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Math Grade 6 Instructional Guide 2011-2012 1 Subject: Math Grade 6 Benchmark Assessments and Instructional Guide Instructional Guides are provided as resource for Alliance classroom teachers. They identify high priority grade-level standards to be taught during each quarter of instruction in the context of proposed units with a suggested amount of time. High priority standards are assessed on quarterly benchmark exams. The secondary curriculum begins in the sixth grade, where the concept of the unknown is solidified through the study of expressions and equations in a wide range of settings. The rational numbers are thoroughly explored in a variety of forms including fraction, decimal, and percent. Examples of rational numbers in a real world context involve the concepts of ratio, proportion, percentage, and rate. The concepts of shape and angle are studied, and formulas for area and volume are discovered. Various methods of representing data are then used, and probabilities of events are calculated. Unit Unit 1: Exploring Integers This course begins with a discussion of the relationship between positive and negative numbers, and the concept of an opposite. Examples of positive and negative numbers in the real world are explored. The set of integers is then defined as the set of whole numbers, , and their opposites. This leads to a discussion of the relationship between the number line and the set of integers, including the meaning of addition and subtraction in terms of the number line. The addition and subtraction of integers is then performed, away from the number line, and rules for adding and subtracting two integers are discussed. The multiplication and division of two integers is also studied. Rules for multiplying or dividing integers are then developed. The concept of positive and negative is revisited in terms of the rules for operations on integers. It is emphasized High Priority Standards And Learning Targets* Number Sense: 2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations that use positive and negative integers and combinations of these operations. Learning Targets 1C Explain how numbers are added or subtracted 1D Add or subtract one positive and one negative integer. 1F Add or subtract more than two integers and explain the process 1G Explain how to add or subtract integers without using a number line and perform these operations. 1H Solve real world problems by adding or subtracting integers. 1I Multiply or divide two positive integers and explain the process 1J Explain the rules of multiplication and division of integers and perform these opera- # CST Items # Q1 Items 6 4 Supporting Medium/Low Priority Standards & Learning Targets Number Sense: 1.1 Students compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line. # CST Items 4.1 3 4.2 4.2 & 4.3 4.4 Learning Targets 1A Compare, explain, and give examples of positive and negative integers. 1B Identify and order positive and negative integers on a number line. Algebra and Functions: 1.0 Students write verbal expressions and sentences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results. Textbook McDougal Littell** 4.2-4.4 4.2-4.4 4.2-4.4 17 (all 1.0) Learning Targets 1H Solve real world problems by adding or subtracting integers. 1Q Write and evaluate an algebraic expression involving adding, subtracting, multiplying and dividing 4.5 & 4.6 4.5 & 4.6 4.5 & 4.6 4.5 & 4.6 4.5 & 4.6 4.5 & 4.6 4.5 & 4.6 * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell Unit that while multiplying or dividing two negative integers gives a positive integer, this is not true for addition and subtraction. A variety of examples are given to justify this conclusion. The end of this unit begins a theme that is studied throughout the course, namely the concepts of variable and unknown. Variables are used to write expressions, and when values are assigned to those variables, the expressions can be evaluated. The meaning of variable in this context is emphasized, namely that these expressions can be evaluated for a variety of numbers (at this level, usually any number) and thus can vary when different numbers are used. On the other hand, an equation is a statement where one or both sides may have an unknown number, and when a number is used for the unknown, the truth of the statement is evaluated. The process of solving an equation is introduced as the process of finding numbers (there may be more than one) that make the equation a true statement. Throughout this discussion the difference between variable and unknown is emphasized, namely that an unknown is a particular value that makes an equation true, while a variable can be a choice of values that make an equation true, or a choice of values that are used to evaluate an expression. Unit 2: Order of Operations High Priority Standards And Learning Targets* # CST Items # Q1 Items tions. 1K Multiply or divide one positive and one negative integer. 1L Multiply or divide two negative integers. 1M Explain the difference between adding and subtracting two negative integers and multiplying/dividing two negative integers. 1N Multiply or divide more than two integers. 1O Solve a real world problem by multiplying or dividing integers. Algebra and Functions: 1.1 Write and solve onestep linear equations in one variable. 6 4 Learning Targets 1R Write and solve one-step linear equations, explaining each step in the process 1S Explain the difference between an expression and an equation and the difference between evaluating and solving. 1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables. Learning Targets 1P Explain the difference between a variable and an unknown. 1Q Write and evaluate an algebraic expression involving adding, subtracting, multiplying and dividing positive and negative integers. Algebra and Functions: 1 2 Math Grade 6 Instructional Guide 2011-2012 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets positive and negative integers. 5.3 & 5.4 1.3 Apply algebraic order of 1 operations and the 5.3 & 5.4 commutative, associative, and distributive 5.3 & 5.4 properties to evaluate expressions; and justify 5.3 & 5.4 each step in the process. Learning Targets 1C Explain how numbers are added or subtracted 1F Add or subtract more than two integers and explain the process 1G Explain how to add or subtract integers without using a number line and perform these operations. 1R Write and solve one-step linear equations, explaining each step in the process Mathematical Reasoning: 1.0 Students make decisions about how to approach problems 2.0 Students use strategies, skills, and concepts in finding solutions. 3.0 Students move beyond a particular problem by generalizing to other situations. All are embedded Number Sense: 1.3-3.6 * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell 2 Unit This unit expands upon the concept of operations on integers from the previous unit, to include multiple operations in one problem. Students explore the correct order of operations, and perform error analysis to solidify this understanding of the correct order. The importance of the organization of steps in solving problems with more than one operation is highlighted. Continuing the theme of a variable, algebraic expressions are written using variables and more than one operation, and they are evaluated for given values of a variable. High Priority Standards And Learning Targets* 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process. # CST Items # Q1 Items 1 2 Learning Targets 2A State the correct order of operations and explain why this rule is necessary 2B Read the solution steps to an order of operations problem, find and explain the error, and rework the problem correctly, justifying each step 2D Evaluate an algebraic expression using order of operations. 1.4 Solve problems manually by using the correct order of operations or by using a scientific calculator. Learning Targets 2E Use the order of operations in solving real world problems that involve more than one operation, including finding the mean of a group of integers. Math Grade 6 Instructional Guide 2011-2012 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets 2.3 Solve addition, subtrac6 tion, multiplication, and 3.6& 4.7 division problems, including those arising in 10.1 concrete situations, that use positive and negative integers and combinations of these operations. Learning Targets 2C Evaluate numerical expressions with more than one operation and explain each step in the process. 2E Use the order of operations in solving real world problems that involve more than one operation, including finding the mean of a group of integers. 1 2 Algebra and Functions: 1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables. 1 Learning Targets 2D Evaluate an algebraic expression using order of operations. Statistics, Data Analysis, and Probability: 1.1 Compute the range, mean, median, and mode of data sets. 1 Learning Targets 2E Use the order of operations in solving real world problems that involve more than one operation, including finding the mean of a group of integers. Mathematical Reasoning: 1.0 Students make decisions about how to approach problems 2.0 Students use strategies, skills, and concepts in finding solutions. Embedded * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell 3 Unit Unit 3: Factors and Multiples of Positive Integers Multiplication plays a critical role in connecting the concepts of the integers and the rational numbers (fractions and decimals). In this unit, multiplication of positive integers is explored in two contrasting ways: writing a positive integer as the product of other, smaller, positive integers (factors), and multiplying a single positive integer by a set of other positive integers (multiples). The concept of a factor is explored first, and used to introduce prime and composite numbers. Factoring is used to write a positive integer as the product of prime factors (i.e. the prime factorization). Common prime factors are then identified amongst a group of positive integers. The product of the common prime factors is defined as the greatest common factor, abbreviated the GCF. This concept is connected with the notion of the greatest common divisor, learned in elementary school. Prime factorization is then used to find the GCF of a group of positive integers. Fractions are defined as an expression for the division of two integers, and the need for studying the GCF is then illustrated through the process of reducing fractions to lowest terms. The GCF is then used to reduce frac- High Priority Standards And Learning Targets* Number Sense: 1.1 Compare and order positive and negative fractions, decimals, and mixed numbers and place them on a number line. # CST Items # Q1 Items 3 2 Learning Targets 3N Apply GCF and LCM to graph a set of fractions on a number line and to order a set of fractions from least to greatest or vice versa. 2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two fractions or to find the reduced form of a fraction). Learning Targets 3A Explain the concept of a factor 3B Explain the difference between prime and composite numbers, and give examples. 3C Find the greatest common factor of a number using prime factorization. 3D Identify whether a fraction is in lowest terms using the GCF. 3E Reduce a fraction to lowest terms using the prime factorization of the numerator and denominator. 3F Explain how the GCF is used in 3 2 Math Grade 6 Instructional Guide 2011-2012 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets 3.0 Students move beyond a particular problem by generalizing to other situations. Number Sense: 1.0 Students compare and order positive and negative fractions, decimals, and mixed numbers. Students solve problems involving fractions, ratios, proportions, and percentages. 15 (all of 1.0) Learning Targets 3C Find the greatest common factor of a number using prime factorization. 3F Explain how the GCF is used in reducing fractions. 3K Explain how the LCM is used in ordering fractions. Mathematical Reasoning: 1.0 Students make decisions about how to approach problems 2.0 Students use strategies, skills, and concepts in finding solutions. 3.0 Students move beyond a particular problem by generalizing to other situations. Embedded * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell 4 Unit tions to lowest terms, and write statements showing the equivalence of the two fractions. Another notion of the multiplication of positive integers is then introduced, namely the concept of a multiple. Lists of multiples are written for given positive integers and using these lists, students identify common multiples from those lists. The smallest of these common multiples is defined as the least common multiple, abbreviated LCM. The need for the least common multiple is also related to fractions by finding the least common denominator (LCD). Students then find the LCD of a group of fractions, rewrite the fractions using the LCD, and then write the fractions in order from least to greatest. High Priority Standards And Learning Targets* # CST Items # Q1 Items Math Grade 6 Instructional Guide 2011-2012 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets reducing fractions. 3G Find the least common multiple of a number using the lists of multiples. 3H Find the least common multiple of a number using prime factorization. 3I Explain the situations you would use each method of finding the LCM. 3J Use the least common multiple to write a set of fractions so that each fraction has the same denominator. 3K Explain how the LCM is used in ordering fractions. 3L Explain the similarities and differences of the least common multiple, and the greatest common factor. 3M Compare and contrast methods for finding the least common multiple, and the greatest common factor. This unit finishes with a careful discussion of the similarities and differences in the concepts of a factor of a positive integer, and a multiple of a positive integer. It is emphasized that there are a finite number of prime factors of a positive integer and that those factors cannot be larger than the original number. Furthermore, it is emphasized that there are an infinite number of multiples of a given positive integer, and that those multiples cannot be smaller than the original number. The methods for finding the GCF and the LCM are also compared and contrasted, as well as the relationship of each value to fractions. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell 5 Math Grade 6 Instructional Guide 2011-2012 Unit Unit 4: Fractions & Decimals The study of the concept of a fraction is continued in this unit, with a discussion of the relationship between fractions and decimals, and how to perform operations on both kinds of numbers. The discussion begins with the addition and subtraction of fractions with like denominators. The notion of adding and subtracting fractions, and why we need to be able to perform those operations on fractions, is illustrated through a variety of real world examples. Connections are then made to the previous unit, as the LCD is used in adding and subtracting fractions with unlike denominators. The connection between multiplication and repeated addition is explored, and used to introduce the concept of the multiplication of two fractions. The relationship between multiplication and division is then highlighted, as the division of two integers is interpreted as multiplication by the reciprocal. The concept of a reciprocal is then used to divide two fractions. The four arithmetic operations on fractions are mastered, and used to solve problems in a real world context. The concept of a fraction as division, from the previous unit, leads into a discussion of the decimal form of a rational number. This discussion begins with fractions whose numera- High Priority Standards And Learning Targets* Number Sense: 2.1 Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation. # CST Items # Q2 Items 1 2 1 2 Textbook McDougal Littell** 10 (all 2.0) 2.3 Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations. 6 Learning Targets 4K Convert between the fraction and decimal form of the number. 4L Add or subtract decimals and explain the algorithm 4M Multiply or divide decimals and explain the algorithm 4N Solve real world problems involving operations on decimals. Learning Targets 4G Explain the reciprocal, and use it to divide fractions. 4H Explain the meaning of dividing by a fraction. 4J Solve the real world problems involving multiplication and division of fractions. 2.4 Determine the least common multiple and the greatest common divisor of whole numbers; use them to solve problems with fractions (e.g., to find a common denominator to add two factions or to find the re- # CST Items Learning Targets 4B Solve real world problems involving the addition and subtraction of fractions. 4L Add or subtract decimals and explain the algorithm 4M Multiply or divide decimals and explain the algorithm Learning Targets 4A Add or subtract fractions with like denominators. 4B Solve real world problems involving the addition and subtraction of fractions. 2.2 Explain the meaning of multiplication and division of positive fractions and perform the calculations. (e.g., 5/8 divided by 15/16= 5/8 x 16/15 = 2/3). Supporting Medium/Low Priority Standards & Learning Targets Number Sense: 2.0 Students calculate and solve problems involving addition, subtraction, multiplication, and division. 3 2 Mathematical Reasoning: 1.0 Students make decisions about how to approach problems 2.0 Students use strategies, skills, and concepts in finding solutions. 3.0 Students move beyond a particular problem by generalizing to other sit- Embedded * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell 6 Unit tor is a multiple of the denominator. The value of these fractions upon division is an integer. The concept of a decimal naturally arises in discussing fractions where the numerator is not a multiple of the denominator. The division process is then used to write the decimal expansion of a fraction of this type. Connections are made between the place value of a decimal and powers of ten in the denominator of a fraction. Using this concept, decimals are written in fraction form. Processes for the four operations on decimals are mastered, and used to solve problems in a real world context. The theme of variable and unknown is once again revisited at the end of this unit. Simple equations involving fraction and decimal coefficients are solved using similar procedures as in earlier units. Simple equations with fraction and decimal solutions are also studied. High Priority Standards And Learning Targets* duced form for a fraction). # CST Items # Q2 Items 6 4 1 2 Learning Targets 4C Explain how to find a least common denominator (LCD). 4D Add or subtract fractions with unlike denominators and explain the process. 4E Explain the differences in adding or subtracting fractions with like denominators, and fractions with unlike denominators. 4F Multiply two or more fractions. 4J Solve the real world problems involving multiplication and division of fractions. Algebra and Functions: 1.1 Write and solve onestep linear equations in one variable. Math Grade 6 Instructional Guide 2011-2012 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets uations. Algebra and Functions: 1.0 Students write verbal 17 expressions and sen(all 1.0) tences as algebraic expressions and equations; they evaluate algebraic expressions, solve simple linear equations, and graph and interpret their results. Learning Targets 4O Solve simple equations with fraction and decimal coefficients. 4P Solve simple equations with fraction and decimal solutions. 1.3 Apply algebraic order of operations and the commutative, associative, and distributive properties to evaluate expressions; and justify each step in the process. Learning Targets 4I Use order of operations for problems with more than one operation on fractions. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell 7 Unit Unit 5: Ratio and Proportion In this unit, the concept of a fraction from previous units is used to establish a relationship between two numbers or quantities. This discussion begins by defining a ratio as a way of writing the relationship between two quantities using a fraction. Other representations of a ratio are also discussed (i.e. a:b, a to b). Ratios are then used to explicitly write the relationships between given quantities, including those arising from a real world context. Variables are also used to write ratios that represent a given relationship. These expressions are evaluated for given values of the variables. Just as a ratio relates two quantities together using a fraction, a proportion also establishes a relationship between two quantities, namely the equivalence of two ratios. The difference between a ratio and proportion is highlighted and compared to the relationship between an expression and an equation. Now that we have established the proportion as an equation, it is natural to discuss a method for solving proportions, namely cross-multiplication. Proportions are then solved using this method, and their solutions are interpreted in a real world context. Finally, ratios and proportions are compared and contrasted, and linked to an understanding of the difference between expressions and equations from previous units (i.e. ratios ex- High Priority Standards And Learning Targets* Number Sense: 1.2 Interpret and use ratios in different contexts ( e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a/b, a to b, a:b) # CST Items # Q2 Items 1 2 Learning Targets 5H Compare ratios and proportions to expressions and equations. 2.1 Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches). Learning Targets 5A Describe a ratio in your own words. 5B Give a variety of examples of ratios written in different ways (a/b, a to b, a:b) 5C Represent a real life situation using a ratio and explain the connection between the situation and the mathematical model 1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Learning Targets 5D Relate two ratios using a proportion. 5E Explain the difference between a ratio and a proportion. 5F Solve proportions by cross multiplication. 5G Solve real world problems using proportions and justify Math Grade 6 Instructional Guide 2011-2012 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets Algebra and Functions: 1.2 Write and evaluate an 1 algebraic expression for a given situation, using up to three variables. 1 Learning Targets 5A Describe a ratio in your own words. 5D Relate two ratios using a proportion. 6 4 2.2 Demonstrate an understanding that rate is measure of one quantity per unit value of another quantity. 6 Learning Targets 5C Represent a real life situation using a ratio and explain the connection between the situation and the mathematical model 2.3 Solve problems involving rates, average speed, distance, and time. 1 Learning Targets 5G Solve real world problems using proportions and justify the mathematical model used to solve the problem. Mathematical Reasoning: 1.0 Students make decisions about how to approach problems 2.0 Students use strategies, skills, and concepts in finding solutions. 3.0 Students move beyond a particular problem by Embedded * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell 8 High Priority Standards And Learning Targets* Unit pressions and proportions equations) # CST Items # Q2 Items 6 4 6 4 the mathematical model used to solve the problem. Algebra and Functions: 1.1 Write and solve onestep linear equations in one variable. Math Grade 6 Instructional Guide 2011-2012 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets generalizing to other situations. Learning Targets 5H Compare ratios and proportions to expressions and equations. Unit 6: Percentages The concepts of ratio and proportion from the previous unit are critical for the study of percentage, as well as the concepts of variable and unknown. A percentage, p%, is first represented as the ratio , and then compared with another ratio using the proportion . Us- ing this relationship, methods of conversion between the fraction form, decimal form and percent form of a number are developed. From here, the focus of the unit shifts to a discussion of two different interpretations of percentage: the percent equation ( ) and percent change. The percent equation is studied and used to answer the following types of questions: what is p% of a number, what number is p% of another number, and a number is what percent of another number? These types of questions are also posed in a real world context. Percent change is then introduced, and used to find the Number Sense: 1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Learning Targets 6A Explain the meaning of per- 1 Learning Targets 6B Explain how percent is related to ratios and proportion. Algebra and Functions: 1.1 Write and solve one-step linear equations in one variable. 6 Learning Targets 6G Use the percent equation to find the percent of a number. 6I Use an algebraic equation to find the base. 6L Increase or decrease a number by a given percent, including sales/taxes, tips, etc. Learning Targets 6B Explain how percent is related to ratios and proportion. 6C Explain how to convert between fractions and percents. 6D Convert between decimal, fraction and percent form. 6E Solve real world problems by converting between fractions, decimals, and percents. 1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interests earned, and tips. Number Sense: 1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b). 5 3 1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables. Learning Targets 6J Find the percent decrease. 6K Find the percent increase. Mathematical Reasoning: 1.0 Students make decisions about how to approach problems 1 Embedded * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell 9 Unit percent increase or decrease of two numbers or quantities. Percent change is then used to increase or decrease a given quantity by a given percent (e.g. increase 10 by 20%). Percent change is also studied in a real world context by solving problems involving growth, discounts, and markup. High Priority Standards And Learning Targets* cents. 6F Find the percentage of a number and explain the connection to 100. 6G Use the percent equation to find the percent of a number. 6H Use the percent equation to find the part of a base and explain what the base represents 6I Use an algebraic equation to find the base. 6J Find the percent decrease. 6K Find the percent increase. 6L Increase or decrease a number by a given percent, including sales/taxes, tips, etc. 6M Explain the difference between using the percent equation, and finding a percent change. # CST Items # Q2 Items Math Grade 6 Instructional Guide 2011-2012 10 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets 2.0 Students use strategies, skills, and concepts in finding solutions. 3.0 Students move beyond a particular problem by generalizing to other situations. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell Math Grade 6 Instructional Guide 2011-2012 11 Unit Unit 7: Rates Continuing the discussion of how quantities are related, and how to represent those relationships, the focus of this unit is to relate two quantities that have different units. A rate is then defined as a special type of ratio that compares two such quantities. A variety of examples of rates are introduced in a real world context. Given that a rate is a special kind of ratio, a proportion is used to show equivalent rates. It is emphasized that equivalent rates must have the same type of units (e.g. miles/hours can never equal gallons/hours). Using proportion, given rates are converted to other, equivalent rates, with the same type of units (e.g. convert miles/hour to feet per second). Very often, rates are written so that the denominator represents one unit of measure, allowing us to write statements such as “per hour”. A method for finding a unit rate from a given rate is discussed, and used in a real world context. Throughout this unit, solutions to problems involving rates are checked using dimensional analysis. High Priority Standards And Learning Targets* Algebra and Functions: 2.1 Convert one unit of measurement to another ( e.g., from feet to miles, from centimeters to inches). # CST Items # Q3 Items 1 2 6 4 Learning Targets 7D Use proportion to find equivalent rates. 7F Use proportion to find unit rates. 2.2 Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity. Learning Targets 7A Compare the meaning of a rate and a ratio. 7B Write a rate to represent a real world situation. 7C Explain the meaning of equivalent rates. 7E Explain the meaning of a unit rate, and give examples. 2.3 Solve problems involving rates, average speed, distance, and time. Learning Targets 7G Solve real world problems involving rates. 1 2 Supporting Medium/Low Priority Standards & Learning Targets Number Sense: 1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b). # CST Items Textbook McDougal Littell** 1 Learning Targets 7A Compare the meaning of a rate and a ratio. 1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/21, find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. 6 Learning Targets 7C Explain the meaning of equivalent rates. 7D Use proportion to find equivalent rates. 1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interests earned, and tips. 5 Learning Targets 7E Explain the meaning of a unit rate, and give examples. Algebra and Functions: 1.1 Write and solve one-step 6 * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell Unit High Priority Standards And Learning Targets* # CST Items # Q3 Items Math Grade 6 Instructional Guide 2011-2012 12 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets linear equations in one variable. 1.2 Write and evaluate an algebraic expression for 1 a given situation, using up to three variables. Learning Targets 7G Solve real world problems involving rates. Unit 8: Measurement and Geometry Continuing the study of how quantities are related even further, this unit focuses on using variables and equations to relate geometric objects and concepts. The ratio continues to be a tool used to describe such a relationship. This unit begins with the definition of the number as the ratio of a circle’s circumference to its diameter. This ratio is explored for a variety of circles, and it is shown that these ratios are all equivalent. A decimal and fractional estimate of the number (3.14, 22/7) is then developed, and it is emphasized that is simply a symbol that represents a certain ratio, and that it is neither a variable nor an unknown. Formulas for area and circumference of a circle Algebra and Functions: 3.0 Students investigate geometric patterns and describe them algebraically. 1 2 Learning Targets 8I Find the perimeter of a triangle or quadrilateral. 8J Find the area of a triangle and a quadrilateral and explain how the formulas work in finding the solution. 8K Solve real world problems involving the area and perimeter of a triangle or quadrilateral. 3.1 Use variables in expressions describing geometric quantities (e.g., P = 2w + 2l, A = 1/2bh, C = d). Mathematical Reasoning: 1.0 Students make decisions about how to approach problems 2.0 Students use strategies, skills, and concepts in finding solutions. 3.0 Students move beyond a particular problem by generalizing to other situations. Number Sense: 1.2 Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations (a/b, a to b, a:b). Embedded 1 Learning Targets 8K Solve real world problems involving the area and perimeter of a triangle or quadrilateral. 1 2 Algebra and Functions: 1.1 Write and solve one-step linear equations in one variable. 6 Learning Targets 8N Describe different types of angle pairs, including vertical, adjacent, complementary and supplementary and use these properties to find missing values of angles. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell Unit are explored in terms of , and the different estimates of are used in the formulas. Estimates of the circumference are then compared with direct measurements. The idea of an equation as a formula is continued in a discussion of triangles and quadrilaterals. Properties of these objects are explored, and the formulas for the area of a triangle and rectangle are discussed and used in a real-world context. The concept of a two-dimensional shape is extended further to threedimensional shapes, or solids. Some types of solids are given special names, including a cube, a prism, and a cylinder. Formulas for the volume of a cube, a triangular prism and a cylinder are found and connected back to the concept of area in two dimensions. The focus of the unit then shifts back to two dimensions with the study of angles. Different types of angles (vertical, adjacent, complementary, supplementary) are defined. The theme of an unknown is then explored geometrically as problems are solved involving finding a missing angle. Angles within a shape are also identified. The measures of the angles of a triangle are explored, and the triangle sum theorem is developed. This theorem is then used to find the missing angle of a triangle, once again solidifying the theme of the unknown. High Priority Standards And Learning Targets* Measurement and Geometry: 1.1 Understand the concept of a constant such as ; know the formulas for the circumference and area of a circle. # CST Items # Q3 Items 3 2 Learning Targets 8A Explain the meaning of pi, and how it relates to the study of ratios. 8C Find the area of a circle. 8D Find the circumference of a circle. 1.2 Know common estimates of (3.14; 22/7) and use these values to estimate and calculate the circumference and the area of circles; compare with actual measurements. 1 2 Learning Targets 8J Find the area of a triangle and a quadrilateral and explain how the formulas work in finding the solution. 8L Use the triangle sum theorem to find the missing angle of a triangle. 8P Explain the relationship between a cylinder and a circle. Mathematical Reasoning: 1.0 Students make decisions about how to approach problems 2.0 Students use strategies, skills, and concepts in finding solutions. 3.0 Students move beyond a particular problem by generalizing to other situations. Learning Targets 8B Use different estimates of pi including 3.14 and 22/7 and explain how these connect to the exact value of pi 8E Directly measure the circumference of a circle and compare these measurements to measurements found using the formula for circumference. 1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base x height); compare these formulas and explain the similarity between them and the formula for the vol- Math Grade 6 Instructional Guide 2011-2012 13 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets 1.4 Solve problems manually by using the correct or1 der of operations or by using a scientific calculator. 3.2 Express in symbolic form simple relation1 ships arising from geometry. 1 Embedded 2 * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell Unit This unit closes by once again comparing and contrasting the different ways a variable and an unknown are used in a geometric context. This is then connected to the notion of variable and unknown from previous units. High Priority Standards And Learning Targets* ume of a rectangular solid. # CST Items # Q3 Items 1 2 4 3 1 2 Math Grade 6 Instructional Guide 2011-2012 14 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets Learning Targets 8O Identify three-dimensional solids, including a prism and cylinder. 8P Explain the relationship between a cylinder and a circle. 8Q Find the volume of a prism and cylinder. 2.1 Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms. Learning Targets 8N Describe different types of angle pairs, including vertical, adjacent, complementary and supplementary and use these properties to find missing values of angles. 2.2 Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle. Learning Targets 8L Use the triangle sum theorem to find the missing angle of a triangle. 8M Explain why the triangle sum theorem is valid. 8N Describe different types of angle pairs, including vertical, adjacent, complementary and supplementary and use these properties to find missing values of angles. 2.3 Draw quadrilaterals and triangles from given information * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell Unit High Priority Standards And Learning Targets* about them (e.g. a quadrilateral having equal sides but no right angles, a right isosceles triangle). # CST Items # Q3 Items Math Grade 6 Instructional Guide 2011-2012 15 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets Learning Targets 8F Identify triangles and quadrilaterals. 8G Classify triangles by angle or by side length. 8H Classify quadrilaterals by angle or by side length. Unit 9: Probability and Statistics This unit begins with a discussion of the concept of a data set. Different statistical measures (range, mean, median, mode) are used to analyze different data sets. Tables and graphs are used to represent data that has been collected through a variety of sampling methods. The focus then shifts to probability, with a discussion of theoretical and experimental probability. Using real world examples, the experimental probability of an event is found. Given a compound event, all possible outcomes are determined and the theoretical probability is calculated for each outcome. This concept is framed in a real world context and used to solve problems involving proportion and probability, continuing the study of the unknown. Statistics, Data Analysis, and Probability: 1.1 Compute the range, mean, median, and mode of data sets. 1 2 3 2 Learning Targets 9A Find the range, mean, median, and mode of a set of different data sets and explain the results 2.2 Identify different ways of selecting a sample (e.g., convenience sampling, responses to a survey, random sampling) and which method makes a sample more representative for a population. Learning Targets 9A Find the range, mean, median, and mode of a set of different data sets and explain the results 1 1 1.3 Understand how the inclusion or exclusion of outliers affect measures of central tendency. Learning Targets 9A Find the range, mean, median, and mode of a set of different data sets and explain the results Learning Targets 9C Collect data using different sampling methods. 2.3 Analyze data displays and explain why the way in which the question was asked might have influenced the results obtained and why the way in which the results were displayed might Statistics, Data Analysis, and Probability: 1.2 Understand how additional data added to data sets may affect these computations of measures of central tendency. 1 2 1.4 Know why a specific measure of central tendency (mean, median, mode) provides the most useful information in a given context. Learning Targets 9B Use tables and graphs to represent data sets and how representation of the data sets can influence conclusions reached. 2.1 Compare different samples of population with 1 1 * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell Unit High Priority Standards And Learning Targets* have influenced the conclusions reached. # CST Items # Q3 Items Learning Targets 9B Use tables and graphs to represent data sets and how representation of the data sets can influence conclusions reached. 3.0 Students determine theoretical and experimental probabilities and use these to make predictions about events: SDP 3.1, SDP 3.2 and SDP 3.3 Learning Targets 9D Solve real world problems involving experimental probability. 9E Find all possible outcomes of a compound event. 9F Find the theoretical probability of the outcome of a compound event. 9G Explain the difference between theoretical and experimental probability. 9H Solve real world problems involving proportion and probability. 9H Solve real world problems involving proportion and probability. 3 2 Math Grade 6 Instructional Guide 2011-2012 16 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets the data from the entire population and identify a situation in which it makes sense to use a sample. Learning Targets 9C Collect data using different sampling methods. 2.4 Identify data that represent sampling errors and explain why the sample (and the display) might be biased. Learning Targets 9B Use tables and graphs to represent data sets and how representation of the data sets can influence conclusions reached. 9C Collect data using different sampling methods. 1 1 3.2 Use data to estimate the probability of future events (e.g., batting averages or number of accidents per mile driven). Learning Targets 9F Find the theoretical probability of the outcome of a compound event. 1 3.4 Understand that the probably of either of two disjoint events occurring is the sum of the two individual probabilities and that the probability of one event following another, in independent trials, is the product of the two probabilities. Learning Targets 9G Explain the difference between theoretical and experimental probability. * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell Unit High Priority Standards And Learning Targets* # CST Items # Q3 Items Math Grade 6 Instructional Guide 2011-2012 17 Supporting Medium/Low Textbook Priority Standards # CST Items McDougal Littell** & Learning Targets Mathematical Reasoning: 1.0 Students make decisions about how to approach problems 2.0 Students use strategies, skills, and concepts in finding solutions. 3.0 Students move beyond a particular problem by generalizing to other situations. Embedded Unit 10: CST Review Unit In this final unit, the connections that have been formed from unit to unit are solidified. The relationships between all the skill sets and concepts that have been learned throughout the course are communicated. As a way of differentiating instruction, understanding of skills and concepts from the course is self-assessed, and an individualized plan is developed to address any deficiencies. Instruction Continues After CST- Q4 Math Culminating Projects * Learning Targets (LT) break the standard down into relevant chunks for teaching and benchmark assessments. The LT number refers to the unit i.e. 1a is Unit 1 learning target A. **Textbook: California Middle School Mathematics Concepts and Skills Course 1 (2001), Ron Larson, Laurie Boswell, Timothy D. Kanold, Lee Stiff. McDougal Littell