
c) Describe the patterns or observations that help you “see” the factors, prime numbers, composite numbers and square
numbers.
d) In the numbers 1-20, label the prime, composite, and square numbers.
e) Describe all the things you notice about the arrays and patterns, but especially discuss what you notice about the
number 1.
*********************************************************************************
Arrays, Factors, and Number Theory
Discussion, suggestions, possible solutions:
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 8 of 19
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Georgia Performance Standards Framework for Mathematics – Grade 6
Number Arrays
1
Facts
1 x1
Factors
1
Patterns, observations
Only one factor, 1. You can always
represent a number using one row.
2
All the ways to make 2 into a rectangular
arrangement
1 x 2, 2x1
1, 2
Two factors – Prime
The different orientations
demonstrate the commutative
property of multiplication.
3
All the ways to make 3 into a rectangular
arrangement
1 x 3, 3 x 1
1, 3
Two factors - Prime
4
All the ways to make 4 into a rectangular
arrangement
1 x 4, 2 x2,
4x1
1, 2, 4
More than two factors – Composite,
and one of the arrays is a square – 4
is a square number
5
All the ways to make 5 into a rectangular
arrangement
1 x 5, 5 x 1
1, 5
Two factors - Prime
6
All the ways to make 6 into a rectangular
arrangement
1 x 6, 2 x3,
3 x2, 6 x 1
1, 2,3,6
More than two factors - composite
7
All the ways to make 7 into a rectangular
arrangement
1x7
1, 7
Two factors - Prime
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 9 of 19
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Georgia Performance Standards Framework for Mathematics – Grade 6
8
All the ways to make 8 into a rectangular
arrangement
1 x 8, 2 x4,
reverse
1, 2, 4, 8
More than two factors - composite
9
All the ways to make 9 into a rectangular
arrangement
1 x 9, 3 x 3
1, 3, 9
More than two factors – composite;
one of the arrays is a square – 9 is a
square number.
10
All the ways to make 10 into a rectangular
arrangement
1 x10, 2 x 5
1, 2, 5, 10
More than two factors - composite
11
All the ways to make 11 into a rectangular
arrangement
1 x 11
1, 11
Two factors - prime
12
All the ways to make 12 into a rectangular
arrangement
1 x 12, 2 x
6. 3 x 4
1, 2, 3, 4,
6, 12
More than two factors - composite
13
All the ways to make 13 into a rectangular
arrangement
1 x 13
1, 13
Two factors - Prime
14
All the ways to make 14 into a rectangular
arrangement
1 x 14, 2 x 7
1, 2, 7, 14
More than two factors - composite
15
All the ways to make 15 into a rectangular
arrangement
1 x 15,3 x 5
1, 3, 5, 15
More than two factors - composite
16
All the ways to make 16 into a rectangular
arrangement
1, 16, 2, x 8,
4x4
1, 2, 4, 8,
16
More than two factors – composite;
one of the arrays is a square – 16 is a
square number.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 10 of 19
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Georgia Performance Standards Framework for Mathematics – Grade 6
17
All the ways to make 17 into a rectangular
arrangement
1 x 17
1, 17
Two factors - Prime
18
All the ways to make 18 into a rectangular
arrangement
1 x 18, 2 x
9, 3, x 6
1, 2,3, 6,
9, 18
More than two factors - composite
19
All the ways to make 19 into a rectangular
arrangement
1 x 19
1, 19
Two factors - prime
20
The arrays for 20 are shown below
1 x20, 2 x
10, 4 x 5, 5
x 4, 10 x2,
20 x 1
1, 2, 4, 5,
10, 20
More than two factors- composite
1
2
1
1
3
4
2
2
5
3
3
6
7
4
4
1
1
1
1
8
9
5
5
2
2
2
2
10
6
6
3
3
3
3
11
7
7
4
4
4
4
12
8
8
13
9
9
14
15
16
17
18
19
20
10
10
5
5
5
5
Observations about 1. The array for 1 is one row by one column, or 1 x 1. It has only one factor, In comparison to all the
other numbers, which either have two factors (making them prime) or more than two factors (making them composite), we can
see that one falls into neither of those categories. One is neither prime nor composite. Also, one is present in ALL arrays as
you can always arrange a number into 1 row, so one of the dimensions of an array is always 1. This can visually prove that
1 is always a factor of any number and explains the identity property of multiplication.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 11 of 19
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Georgia Performance Standards Framework for Mathematics – Grade 6
• Common Factors
Common Factors
Use your previous work from Arrays, Factors and Number Theory
a) List all the factors in order from least to greatest, for each number 1-20.
b) Choose any two numbers from your list of 1-20. What factors are in both lists?
c) What is the largest factor that they have in common?
d) Try this on several other pairs of numbers from 1-20.
e) Can you do it for 3 of the numbers? Try it for 3 numbers.
f) When would it be useful to know the common factors or the greatest common factor of two or more
numbers?
g) What advice would you offer to a friend who was having trouble finding all the factors of any number?
*********************************************************************************
Common Factors
Discussion, suggestions, possible solutions:
a) All of the factors listed in order from least to greatest can be seen in the previous chart.
b) Answers may vary.
c) Answers may vary
d) Answers may vary
e) Answers may vary
f) Knowing common factors can help us when we want to simplify fractions. Students may provide
other ideas.
g) Answers may vary, but might include making or visualizing the number in all possible rectangles or
arrays, and recording the dimensions ( or numbers of rows and columns.)
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 12 of 19
Copyright 2006 © All Rights Reserved
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Georgia Performance Standards Framework for Mathematics – Grade 6
The Square Game
The Square Game
The Square Game: (from Achieve, Inc.)
In a party, the following game is played: Each player is given a rectangular piece of graph paper that is 56 centimeters long
and 84 centimeters wide. The horizontal and vertical lines are spaced one centimeter apart.
The paper is to be cut along the grid (graph) lines into square pieces that are all the same size without having any paper left
over. The winner is the one who cuts the largest square pieces of paper. What would be the length in centimeters of the side
of each winning square?
*********************************************************************************
• You are the Teacher! Give ‘em homework!
You are the Teacher! Give ‘em homework!
Your teacher’s favorite method for assigning homework problems is to assign the factors or multiples of some of the number
of problems in your book. For example, he might say, “On page 78, out of the 32 problems that are there, do the problems
that are the factors of 24.” On another day, he might say, “On page 84, out of the 32 problems that are there, do the
problems that are the multiples of 3”. In which case would you do more problems? Explain how you figured it out. Suggest
another use of factors, multiples, or primes to your teacher to use when assigning problems. Explain why you chose this
method.
*********************************************************************************
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 13 of 19
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Georgia Performance Standards Framework for Mathematics – Grade 6
•
Three Number Theory Challenges
This culminating task has three parts that together represent the level of depth and rigor and complexity expected of all 6th
grade students to demonstrate evidence of learning.
UNIT TWO TASK: “Three Number Theory Challenges
Part I: Hi Mike! Let me tell you about the Fundamental Theorem of Arithmetic!”
The fundamental theorem of arithmetic states that every natural number greater than one is either prime or can be written as
a unique product of prime factors. What does this mean? Refer to the work you did in previous problems to help you
explain the fundamental theorem of arithmetic to your friend, Mile, who has been absent. Be sure to include the following
terms: factor, multiple, divisible, prime, composite, prime factorization and exponents.
Part II: Secret Number
Juanita has a secret number. Read her clues and then answer the questions that follow:
Juanita says, “Clue 1” My secret number is a factor of 60.”
1. Can you tell what Juanita’s secret number is? Explain your reasoning.
2. Daren said that Juanita’s number must also be a factor of 120. Do you agree or disagree with Daren? Explain your
reasoning.
3. Malcolm says that Juanita’s number must also be a factor of 15. Do you agree or disagree with Malcolm? Explain
your reasoning.
4. What is the smallest Juanita’s number could be? Explain.
5. What is the largest Juanita’s number could be. Explain.
Suppose for Juanita’s second clue she says, “ Clue 2: My number is prime.”
6. Can the class guess her number and be certain? Explain your answer.
Suppose for Juanita’s third clue she says, “Clue 3: 15 is a multiple of my secret number.”
7. Now can you tell what her number is? Explain your reasoning.
8. Your secret number is 36. Write a series of interesting clues using factors, multiples, and other number properties
needed for somebody else to identify your number.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 14 of 19
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Georgia Performance Standards Framework for Mathematics – Grade 6
This task is from Balanced Assessment for the Mathematics Curriculum, Middle Grades Assessment Package 2 Berkely,
Harvard, Michigan State, Shell Centre, Dayle Seymour Publications, Copyright 2000, pages 189-200.
Part III: Slammin’ Lockers
Georgia Middle School has 100 students with lockers numbered 1 through 100. One day, Sally walks down the hall and
opens all the lockers. Dick goes behind her and closes all the lockers with even number. Then, Jane changes the situation of
the lockers with numbers that are multiples of 3. (Changing the situation means that a closed locker is opened and an open
locker is closed.) if this pattern continues FOR ALL 100 STUDENTS, which lockers will remain open after the 100th
student walks down the hall?
Explain your thinking giving details, and using both appropriate mathematical models and language.
What if there were 500 students and 500 lockers? 1000?
Can you find a rule for any number of students and lockers? Explain why your rule works.
Standards Addressed in this Task
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.
M6P3a: Organize and consolidate their mathematical thinking through communication.
M6P3b: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
M6P3d: Use the language of mathematics to express mathematical ideas precisely.
M6P4a: Recognize and use connections among mathematical ideas.
M6P4b: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 15 of 19
Copyright 2006 © All Rights Reserved
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Georgia Performance Standards Framework for Mathematics – Grade 6
Concepts/Skills to Maintain
•
•
Multiples and factors
Perimeter, capacity and area of geometric figures
Suggestions for Classroom Use
While this task may serve as a summative assessment, it also may be used for teaching and learning. It is important that all
elements of the task be addressed throughout the learning process so that students understand what is expected of them.
•
•
•
Peer Review
Display for parent night
Place in portfolio
Discussion, Suggestions and Possible Solutions
Part I: Hi Mike! Let me tell you about the Fundamental Theorem of Arithmetic!”
The fundamental theorem of arithmetic states that every natural number greater than one is either prime or can be written as
a unique product of prime factors. What does this mean? Refer to the work you did in previous problems to help you
explain the fundamental theorem of arithmetic to your friend, Mile, who has been absent. Be sure to include the following
terms: factor, multiple, divisible, prime, composite, prime factorization and exponents.
Solutions will vary therefore it would be useful for students to conduct a peer review of each other’s work asking and
answering questions; justifying and defending work and providing opportunities for revision.
Part II: Secret Number
Juanita has a secret number. Read her clues and then answer the questions that follow:
Juanita says, “Clue 1” My secret number is a factor of 60.”
a. Can you tell what Juanita’s secret number is? Explain your reasoning.
b. Daren said that Juanita’s number must also be a factor of 120. Do you agree or disagree with Daren? Explain
your reasoning.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 16 of 19
Copyright 2006 © All Rights Reserved
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Georgia Performance Standards Framework for Mathematics – Grade 6
c. Malcolm says that Juanita’s number must also be a factor of 15. Do you agree or disagree with Malcolm?
Explain your reasoning.
d. What is the smallest Juanita’s number could be? Explain.
e. What is the largest Juanita’s number could be. Explain.
Suppose for Juanita’s second clue she says, “ Clue 2: My number is prime.”
f. Can the class guess her number and be certain? Explain your answer.
Suppose for Juanita’s third clue she says, “Clue 3: 15 is a multiple of my secret number.”
g. Now can you tell what her number is? Explain your reasoning.
Solutions will vary therefore it would be useful for students to conduct a peer review of each other’s work asking and
answering questions; justifying and defending work and providing opportunities for revision.
This task is from Balanced Assessment for the Mathematics Curriculum, Middle Grades Assessment Package 2 Berkely,
Harvard, Michigan State, Shell Centre, Dayle Seymour Publications, Copyright 2000, pages 189-200.
Part III: Slammin’ Lockers
Georgia Middle School has 100 students with lockers numbered 1 through 100. One day, Sally walks down the hall and
opens all the lockers. Dick goes behind her and closes all the lockers with even number. Then, Jane changes the situation of
the lockers with numbers that are multiples of 3. (Changing the situation means that a closed locker is opened and an open
locker is closed.) if this pattern continues FOR ALL 100 STUDENTS, which lockers will remain open after the 100th
student walks down the hall?
Explain your thinking clearly, using appropriate mathematical models and mathematical language.
What if there were 500 students and 500 lockers? 1000?
Can you find a rule for any number of students and lockers? Explain why your rule works.
Solutions will vary therefore it would be useful for students to conduct a peer review of each other’s work asking and
answering questions; justifying and defending work and providing opportunities for revision.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 17 of 19
Copyright 2006 © All Rights Reserved
One Stop Shop For Educators
Georgia Performance Standards Framework for Mathematics – Grade 6
STUDENT WORK:
Part I: Hi Mike! Let me tell you about the Fundamental Theorem of Arithmetic!”
The fundamental theorem of arithmetic states that every natural number greater than one is either prime or can be written as
a unique product of prime factors. What does this mean? Refer to the work you did in previous problems to help you
explain the fundamental theorem of arithmetic to your friend, Mile, who has been absent. Be sure to include the following
terms: factor, multiple, divisible, prime, composite, prime factorization and exponents.
No student work is available at this time.
Part II: Secret Number
This task is from Balanced Assessment for the Mathematics Curriculum, Middle Grades Assessment Package 2 Berkely,
Harvard, Michigan State, Shell Centre, Dayle Seymour Publications, Copyright 2000, pages 189-200.
Juanita has a secret number. Read her clues and then answer the questions that follow:
Juanita says, “Clue 1” My secret number is a factor of 60.”
h. Can you tell what Juanita’s secret number is? Explain your reasoning.
i. Daren said that Juanita’s number must also be a factor of 120. Do you agree or disagree with Daren? Explain
your reasoning.
j. Malcolm says that Juanita’s number must also be a factor of 15. Do you agree or disagree with Malcolm?
Explain your reasoning.
k. What is the smallest Juanita’s number could be? Explain.
l. What is the largest Juanita’s number could be. Explain.
Suppose for Juanita’s second clue she says, “ Clue 2: My number is prime.”
m. Can the class guess her number and be certain? Explain your answer.
Suppose for Juanita’s third clue she says, “Clue 3: 15 is a multiple of my secret number.”
n. Now can you tell what her number is? Explain your reasoning.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 18 of 19
Copyright 2006 © All Rights Reserved
One Stop Shop For Educators
Georgia Performance Standards Framework for Mathematics – Grade 6
This student’s work
meets the essential
demands of the task.
Almost all questions were
correctly answered and
she provides good
explanations. For
example, she lists all the
factors of 60 and offers a
counterexample when
disagreeing with
Malcolm. She makes an
error in question 6,
forgetting 2 as a prime
factor of 60. In response
to question 8, she gives
clues that enable the
reader to guess her secret
number precisely and all
her clues are needed.
Part III: Slammin’ Lockers
Solutions will vary therefore it would be useful for students to conduct a peer review of each other’s work asking and
answering questions; justifying and defending work and providing opportunities for revision.
No student work is available at this time.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Unit 2 Organizer y FUN AND GAMES: EXTENDING AND APPLYING NUMBER THEORY
September 20, 2006 y Page 19 of 19
Copyright 2006 © All Rights Reserved