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Standard: 6-3 The student will demonstrate through the mathematical processes an understanding of writing, interpreting, and using mathematical expressions, equations, and inequalities. Indicator 6-3.1 Analyze numeric and algebraic patterns and pattern relationships Continuum of Knowledge: The study of patterns is extensive throughout elementary school. Students begin the process of transitioning from the concrete to the abstract and symbolic in the 5th grade, and learn to represent patterns in words, symbols, and algebraic expressions/equations for the first time (5-3.2). In 6th grade, students analyze numeric and algebraic patterns and pattern relationships (6-3.1) and represent algebraic relationships with variables in expressions, simple equations, and simple inequalities (6-3.3). This will lay the foundation for the study of slope in the 7th grade (7-3.2). Taxonomy Level Cognitive Dimension: Analyze Knowledge Dimension: Conceptual Key Concepts

• Expressions • Equations • Inequalities • Function • Rule • Patterns • Linear function

Instructional Guidelines For this indicator, it is essential for students to:

• Solve arithmetic (adding/subtracting) and geometric (multiplying by common ratio) sequences

• Represent patterns using tables, graphs, and equations. • Write mathematical rules for patterns from numeric and pictorial patterns • Determine which representation makes it easier to describe, extend, and or

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make predictions using the patterns For this indicator, it is not essential for students to:

• Perform multiplication/division with fractions or decimals for geometric patterns

• To solve pattern problem involving shapes (actually drawing pictures to complete the pattern)

Student Misconceptions/Errors Geometric patterns are sequences that involve multiplying by a common ratio not based on shapes. Instructional Resources and Strategies

• Students will need an in-depth experience discussing real world patterns and patterns which provide concrete examples before they can begin to represent them symbolically. Teachers will need to model many examples that involve moving from the concrete to the symbolic.

• This pattern may be represented using concrete models. Which of the

following numeric patterns best represents the geometric pattern below? Explain your reasoning.

 

 

 

a. 1, 3, 9, 12 b. 1, 2, 4, 8

c. 1, 3, 6, 9 d. 1, 3, 6, 10

• Students should explain their observations of a pattern in their own words.

This verbalization will enable students to begin to write a mathematical rule for a pattern later.

Assessment Guidelines The objective of this indicator is to analyze, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. Conceptual knowledge is not bound by specific examples; therefore, the student’s conceptual knowledge of these patterns relationships (words, table and graph) should be explored using a variety of

Algebra Sixth Grade

 

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examples. The learning progression to analyze requires students to recall the structure of a function table and a graph. Students generalize connections (6-1.7) among the multiple representations and generate descriptions and mathematical statements about pattern relationships using correct and clearly written and spolen words (6-1.6). Students prove or disprove their answer (6-1.2) and place an emphasis on the similar meaning that is conveyed by each representation.

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Standard: 6-3 The student will demonstrate through the mathematical processes an understanding of writing, interpreting, and using mathematical expressions, equations, and inequalities. Indicator 6-3.2 Apply order of operations to simplify whole-number expression Continuum of Knowledge: Sixth grade is the first time students are introduced to using order of operations to evaluate a numerical expression (6-3.2). In 7th grade, students will use inverse operations to solve two-step equations and two-step inequalities. (7-3.4) Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts Vocabulary Order of operations Exponents Symbols All grouping symbols { [ ( ) ] } Instructional Guidelines For this indicator, it is essential for students to:

• Solve problems that involve all operations with whole numbers • Work with whole numbers expressions only • Understand the reasoning behind order of operations

For this indicator, it is not essential for students to:

• Include negatives, fractions or decimals. Student Misconceptions/Errors Many students are simply introduced to the concept with the phrase “Please Excuse

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My Dear Aunt Sally”, often referred to as PEMDAS. While this is a helpful mnemonic devise, it can easily lead to some common misconceptions. Many students come to believe that multplication is always done before division and that addition is always done before subtraction. By being taught this mnemonic device, students do not fully understand that the operations of multiplication and division (or addition and subtraction) are performed in the order that they appear, from left to right. Instructional Resources and Strategies

• However, students did not evaluate expressions. "Evaluating an expression", “Solve the expression” and “Find the solution to the expression” each with the same meaning will be new and important phrases for students to understand.

• Solving problems in context are useful to help students better understand the

concept. For example, Jay shot 4 arrows at the target. His total score was 45. Which of these scores is not a possible result of Jay’s 4 shots? How do you know?

a. 25 + (2 x 5) + 10 b. 15 + (3 x 10) c. (2 x 15) + 10 + 5 d. 25 + 5 + (2 x 10)

• After students have been given opportunites to discover why an agreement

for the order of operations is necessary, it is sugggested that order of operations be introduced using a table format with students being taught that the higher in the table an operation is, the more important it is and must be done first.

Level 1

{ [ ( ) ] } All Grouping Symbols

Level 2

Exponents

Level 3

Multiplication and Division

Proceeding from Left to Right

Addition and Subtraction

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Assessment Guidelines The objective of this indicator is apply, which is in the “apply procedural” of the Revised Taxonomy. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem or problem situation. Although the focus is to gain computational fluency with order of operations, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression to apply requires student to be fluently in all whole number operations. Given an expression, students explore various ways to simplify the expression. Students explain and justify their process of simplifying to their classmates and their teacher. They use inductive reasoning to generalize connections among strategies with an emphasis on the need for a common process to simplify. Students analyze the order of operations and gain of understanding of the structure and purpose of each level. They use this understanding to generate and solve more complex problems (6-1.1).

Level 4 Proceeding from Left to Right

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Standard: 6-3 The student will demonstrate through the mathematical processes an understanding of writing, interpreting, and using mathematical expressions, equations, and inequalities. Indicator 6-3.3 Represent algebraic relationships with variables in expressions, simple equations, and simple inequalities Continuum of Knowledge: In fourth grade, students translated among, letter, symbols and words to represent quantities in simple mathematical expression or equations (4-3.4). In fifth grade, students represented numeric, algebraic and geometric pattern in words, symbols, algebraic expression and algebraic equations (5-3.1). In sixth grade, students represent algebraic relationships with variables in expressions, simple equations, and simple inequalities (6-3.3). In seventh grade, students represent proportional relationships with graphs, tables, and equations (7-3.6) and represent algebraic relationships with equations and inequalities (8-3.2). Taxonomy Level Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts Vocabulary Expression Equations Inequality Variable Equivalency Algebraic relationships Symbols <, >, = ≤, ≥ Instructional Guidelines For this indicator, it is essential for students to:

• Write an equation or inequality from a picture

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• Write an equation or inequality from a word problem • Understand inequality symbols • Understand the concept of equivalency • Understand that algebraic relationships can be in the form of words, tables or

graphs

For this indicator, it is not essential for students to:

• Solve or graph equations or inequalities Student Misconceptions/Errors Many students have a common misconception that different variables represent different numbers. Students also misunderstand the concept of equivalence. They must establish that the equal sign plays different roles based on the situation. In this instance, it does not mean do something. It means that there is a relationship of equivalence on either side of the equal sign. Instructional Resources and Strategies

• Please note that a more in depth understanding of the concept of inequality is crucial in the 6th grade. Students have been using the inequality symbols > and < since the 2nd grade in grade appropriate applications. It is imperative at this level that students' think of an inequality as much more than "the alligator eats the biggest piece". Students must be encouraged to view inequalities as a way to describe and represent a relationship between/among quantities. In sixth grade, students are introduced to the symbols ≤ and ≥.

• Students are translating from one to another; therefore, their understanding of the multiple representations is essential. For example,

o Which of these problems could be solved by using the open sentence?

A – 5 = ?

a) Janis is 5 years older than Seth. If A is Seth’s age, how old is Janis? b) Todd is 5 years younger than Amelia. If A is Amelia’s age in years,

how old is Todd? c) Isaac is 5 times as old as Bert. If A is Bert’s age in yars, how old is

Isaac? d) Nathan is one-fifth as old as Leslie. If A is Nathan’s age, how old is Leslie?

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o The two number sentences shown below are true.

 −   = 6    +             = 2 

If both equations shown above are true, which of the following equations must also be true? Circle your choice and explain why. (Students should circle the first equation)

           X                    =   

      X        2       =         

     +                  =      12 

     +                  =   

Assessment Guidelines The objective of this indicator is to represent which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To understand means to construct meaning; therefore, the students’ focus is on building conceptual knowledge of the relationships between the forms. The learning progression to represent requires students to understand the concepts of equivalency and inequalities. Students analyze algebraic relationships (words, tables and graphs) to determine known and unknown values and the operations involved. They generate descriptions of the observed relationship and generalize the connection (6-1.7) between their description and structure of expression, equations or inequalities. Students explain and justify their ideas with their classmates and teachers using correct and clearly written or spoken words, variables and notation to communicate their ideas (6-1.6). Students then compare the relationships (words, tables and graphs) to their equation, inequality or expression to verify that each form conveys the same meaning.

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Standard: 6-3 The student will demonstrate through the mathematical processes an understanding of writing, interpreting, and using mathematical expressions, equations, and inequalities. Indicator 6-3.4 Use the commutative, associative, and distributive properties to show that two expressions are equivalent. Continuum of Knowledge: In fifth grade, students will identify applications of commutative, associative, and distributive properties with whole numbers (5-3.4). In sixth grade, students use the commutative, associative, and distributive properties to show that two expressions are equivalent (6-3.4). In eighth grade, students use commutative, associative, and distributive properties to examine the equivalence of a variety of algebraic expressions (8-3.3). Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts Vocabulary Commutative property Associative property Distributive property Instructional Guidelines For this indicator, it is essential for students to:

• Gain a conceptual understanding each rule (what it can and can’t do) • Verbalize each rule using appropriate terminology • Perform whole number computations

For this indicator, it is not essential for students to:

• Use properties in situations that involve multiplication/division of fractions and decimals

• Create a formal rule for each property using variables. For example, a + b = b + a

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Student Misconceptions/Errors Students might have difficulty naming the property that they use. The most important point here is that they understand what the different properties allow them to do and not do. Instructional Resources and Strategies

• The focus is for students to understand how these properties can be used to create equivalent expressions. Students should verbalize their understanding of these properties using correct and clear mathematical language but it is not necessary for them to recite or write formal rules. Students should demonstrate a clear understanding of the concepts of equivalence by using the commutative, associative, and distributive properties. These properties should be used in situations that involve all operations with whole numbers, addition and subtraction of fractions and decimals, and powers of 10 through 106.

• Using problem situations to explore these concepts will support retention. Streamline video:

• Power of Algebra, The Basic Properties o The Commutative Properties of Addition and Multiplication (02:21) o The Associative Properties of Addition and Multiplication (00:46) o The Distributive Properties of Multiplication over Addition (01:26)

Assessment Guidelines The objective of this indicator is to use which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Although the focus of the indicator is to use which is a knowledge of specific steps and details, learning experiences should integrate both memorization and concept building strategies to support retention. The learning progression to use requires student to explore a variety of examples of these number properties using a various types of numbers. They analyze these examples and generalize connections (6-1.7) about what they observe using correct and clearly written or spoken language (6-1.6) to communicate their understanding. Students do not generalize these connections using formal rules involving variables. Students connect these statements to the terms commutative, associative and distributive. Students then develop meaningful and personal strategies that enable them to recall these relationships.

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Standard: 6-3 The student will demonstrate through the mathematical processes an understanding of writing, interpreting, and using mathematical expressions, equations, and inequalities. Indicator 6-3.5 Use inverse operations to solve one-step equations that have whole-number solutions and variables with whole-number coefficients. Continuum of Knowledge: In fourth grade, students apply procedures to find the value of an unknown letter or symbol in whole number equations (4-3.5). In sixth grade, students use inverse operations to solve one-step equations that have whole-number solutions and variables with whole-number coefficients (6-3.5). In seventh grade, students will solve two-step equations and two-step inequalities. (7-3.4) Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts Vocabulary

• Inverse operation • Coefficient • Evaluate • Solve • Solution • Additive inverse • Multiplicative inverse

Instructional Guidelines For this indicator, it is essential for students to:

• Add, subtract, multiply, and divide whole numbers • Understand the concept of a variable and how to solve for it • Understand additive inverse (the sum of a number and its opposite is 0) • Use manipulatives to model equations and the process of solving one-step

equation

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• Solving equations using inverse operations • Check their solutions

For this indicator, it is not essential for students to:

• Include negatives, fractions, or decimals

Student Misconceptions/Errors Students think that an equal sign means provide an answer rather than seeing it as an indicator of equality. Students often solve equation and do not understand why they are doing it. A common question is “is my answer right?” Students who ask this lack a conceptual understanding of the concept of equivalency and the purpose of the procedure of solving. Instructional Resources and Strategies

• To solve an equation means to find value (s) for the variable that make the equation true. Pan balance may be used to develop skills in solving equations with one variable. “The balance makes it reasonably clear to students that if you add or subtract a value from one side, you must add or subtract a like value from the other side to keep the scales balanced” (p. 280). Show a balance with variable expressions in each side. Use only one variable. Make the tasks such that a solution by trial and error is not reasonable. For example, the solution to 3x + 2 = 11 – x is not a whole number. (Use whole numbers only!) Suggest that adjustments be made to the quantities in each pan as long as the balance is maintained. Begin with simple equations, such as 12 + n = 27 in order to help students develop skills and explain their rationale. Students should also be challenged to devise a method of proving that their solutions are correct. (Solutions can be tested by substitution in the original equation.)

• http://illuminations.nctm.org/LessonDetail.aspx?ID=U170 Everything Balances Out In The End (use to reinforce the concept of equality)

• http://illuminations.nctm.org/LessonDetail.aspx?id=L247 Building Bridges Assessment Guidelines The objective of this indicator is use which is in the “apply procedural” of the Revised Taxonomy. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem or problem situation. Although the focus is to gain computational fluency with solving one step equations, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression to use requires students to explore the concepts of equivalency and variables using concrete models such as balance

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scales. Student use this understanding of balance to analyze a variety of examples of simple one step equations. Students use inductive reasoning (6-1.3) to generalize connections (6-1.7) among types of equations (addition, subtraction, multiplication and division) and generate mathematical statements (6-1.5) related to how these equations can be solved. Students engage in repeated practice to support retention and check their answers.