Grade 8 Math Curriculum Notebook

8th Grade Math Curriculum Notebook
2017


Curriculum Notebook Table of Contents
Standards
Standards indicate the broad goals for a student to master in a course. Standards are typically set by a state or district school board.
Mathematical Practice Standards ......................................................................................................... Page 4 The Number System .............................................................................................................................. Page 7 Expressions and Equations .................................................................................................................... Page 7 Functions ............................................................................................................................................... Page 8 Geometry .............................................................................................................................................. Page 9 Statistics and Probability ....................................................................................................................... Page 9
Essential Learning Standards
Particular standards/objectives/indicators that a school/district defines as critical for student learning. In fact, they are so critical that students will receive intervention if they are not learned. Essentials are chosen because they 1. have endurance, 2. have leverage, and 3. are important for future learning.
Math .............................................................................................................................................................. Page 10
Curriculum Resources
The materials teachers use to plan, prepare, and deliver instruction, including materials students use to learn about the subject. Such materials include texts, textbooks, tasks, tools, and media. Sometimes organized into a comprehensive program format, they often provide the standards, units, pacing guides, assessments, supplemental resources, interventions, and student materials for a course.
5 Strands of Mathematical Proficiency ......................................................................................................... Page 12 USBE Core Content Guides............................................................................................................................ Page 13
Pacing Guide
The order and timeline of the instruction of standards, objectives, indicators, and Essentials over the span of a course (semester or year).
Math Pacing Guide ......................................................................................................................................... Page 66 Detailed Pacing Guide ................................................................................................................................... Page 67
Units
A plan for several weeks of instruction, usually based on a theme, that includes individual lesson plans. Units often also include: Standards, learning targets/goals, skills, formative and summative assessment, student materials, essential questions, big ideas, vocabulary, questions, and instructional methods.
Understanding By Design .............................................................................................................................. Page 72 Math .............................................................................................................................................................. Page
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Assessment Standards
A set of criteria to guide the assessment of student learning in a course that is based on Standards/Essentials of the course; this might include formative assessment practices, summative assessments/practices, common assessment plans, feedback practices, and a schedule for testing.
SAGE ...................................................................................................................................................... Page 73 Ethics ..................................................................................................................................................... Page 74
Intervention Standards
A set of criteria to guide teachers to provide additional instruction to students who did not master the content in Tier 1 instruction. This might include: commercial intervention programs, teacher-developed intervention materials, diagnostic testing, RTI/MTSS processes, and a list of essential knowledge/skills that will prompt intervention if the student does not demonstrate mastery.
RTI ......................................................................................................................................................... Page 77 MTSS...................................................................................................................................................... Page 79
Supplemental Resources
Instructional materials, beyond the main curricular materials, used to strategically fill gaps/weaknesses of the core program materials.
Provo Way Instructional Model ............................................................................................................ Page 81 Mathematics ......................................................................................................................................... Page 84
Evidence-based Pedagogical Practices
A list of teaching strategies that are supported by adequate, empirical research as being highly effective.
John Hattie ............................................................................................................................................ Page 85
Glossary
Terms and acronyms used in this document ........................................................................................ Page 86
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Mathematics
Standards for Mathematical Practice
The Standards for Mathematical Practice in Eighth Grade describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes.
1 Make sense of problems and persevere in solving them.
Explain the meaning of a problem and look for entry points to its solution. Analyze givens, constraints, relationships, and goals. Make conjectures about the form and meaning of the solution, plan a solution pathway, and continually monitor progress asking, “Does this make sense?” Consider analogous problems, make connections between multiple representations, identify the correspondence between different approaches, look for trends, and transform algebraic expressions to highlight meaningful mathematics. Check answers to problems using a different method.
2 Reason abstractly and quantitatively
Make sense of the quantities and their relationships in problem situations. Translate between context and algebraic representations by contextualizing and decontextualizing quantitative relationships. This includes the ability to decontextualize a given situation, representing it algebraically and manipulating symbols fluently as well as the ability to contextualize algebraic representations to make sense of the problem.
3 Construct viable arguments and critique the reasoning of others
Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Make conjectures and build a logical progression of statements to explore the truth of their conjectures. Justify conclusions and communicate them to others. Respond to the arguments of others by listening, asking clarifying questions, and critiquing the reasoning of others.
4 Model with mathematics
Apply mathematics to solve problems arising in everyday life, society, and the workplace. Make assumptions and approximations, identifying important quantities to construct a mathematical model. Routinely interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
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5 Use appropriate tools strategically
Consider the available tools and be sufficiently familiar with them to make sound decisions about when each tool might be helpful, recognizing both the insight to be gained as well as the limitations. Identify relevant external mathematical resources and use them to pose or solve problems. Use tools to explore and deepen their understanding of concepts.
6 Attend to precision
Communicate precisely to others. Use explicit definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. Specify units of measure and label axes to clarify the correspondence with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.
7 Look for and make use of structure
Look closely at mathematical relationships to identify the underlying structure by recognizing a simple structure within a more complicated structure. See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot
be more than 5 for any real numbers x and y.
8 Look for and express regularity in repeated reasoning
Notice if reasoning is repeated, and look for both generalizations and shortcuts. Evaluate the reasonableness of intermediate results by maintaining oversight of the process while attending to the details.
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Grade 8 Mathematics
In Grade 8, instructional time should focus on three critical areas:
1. Formulatingandreasoningaboutexpressionsandequations;
2. Grasping the concept of functions and using functions to describe
quantitative relationships; and
3. Analyze two- and three-dimensional space and figures using distance,
angles, similarity and congruence, and understanding and applying the Pythagorean Theorem.
The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers.
Expressions and Equations
Work with radical and integer exponents. Understand the connections between proportional relationships, lines, and linear relationships. Analyze and solve linear equations and inequalities and pairs of simultaneous linear equations.
Functions
Define, evaluate, and compare functions. Use
functions to model relationships between quantities.
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software. Understand and apply the Pythagorean Theorem and its converse. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres
Statistics and Probability
Investigate patterns of association in bivariate data.
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
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The number system NS
1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers, show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
3. Understand how to perform operations and simplify radicals with emphasis on square roots.
Expressions and equations EE
Work with radicals and integer exponents
1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 x 3-5 = 3-3 = 1/33 = 1/27.
2. Use square root and cube root symbols to represent solutions to equations of the form x2=p and x3=p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.
4. Perform operations with numbers expressed in scientific notation including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
5. Graph proportional relationships, interpreting the unit rates as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for al ine through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
Know that there are numbers that are not rational, and approximate them by
rational numbers
Understand the connections between proportional relationships, lines, and linear
equations
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Analyze and solve linear equations and pairs of simultaneous linear equations
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many
solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different number).
b. Solve single-variable linear equations and inequalities with rational number coefficients, including equations and inequalities whose solutions require expanding expressions using the distributive property and collecting like terms.
c. Solve single-variable absolute value equations.
8. Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables graphically, approximating when solutions are not integers and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two variables graphically.
Functions F
Define, evaluate, and compare functions
1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output (Function notation is not required in grade 8).
2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables or by verbal descriptions).
3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
Use functions to model relationships between quantities
4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship of from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or table of values.
5. Describe qualitatively the function relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
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Geometry G
1. Verify experimentally the properties of rotations, reflections and translations.
a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure.
c. Parallel lines are taken to parallel lines.
2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
3. Observe the orientation of the plane in preserved in rotations and translations, but not with reflections. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
5. Use informal arguments to establish facts about the angle sum & exterior angle of triangles, about angles created by parallel lines are cut by a transversal, and the angle- angle criterion for similarity of triangles.
Understand and apply the Pythagorean Theorem
6. Explore and explain proofs of the Pythagorean and its converse.
7. Apply the Pythagorean theorem to determine unknown side lengths in right triangles in
real-world and mathematical problems in two- and three-dimensions.
8. Apply the Pythagorean theorem to find the distance between two points in a coordinate
system.
9. Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.
Statistics and Probability SP
Investigate patterns of association in bivariate data
1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.
Understand congruence and similarity using physical models, transparencies, or
geometry software
Solve real-world and mathematical problems involving volume of cylinders, cones,
and spheres
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Mathematics Essential Learning Standards
Essential Skills from Standards for Mathematical Content
Grade 8
In grade 8 instructional time should focus on three critical areas:
(1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; and (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
The Number System (8NS)
A. Know that there are numbers that are not rational, and approximate them by rational numbers.
8.NS.1 Know that numbers that are not rational are called irrational. Understand
informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
8.NS.3 Understand how to perform operations and simplify radicals with emphasis on square roots.
Expressions and Equations (8EE)
A. Work with radicals and integer exponents
8.EE.1 Know and apply the properties of integer exponents to generate equivalent
numerical expressions
B. Understand the connections between proportional relationships, lines, and linear equations.
8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the
graph. Compare two different proportional relationships represented in different
ways
C. Analyze and solve linear equations and pairs of simultaneous linear equations Solve linear equations in one variable
Analyze and solve pairs of simultaneous linear equations
Functions (8F)
A. Define, evaluate, and compare functions
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line
B. Use functions to model relationships between quantities
8.EE.7 8.EE.8
8.F.2 8.F.3
8.F.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values
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Geometry (8G)
A. Understand congruence and similarity using physical models, transparencies, or geometric software.
8.G.5 Use informal arguments to establish facts about the angle sum and exterior
angle of triangles, about the angles created when parallel lines are cut by a
transversal, and the angle-angle criterion for similarity of triangles.
B. Understand and apply the Pythagorean Theorem
8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right
triangles in real-world and mathematical problems in two and three dimensions. 8.G.8
C. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use
them to solve real-world and mathematical problems.
Statistics and Probability (8SP)
A. Investigate patterns of association in bivariate data.
8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
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Curriculum Resources
5 Strands of Mathematical Proficiency from NRC’s Adding It Up
Conceptual understanding: Comprehension of mathematical concepts, operations, and relations
Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently and appropriately
Strategic competence: ability to formulate, represent, and solve mathematical problems
Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification
Productive disposition: habitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one's own efficacy
Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.
The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut
. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.
In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
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USBE Core Content Guides
Click on a standard to go to the USBE Core Content Guide
8NS1 Know irrational numbers and that every number has a decimal expression
8NS2 Use rational approximations of irrational numbers for comparison
8EE1 Know and apply integer exponents to generate equivalent numerical expressions
8EE2 Use square root and cube root symbols to represent solutions to equations
8EE3 Use numbers expressed in the form of a single digit times an integer power of 10
8EE4 Perform operations with numbers expressed in scientific notation
8EE5 Graph proportional relationships, interpreting the unit rates as the slope of the graph
8EE6 Use similar triangles to explain why the slope is the same between any two points
8EE7 Solve linear equations in one variable
8EE8 Analyze and solve pairs of simultaneous linear equations
8F1 Understand that a function is a rule that assigns to each input exactly one output
8F2 Compare properties of two functions each represented in a different way
8F3 Interpret the equation y = mx + b as defining a linear function
8F4 Construct a function to model a linear relationship between two quantities
8F5 Describe qualitatively the function relationship between 2 quantities by their graphs
8G1 Verify experimentally the properties of rotations, reflections and translations
8G2 Understand that a two-dimensional figure is congruent to another
8G3 Describe the effect of dialations, translations, rotations, and reflections on figures
8G4 Understand that a two-dimensional figure is similar to another
8G5 Use informal arguments to form facts about the angle sum & exterior angle of triangles
8G6 Explain a proof of the Pythagorean and its converse
8G7 Apply the Pythagorean theorem to determine unknown side lengths in right triangles
8G8 Apply Pythagorean theorem for the distance between 2 points in a coordinate system
8G9 Know the formulas for the volumes of cones, cylinders, and spheres
8SP1 Construct and interpret scatter plots for bivariate measurement data
8SP2 Know straight lines are widely used to model relationships of quantitative variables
8SP3 Use linear equation to solve problems in the context of bivariate measurement data
8SP4 Understand patterns can be seen in bivariate data by frequencies & relative frequencies
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The Number System 8NS1
Core Content
Standard 1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers, show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number
MASTERY Patterns of Reasoning:
Conceptual:
• Know that there are numbers that are not rational
• Know that numbers that are not rational are called irrational
• Understand informally that every number has a decimal expansion, for rational
numbers, show that the decimal expansion repeats eventually
Procedural:
• Convert a decimal expansion which repeats into a rational number Representational:
• Graph the approximate value of an irrational number on a number line Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the subsets of the real number system (natural numbers, whole numbers, integers, rational numbers)
Procedural:
• Convert rational numbers to decimals using long division (terminating and repeating) 7NS2d
Representational:
• Graph rational numbers on a number line
Academic Vocabulary and Notation
Decimal expansion, repeating decimals, terminating decimals, rational, irrational, square root, radical notation, pi
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Instructional Strategies Used Resources Used
• Use the Pythagorean Theorem with non-http://www.learnzillion.com/lessons/222-
perfect squares to introduce irrational
numbers 8G7
• Use the powers of ten technique:
Assessment Tasks Used
Skill-based Task:
Convert 0.352 (where the two repeats infinitely) to a fraction
Group the following numbers based on what you know about the number system:
5.3, 1.7 (where the seven repeats infinitely), square root of 10, 2, pi, 4.010010001. . .
compare-irrational-and-rational-numbers
http://www.learnzillion.com/lessons/223- convert-repeating-decimals-into-fractions
http://www.learnzillion.com/lessons/221- distinguish-between-rational-and-irrational- numbers
http://www.learnzillion.com/lessons/220- understand-and-apply-the-definition-of- irrational-numbers
http://www.learnzillion.com/lessons/219- understand-and-apply-the-definition-of- rational-numbers
Problem Task:
Suppose you have a fraction with a denominator of 7. What is the longest string of non-repeating digits that will occur in the decimal expansions of the number? (hint: Use the long division algorithm to show that for a denominator of n, there are only n possible remainders, 0 to n – 1)
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The Number System 8NS2
Core Content
Standard 2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., pi squared). For example, by truncating the decimal expansion of the square root of two, show that the square root of two is between 1 and 2, then between 1.4 and 1.5, explain how to continue on to get better approximations.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that irrational numbers can be found between rational numbers
• Understand that truncating decimal expansions of irrational numbers can be used as
approximate values for irrational numbers
Procedural:
• Compare and order irrational numbers
• Use approximations of irrational numbers to estimate the value of expressions Representational:
• Represent approximate values of irrational numbers on a number line diagram Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand that two rational numbers can be placed on a number line with infinite values between them
• Understand what rational numbers are Procedural:
• Compare rational numbers
• Approximate irrational numbers as fractions or decimals Representational:
• Represent values of rational numbers on a number line diagram
Academic Vocabulary and Notation
Rational, irrational, decimal expansion, square root, radical notation, pi, truncating, rounding Instructional Strategies Used Resources Used
• Construct the Wheel of
Theodorus to create physical http://www.learnzillion.com/lessons/224-place- lengths of the square roots of the nonperfect-square-roots-between-2-integers counting numbers. Transfer
those lengths onto a number line
• Find increasingly accurate estimations for square roots of numbers by guess-and-check with a calculator
Cluster Title: Know that there are numbers that are not rational, and approximate them by rational numbers
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Assessment Tasks Used
Skill-based Task: Problem Task:
Place the following numbers on a number Explain when each approximation of pi (3.14, 3, and line: 22/7) is useful in calculation the circumference of a A5.3, 1.7 where the seven repeats, square circle. Compare the answers you would get with root of 10, 2 and pi halves. each approximation. (Extension: research how
Find the two integers between which the square root of 42 lies.
different cultures have approximated pi.)
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Expressions and Equations 8EE1
Core Content
Cluster Title: Work with radicals and integer exponents
Standard 1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example:
32 x3-5 =3-3 =1/33 =1/27
MASTERY Patterns of Reasoning:
Conceptual:
• Know the properties of integer exponents. Procedural:
• Apply the properties of integer exponents to simplify and evaluate numerical expressions.
Representational:
• Model the properties of integer exponents as multiple multiplications. Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand exponents as repeated multiplication. (6.EE.1) Procedural:
• Compute fluently with integers (add, subtract, and multiply). Representational:
• Model multiplication of integers
Academic Vocabulary and Notation
exponent, base, power, integer, exponential notation such as: or n^x, negative exponential notation or n ^ -x
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Instructional Strategies Used
• Use repeated multiplication and division to informally derive the exponent rules
• Have students examine equivalent numerical expressions with exponents
Resources Used
Birch, David. The King’s Chessboard http://www.learnzillion.com/lessons/178-apply-
exponents-to-negative-bases
http://www.learnzillion.com/lessons/180-convert- from-standard-to-scientific-notation
http://www.learnzillion.com/lessons/174-divide- exponential-expressions-part-1
http://www.learnzillion.com/lessons/175-divide- exponential-expressions-part-2
http://www.learnzillion.com/lessons/179-evaluate- expressions-involving-exponents
http://www.learnzillion.com/lessons/171-evaluate- expressions-using-the-definition-of-exponent-and- base
http://www.learnzillion.com/lessons/181-multiply- numbers-expressed-in-scientific-notation
http://www.learnzillion.com/lessons/173-raise- exponential-expressions-to-powers
http://www.learnzillion.com/lessons/172-simplify- exponential-multiplication-expressions
http://www.learnzillion.com/lessons/176-simplify- expressions-with-0-and-negative-exponents-part-1
http://www.learnzillion.com/lessons/177-simplify- expressions-with-0-and-negative-exponents-part-2
Problem Task:
Explain why 35 x 32 = 37 and not 97.
Write three expressions equivalent to 32 x 92
Assessment Tasks Used
Skill-based Task:
Simplify:
57 (((2)2)2)2 53
-50
2-2 x 4
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Expressions and Equations 8EE2
Core Content
Cluster Title: Work with radicals and integer exponents.
Standard 2. Use square root and cube root symbols to represent solutions of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that all non-perfect square roots and cube roots are irrational Procedural:
• Evaluate the square roots of small perfect squares and cube roots of small perfect cubes Representational:
• Represent the solutions to equations using square root and cube root symbols Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand definition of inverse operations
Procedural:
• Calculate area with square units and volume with cubed units
• Solve equations with inverse operations
Representational:
• Represent area and volume with appropriate units
• Model inverse operations
Academic Vocabulary and Notation
Square, square root, cube, cube root, radical notation, radical
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Instructional Strategies Used
• Use the geometric representation of square area and cube volumes and their relation to the side length
• Use the idea of inverse operations to introduce the concept of roots
Resources Used
http://www.learnzillion.com/lessons/183-find-square- roots-of-perfect-squares
http://www.learnzillion.com/lessons/187-find-the- side-lengths-of-a-square-given-its-area
http://www.learnzillion.com/lessons/188-identify- perfect-cubes-and-find-cube-roots
http://www.learnzillion.com/lessons/182-identify- perfect-squares-and-find-square-roots
http://www.learnzillion.com/lessons/184-simplify- expressions-involving-square-roots
http://www.learnzillion.com/lessons/189-solve- equations-with-cubes-and-cube-roots
http://www.learnzillion.com/lessons/186-solve- equations-with-squares-and-square-roots
http://www.learnzillion.com/lessons/185-solve- problems-with-inverse-of-squares-and-square-roots
Problem Task:
Is the square root of a number always smaller than the number itself? Explain
Assessment Tasks Used
Skill-based Task:
If a square has an area of 9/16 square inches, what is the length of a side?
If a cube has a volume of 0.125 cubic meters, what are the dimensions of the cube?
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Expressions and Equations 8EE3
Core Content
Cluster Title: Work with radicals and integer exponents
Standard 3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than another. For example, estimate the population of the United States as and the population of the world as , and determine that the world population is more than 20 times larger. MASTERY Patterns of Reasoning:
Conceptual:
• Understand the approximate value of very large and very small numbers can be expressed as a product of a single digit and a power of 10
• Understand that a number expressed as the product of a single digit and a power of ten can be used as a scale factor for comparison
Procedural:
• Estimate numbers as a product of a single digit and a power of ten
• Compare numbers expressed as a product of a single digit and a power of ten by a scale
factor
Representational:
• Model the comparison of numbers expressed as a product of a single digit and a power of ten
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand properties of integer exponents
• Understand powers of ten
• Understand place value
Procedural:
• Use powers of ten.
• Use place value
• Estimate and round numbers
Representational:
• Model properties of integer exponents
• Use models to show place value Academic Vocabulary and Notation
Powers of ten, estimate, exponential notation, ^
22


Instructional Strategies Used
Use large and small real-life numbers (e.g. national debt).
Use base 10 blocks
Write numbers as repeated multiplication or as the value of 10 (e.g. )
Assessment Tasks Used
Skill-based Task:
Resources Used
www.Powersof10.com/film Eames Office, 2010
• The mass of earth is The mass of the moon is How many times bigger is the mass of the earth than the mass of the moon?
• A proton has mass and an electron has mass How many times smaller is the electron than the proton?
Problem Task:
Provide two (2) original (not teacher-given) real- life situations that could be illustrated using powers of ten, one that describes a very small number and one that describes a very larger number. Estimate how much larger one is than the other.
23


Expressions and Equations 8EE4
Core Content
Cluster Title: Work with radicals and integer exponents
Standard 4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology
MASTERY Patterns of Reasoning:
Conceptual:
• Recognize appropriate units for representing very large and very small quantities Procedural:
• Add, subtract, multiply and divide with numbers expressed in scientific notation and decimal notation
• Convert between decimal notation and scientific notation
• Interpret numbers expressed in scientific notation, including numbers generated by
technology
Representational:
• Represent very large and very small quantities in scientific notation and use appropriate units
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand powers of ten and place value Procedural:
• Use rules of exponents Representational:
• Model place value with manipulatives
Academic Vocabulary and Notation
Scientific notation, decimal notation, power of ten, units of measure Instructional Strategies Used Resources Used
• Use examples found in science to create authentic reasons to use scientific notation
24


Assessment Tasks Used
Skill-based Task:
Evaluate and express your answers in scientific notation
3 x 105 + 5.54 x 107
4.2 x 10-2 – 7.4 x 102
(3 x 108)(500) 30 ‘
1.5 x 10-4
Multiply 345,328,004 x 234 on your calculator and write the answer in scientific notation
Problem Task:
Express your age at your last birthday in each of the following units: year, months, days, hours, minutes, and seconds. Which values would be useful to write in scientific notation? Justify your reasoning
Compare your age to that of the 4.5 billion year old earth
25


Expressions and Equations 8EE5
Core Content
Standard 5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand the connections between proportional relationships, lines and linear equations
• Understand that the unit rate is the slope of a linear graph Procedural:
• Recognize unit rate as slope and interpret the meaning of the slope in context
• Recognize that proportional relationships include the point (0,0)
• Compare different representations of two proportional relationships represented as
contextual situations, graphs, or equations
Representational:
• Represent proportional relationships graphically when given a table, equation or contextual situation
• Model proportional relationships with manipulatives Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand unit rates
Procedural:
• Use an equation to create a table
• Calculate unit rates
Representational:
• Represent values by plotting them on the coordinate axes
Academic Vocabulary and Notation
Slope, unit rate, rate of change, m
Cluster Title: Understand the connections between proportional relationships, lines and linear equations
26


Instructional Strategies Used Resources Used
www.Illuminations.nctm.org/ActrivityDetail.aspx?ID=124. • Categorize linear relationships NCTM. (9-12 Activity: Tow terrains)
represented in multiple ways as either proportional or not proportional
• Plot relationships generated from real-life proportional examples (e.g., shopping) and interpret the slope in context of the situation
Assessment Tasks Used
Skill-based Task:
This is a graph of Susie’s trip to John’s cabin. John made the same trip in 4 hours. Compare their rates. Who traveled at a faster rate? How do you know?
Problem Task:
Give examples of relationships that are proportional and relationships that are linear, but not proportional.
27


Expressions and Equations 8EE6
Core Content
Standard 6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. MASTERY Patterns of Reasoning:
Conceptual:
• Understand why the slope is the same between any two distinct points on a non-vertical line
Procedural:
• Explain why the slope is the same between any two distinct points on a non-vertical line using similar right triangles
• Write an equation in the form y = mx + b from a graph of a line on the coordinate plane
• Determine the slope of a line as the ratio of the leg lengths of similar right triangles Representational:
• Represent similar right triangles on a coordinate plane to show equivalent slopes Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand triangle similarity requires proportionality Procedural:
• Recognize similar triangles Representational:
• Model similar triangles on a coordinate plane
Academic Vocabulary and Notation
Similar triangles, m, b, linear, right triangle, origin, rise, run, slope, y-intercept
Cluster Title: Understand the connections between proportional relationships, lines and linear equations
Instructional Strategies Used
• Have students draw many right triangles with the hypotenuse on the line and compare the ratio of the leg lengths
• Discuss the value of choosing easy to read points when determining slope
• Relate negative slopes to the change in y as x increases
Resources Used
www.illuminations.nctm.org/ActivitiesDetail.aspx?ID=144
NCTM (3-8 Activity: chairs)
28


Assessment Tasks Used
Skill-based Task:
Points A, D, B and E are collinear. Show that segment AB and segment DE have the same slope
Problem Task:
How is it possible to have similar triangles that do not yield the same slope?
29


Expressions and Equations 8EE7
Core Content
Cluster Title: Analyze and solve linear equations and pairs of simultaneous linear equations
Standard 7. Solve linear equations in one variable
a) Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
b) Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that linear equations in one variable can have a single solution, infinitely many solutions or no solutions
• Understand how to expand expressions using the distributive property and collecting like terms
Procedural:
• Identify and provide examples of equations that have one solution, infinitely many solutions, or no solutions
• Solve multistep linear equations with rational coefficients and variables on both sides of the equation
Representational:
• Model examples of equations that have a single solution, infinitely many solutions, or no solutions
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand properties of algebra necessary for simplifying algebraic expressions Procedural:
• Solve one- and two-step equations (7EE4a)
• Use properties of algebra to simplify algebraic expressions Representational:
• Use manipulatives to model the solving of one-step and two-step equations Academic Vocabulary and Notation
Solve, variable, order of operations, solution, like terms, distributive property
30


Instructional Strategies Used Resources Used
http://www.learnzillion.com/lessons/148-use-inverse- • Build on the equations solved in ops-to-solve-equations-of-form-axbc
seventh grade and move toward
increased fluency and proceduralhttp://www.learnzillion.com/lessons/152-solve-
skill in solving more complex
linear equations
• Examine solutions in the context
of the original equation
• Consider teaching unique
solutions, no solutions and infinitely many solution with 8EE8
Assessment Tasks Used
Skill-based Task:
Solve the following equations and identify the number of solutions
3(x + 7) = 10
2(x – 5) = 1⁄2 (4x + 6) 2(x + 3) = 2x + 6
equations-with-infinite-solutions
http://www.learnzillion.com/lessons/151-solve- equations-with-no-solutions
http://www.learnzillion.com/lessons/150-solve- equations-with-variables-on-both-sides
http://www.learnzillion.com/lessons/147-use-inverse- operations-to-solve-equations
http://www.learnzillion.com/lessons/149-solve- multistep-equations-using-the-distributive-property
Problem Task:
Create equations that would result in one solution, no solutions, or infinitely many solutions. What is it about the structure of the original equation that reveals the number of solutions?
31


Expressions and Equations 8EE8
Core Content
Cluster Title: Analyze and solve linear equations and pairs of simultaneous linear equations
Standard 8. Analyze and solve pairs of simultaneous linear equations
a. Understand that solutions to a system of two linear equations in two variables
correspond to points of intersection of their graphs, because points of intersection satisfy
both equations simultaneously
b. Solve systems of two linear equations in two variables algebraically, and estimate
solutions by graphing the equations. Solve simple cases by inspections. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously
Procedural:
• Solve systems of two linear equations in two variables algebraically
• Estimate solutions by graphing the equations
• Solve simple cases by inspections
• Solve real-world and mathematical problems leading to two linear equations in two
variables
Representational:
• Model solutions of equations that have a single solution, infinitely many solutions, or no solutions
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand what a solution to a linear equation is
Procedural:
• Solve a one variable equation
• Solve for a specified variable in an equation
Representational:
• Represent linear equations graphically
Academic Vocabulary and Notation
Elimination, substitution, solution, intersection, solve, system of linear equations
32


Instructional Strategies Used
• Compare cell phone plans, bike rentals rates, or repair costs using equations
• Explore cases where on solution strategy is more efficient than another
Resources Used
http://www.learnzillion.com/lessons/154-solve- systems-of-equations-graphing-1
http://www.learnzillion.com/lessons/156-solve- systems-of-equations-graphing-2
http://www.learnzillion.com/lessons/159-understand- elimination-method-when-solving-systems
http://www.learnzillion.com/lessons/160-solve- systems-of-equations-using-elimination-2
http://www.learnzillion.com/lessons/161-solve- systems-of-equations-using-elimination-3
http://www.learnzillion.com/lessons/162-solve- systems-of-equations-using-elimination-4
http://www.learnzillion.com/lessons/157-solve- systems-of-equations-using-substitution-1
http://www.learnzillion.com/lessons/158-solve- systems-of-equations-using-substitution-2
http://www.learnzillion.com/lessons/164-determine- whether-2-lines-intersect
http://www.learnzillion.com/lessons/153-determine- whether-a-point-is-a-solution-to-a-linear-equation
http://www.learnzillion.com/lessons/155-estimate- solutions-of-equations-graphing
http://www.learnzillion.com/lessons/163-identify- systems-with-no-solutions
Problem Task:
You have been hired by a cell phone company to create two rate plans for customers, one that benefits customers with low usage and that benefits customers with high usage. At 500 minutes, both plans should be within $5 of each other. Design a presentation showing two plans that will meet these requirements, including graphs and equations
Assessment Tasks Used
Skill-based Task:
Solve the system of equations
2x + 3y = 4 -x + 4y = -13
33


Functions 8F1
Core Content
Cluster Title: Define, evaluate, and compare functions
Standard 1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output
MASTERY Patterns of Reasoning:
Conceptual:
Understand that a function is a rule that assigns to each input exactly one output
Procedural:
• Recognize a graph of a function as the set of ordered pairs consist Representational:
• Model solutions of equations that have a single solution, infinitely many solutions, or no solutions
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand what a solution to a linear equation is Procedural:
• Evaluate expressions for a given value Representational:
• Graph ordered pairs on the coordinate plane Academic Vocabulary and Notation
Function, input, output, dependent, independent Instructional Strategies Used Resources Used
www.illuminations.nctm.org/ActivityDetail.aspx?ID=215
• Explore functions that ariseNCTM (9-12 Activity: function matching)
from real-life relationships
where one variable Lloyd, Gwendolyn, et al. Developing essential understanding determines a unique value of expressions, equations, and functions grades 6-8 NCTM, of another 2011
• Use a variety of representations to have students identify functions and relations that are not functions
34


Assessment Tasks Used
Skill-based Task:
Does the set of ordered pairs (2, 5), (3, 5), (4, 6), (2, 8), and (6, 7) represent a function?
Problem Task:
Could the set of ordered pairs, (2, 5), (3, 5), (4, 6), (2, 8), and (6, 7) describe the number of seconds since you left home and the number of meters you’ve walked? Is this a function? Justify your answers.
Does the set of students in the classroom and their birthdays represent a function? Justify your answer.
Find three examples of relationships in the real world that can be represented by functions and three relationships that are not functions.
35


Functions 8F2
Core Content
Cluster Title: Define, evaluate, and compare functions
Standard 2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change
MASTERY Patterns of Reasoning:
Conceptual:
Recognize comparable properties such as slopes, y-intercepts and values of two functions
Procedural:
• Compare two linear functions each represented a different way and describe similarities and differences in slopes, y-intercepts, and values
Representational:
• Model the comparison of two linear functions each represented a different way
• Represent two linear functions in multiple ways Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know what a slope and y-intercept are
• Know the difference between linear and non-linear
Procedural:
• Determine slopes and y-intercepts of linear equations Representational:
• Graphically represent the slopes and y-intercepts of linear equations
Academic Vocabulary and Notation
Slope, intercept, rate of change, function, linear, non-linear
Instructional Strategies Used Resources Used
• Given one representation of a
function, create the others Lloyd, Gwendolyn, et al. Developing essential
• Put students in small understanding of expressions, equations, and functions groups. Give groups scenarios grades 6-8 NCTM, 2011
and ask each group to create a
different representation of the http://www.learnzillion.com/lessons/290-compare- scenario (table, equation, graph) distancetime-graphs-with-distancetime-equations
• Identify attributes (slope, y- intercept, values) of a function in its equation, graph or a table
36


Assessment Tasks Used
Skill-based Task:
Is y = 2(x + 5) the same as the function described as “twice a quantity plus 5”?
Problem Task:
Billy argues that the equation y = 4x + 5 is equivalent to the equation of the line that goes through (2, 6) and (3, 10). How did he arrive at this conclusion? Is he correct? Justify your answer.
37


Functions 8F3
Core Content
Cluster Title: Define, evaluate, and compare functions
Standard 3. Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; given examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4) and (3,9), which are not on a straight line
MASTERY Patterns of Reasoning:
Conceptual:
Recognize functions written in the form y = mx + b are linear and that every linear
function can be written in the form y = mx + b Procedural:
• Distinguish between linear and non-linear functions given their algebraic expression, a table or a graph
Representational:
• Represent linear and non-linear functions graphically Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand linear slope as a constant rate of change Procedural:
• Generate ordered pairs from an equation Representational:
• Graphically represent ordered pairs obtained from an equation Academic Vocabulary and Notation
Collinear, linear, nonlinear
Instructional Strategies Used
• Examine constant and non-constant rates of change in tables of values
• Explore growing patterns generated from a variety of contexts to explore linear and nonlinear relationships
Assessment Tasks Used
Skill-based Task:
Determine which of the following equations are linear
y=x2 +5x+6
y = x3
y=1/x y=x/2 y1357
y=x(2+x) y = 7x + 6
Resources Used
Problem Task:
Hermione argues that the table below represents a linear function. Is she correct? How can you know?
x
2
4
8
16
38


Functions 8F4
Core Content
Cluster Title: Use functions to model relationships between quantities
Standard 4. Construct a function to model a linear relationship between two
quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a
graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values
MASTERY Patterns of Reasoning:
Conceptual:
• Know how to determine the initial value and rate of change given two points, a graph, a table of values, a geometric representation, or a story problem
Procedural:
• Determine the initial value and rate of change given two points, a graph, a table of values, a geometric representation, or a story problem
• Write the equation of a line given two points, a graph a table of values, a geometric representation, or a story problem (verbal description) of a linear relationship
Representational:
• Model relationships between quantities Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the meaning of slope and y-intercept Procedural:
• Write an equation as y = mx + b given a graph Representational:
• Graphically represent linear equations
Academic Vocabulary and Notation
Linear relationship, y-intercept, slope
Instructional Strategies Used
Use real world application to generate a table of values. Use the table to construct a function that models the relationship
• Connect to other standards in the Expressions and equation domain
Resources Used
Illuminations.nctm.org NCTM
http://www.learnzillion.com/lessons/288-construct- linear-functions-from-a-graph
http://www.learnzillion.com/lessons/289-construct- linear-functions-from-a-situation
http://www.learnzillion.com/lessons/287-construct- linear-functions-from-tables
39


Assessment Tasks Used
Skill-based Task:
Find the equation of the line that goes through (3, 5) and (-5, 7)
Problem Task:
The student council is planning a ski trip to Sundance. There is a $200 deposit for the lodge and the tickets will cost $70 per student. Construct a function, build a table, and graph the data showing how much it will cost for the students’ trip
Wally created the table below for a function he knows to be linear. He thinks something must be wrong with his table because he can’t find the original function from the table. Find the error and the original function. Explain your strategy for finding the error
17.8 30.6 43.4 56.2 66 81.8 94.6 107.4
3.2
6.4
9.6
12.8
16
19.2
22.4
25.6
40


Functions 8F5
Core Content
Cluster Title: Use functions to model relationships between quantities
Standard 5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally
MASTERY Patterns of Reasoning:
Conceptual:
• Understand the qualitative nature of the functional relationship between two quantities analyzed in a graph
Procedural:
• Sketch a graph that exhibits the qualitative features of a function that has been described verbally
• Describe attributes of a function by analyzing a graph Representational:
• Model relationships between quantities Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the basic graphing conventions Procedural:
• Graph linear equations Representational:
• Graphically represent linear equations
Academic Vocabulary and Notation
Increasing and decreasing rates of change, linear, nonlinear, initial value Instructional Strategies Used Resources Used
Friel, Susan, et al. Navigating through
• Tell a story based on a given graph algebra. NCTM, 2001
• Given a story, draw a graph
41


Assessment Tasks Used
Skill-based Task:
When is the graph increasing? When is the graph decreasing?
Problem Task:
The following is a graph of the heart rate of a man running on a treadmill. When is his heart rate changing at the greatest rate? What is happening when the graph is horizontal? Does the decreasing graph show a constant rate of change?
42


Geometry 8G1
Core Content
Standard 1. Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length
b. Angles are taken to angles of the same measure
c. Parallel lines are taken to parallel lines
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that congruence of line segments and angles is maintained through rotation, reflection, and translation
• Understand that lines remain lines through rotation, reflection, and translation
• Understand that when parallel lines are rotated, reflected, or translated, each in the
same way, they remain parallel lines
Procedural:
• Verify through experience that congruence of line segments and angles is maintained through rotation, reflection, and translation
• Verify through experience that lines remain lines through rotation, reflection, and translation
• Verify through experience that when parallel lines are rotated, reflected, or translated, each in the same way, they remain parallel lines
Representational:
• Model congruence of line segments and angles is maintained through rotation, reflection, and translation
• Model lines remain lines through rotation, reflection, and translation
• Model when parallel lines are rotated, reflected, or translated, each in the same way,
they remain parallel lines Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know definitions and properties of angles, segments, lines and parallel lines Procedural:
• Measure angles and line segments Representational:
• Represent angles, lines, line segments and parallel lines
Academic Vocabulary and Notation
Line, angle, line segment, parallel lines, rigid motion, congruent, center of rotation, line of reflection, rotation, reflection, translation, transformation, symbols for parallel, congruent, angle and segment
Cluster Title: Understand congruence and similarity using physical models, transparencies, or geometry software
43


Instructional Strategies Used
• Use dynamic geometry software or excel to explore properties of rotations, reflections, and translations
• Use a coordinate grid and apply rules such as (-x, y) or (x, y + 7) to the coordinates of a given figure. Compare the resulting image to the original
Assessment Tasks Used
Skill-based Task:
Verify that a triangle when rotated remains a triangle
Resources Used
www.geogebra.com www.Illuminations.nctm.org/LessonDetail.aspx?ID=U139
NCTM (9 – 12 Lessons: Symmetries II)
Problem Task:
Create a tessellation using rotations, reflections, and translations
44


Geometry 8G2
Core Content
Standard 2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that congruence of two figures is maintained while undergoing rigid transformations
Procedural:
• Describe the transformation of a figure as a rotation, reflection, translation or a combination of transformations
Representational:
• Model congruence of two, two-dimensional figures as one experiences multiple transformations
• Model the sequence of transformations that exhibits the congruence between two figures that were originally congruent
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know definitions and rotations, reflections, and translations Procedural:
• Identify rotations, reflections and translations of two-dimensional objects Representational:
• Model rotations, reflections and translations of two-dimensional objects Academic Vocabulary and Notation
rotation, reflection, translation, congruent, center of rotation, line of reflection, angle of rotation
Instructional Strategies Used Resources Used
• Use dynamic geometry www.geogebra.com
software to explore rotations,
reflections, and translations of www.Illuminations.nctm.org/LessonDetail.aspx?ID=U139
Cluster Title: Understand congruence and similarity using physical models, transparencies, or geometry software
two-dimensional figures
• Use digital photographs with
at least two congruent shapes and discuss the needed transformations to map one onto the other
NCTM (9 – 12 Lessons: Symmetries II)
45


Assessment Tasks Used
Skill-based Task:
Triangle x was transformed to
x’. Describe the sequence of transformations that was used to show that triangle x is congruent to triangle x’.
Problem Task:
Find at least two different ways to describe the transformations that map the first figure onto the second
46


Geometry 8G3
Core Content
Standard 3. Describe the effect of dilations, translations, rotations, and reflections on two- dimensional figures using coordinates
MASTERY Patterns of Reasoning:
Conceptual:
• Understand how to dilate, translate, rotate, and reflect two-dimensional figures on the coordinate plane
Procedural:
• Describe the effects of dilations, reflections, translations and rotations using coordinate notation
• Given an image and its transformed image, use coordinate notation to describe the transformation
Representational:
• Model transformations on a coordinate plane Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know coordinate notation Procedural:
• Plot points on a coordinate plane
• Identify points on a coordinate plane Representational:
• Represent location on a coordinate plane
Academic Vocabulary and Notation
Transformation, rotation, reflection, translation, congruent, center of rotation, line of reflection, angle of rotation, coordinate, dilation, image, symbols for similarity and congruence, similar, congruent
Instructional Strategies Used
oft
Cluster Title: Understand congruence and similarity using physical models, transparencies, or
geometry s
• Have students take pictures of
transformations they see in the world
around them, then overlay the picture
with the coordinate plane and describe shodor. The transmographer the transformation using coordinate
notation
• Use a coordinate grid and apply rules such as (-x, y) or (x, y + 7) to the coordinates of a given figure. Compare the resulting image to the original
Resources Used
www.geogebra.com www.Shodor.org
47
w


Assessment Tasks Used
Skill-based Task: Problem Task:
Given a triangle with vertices at (5, 2), (-7, 8) Given an original shape and its image on a and (0, 4) find the new vertices of the triangle coordinate plane, determine the rule or rules after undergoing the transformation described that translated the original to the resulting as follows: image.
The vertices of triangle A are (1, 0), (1,1), (0, 0) and triangle A’ are (2, 1), (2, 2), (3, 1). Describe the series of transformations performed on triangle A that result in triangle A’
48


Geometry 8G4
Core Content
Standard 4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that any combination of transformations will result in similar figures Procedural:
• Describe the sequence of transformations needed to show how one figure is similar to another
Representational:
• Model dilations of figures by a given scale factor Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand proportions Procedural:
• Rotate, translate, reflect and dilate two-dimensional figures Representational:
• Represent rotations, reflections, translations, and dilations graphically Academic Vocabulary and Notation
Similar, similarity, dilation, rotation, reflection, translation, transformation
Cluster Title: Understand congruence and similarity using physical models, transparencies, or geometry software
Instructional Strategies Used
• Use patty paper transformations
• Use dynamic geometry software to make transformations and compare transformations
Assessment Tasks Used
Skill-based Task:
Which of the following transformationswillresultinasimilar figure?
Resources Used
Similar triangles lessons and materials www.geogebra.com
Problem Task:
List the sequence of transformations that verifies the similarityofthetwofiguresshownhere.
49


Geometry 8G5
Core Content
Standard 5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle- angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that the measure of an exterior angle of triangle is equal to the sum of the measures of the non-adjacent angles
• Know that the sum of the angles of a triangle equals 180 degrees.
• Recognize that if two triangles have two congruent angles, they are similar (A-A
similarity)
• Know what a transversal is and its properties in relation to parallel lines and pairs of
angles
Procedural:
• Determine the relationship between corresponding angles, alternate interior angles, alternate exterior angles, vertical angle pairs, and supplementary pairs when parallel lines are cut by a transversal
• Use transversals and their properties to argue three angles of a triangle create a line Representational:
• Model A-A similarity
• Model the sum of three angles of a triangle form a line Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand definition of similar figures Procedural:
• Measure angles Representational:
• Model adjacent angles
Academic Vocabulary and Notation
Corresponding angles, alternate interior angles, alternate exterior angles, consecutive interior angles, supplementary pairs, vertical pairs, transversal, adjacent, non-adjacent, exterior angle of a triangle, remote interior angles of a triangle
Instructional Strategies Used
Cluster Title: Understand congruence and similarity using physical models, transparencies, or geometry software
Use a series of transformations of a triangle to produce parallel lines and examine the properties of resultant angles and triangles
Resources Used
www.geogebra.com
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