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 TITLE : Unit 01: Value and Magnitude of Rational Numbers SUGGESTED DURATION : 6 days

#### Unit Overview

Introduction
This unit bundles student expectations that address sets and subsets of rational numbers, ordering rational numbers, and converting between standard decimal notation and scientific notation. According to the Texas Education Agency, mathematical process standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.” Additionally, the availability of graphing technology is required during STAAR testing.

Prior to this Unit
In Grade 5, students extended their understanding of the base-10 place value system of whole numbers through the billions and decimals through the thousandths. Students examined the relationships within the base-10 place value system that moving left across the places, the values are 10 times the position to the right; whereas, moving right across the places, the values are one-tenth the value of the place to the left. In Grade 6, students ordered a set of rational numbers arising from mathematical and real-world contexts. In Grade 7, students extended previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers. These subsets included counting (natural) numbers, whole numbers, integers, and rational numbers.

During this Unit
Students continue to 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. Rational numbers are the focus of this unit as students order a set of rational numbers that arise from mathematical and real-world situations. Students extend previous understandings of the relationships within the base-10 place value system as they convert between standard decimal notation and scientific notation. Both positive and negative numbers are represented with standard decimal notation and scientific notation, including values greater than and less than one.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 8

After this Unit
In Unit 08, students will extend previous knowledge of sets and subsets to order and describe relationships between sets of real numbers, which includes rational numbers and their subsets as well as irrational numbers. In Algebra 2, students will be introduced to the complex number system.

In Grade 8, ordering a set of real numbers is identified as STAAR Readiness Standard 8.2D. Describing the relationships between sets and subsets of real numbers using a visual representation and converting between standard decimal notation and scientific notation are STAAR Supporting Standards 8.2A and 8.2C. All of these standards are subsumed under the Grade 8 STAAR Reporting Category 1: Numerical Representations and Relationships and the Texas Response to Curriculum Focal Points (TxRCFP): Grade Level Connections. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A1, B1, B2; II. Algebraic Reasoning D1, D2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to the National Council of Teachers of Mathematics (NCTM) “Instructional programs from prekindergarten to grade 12 should enable all students to understand numbers, ways of representing numbers, relationships among number, and number systems” (2000, p. 148). When students describe the number system as a hierarchy of sets and subsets, they are able to extend their understandings of how numbers are interrelated. “Many problems involving ordering capture children’s interest because they want to know which is more, which is shorter, which is larger, and so on” (Reyes, Lindquist, Lambdin & Smith, 2012, p. 263). The concept of comparing numbers is embedded within the concept of ordering numbers. An additional representation of numbers is with scientific notation.

“[Students] should develop a sense of the magnitude of very large numbers (millions and billions) and become proficient at reading and representing them…Contexts in which large numbers arise naturally are found in other school subjects as well as in everyday life. Students’ experience in working with very large [and very small] numbers and in using the idea of orders of magnitude will also help build their facility with proportionality” (NCTM, 2000, p. 217).

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
Reyes, R. E., Lindquist, M., Lambdin, D. V., & Smith, N. L. (2012). Helping children learn mathematics. Hoboken, NJ: Wiley.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place.  How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Rational numbers create a more sophisticated number system where new relationships exist within and between sets and subsets of numbers (counting numbers; whole numbers; integers; rational numbers).
• What representations can be used to visually demonstrate relationships between sets and subsets of numbers?
• How does organizing numbers in sets and subsets aid in understanding the relationships between rational numbers?
• What relationships exist between sets and subsets of numbers?
• How are the elements in counting (natural) numbers, whole numbers, integers, and rational numbers related?
• How can a number belong to the same set of numbers but not necessarily the same subset of numbers?
• What relationship exists between rational numbers and the other number sets?
• Quantities are compared and ordered to determine magnitude of number and equality or inequality relations (rational numbers such as fractions, decimals, percents).
• Why is it important to identify the unit or attribute being described by numbers before comparing or ordering the numbers?
• How can …
• place value
• numeric representations
• concrete representations
• pictorial representations
… aid in the comparison and/or ordering of numbers?
• How can the comparison of two numbers be described and represented?
• How are quantifying descriptors used to determine the order of a set of numbers?
• Number and Operations
• Number
• Counting (Natural) numbers
• Whole numbers
• Integers
• Rational numbers
• Number Representations
• Sets and subsets
• Compare and Order
• Comparative language
• Comparison symbols
• Relationships and Generalizations
• Numerical
• Equivalence
• Representations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place.  How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy? Understanding and generalizing operational relationships leads to more sophisticated representations and solution strategies in order to investigate or solve problem situations in everyday life. What relationships exist within and between mathematical operations? How does generalizing operational relationships lead to developing more flexible, efficient representations and/or solution strategies? Why is understanding the problem solving process an essential part of learning and working mathematically?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• The ability to recognize and represent numbers in various forms develops the understanding of equivalence and allows for working flexibly with numbers in order to communicate and reason about the value of the number (positive and negative decimals).
• Why is scientific notation powerful?
• How is the base-10 place value system related to the positive or negative exponent used in scientific notation?
• What is the process to convert a number from
• standard form to scientific notation?
• scientific notation to standard form?
• How can it be determined if two expressions are equivalent?
• How are properties and operational understandings used to generate equivalent expressions?
• Why can it be useful to simplify expressions?
• Number and Operations
• Base-10 Place Value System
• Number Representations
• Standard decimal notation
• Scientific notation
• Relationships and Generalizations
• Numerical
• Equivalence
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that a negative power of ten may imply a negative number, rather than realizing the relationships within the base-10 place value system and powers of 10.
• Some students may think that multiplying by a power of ten means to add zeros to the end of the number.
• Some students may think that the number of zeros in a number in standard decimal form translates to the digit exponent when converting to scientific notation (e.g., the number 1,254,000,000,000 has nine zeros, however the exponent of the power of 10 when the number is written is scientific notation is 12, not 9.).

Underdeveloped Concepts:

• Some students may think that a number can only belong to one set (counting [natural] numbers, whole numbers, integers, or rational numbers) rather than understanding that some sets of numbers are nested within another set as a subset.

#### Unit Vocabulary

• Base – the number in an expression or equation which is raised to a power or exponent
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Decimal notation – a representation of a real number, not including counting (natural) numbers, which uses a decimal point to show place values that are less than one, such as tenths and hundredths (e.g., 0.023, etc.)
• E – a symbol used in a calculator to indicate that the preceding number should be multiplied by ten raised to the number that follows
• Integers – the set of counting (natural numbers), their opposites, and zero {–n, …, –3, –2, –1, 0, 1, 2, 3, ..., n}. The set of integers is denoted by the symbol Z.
• Order numbers – to arrange a set of numbers based on their numerical value
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
• Powers – denoted by a number or variable in the superscript place of the base, which designates how many times the base will be multiplied by itself if it is positive or by its inverse if it is negative. If the power is 1, the base will be multiplied by 1 and will not change. If the power is 0, the simplified form will equal 1.
• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Scientific notation – a representation of a number by using a method to write very large or very small numbers using powers of ten that is written as a decimal with exactly one nonzero digit to the left of the decimal point, multiplied by a power of ten (e.g., 2.3 × 10–2, etc.)
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Ascending Base-10 place value system Descending Exponent Multiplicative identity Number line Repeating decimal Set of numbers Subset of numbers Terminating decimal
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 8 Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
8.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
8.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
8.1B 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.
Process Standard

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

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
8.1C

Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Process Standard

Select

TOOLS, INCLUDING PAPER AND PENCIL AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Paper and pencil
• Technology
• Techniques
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
8.1D

Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Process Standard

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
8.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
8.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
8.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
8.2 Number and operations. The student applies mathematical process standards to represent and use real numbers in a variety of forms. The student is expected to:
8.2A

Extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers.

Supporting Standard

Extend

PREVIOUS KNOWLEDGE OF SETS AND SUBSETS USING A VISUAL REPRESENTATION

Including, but not limited to:

• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Integers – the set of counting (natural numbers), their opposites, and zero {–n, …, –3, –2, –1, 0, 1, 2, 3, ..., n}. The set of integers is denoted by the symbol Z.
• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Visual representations of the relationships between sets and subsets of real numbers

To Describe

RELATIONSHIPS BETWEEN SETS OF RATIONAL NUMBERS

Including, but not limited to:

• All counting (natural) numbers are a subset of whole numbers, integers, and rational numbers.
• All whole numbers are a subset of integers and rational numbers.
• All integers are a subset of rational numbers.
• All counting (natural) numbers, whole numbers, and integers are a subset of rational numbers.
• Not all rational numbers are integers, whole numbers, or counting (natural) numbers.
• Terminating and repeating decimals are rational numbers but not integers, whole numbers, or counting (natural) numbers.

Note(s):

• Grade 5 classified two-dimensional figures in a hierarchy of sets and subsets using graphic organizers based on their attributes and properties.
• Grade 6 classified whole numbers, integers, and rational numbers using a visual representation such as a Venn diagram to describe relationships between sets of numbers.
• Grade 7 extended previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers.
• Grade 8 introduces the set of irrational numbers as a subset of real numbers.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Grade Level Connections (reinforces previous learning and/or provides development for future learning)
8.2C Convert between standard decimal notation and scientific notation.
Supporting Standard

Convert

BETWEEN STANDARD DECIMAL NOTATION AND SCIENTIFIC NOTATION

Including, but not limited to:

• Decimal notation – a representation of a real number, not including counting (natural) numbers, which uses a decimal point to show place values that are less than one, such as tenths and hundredths (e.g., 0.023, etc.)
• Scientific notation – a representation of a number by using a method to write very large or very small numbers using powers of ten that is written as a decimal with exactly one nonzero digit to the left of the decimal point, multiplied by a power of ten (e.g., 2.3 × 10–2, etc.)
• Powers – denoted by a number or variable in the superscript place of the base, which designates how many times the base will be multiplied by itself if it is positive or by its inverse if it is negative. If the power is 1, the base will be multiplied by 1 and will not change. If the power is 0, the simplified form will equal 1.
• Base – the number in an expression or equation which is raised to a power or exponent
• E – a symbol used in a calculator to indicate that the preceding number should be multiplied by ten raised to the number that follows
• Relationship between place value and scientific notation
• Format of scientific notation
• Powers of 10
• Positive or negative integer exponents
• Negative exponents move the decimal to the left the same number of places as the absolute value of the exponent.
• Positive exponents move the decimal to the right the same number of places as the exponent.
• Positive or negative decimal with exactly one nonzero digit to the left of the decimal point
• Multiplicative identity
• Decimal notation to scientific notation and vice versa

Note(s):

• Grade 8 introduces converting between standard decimal notation and scientific notation.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Grade Level Connections (reinforces previous learning and/or provides development for future learning)
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
8.2D

Order a set of real numbers arising from mathematical and real-world contexts.

Order

A SET OF RATIONAL NUMBERS ARISING FROM MATHEMATICAL AND REAL-WORLD CONTEXTS

Including, but not limited to:

• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Various forms of rational numbers
• Rational numbers (positive or negative)
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, ten thousands, etc.
• Order numbers – to arrange a set of numbers based on their numerical value
• Number lines (horizontal/vertical)
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Quantifying descriptor in mathematical and real-world problem situations (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)

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