Download Algebra 1 Unit 3: Systems of Equations

6th Grade Unit 2 Prime Time
Enduring understanding (Big Idea): 1. Determine the factors of whole numbers. 2. Find greatest common factors and
generate multiples. 3. Determine prime factorization. 4. Classify numbers as prime or composite.
Essential Questions: Will breaking a number into factors help me solve the problem? What relationships will doing that help
me see? What do the factors and multiples of the numbers tell me about the situation? How can I find the factors of the
numbers? How can I find the multiples? What common factors and common multiples do the numbers have?
BY THE END OF THIS UNIT:
Students will know GCF, LCM; the difference
Between factors and multiples; how to factor numbers
Vocabulary: common factor, common multiple, composite number,
Conjecture, Dimensions, (whole number) divisor, even number,
exponent, (whole number) factor, factor pair, factorization, greatest
common factor, least common multiple, odd number, prime
factorization, prime number, proper factor, square number, Venn
diagram
Unit Resources
Students will be able to…
 Find relationships among factors, multiples, divisors,
and products
 Find whole-number patterns and relationships and use
the patterns to reason about future occurrences.
 Find common multiples and common factors
 Factor and find prime factorizations
Mathematical Practices in Focus:
1. Make sense of problems and persevere in solving them.
Prime Time CMP2
4. Model with Mathematics.
5. Use appropriate tools strategically.
7. Look for and make use of structure
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
6th Grade Unit 2 Prime Time
Prime Time: Factor and Multiples
Unit Plans
Standards 6.EE.2.b,
Investigation 1 Factors and Multiples
Standard 6.NS.4
Investigation 2 Whole Number Patterns
and Relationships
Standard 6.NS.4, 6.EE.2b
Investigation 3 Common Multiples and
Common Factors
Standards 6.EE.1, 6.EE.2.b
Investigation 4 Factorizations
Investigation
Suggested ACE Questions
Problem 1.1 Playing the Factor Game
Problem 1.2 Playing to Win the Factor
Game
Problem 1.3 The Product Game
Mathematical Reflections
1.1 ACE #1-7, 28-31, 39
1.2 ACE #8-16, 32-34, 40-44
1.3 ACE #17-27, 35-38, 45-49
Problem 2.1 Finding Patterns
Problem 2.3 Classifying Numbers
Mathematical Reflections
2.1 ACE #1-9, 22-24; 31, 32
2.3 ACE #17-21, 26-30
Problem 3.1 Riding a Ferris Wheel
Problem 3.2 Looking at Cicada Cycles
Problem 3.3 Bagging Snacks
Problem 3.4 Planning a Picnic
Mathematical Reflections
3.1 ACE
3.2 ACE
3.3 ACE
3.4 ACE
Problem 4.1 Product Puzzle
Problem 4.2 Finding the Longest Factor
String
Problem 4.3 Using Prime Factorization
Mathematical Reflections
4.1ACE #1-3,25
4.2 ACE #4-18,27-29
4.3ACE #19-24,32
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
#1-8, 31, 32
#9-15
#16-23
#24-30
6th Grade Unit 2 Prime Time
CORE CONTENT
Cluster Title: Apply and extend previous understandings of arithmetic to algebraic expressions.
Standard 6.EE.2.b Write, read, and evaluate expressions in which letters stand for numbers.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or
more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view
(8 + 7) as both a single entity and a sum of two terms.
Concepts and Skills to Master
 Understand that terms are values in an expression separated by addition and subtraction (e.g., x + 3 contains two terms;
2x – 5 contains two terms).
 Understand that multiplication and division in an expression represent a single term (e.g., 3y is a term, ½x is a term).
 Know that a coefficient is a number that multiplies a variable (e.g., 3 is the coefficient of 3y).
 Read an expression correctly using appropriate mathematical terms (e.g. 2(8+7) is read as the product of 2 times the
quantity or sum of 8+7).
 Identify parts of an expression (e.g., in the expression 2x – 5, 2x is the first term where 2 is the coefficient of the variable
x and 5 is the second term, a constant).
 Represent the parts of an expression with manipulatives, diagrams, or coordinate plane.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Recognize that variables represent unknown quantities.
 Know the meaning of the terms sum, product, factor, quantity, and quotient.
 Read expressions of single operations (e.g., 3y, x + 7, 4 + 3).
 Represent the four operations (addition, subtraction, multiplication and division) with manipulatives, diagrams, number lines,
and story context.
Academic Vocabulary
Coefficient, Dividend, Divisor, Equation, Expression, Factor, Multiplier, Product, Quotient, Sum, Term (optional term – Constant),
Parentheses
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
6th Grade Unit 2 Prime Time
Suggested Instructional Strategies
Resources
1. Give an example of an expression, such as 4x + 7 + x, and identify the parts. Textbook Correlation
The terms are 4x, 7 and x. The coefficients are 4 and 1 (since there is one x).
 Prime Time Investigation 1,3,4
The values 4 and x are both factors of the product 4x. Give additional
examples to students to identify parts of expressions in small groups.
http://nlvm.usu.edu/en/nav/category_g_3_t_2.html
2. Give the students word problems such as the following and have them find
the solutions:
Algebra Tiles
The sum of twice a number plus 13 is 75. Write the equation and find the
number.
 Math Goodies - Writing Algebraic
3. Use the algebra tiles to represent terms in an expression (e.g., 2x requires
Expressions - Tutorial and Practice Exercises
two x pieces)
Game 1:
http://www.mathgoodies.com/lessons/vol7/e
Give students two sets of cards. The first set contains equations and the
xpressions.html
second set contains the academic vocabulary. Students turn over an equation
 Study Guides and Tutorials - Teacher
card. Then students match academic vocabulary to the parts of the equation.
Tutorial Game 2:
http://www.studygs.net/mathproblems.htm
Start with a small expression, such as 2x - 4.
 Mathwire - I Have, Who Has - Game Break the expression into distinct parts. Write on separate cards 2, x, -, and 4.
http://mathwire.com/whohas/whalgA.pdf
Assign a child a card to hold and line the children in the front of the
 Math Play - Who Wants to Be a Millionaire classroom in the order of the expression.
Game –
To identify parts of the expression, have students sit down when you say
http://www.math-play.com/Algebraicthe vocabulary word that corresponds with their card.
ExpressionsMillionaire/algebraicexpressions-millionaire.html
Example: Teacher says “coefficient.” Student holding the 2 sits down. Teacher
says “term.” Students holding 2 and x may link arms and sit, as well as student
holding 4.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
6th Grade Unit 2 Prime Time
Sample Formative Assessment Tasks
Skill-based task
Provide an expression (e.g., 6x + 4) for students to identify
the parts.
Write an expression that has two terms and a product.
Write an expression that has a coefficient and a sum.
Problem Task
This is a skill-based standard. Therefore, there is no problem task.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
6th Grade Unit 2 Prime Time
CORE CONTENT
Cluster Title: Apply and extend previous understandings of arithmetic to algebraic expressions
Standard 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.
Concepts and Skills to Master
 Understand the meaning of exponents.
 Understand exponential notation (e.g., and ). 3^4=34
 34=3×3×3×3
 Understand that exponential notation represents multiplication where the base always remains the same.
 Find the value of an expression using exponential notation (e.g., ). 34=81
 Write exponential notation (e.g., and ). 3^4=34
 34=3×3×3×3
 Use multiple representations that illustrate geometric dimensions and collections of quantities such as:
 Represent the power of 1 as a linear model. This results in a one dimensional representation.
 Represent the power of 2 as area with square units. This results in a two dimensional representation.
 Represent the power of 3 as volume with cubic units. This results in a three dimensional representation.
 Represent the accumulation of quantities.
 Show that an exponent is notation representing repeated multiplication
SUPPORTS FOR TEACHERS
Critical Background Knowledge
 Repeated multiplication of similar “base” quantities (e.g., 2×2=4 2×4=8 2×8=16
 Skip counting
 Multiplication facts
 Model repeated multiplication with manipulatives, arrays, and number lines.
Academic Vocabulary
^, base number, expressions, exponents, power, superscripted numbers
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
6th Grade Unit 2 Prime Time
Suggested Instructional Strategies
1. Have the students use manipulatives to show the number 3 after it
has been doubled, which would represent 32=9 square units. Then
have them show 33 = 27 cubic units with the manipulatives.
2. Play memory games with matching numerical expressions.
Example: 2^3 matches with 23, which matches with 2 x 2 x 2, which
matches with 8.
3. “I Have, Who Has?” game.
Example:
The first student says: “I have 2^3.”
The second student says: “I have 8. Who has 4^2?”
The third student says: “I have 16. Who has the expression equal to
27?” Etc…
Extension: Every positive integer can be expressed as the sum of
four or fewer squared numbers (Note: You can use the same integer
more than once.)
Examples:
5 = 12 + 22 101 = 42 + 62 + 72 170 = 22 + 62 + 72 + 92
Write 83 as the sum of three square numbers
Sample Formative Assessment Tasks
Skill-based task
Evaluate:
32
Write five cubed times four as a numerical expression.
Resources
Textbook Correlation
 Prime Time Investigation 4
The King’s Chessboard, by David Burch
One Grain of Rice, by Demi
http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Martin/i
nstructional%20unit/day4.exponential/excel/grainofrice.html
Minnie’s Diner, by Dale Ann Dodds
http://www.wimp.com/makingnoodles
http://cca6summer2011.wikispaces.com/file/detail/CCA6201
1RP3_NS3_EE1.doc
http://cca6summer2011.wikispaces.com/file/detail/EE1+Expo
nents+6th+grade.doc
Problem Task
Certain biological cells quadruple each hour. Start with one
cell at 2:00 and find out how many cells there will be by 5:00.
Create a diagram to represent the cell growth. Include an
equation using exponential notation.
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
6th Grade Unit 2 Prime Time
CORE CONTENT
Cluster Title: Compute fluently with multi-digit numbers and find common factors and multiples.
Standard 6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two
whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor
as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9 + 2).
Concepts and Skills to Master
 Understand that greatest common factor and least common multiple are ways to discuss number relationships in multiplication and
division.
 Understand the distributive property using sums and its use in adding numbers 1-100 with a common factor (e.g., 20 + 24 = 4(5 + 6)).
 Compute fluently using the distributive property of multiplication over addition.
 Find greatest common factor of two whole numbers less than or equal to 100.
 Find the least common multiple of two whole numbers less than or equal to 12.
 Model factorization of whole numbers 1-100 using a number line and manipulatives.
SUPPORTS FOR TEACHERS
Critical Background Knowledge
Understand that a factor is a whole number that divides without a remainder into another number; Understand that a multiple is a whole
number that is a product of the number and any other factor; Know the distributive property; Compute using the distributive property;
Find factors and multiples of a given number; Model the distributive property of multiplication over addition using manipulatives,
diagrams, and story contexts
Academic Vocabulary
distributive property, factor, greatest common factor (GCF), least common multiple (LCM), multiple
Suggested Instructional Strategies
Resources
Textbook Correlation: Prime Time Investigation 2,3
Solve for LCM and/or GCF using factor
http://cca6summer2011.wikispaces.com/file/detail/NS4+-+Distributive+Property+towers, Venn diagrams, and factor trees.
+Equal+groups.doc
Use a model to show that 4(9 + 2) is four
Distributing and Factoring Using Area
groups of 9 and four groups of 2.
http://illuminations.nctm.org/LessonDetail.aspx?id=L744
Online Practice for the distributive Property
http://www.ixl.com/math/grade-6/distributive-property
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.
6th Grade Unit 2 Prime Time
Sample Formative Assessment Tasks
Skill-based task
Find the greatest common factor of 24 and 60.
Find the least common multiple of 6 and 10.
Use the distributive property to show 15 + 75
Problem Task
Hot dogs come in packs of 8. Buns come in packs of 12. How many
packs of hot dogs and bags of buns would you have to buy to have an
equal number of hot dogs and buns?
You need to make gift bags for a party with the same number of
balloons and candy in each bag. One package of candy has 24 pieces.
One package of balloons has 20 balloons. You need to use all the
candy and all the balloons. What is the greatest number of gift bags
that you can make containing an equal number of items?
Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order
not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.