ALEX Classroom Resource

  

Grade 6 Mathematics Module 4, Topic D: Expanding, Factoring, and Distributing Expressions

  Classroom Resource Information  

Title:

 Grade 6 Mathematics Module 4, Topic D: Expanding, Factoring, and Distributing Expressions

URL:

 https://www.engageny.org/resource/grade-6-mathematics-module-4-topic-d-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 4, Topic D, students become comfortable with new notations of multiplication and division and recognize their equivalence to the familiar notations of the prior grades. The expression 2 × b is exactly the same as 2 · b and both are exactly the same as 2b.  Similarly, 6 ÷ 2 is exactly the same as 6/2. These new conventions are practiced to automaticity, both with and without variables. Students extend their knowledge of the greatest common factor and the distributive property from Module 2 to expand, factor and distribute expressions using new notation (6.NS.B.4). In particular, students are introduced to factoring and distributing as algebraic identities.  These include:  a + a = 2 · a = 2a, (a + b) + (a + b) = 2 · (a + b) = 2(a + b) = 2a + 2b, and a ÷ b = a/b.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 6
7. Use the distributive property to express the sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Use the distributive property to write an equivalent expression for the sum of the two numbers as the product of the greatest common factor of the two numbers, and the sum of two whole numbers with no common factor. [if the two whole numbers are 36 and 8, 36+8 = 4(9+2)].
Teacher Vocabulary:
  • Greatest common factor
  • Distributive property
  • Parentheses
  • Decompose
Knowledge:
Students know:
  • Distributive property of multiplication over addition.
  • Strategies to express the sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor by decomposing the numbers.
Skills:
Students are able to:
  • Use and model the distributive property to express the sum of two whole numbers with a common factor as a multiple of a sum of two whole numbers with no common factor by decomposing the numbers.
Understanding:
Students understand that:
  • Multiplication is distributive over addition.
  • Composing and decomposing numbers provides insights into relationships among numbers.
  • Quantities can be represented using a variety of equivalent expressions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.7.1: Define greatest common factor, least common multiple, and the distributive property.
M.6.7.2: Design problems using greatest common factor and the distributive property.
M.6.7.3: Show an understanding of how to solve a problem using the distributive property, with or without the use of a calculator.

Prior Knowledge Skills:
  • Identify factor and product.
  • Explain why addition and subtraction strategies work, using place value and the properties of operations.
  • Apply properties of operations as strategies to multiply and divide.
Mathematics
MA2019 (2019)
Grade: 6
8. Find the greatest common factor (GCF) and least common multiple (LCM) of two or more whole numbers.

a. Use factors and multiples to determine prime factorization.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given any two or more whole numbers,
  • Strategically select and apply strategies for finding the greatest common factor of the two numbers and justify that the strategy used does produce the correct value for the greatest common factor.
  • Strategically select and apply strategies for finding the least common multiple of the two numbers and justify that the strategy used does produce the correct value for the least common multiple.
  • Use the relationship between factors and multiples to determine prime factorization.
Teacher Vocabulary:
  • Greatest common factor
  • Least common multiple
  • Exponential Form
  • Prime Factorization
  • Factors
  • Multiples
  • Prime
  • Relatively Prime
  • Composite
Knowledge:
Students know:
  • Strategies for determining the greatest common factor of two or more numbers,
  • Strategies for determining the least common multiple of two or more numbers,
  • Strategies for determining the prime factorization of a number.
Skills:
Students are able to:
  • Apply strategies for determining greatest common factors and least common multiples.
  • Apply strategies for determining the product of a number's prime factors in multiple forms which include exponential form and standard form.
Understanding:
Students understand that:
  • Determining when two numbers have no common factors other than one means they are considered relatively prime.
  • Composing and decomposing numbers provides insights into relationships among numbers.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.8.1: Identify the least common multiple of a given set of numbers, with or without the use of a calculator.
M.6.8.2: List multiples of any given whole number, with or without the use of a calculator.
M.6.8.3: Identify the greatest common factors of a given set of numbers, with or without the use of a calculator.
M.6.8.4: Define prime factorization.
M.6.8.5: List common factors of given whole numbers, with or without the use of a calculator.
M.6.8.6: Identify the prime factorization of a single digit number, with or without the use of a calculator.
M.6.8.7: Identify the prime factorization of any two digit whole number, with or without the use of a calculator.

Prior Knowledge Skills:
  • Define Multiple.
  • Name the first ten multiples of each one-digit natural number.
  • Name the first 10 multiples of each one-digit natural number.
    Example: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70.
  • Count within 1000; skip-count by 5s, 10s, and 100s.
  • Recall basic multiplication facts.
  • Name the first ten multiples of each one-digit natural number.
  • Identify all factor pairs for a whole number in the range 1-20.
  • Apply properties of operations as strategies to multiply and divide.
  • Define factors, prime number, and composite number.
Mathematics
MA2019 (2019)
Grade: 6
15. Write, read, and evaluate expressions in which letters represent numbers in real-world contexts.

a. Interpret a variable as an unknown value for any number in a specified set, depending on the context.

b. Write expressions to represent verbal statements and real-world scenarios.

c. Identify parts of an expression using mathematical terms such as sum, term, product, factor, quotient, and coefficient.

d. Evaluate expressions (which may include absolute value and whole number exponents) with respect to order of operations.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given contextual or mathematical problems both when known models exist (for example formulas) or algebraic models are unknown,
  • Interpret the parts of the model in the original context.
  • Create the algebraic model of the situation when appropriate.
  • Use appropriate mathematical terminology to communicate the meaning of the expression.
  • Evaluate the expressions for values of the variable including finding values following conventions of parentheses and order of operations.
Teacher Vocabulary:
  • Expressions
  • Term
  • Coefficient
  • Sum
  • Product
  • Factor
  • Quotient
  • Variable
  • Constant
  • Difference
  • Evaluate
  • Order of Operations
  • Exponent
  • Absolute Value
Knowledge:
Students know:
  • Correct usage of mathematical symbolism to model the terms sum, term, product, factor, quotient, variable, difference, constant, and coefficient when they appear in verbally stated contexts.
  • Conventions for order of operations.
  • Convention of using juxtaposition (5A or xy) to indicate multiplication.
Skills:
Students are able to:
  • Translate fluently between verbally stated situations and algebraic models of the situation.
  • Use operations (addition, subtraction, multiplication, division, and exponentiation) fluently with the conventions of parentheses and order of operations to evaluate expressions for specific values of variables in expressions.
  • Use terminology related to algebraic expressions such as sum, term, product, factor, quotient, or coefficient, to communicate the meanings of the expression and the parts of the expression.
Understanding:
Students understand that:
  • The structure of mathematics allows for terminology and techniques used with numerical expressions to be used in an analogous way with algebraic expressions, (the sum of 3 and 4 is written as 3 + 4, so the sum of 3 and y is written as 3 + y).
  • When language is ambiguous about the meaning of a mathematical expression grouping, symbols and order of operations conventions are used to communicate the meaning clearly.
  • Moving fluently among representations of mathematical situations (words, numbers, symbols, etc.), as needed for a given situation, allows a user of mathematics to make sense of the situation and choose appropriate and efficient paths to solutions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.15.1: Define algebraic expression and variable.
M.6.15.2: Convert mathematical terms to mathematical symbols and numbers.
M.6.15.3: Translate verbal and numerical expression using all operations.
M.6.15.4: Define coefficient, constant and term.
M.6.15.5: Match mathematical terms with correct mathematical symbols.
M.6.15.6: Convert mathematical terms to mathematical symbols and numbers.
M.6.15.7: Calculate an expression in the correct order. with or without a calculator (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
M.6.15.8: Choose the correct value to replace each variable in the algebraic expression (Substitution).
M.6.15.9: Calculate a numerical expression, with or without a calculator (Ex. V=4x4x4).
M.6.15.10: Recognize the correct order to solve expressions with more than one operation.

Prior Knowledge Skills:
  • Recognize key terms to solve word problems.
    Examples: times, every, at this rate, each, per, equal/equally, in all, total.
  • Recognize key terms to solve word problems.
    Examples: times, every, at this rate, each, per, equal/equally, in all, total.
  • Define simple expression.
  • Recall simple equations.
  • Recognize properties of addition and multiplication.
  • Recall addition, subtraction, multiplication, division symbols.
  • Define parentheses, braces, and brackets.
  • Define numerical expression.
  • Recognize expressions.
  • Apply properties of operations as strategies to add and subtract.
  • Recall properties of operations as strategies to add and subtract.
  • Represent addition and subtraction with objects, mental images, drawings, expressions, or equations.
  • Use addition, subtraction, multiplication and division to solve one- and two-step word problems.
  • Recognize key terms to solve word problems.
  • Apply properties of operations as strategies to multiply and divide.
  • Apply the area and perimeter formulas for rectangles in real-world and mathematical problems.
  • Recall the formula for area (L × W).
  • Recognize that unit squares are equal.
  • Recall the formula for perimeter (P= L+L+W+W or P=2L + 2W).
  • Recall basic addition and multiplication facts.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.6.15 Evaluate algebraic expressions when given specific values for the variables (e.g. x + 2, where x = 4).
Mathematics
MA2019 (2019)
Grade: 6
16. Generate equivalent algebraic expressions using the properties of operations, including inverse, identity, commutative, associative, and distributive.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given contextual or mathematical problems which may be modeled by algebraic expressions, use properties of the operations to produce combined and re-written forms of the expressions that are useful in resolving the problem.
Teacher Vocabulary:
  • Properties of operations
  • Distributive property
  • Inverse property
  • Identity property
  • Commutative property
  • Associative property
  • Equivalent algebraic expressions
Knowledge:
Students know:
  • the properties of operations, including inverse, identity, commutative, associative, and distributive and their appropriate application to be able to generate equivalent algebraic expressions.
Skills:
Students are able to:
  • Accurately use the properties of operations on algebraic expressions to produce equivalent expressions useful in a problem solving context.
Understanding:
Students understand that:
  • The properties of operations used with numerical expressions are valid to use with algebraic expressions and allow for alternate but still equivalent forms of expressions for use in problem solving situations.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.16.1:Define equivalent, simplify, term, distributive property, associative property of addition and multiplication, and the commutative property of addition and multiplication.
M.6.16.2: Simplify expressions with parentheses (Ex. 5(4 + x) = 20 + 5x).
M.6.16.3: Combine terms that are alike of a given expression.
M.6.16.4: Recognize the property demonstrated in a given expression.
M.6.16.5: Simplify an expression by dividing by the greatest common factor.
Example: 18x + 6y = 6(3x + y).
M.6.16.6: Determine the greatest common factor in an algebraic expression.

Prior Knowledge Skills:
  • Define parentheses, braces, and brackets.
  • Define numerical expression.
  • Recognize expressions.
  • Apply properties of operations as strategies to add and subtract.
  • Recall properties of operations as strategies to add and subtract.
  • Represent addition and subtraction with objects, mental images, drawings, expressions, or equations.
  • Define simple expression.
  • Recall simple equations.
  • Recognize properties of addition and multiplication.
  • Recall addition, subtraction, multiplication, division symbols.
  • Use addition, subtraction, multiplication and division to solve one- and two-step word problems.
  • Apply properties of operations as strategies to multiply and divide.
Tags: algebra, distributive property, equivalent, evaluate, expressions, factor, greatest common factor, operations, properties
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There are six lessons on this topic.

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Author: Hannah Bradley
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