6th Grade
6th Grade
Embedded Files
Mathematics
Mathematics
Introduction
Introduction
(1) The desire to achieve educational excellence is the driving force behind the Texas essential knowledge and skills for mathematics, guided by the college and career readiness standards. By embedding statistics, probability, and finance, while focusing on computational thinking, mathematical fluency, and solid understanding, Texas will lead the way in mathematics education and prepare all Texas students for the challenges they will face in the 21st century.
(2) The process standards describe ways in which students are expected to engage in the content. The placement of the process standards at the beginning of the knowledge and skills listed for each grade and course is intentional. The process standards weave the other knowledge and skills together so that students may be successful problem solvers and use mathematics efficiently and effectively in daily life. The process standards are integrated at every grade level and course. When possible, students will apply mathematics to problems arising in everyday life, society, and the workplace. Students will use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. Students will select appropriate tools such as real objects, manipulatives, algorithms, paper and pencil, and technology and techniques such as mental math, estimation, number sense, and generalization and abstraction to solve problems. Students will effectively communicate mathematical ideas, reasoning, and their implications using multiple representations such as symbols, diagrams, graphs, computer programs, and language. Students will use mathematical relationships to generate solutions and make connections and predictions. Students will analyze mathematical relationships to connect and communicate mathematical ideas. Students will display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
(3) The primary focal areas in Grade 6 are number and operations; proportionality; expressions, equations, and relationships; and measurement and data. Students use concepts, algorithms, and properties of rational numbers to explore mathematical relationships and to describe increasingly complex situations. Students use concepts of proportionality to explore, develop, and communicate mathematical relationships. Students use algebraic thinking to describe how a change in one quantity in a relationship results in a change in the other. Students connect verbal, numeric, graphic, and symbolic representations of relationships, including equations and inequalities. Students use geometric properties and relationships, as well as spatial reasoning, to model and analyze situations and solve problems. Students communicate information about geometric figures or situations by quantifying attributes, generalize procedures from measurement experiences, and use the procedures to solve problems. Students use appropriate statistics, representations of data, and reasoning to draw conclusions, evaluate arguments, and make recommendations. While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.
Unit 01: Equivalent Forms of Fractions, Decimals, and Percents
Unit 01: Equivalent Forms of Fractions, Decimals, and Percents
(10 classes for the entire unit)
(10 classes for the entire unit)
Students extend their mathematical foundations of equivalency within rational numbers, including percents. Concrete and pictorial models, including 10 by 10 grids, strip diagrams, and number lines are used to represent multiples of benchmark fractions and percents. Additionally, percents are represented with concrete and pictorial models, fractions, and decimals. Students continue their understanding of equivalency by generating and using equivalent forms of fractions, decimals, and percents to solve real-world problems, including those involving money. Percents less than or greater than 100%, including percents with fractional or decimal values such as 8.25% or 8 1/4% are encompassed within this unit. Students apply their understandings of percents to solve real-world problems that involve finding the whole given a part and the percent, the part given the whole and a percent, and the percent given the part and the whole. Methods for solving real-world problem situations involving percents, such as the use of proportions or scale factors between ratios, are not included in this unit. Computations within this unit are restricted to operational capabilities from Grade 5 which include sums and differences with any positive rational numbers, products with factors limited to a whole number by a whole number, a decimal by a decimal, or a whole number by a fraction, and quotients limited to whole number dividends and divisors, a decimal dividend by a whole number divisor, or whole number and unit fraction dividends and divisors.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.4E, 6.4F, 6.4G, 6.5B, 6.5C
Unit 02: Ordering Fractions, Decimals, and Integers
Unit 02: Ordering Fractions, Decimals, and Integers
(5 classes for the entire unit)
(5 classes for the entire unit)
Students continue their understanding of equivalency by generating and using equivalent forms of fractions, decimals, and percents to solve real-world problems, including problems involving money. The negative aspect of the number line is introduced as students explore the concept of integers and negative rational numbers. Students locate an integer or rational number on a number line and use its location to compare and order a set of numerical values, which may be presented in various forms. The number line may be used as a tool to assist in comparing or ordering a set of numbers; however, students are also expected to order a set of rational numbers arising from mathematical and real-world contexts using any strategy, such as place value, number sense, or comparisons to benchmarks. Students examine the sets and subsets of rational numbers and use a visual representation, such as a Venn diagram, to describe the relationships between the sets and subsets. This is the first time students classify a number as a natural (counting) number, whole number, integer, or rational number. Although the focus of operations in Grade 6 is with integers and positive rational numbers, students are expected to classify, compare, and order both positive and negative numerical values.
TEKS in this unit: 6.1A, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.2A, 6.2C, 6.2D, 6.4G
Unit 03: Operations with Positive Fractions and Decimals
Unit 03: Operations with Positive Fractions and Decimals
(10 classes for the entire unit)
(10 classes for the entire unit)
Students expand their understanding of the representations for division to include fraction notation such as a/b, which is equivalent to a ÷ b where b ≠0. Students recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values as well as determine whether a quantity is increased or decreased when multiplied by a fraction greater than or less than one. Exposure to solving mathematical and real-world situations assists students in generalizing operations with positive fractions and decimals, which builds fluency and reasonableness of solutions. All of these standards are encompassed within the additional expectation in this unit, which is for students to multiply and divide positive rational numbers fluently.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.2E, 6.3A, 6.3B, 6.3E
Unit 04: Operations with Integers
Unit 04: Operations with Integers
(10 classes for the entire unit)
(10 classes for the entire unit)
Students examine number relationships involving identifying a number, its opposite, and absolute value. Previous work with number lines transitions to the understanding that absolute value can be represented on a number line as the distance a number is from zero. This builds to the relationship that since distance is always a positive value or zero, then absolute value is always a positive value or zero. Although students have been introduced to the concept of integers, this is the first time students are exposed to operations with negative counting (natural) numbers, which is a subset of integers. The development of integer operations with concrete and pictorial models is foundational to student understanding of operations with integers. Forgoing the use of concrete and pictorial models as a development of integer operations could be detrimental to future success with computations involving negative quantities, such as negative fractions and decimals. The use of concrete and pictorial models for integer operations is intended to be a bridge between the abstract concept of operations with integers and their standardized algorithms. It is expected that once the concept of integer operations has been sufficiently developed and connected to the standardized algorithms, students should add, subtract, multiply, and divide integers fluently.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.2B, 6.3C, 6.3D
Unit 05: Proportional Reasoning with Ratios and Rates
Unit 05: Proportional Reasoning with Ratios and Rates
(15 classes for the entire unit)
(15 classes for the entire unit)
Students are formally introduced to proportional reasoning with the building blocks of ratios, rates, and proportions. Students examine and distinguish between ratios and rates as they give examples of ratios as multiplicative comparisons of two quantities describing the same attribute and examples of rates as the comparison by division of two quantities having different attributes. Students extend previous work with representing percents using concrete models and fractions. Additionally, students are introduced to generating equivalent forms of fractions, decimals, and percents using ratios, including problems that involve money. Students solve and represent problem situations involving ratios and rates with scale factors, tables, graphs, and proportions. Students also represent real-world problems involving ratios and rates, including unit rates, while converting units within a measurement system. These representations allow students to develop a sense of covariation when using proportional reasoning to solve problems, which means they are able to determine and analyze how related quantities change together. Students use both qualitative and quantitative reasoning to make both predictions and comparisons in problem situations involving ratios and rates. Students revisit solving real-world problems to find the whole given a part and the percent, to find the part given the whole and the percent, and to find the percent given the part and the whole, including the use of concrete and pictorial models. Methods for solving real-world problem situations involving percents, such as the use of proportions (using a letter standing for the unknown quantity) or scale factors between ratios, are included within this unit. Extensive and deliberate development of proportional reasoning skills is foundational for all future mathematics coursework, much of which concentrates on the concept of proportionality.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.4B, 6.4C, 6.4D, 6.4E, 6.4G, 6.4H, 6.5A, 6.5B
Unit 06: Equivalent Expressions and One-Variable Equations
Unit 06: Equivalent Expressions and One-Variable Equations
(13 classes for the entire unit)
(13 classes for the entire unit)
Students transition from using order of operations without exponents to simplifying numerical expressions using order of operations with exponents and to generating equivalent numerical expressions. Previous work with prime and composite numbers is extended to introduce prime factorization as a means to generate equivalent numerical expressions. Students should recognize that when a number is decomposed into prime and composite factors, the product of the factors is equivalent to the original number. Previously, students have only utilized numerical expressions, but in this unit they are formally introduced to algebraic expressions. Students investigate generating equivalent numerical and algebraic expressions using the properties of operations that include the inverse, identity, commutative, associative, and distributive properties. Concrete models, pictorial models, and algebraic representations are used to determine if two expressions are equivalent. Although students have experienced equations in previous coursework, they are now expected to bridge their understandings of expressions and equations in order to differentiate between the two. Students will be expected to explain the defining characteristics of an expression and the defining characteristics of an equation. Previously, students have used a letter for an unknown quantity. In this unit, students transition to using the term variable when referencing a letter that represents an unknown quantity. Equations within this unit are limited to one-variable, one-step equations. Constants or coefficients of one-variable, one-step equations include positive rational numbers or integers. Students are expected to analyze constraints or conditions within a problem situation and write a one-variable, one-step equation as well as write a corresponding real-word problem when given a one-variable, one-step equation. Concrete models, pictorial models, and algebraic representations are used again as students model and solve one-variable, one-step equations that represent problems, including geometric concepts. Although certain models, such as algebra tiles, may limit a student’s ability to model solving equations with whole number or integer constants or coefficients, students should also solve equations with positive rational number constants or coefficients with an algebraic model. Students are expected to represent their solution on a number line as well as determine if a given value(s) make(s) the one-variable, one-step equation true.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.7A, 6.7B, 6.7C, 6.7D, 6.9A, 6.9B, 6.9C, 6.10A, 6.10B
Unit 07: One-Variable Inequalities
Unit 07: One-Variable Inequalities
(12 classes for the entire unit)
(12 classes for the entire unit)
Students transition from one-variable, one-step equations to one-variable, one-step inequalities. This is the first time students are formally introduced to algebraic inequalities. Constants or coefficients of one-variable, one-step inequalities may include positive rational numbers or integers. Students are expected to analyze constraints or conditions within a problem situation and write a one-variable, one-step inequality to represent the situation. Students are also expected to write a corresponding real-word problem when given a one-variable, one-step inequality. Concrete models, pictorial models, and algebraic representations are used again as students model and solve one-variable, one-step inequalities that represent problems, including geometric concepts. Although certain models, such as algebra tiles, may limit a student’s ability to model solving an inequality with whole number or integer constants or coefficients, students should also solve inequalities with positive rational number constants or coefficients with an algebraic model. Students are expected to represent their solution on a number line as well as determine if a given value(s) make(s) the one-variable, one-step inequality true.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.9A, 6.9B, 6.9C, 6.10A, 6.10B
Unit 08: Algebraic Representations of Two-Variable Relationships
Unit 08: Algebraic Representations of Two-Variable Relationships
(12 classes for the entire unit)
(12 classes for the entire unit)
Students extend previous knowledge of graphing positive rational numbers on the coordinate plane to graphing both positive and negative rational numbers in all four quadrants of the coordinate plane. This skill is foundational for the other algebraic concepts in this unit. Although students have recognized the differences between additive and numerical patterns in tables and graphs and generated numerical patterns when given a rule in the form y = ax or y = x + a, students are now expected to compare two rules verbally, graphically, or symbolically in order to differentiate between additive and multiplicative relationships and explain their reasoning. Students should recognize the characteristics of an additive or multiplicative relationship when given a verbal description, graph, or equation. Relationships within a graph are examined as students identify independent and dependent relationships and quantities. Within this unit, students represent a given situation using verbal descriptions, tables, graphs, and equations, as well as demonstrate understanding that each representation models the same data. Also, given one representation, students should be able to create one or all of the different representations for the problem situation. For this grade level, problem situations for additive relationships may include both positive and negative rational numbers, whereas multiplicative relationships may only include integers or positive fractions or decimals.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.4A, 6.6A, 6.6B, 6.6C, 6.11A
Unit 09: Geometry and Measurement
Unit 09: Geometry and Measurement
(13 classes for the entire unit)
(13 classes for the entire unit)
Students extend their knowledge of triangles and their properties to include the sum of the angles of a triangle, and how those angle measurements are related to the three side lengths of the triangle. Students examine and analyze the relationship between the three side lengths of a triangle and determine whether three side lengths will form a triangle using the Triangle Inequality Theorem. Students also decompose and rearrange parts of parallelograms (including rectangles), trapezoids, and triangles in order to model area formulas for each of the figures. Students write equations for and determine solutions to problems related to the area of rectangles, parallelograms, trapezoids, and triangles. Problems include situations where the equation represents the whole area of the shape or partial area of the shape. Writing equations and determining solutions is extended to the volume of right rectangular prisms. Positive rational numbers should be used in problem situations for this unit. Students expand previous knowledge of converting units within the same measurement system when determining solutions to problems involving length. Conversion processes for measurement extend beyond the use of proportions to now include dimensional analysis and conversions graphs.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.4H, 6.8A, 6.8B, 6.8C, 6.8D
Unit 10: Data Analysis
Unit 10: Data Analysis
(20 classes for the entire unit)
(20 classes for the entire unit)
Students extend previous knowledge of data representations including dot plots and stem-and-leaf plots, and are formally introduced to histograms, box plots, and percent bar graphs. Students use graphical representations to describe the shape, center, and spread of the data distribution. Descriptions of shape, center, and spread include skewed right, skewed left, symmetric, mean, median, mode, range, and interquartile range. Students also summarize numeric data with numerical summaries, including the measures of center and the measures of spread. Categorical data is summarized numerically with the mode and a relative frequency table and summarized graphically with a percent bar graph. Students are required to distinguish between situations that yield data with and without variability.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.12A, 6.12B, 6.12C, 6.12D, 6.13A, 6.13B
Unit 11: Personal Financial Literacy
Unit 11: Personal Financial Literacy
(5 classes for the entire unit)
(5 classes for the entire unit)
Students compare the features and costs of checking accounts and debit cards offered by different local financial institutions. Students examine fees associated with both checking accounts and debit cards, and they balance a check register associated with a checking account. Students distinguish between credit cards and debit cards. The information included within a credit report is examined and students are expected to explain why it is important to have a positive credit history. Students describe the value of credit reports to borrowers and to lenders. The salaries of various career choices are explored and the correlation between the salary and required levels of education is analyzed. Students consider this relationship as they calculate the effects of different annual salaries on lifetime income. Various methods to pay for college are examined, including savings, grants, scholarships, student loans, and work-study.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.14A, 6.14B, 6.14C, 6.14D, 6.14E, 6.14F, 6.14G, 6.14H
Unit 12: Essential Understanding of Proportionality
Unit 12: Essential Understanding of Proportionality
(10 classes for the entire unit)
(10 classes for the entire unit)
Students revisit and solidify essential understandings of proportionality. Students solve and represent problem situations involving ratios and rates with scale factors, tables, graphs, and proportions. They represent real-world problems involving ratios and rates, including unit rates, while converting units within a measurement system. Students use both qualitative and quantitative reasoning to make both predictions and comparisons in problem situations involving ratios and rates. They solve real-world problems to find the whole given a part and the percent, to find the part given the whole and the percent, and to find the percent given the part and the whole, including the use of concrete and pictorial models.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.4B, 6.4G, 6.4H, 6.5A, 6.5B
Unit 13: Essential Understanding of Equations
Unit 13: Essential Understanding of Equations
(10 classes for the entire unit)
(10 classes for the entire unit)
Students revisit and solidify essential understandings of equations. Students represent two-variable algebraic relationships, including additive and multiplicative relationships, in the form of verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b, and model and solve one-variable, one-step equations that represent problems, including geometric concepts. Although certain models, such as algebra tiles, may limit a student’s ability to model solving equations with whole number or integer constants or coefficients, students should also solve equations with positive rational number constants or coefficients with an algebraic model. Students apply their knowledge of triangles and their properties to include the sum of the angles of the triangle and how those angle measurements are related to the three side lengths of the triangle. Students write equations and determine solutions to problems related to area of rectangles, parallelograms, trapezoids, and triangles. Problems include situations where the equation represents the whole area of the shape or partial area of the shape. Students extend concepts of equations to determining the volume of right rectangular prisms.
TEKS in this unit: 6.1A, 6.1B, 6.1C, 6.1D, 6.1E, 6.1F, 6.1G, 6.6C, 6.8A, 6.8C, 6.8D, 6.10A
Texas Essential Knowledge & Skills (TEKS)
Texas Essential Knowledge & Skills (TEKS)
TEKS - Math - G6.pdfPage updated
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