The mathematics curriculum provides sequential and comprehensive K-12 instruction in a collaborative, student-centered learning environment that fosters critical thinking, creativity, skillful problem-solving, and effective communication in order to enable all students to adapt to an ever-changing, global society and increase college and career readiness. An emphasis has been placed on conceptual understanding, higher-order thinking, and problem solving skills to prepare students for 21st century careers. This is further embedded through the integrated use of technology into daily lessons. Instruction focuses on meaningful development of mathematical ideas at each grade level where students are given the opportunity to explore, engage, and take risks with content as they build and expand their knowledge and understanding of mathematics. Students will experience mathematics as a coherent and useful subject within the context of real-life situations. In all, the curriculum aims to reach high standards while encouraging curiosity and building confidence in a collaborative atmosphere.
Grade 6 Description
The Grade 6 Mathematics course will provide students with the information and tools necessary to be active in their learning of the basics of algebra. Students will learn algebraic skills, problem solving skills, geometry concepts, and what rational numbers are. Students will also strive to develop an importance for algebra by studying its relation to real world applications. These topics will prepare them for 7th grade math and beyond.
Grade 6Units:
- Topic 1: Use Positive Rational Numbers
- Topic 2: Integers and Rational Numbers
- Topic 3: Numeric and Algebraic Expressions
- Topic 4: Represent and Solve Equations and Inequalities
- Topic 5: Understand and Use Ratio and Rate
- Topic 6: Understand and Use Percent
- Topic 7: Solve Area, Surface Area, and Volume Problems
- Topic 8: Display, Describe, and Summarize Data
Subject: Mathematics / Grade: 6 / Suggested Timeline: 24 class periods
UnitTitle:
Topic 1: Use Positive Rational Numbers
Unit Overview/Essential Understanding:
- Algorithms can be used to add, subtract, and multiply decimal numbers.
- An algorithm can be used to divide whole numbers and decimals fluently.
- Visual models, such as area models and number lines, can be used to multiply fractions.
- The product of two fractions can be found by multiplying the numerators and then the denominators.
- Multiplying mixed numbers is an extension of multiplying fractions.
- Many real-world problem situations can be represented with a mathematical model, but that model may not represent a real-world situation exactly.
- Visual models, such as number lines and area models, and equations can be used to represent and solve problems that involve division of fractions.
- Dividing a whole number by a fraction is equivalent to multiplying the whole number by the reciprocal of the fraction.
- Visual models, such as number lines and area models, can be used to represent and solve problems that involve division of a fraction by a fraction.
- Dividing by a fraction is equivalent to multiplying by the fraction’s reciprocal.
- The quotient of mixed numbers can be found by writing the mixed numbers as fractions and multiplying the dividend by the divisor’s reciprocal.
- Multistep problems require students to carefully plan the steps they follow to find the solution.
Unit Objectives:
At the end of this module, students will be able to independently use their learning to:
- add, subtract and multiply decimals with precision
- add, subtract and multiply decimals to solve real word problems
- use place value structure to divide whole numbers and decimals
- divide decimals and whole numbers to solve real world problems
- use models to multiply fractions
- multiply the numerators and then the denominators to find the product of two fractions
- multiply mixed numbers
- use models to divide with fractions
- use equations to divide with fractions
- Use models and algorithms to divide fractions by fractions
- divide with mixed numbers
- estimate the quotient of mixed numbers
- solve multistep problems with fractions and decimals
Focus Standards Addressed in this Unit:
CC.2.1.6.E.2 – Identify and choose appropriate processes to compute fluently with multi-digit numbers
CC.2.1.6.E.1 – Apply and extend previous understandings of multiplication and division to divide fractions by fractions
Important Standards Addressed in this Unit:
N/A
Misconceptions:
- Where to place the decimal when multiplying decimal numbers together.
- When the divisor is multiplied by a power of 10, to make it a whole number, the dividend must be multiplied by the same power of 10.
- To find the reciprocal of a whole number, the whole number must be turned to a fraction with 1 as the denominator.
- When multiplying fractions, the reciprocal of the divisor must be found first even when it is a whole number.
Concepts/Content:
- Ratios
- Proportions
- Percent
- Number Theory Concepts and Operations
- Interpret and compute quotients of fractions
- Solve Problems and compute fluently with whole numbers and decimals
- Solve and Discuss It/Explain It
- Teacher Modeled examples
- Guided practice
- Individual practice
Assessments:
Summative Assessments:
- Mid-topic Assessment
- Mid-topic Performance Task
- End of topic Assessment
- End of topic Performance Assessment
- Lesson Quizzes
- Online Homework (Math XL)
Interdisciplinary Connections:
- STEM Activity – “Improve Your School”
- 3-Act Math – “Stocking Up”
- Pearson, enVisionmath2.0
Subject: Mathematics / Grade: 6 / Suggested Timeline: 20 class periods
Unit Title:
Topic 2: Integers and Rational Numbers
Unit Overview/Essential Understanding:
- Integers are counting numbers, their opposites, and zero.
- Integers can be compared, ordered, and used to describe real-world contexts.
- Each rational number can be associated with a unique point on the number line.
- A number to the right of another on the number line is the greater number
- The absolute value of a number can be described as the number’s distance from zero on the number line.
- A coordinate plane is formed by a horizontal number line, the x-axis, and a vertical number line, the y-axis, that intersect at a point called the origin.
- An ordered pair (x, y) locates a point on the coordinate plane.
- Many real-world problem situations can be represented with a mathematical model, but that model may not represent a real-word situation exactly
- The distance between two points on a coordinate plane with the same first coordinate or the same second coordinate can be found by adding or subtracting the absolute values of the coordinates that differ.
- The coordinates of the vertices of a polygon on the coordinate plane can be used to find the lengths of the sides of the polygon and its perimeter.
Unit Objectives:
At the end of this module, students will be able to independently use their learning to:
- identify opposites of integers
- compare and order integers
- use integers to represent real-world quantities and explain the meaning of zero in each context
- plot rational numbers on a number line
- compare and order rational numbers
- use rational numbers to represent real-word quantities
- use absolute value to represent a number’s distance from zero
- interpret absolute value in real-world situations
- identify and graph points with rational coordinates on the coordinate plane
- reflect points with rational coordinates across both axes
- use mathematical modeling to represent a problem situation and to propose a situation
- test and verify the appropriateness of their math models
- explain why the results from their mathematical models may not align exactly with the problem situation
- use absolute value to find the distance between two points that lie on the same horizontal or vertical line on a coordinate plane
- solve real-world and mathematical problems involving distances on the coordinate plane
- find side length of polygons on the coordinate plane
- find the perimeter of polygons on the coordinate plane
Focus Standards Addressed in this Unit:
CC.2.1.6.E.4 – Apply and extend previous understandings of numbers to the system of rational numbers
CC.2.3.6.A.1 – Apply appropriate tools to solve real-world and mathematical problems involving area, surface area, and volume
Important Standards Addressed in this Unit:
N/A
Misconceptions:
- Numbers that are not preceded by a sign are presumed to be positive.
- Because negative numbers increase from right to left, students may place mixed numbers to the right of the whole number rather than the left.
- Students may confuse the account balances with their absolute values.
- Students may think that they can find the distance between any two coordinates by adding or subtracting the absolute values of coordinates.
Concepts/Content:
- Integers
- Rational numbers
- Area
- Surface Area
- Volume
- Use positive and negative numbers to represent .quantities in real-word contexts
- Plot integers and other rational numbers on a number line and on a coordinate graph
- Interpret the opposite and absolute value of an integer as its distance from zero on a number line
- Compare and order rational numbers
- Solve and Discuss It/Explain It
- Teacher Modeled examples
- Guided practice
- Individual practice
Assessments:
Summative Assessments:
- Mid-topic Assessment
- Mid-topic Performance Task
- End of topic Assessment
- End of topic Performance Assessment
- Lesson Quizzes
- Online Homework (Math XL)
Interdisciplinary Connections:
- STEM Activity – “Improve Your School”
- 3-Act Math – “The Ultimate Throw”
- Pearson, enVisionmath2.0
Subject: Mathematics / Grade: 6 / Suggested Timeline: 25 class periods
Unit Title:
Topic 3: Numeric and Algebraic Expressions
Unit Overview/Essential Understanding:
- A whole number exponent can be used to represent repeated multiplication of a number.
- Any number can be written as its prime factorization.
- The greatest common factor (GCF) is the greatest factor that two or more numbers have in common.
- The least common multiple (LCM) is the smallest multiple that two or more non-zero numbers have in common.
- There is an agreed-upon order in which operations are carried out in a numerical expression.
- Algebraic expressions use variables to describe situations in which some of the information is not known.
- Parts of expressions can be described using words such as term, coefficient, product, and factor.
- The value of an algebraic expression can be found by replacing the variables with given numbers and doing the calculation that results.
- Many real-world problem situations can be represented with a mathematical model, but that model may not represent a real-world situation exactly.
- The Distributive Property and other properties of operations are used to identify and write equivalent expressions.
- Algebraic expressions can be simplified using the properties of operations to combine like terms and generate equivalent expressions.
Unit Objectives:
At the end of this module, students will be able to independently use their learning to:
- write expressions using whole number exponents to represent real-world and mathematical problems
- evaluate expressions with whole number exponents
- find the prime factorization of a whole number
- find the greatest common factor (GCF) and the least common multiple (LCM) of two whole numbers
- use the GCF and the distributive property to add
- use the GCF and LCM to solve problems
- evaluate expressions using the order of operations
- insert grouping symbols in a numerical expression to affect the value of the expression
- write an algebraic expression to model a pattern
- write an algebraic expression from a word phrase
- use precise mathematical language when identifying parts of an expression
- evaluate algebraic expressions, including those with whole numbers, decimals, and fractions
- use mathematical modeling to represent a problem situation and to propose a solution
- test and verify the appropriateness of their math models
- explain why the result from their mathematical models may no align exactly to the problem situation
- write equivalent algebraic expressions
- identify equivalent algebraic expressions
- justify whether two expressions are equivalent
- use properties of operations to simplify algebraic expressions by combining like terms
Focus Standards Addressed in this Unit:
CC.2.2.6.B.1 – Apply and extend previous understandings of arithmetic to algebraic expressions
CC.2.1.6.E.3 – Develop and/or apply number theory concepts to find common factors and multiples
CC.2.2.6.B.2 – Understand the process of solving a one-variable equation or inequality and apply the real-world and mathematical problems
Important Standards Addressed in this Unit:
N/A
Misconceptions:
- The exponent tells you the number of times the base is a factor, not the number of times it is multiplied
- When completing an order of operations problem, multiplication and division are done together from left to right.
- Parts of an algebraic expression are not always written in the same order.
- You can create equivalent expressions with Commutative, Associative, and Distributive properties.
Concepts/Content:
- Algebraic Expressions
- Number Theory Concepts and Operations
- Write, identify and evaluate numerical expressions involving exponents
- Write, read and evaluate algebraic expressions
- Apply the properties of operations to generate equivalent expressions
- Solve problems and compute fluently with whole numbers and decimals
- Find common multiples and factors including greatest common factor and least common multiple
- Use the distributive property to express a sum of two numbers.
- Solve and Discuss It/Explain It
- Teacher Modeled examples
- Guided practice
- Individual practice
Assessments:
Summative Assessments:
- Mid-topic Assessment
- Mid-topic Performance Task
- End of topic Assessment
- End of topic Performance Assessment
- Lesson Quizzes
- Online Homework (Math XL)
Interdisciplinary Connections:
- STEM Activity – “Design a Bridge”
- 3-Act Math – “The Field Trip”
- Pearson, enVisionmath2.0
Subject: Mathematics / Grade: 6 / Suggested Timeline: 27 Class Periods
Unit Title:
Topic 4: Represent and Solve Equations and Inequalities
Unit Overview/Essential Understanding:
- A solution of an equation is a value for the variable that makes the equation true.
- An equation is true when the expressions or numbers on both sides of the equal sign have the same value.
- The same number can be added to, subtracted from, or multiplied on both sides of an equation and equality is maintained.
- Dividing on both sides of an equation by the same non-zero number also maintains equality.
- A problem situation can be represented by an equation with a variable.
- An equation can be solved by using the inverse operation and a property of equality.
- A multiplication or division problem situation can be represented by an equation with a variable.
- Inverse relationships and properties of equality can be used to solve equations with fractions, mixed numbers and decimals.
- An inequality is a mathematical sentence that contains the inequality symbol < (is less than), > (is greater than), ≤ (is less than or equal to), ≥ (is greater than or equal to), or ≠ (is not equal to).
- An inequality describes a situation that has an infinite number of numerical possibilities.
- Many real-world problem situations can be represented with a mathematical model, but that model may not represent a real-world situation exactly.
- Variables can be used to represent quantities that change in relationship to one another.
- The dependent variable changes in response to the independent variable.
- Patterns can be used to identify the relationship between quantities and write an equation that describes the relationship.
- Tables, graphs, and equations can be used to analyze the relationship between dependent and independent variables.
Unit Objectives:
At the end of this module, students will be able to independently use their learning to:
- identify equations and variables
- use substitution to find solutions to equations
- use the properties of equality to keep both sides of an equation equal
- identify which properties or equality are used to write equivalent expressions
- write one-variable addition and subtraction equations
- use inverse relationships and properties of equality to solve one-step addition and subtraction equations
- write one-variable multiplication and division equations
- use inverse relationships and properties of equality to solve one-step multiplication and division equations
- write and solve equations that involve fractions, decimals, and mixed numbers
- understand the symbols required to write an inequality
- write inequalities to describe mathematical or real-world situations
- describe solutions to an inequality
- represent solutions to an inequality on a number line
- use mathematical modeling to represent a problem situation and to propose a solution
- identify dependent variables
- identify independent variables
- analyze the relationships between variables by using tables
- write equations to represent the relationships between variables
- analyze the relationship between dependent and independent variables using tables, graphs, and equations
Focus Standards Addressed in this Unit:
CC.2.2.6.B.1 – Apply and extend previous understandings of arithmetic to algebraic expressions
CC.2.2.6.B.2 – Understand the process of solving a one-variable equation or inequality and apply the real-world and mathematical problems
CC.2.2.6.B.3 – Represent and analyze quantitative relationships between dependent and independent variables
Important Standards Addressed in this Unit:
N/A
Misconceptions:
- The exponent tells you the number of times the base is a factor, not the number of times it is multiplied
- When completing an order of operations problem, multiplication and division are done together from left to right.
- Parts of an algebraic expression are not always written in the same order.
- You can create equivalent expressions with Commutative, Associative, and Distributive properties.
Concepts/Content:
- Algebraic Expressions
- Algebraic Equations
- Write, identify and evaluate numerical expressions involving exponents
- Write, read and evaluate algebraic expressions
- Apply the properties of operations to generate equivalent expressions
- Represent and analyze quantitative relationships between independent and dependent variables
- Solve and interpret one variable equations or inequalities in real world mathematical problems