Access M/J Grade 8 Pre-Algebra (#7812030)

September 2014Revised January 2017

Access Pre-Algebra  (#7812030)Course Number: 7812030

Course Section: Exceptional Student Education

Course Type: Core

Course Status : Course Approved

Grade Level(s) Version: 6,7,8

Course Path: Section: Exceptional Student Education > Grade Group: Middle/Junior High > Subject: Academics - Subject Areas >

Abbreviated Title: ACC M/J PRE-ALG

Course Length: Year (Y)

GENERAL NOTES

Access Courses: Access courses are intended only for students with a significant cognitive disability. Access courses are designed to provide students with access to the general curriculum. Access points reflect increasing levels of complexity and depth of knowledge aligned with grade-level expectations. The access points included in access courses are intentionally designed to foster high expectations for students with significant cognitive disabilities.

Access points in the subject areas of science, social studies, art, dance, physical education, theatre, and health provide tiered access to the general curriculum through three levels of access points (Participatory, Supported, and Independent). Access points in English language arts and mathematics do not contain these tiers, but contain Essential Understandings (or EUs). EUs consist of skills at varying levels of complexity and are a resource when planning for instruction.

RESOURSES:For information related to the resources imbedded in this document please click here. Additional resources may be found by clicking here

Yellow Highlights indicate they are standards on the FSAA Blueprint.

Course Standards

MAFS.8.EE.1.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3²×3-5=3-3=1/3³=1/27Related Access Points

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MAFS.8.EE.1.AP.1a: Use properties of integer exponents to produce equivalent expressions.

EUs

Concrete: Add and subtract integers (e.g., use manipulatives, a number

line or calculator to add 2 + -5). Use manipulatives to demonstrate what an exponent

represents (e.g., 8³ = 8 × 8 × 8). Produce the correct amount of base numbers to be

multiplied given a graphic organizer or template.Representation: Understand the following concepts, symbols, and vocabulary:

base number, exponent, integer. Select the correct expanded form of what an exponent

represents (e.g., 8³ = 8 × 8 × 8). Identify the number of times the base number will be

multiplied based on the exponent. Understand that a negative exponent will result in a fraction

with a numerator of 1 (for example, 5−2¿

¿ = 1/25).

Resources Element Card 8th: Click here

MAFS.8.EE.1.2: 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.Related Access Points

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MAFS.8.EE.1.AP.2a: Use appropriate tools to calculate square root and cube root.

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EUs

Concrete: Use manipulatives to make a square. The area of the square

is the perfect square. The length of each side is the square root.

Use manipulatives to make a cube. The volume of the cube is the perfect cube. The length of each side is the cube root.

Use a multiplication table to identify perfect squares.Representation: Identify the square root and cube root function button on a

calculator.Resources Element Card 8th: Click here

MAFS.8.EE.1.AP.2b: Find products when bases from -6 to 6 are squared and cubed, using a calculator.

EUs

Concrete: Use a calculator to determine the squares and cubes of

numbers ranging from -6 to 6. Use a calculator to determine the square roots of numbers

ranging from 0 to 36. Use a calculator to determine the cubed roots of numbers

ranging from -216 to 216.Representation: Identify the square and cube functions on a calculator. Identify the square root and cube root function button on a

calculator.Resources Element Card 8th: Click here

MAFS.8.EE.1.AP.2c: Identify perfect squares from 0 to100 by modeling them on graph paper or building with tiles.

EUs

Concrete: Use manipulatives to construct squares with side lengths up

to 10. Use graph paper to construct squares with side lengths up to

10.

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Representation: Draw a square and label the side lengths with measurements

up to 10.Resources Element Card 8th: Click hereMAFS.8.EE.1.AP.2d: Identify squares and cubes as perfect or non-perfect.

EUs

Concrete: Use a calculator to determine the squares and cubes of

numbers. Use a calculator to determine the square roots of numbers. Use a calculator to determine the cubed roots of numbers.Representation: Identify a whole number (perfect). Identify a decimal number (non-perfect).

Resources Element Card 8th: Click hereMAFS.8.EE.1.AP.2e: Recognize that non-perfect squares/cubes are irrational.

EUsConcrete: Use base ten manipulatives to divide numbers. Match characteristics of irrational and rational numbers.Representation: Identify the characteristics of an irrational number. Identify irrational decimal quotients Identify non-perfect square roots Identify non-perfect cube roots

Resources Element Card 8th: Click here

MAFS.8.EE.1.3: 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. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.Related Access Points

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MAFS.8.EE.1.AP.3a: Multiply single digits by the power of 10 using a calculator.

EUsConcrete: Use base ten blocks to multiply a single digit number by 10,

100, 1000, etc.Representation: Use a calculator to multiply a single digit number by the

powers of 10.Resources Element Card 8th: Click hereMAFS.8.EE.1.AP.3b: Identify the products of powers of 10 (through 108).

EUsConcrete: Identify a base ten bundle (e.g., tens, hundreds, thousands). Select the appropriate base ten bundle to represent the

number expressed in scientific notation.Representation: Understand the following concepts, symbols, and vocabulary:

scientific notation, base number, power, and exponent. Select the correct numeric representation. Identify the correct notation used when converting to

scientific notation. Identify the correct place value when converting from

scientific notation.

Resources Content Module Radicals and Exponents: Click here Element Card 8th: Click here

MAFS.8.EE.1.4: 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.Related Access Points

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MAFS.8.EE.1.AP.4a: Perform operations with numbers expressed in scientific notation using a calculator.

EUs

Concrete: Select the appropriate base ten bundle to represent the

number expressed in scientific notation. Match exponent to decimal number. Use base ten bundles to perform operations on numbers

expressed in scientific notation.Representation: Understand the following concepts, symbols, and vocabulary:

scientific notation, base number/digit term, exponent, positive and negative numbers.

Select the correct numeric representation for a given question (e.g., 5 × 10-9).

Identify the correct symbol used when converting scientific notation.

Use a calculator to perform operations on numbers expressed in scientific notation.Click here

Resources Element Card 8th: Click here

MAFS.8.EE.2.5: 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.

Remarks/Examples:Examples of Opportunities for In-Depth Focus When students work toward meeting this standard, they build on grades 6–7 work with proportions and position themselves for grade 8 work with functions and the equation of a line.

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MAFS.8.EE.2.AP.5a: Define rise/run (slope) for linear equations plotted on a coordinate plane.

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EUs

Concrete: Identify parts of a line graph. Identify the two coordinates of a point on a line graph. Count the distance up/down between two points on the

coordinate plane (rise). Count the distance to the right, between two points on the

coordinate plane (run). Use a template (rise/run) to determine the slope. Recognize a relationship between two points on a graph. Examine the values of the x variable or y variable to look for

a pattern.Representation: Understand the following concepts, vocabulary, and symbols:

coordinates, ordered pairs (x, y), intercept, grid, axis, point, proportion, line, slope.

Graph a series of coordinates on a graph. Identify given coordinates (x, y) as a point on a graph. Use subtraction to determine the change in y and the change

in x (m=y2-y1/x2-x1 ).

ResourcesContent Module Ratio and Proportions: Click here Curriculum Resource Guide Equations: Click here Element Card 8th: Click here

MAFS.8.EE.2.6: 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.Related Access Points

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MAFS.8.EE.2.AP.6a:Define y = mx by identifying the coordinates (x, y) of a point and rise/run (m) for a linear equation plotted on a coordinate plane that passes through the origin.

EUs Concrete:

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Identify parts of a line graph. Graph a series of coordinates on a coordinate plane. Identify the two coordinates of a point on a line graph. Examine the values of the x variable or y variable to look for

a pattern. Count the distance up/down between two points on the

coordinate plane (rise). Count the distance to the right, between two points on the

coordinate plane (run). Use a template (rise/run) to determine the slope. Recognize a positive or negative relationship between two

points.Representation: Understand the following concepts, vocabulary, and symbols:

coordinates, ordered pairs (x, y), intercept, grid, axis, point, proportion, line, slope.

Graph a series of coordinates on a graph. Identify given coordinates (x, y) as a point on a graph. Identify the intercept(s) on a graph. Identify the slope using the equation. Identify the slope using the graph. Use the format to write an equation for a line.

Resources Element Card 8th: Click here

MAFS.8.EE.3.7: 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.

Remarks/Examples:Fluency Expectations or Examples of Culminating Standards Students have been working informally with one-variable linear equations since as early as kindergarten. This important line of development culminates in grade 8 with the solution of general one-variable linear equations, including cases with infinitely many solutions or no solutions as well as cases requiring algebraic manipulation using properties of operations. Coefficients and constants in these equations may be any rational numbers.

Examples of Opportunities for In-Depth Focus This is a culminating standard for solving one-variable linear equations.Related Access Points

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MAFS.8.EE.3.AP.7a: Simplify linear equations and solve for one variable.

EUs

Concrete: Use manipulatives or a graphic organizer to set up a

problem. Identify the reciprocal operation in order to solve one-step

equations. Use manipulatives or a graphic organizer to illustrate the

distributive property. Use manipulatives or a graphic organizer to solve a problem.Representation: Understand the following concepts, vocabulary, and symbols:

+, -, ×, ÷, =, variable, equation. Draw a picture of a simple equation to translate wording to

solve for x or y. Simplify equations by combining like terms Simplify equations using the distributive property. Identify solutions of an equation after solving.

ResourcesContent Module Equations: Click here Curriculum Resource Guide Equations: Click here Element Card 8th: Click hereMASSI: Click here

MAFS.8.EE.3.8: 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 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. 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.

Remarks/Examples:Examples of Opportunities for In-Depth Focus When students work toward meeting this standard, they build on what they know about two-variable linear equations, and they enlarge the varieties of real-world and mathematical problems they can solve.Related Access Points

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MAFS.8.EE.3.AP.8a: Identify the coordinates of the point of intersection for two linear equations plotted on a coordinate plane.

EUs

Concrete: Identify the solution to a system (i.e., find when the two lines

on the same graph cross). Graph a line on a coordinate plane. Use manipulatives or graphic organizer to solve a problem.Representation: Understand the following concepts, vocabulary, and symbols:

+, -, ×, ÷, =, variable, equation. Identify a coordinate that represents the solution.

Resources Element Card 8th: Click here

MAFS.8.EE.3.AP.8b:Given two sets of coordinates for two lines, plot the lines on a coordinate plane and define the rise/run (m) for each line to determine if the lines will intersect or not.

EUs

Concrete: Identify the solution to a system (i.e., find when the two lines

on the same graph cross). Graph a line on a coordinate plane. Use manipulatives or graphic organizer to solve a problem. Use substitution to generate values to graph a line.Representation: Understand the following concepts, vocabulary, and symbols:

+, -, ×, ÷, =, variable, equation. Use the slope and the origin of the line, determine if the lines

will intersect. Select the solution of the equation algebraically using

substitution. Generate the solution of the equation algebraically using

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substitution.Resources Element Card 8th: Click here

MAFS.8.F.1.1: 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.Related Access Points

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MAFS.8.F.1.AP.1a: Graph the points of a function given the rule of a simple function and identifying four values of x and y.

EUsConcrete: Use the vertical line test to determine whether a line is a

function or non-function. Identify if a function exists given a table.Representation: Understand the following concepts, vocabulary, and symbols:

function, input, output. Locate input and output on a T-chart or function table.

Understand that an input will only have one output. Use a T-chart or function table to determine at least four

values of an equation. Using the values of the T-chart, graph the points.

MAFS.8.F.1.2: 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.

Remarks/Examples:Examples of Opportunities for In-Depth Focus Work toward meeting this standard repositions previous work with tables and graphs in the new context of input/output rules.

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MAFS.8.F.1.AP.2a: Compare the rise/run (m) of two simple linear functions.

EUs

Concrete: Identify properties of a function on a graph (e.g., slope,

increasing or decreasing, where does it cross the x- and y-axis).

Identify if a function exists given a table. Given an equation, determine the slope. Given the graphs of two simple linear equations, compare the

slopes to determine if they are equivalent. Given the equation, compare the slopes to determine if they

are equivalent.Representation: Understand the following concept, vocabulary, and symbol:

function. Identify properties of a function given a graph, table, or

equation.

MAFS.8.F.1.3: 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. 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.Related Access Points

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MAFS.8.F.1.AP.3a: Identify graphed functions as linear or not linear.EUs Concrete:

Identify a linear function on a graph as one that forms a straight line.

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Identify a non-linear function on a graph as one that does not make a straight line.

Representation: Understand the following concepts, vocabulary and symbols:

linear, non-linear, function. Label a function on a graph as being either linear or non-

linear. Identify functions as linear or non-linear given a table or

graph.

Resources Content Module Equations: Click hereElement Card 8th: Click here

MAFS.8.F.2.4: 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.Related Access Points

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MAFS.8.F.2.AP.4a: Identify rise/run (m) as slope and identify the coordinates of the y-intercept.

EUs

Concrete: Indicate the point on a line that crosses the y-axis. Describe the rate of change qualitatively (e.g., steep =

rapidity of change). (E.g., compare the incline of an escalator to the incline of a wheelchair ramp).

Count the distance up/down between two points on the coordinate plane (rise).

Count the distance to the right, between two points on the coordinate plane (run).

Representation: Understand the following concepts and vocabulary: x-axis, y-

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axis, x-intercept, y-intercept, line, rise, fall, slope, rate of change.

Interpret/define a line graph with coordinates for multiple points.

Identify coordinates (points) on a graph.

Resources Content Module Equations: Click hereElement Card 8th: Click here

MAFS.8.F.2.5: 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.Related Access Points

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MAFS.8.F.2.AP.5a: Sketch a graph that exhibits the slope and y-intercept provided.

EUs

Concrete: Identify the slope and the y-intercept of a graph. Identify where the function increases or decreases on a graph. Draw a sketch given a point and a y-intercept. Draw a sketch given a slope and a y-intercept.Representation: Given a verbal description, determine whether the slope is

increasing or decreasing. Match the graph to a given slope and y-intercept.

MAFS.8.F.2.AP.5b: Identify the slope coordinates of one point and the y-intercept.

EUsConcrete: Identify the slope and the y-intercept of a graph. Identify where the function increases or decreases on a graph. Match a description to a graph.

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Representation: Using appropriate tools, create a linear function on a graph

from a description, written or verbal.MAFS.8.F.2.AP.5c: Describe or select the relationship between two plotted graphs.

EUs

Concrete: Identify characteristics of a graph. Match a description to a graph. Use manipulatives to represent the relationship between two

graphs (E.g., use uncooked spaghetti to describe the slope between an escalator (incline) and an elevator shaft (vertical)).

Representation: Interpret/define a line graph with coordinates for multiple

points. Identify coordinates (points) on a graph. Describe the situation that may account for the characteristics

in the graph. Using appropriate tools, create a linear function on a graph

from a description, written or verbal.

ResourcesContent Module Equations: Click hereCurriculum Resource Guide Ratio and Proportion: Click here MASSI: Click here

MAFS.8.G.1.1: 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.

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MAFS.8.G.1.AP.1a: Perform rotations, reflections, and translations using pattern blocks.

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EUsConcrete: Use manipulatives to demonstrate rotations, reflections, or

translations.Representation: Understand the following vocabulary: reflection, rotations,

and translation. Match or identify when a two-dimensional drawing has been

rotated, reflected, or translated. Tracing a figure and slide it over to translate the figure. Tracing a figure and rotating it to create a rotation. Tracing a figure and flip it to create a reflection.

ResourcesContent Module Coordinate Plane: Click hereCurriculum Resource Guide Measurement and Geometry: Click hereElement Card 8th: Click here

MAFS.8.G.1.AP.1b: Draw rotations, reflections, and translations of polygons.

EUs

Concrete: Use manipulatives to demonstrate translations (sliding

object). Use manipulatives to demonstrate rotation (rotating figure). Use manipulatives to demonstrate reflections (flipping

object).Representation: Tracing a figure and slide it over to translate the figure. Tracing a figure and rotating it to create a rotation. Tracing a figure and flip it to create a reflection.

Resources Element Card 8th: Click here

MAFS.8.G.1.2: 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.

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MAFS.8.G.1.AP.2a:Demonstrate that two-dimensional polygons that are rotated, reflected, or translated are still congruent using area, perimeter, and length of sides on a coordinate plane.

EUsConcrete: Model a rotation, reflection, or translation on the coordinate

plane using manipulatives.Representation: Identify a rotation, reflection, or translation when it occurs on

the coordinate plane.

Resources Content Module Coordinate Plane: Click hereContent Module Perimeter, Area and Volume: Click here

MAFS.8.G.1.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Related Access Points

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MAFS.8.G.1.AP.3a: Dilate common polygons using graph paper and identifying the coordinates of the vertices.

EUs Concrete: Use manipulatives (cookie dough, play doh, clay, etc.) to

demonstrate that the manipulative’s size can be changed by flattening, stretching, and rolling it (dilation).

Using graph paper and manipulatives, demonstrate that a polygon can be dilated. E.g., Use manipulatives (cookie dough, play doh, clay, etc.) to trace a figure, then flatten, stretch or roll the figure and trace it again. Identify the different coordinates.

Using manipulatives, identify two figures that are the same shape and size.

Using manipulatives, identify two figures that are different

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sizes but the same shape. Use the two figures to find the coordinates of the vertices.

Representation: Understand the following vocabulary: dilate (dilation),

polygon, coordinate, vertices, increasing and decreasing of size and scale.

Draw two figures that are different sizes but the same shape on graph paper. Identify the different coordinates.

MAFS.8.G.1.AP.3b: Given two figures on a coordinate plane, identify if the image is dilated, translated, rotated, or reflected.

EUsConcrete: Model a rotation, reflection, translation and dilation on the

coordinate plane using manipulatives.Representation: Understand the following concepts and vocabulary: rotation,

reflection, translation, dilation. Identify a rotation, reflection, translation, and dilation when it

occurs on the coordinate plane.

Resources Curriculum Resource Guide Measurement and Geometry: Click here

MAFS.8.G.1.4: 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.Related Access Points

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MAFS.8.G.1.AP.4a: Recognize congruent and similar figures.EUs Concrete:

Recognize corresponding points and sides in figures (e.g.,

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match concrete examples of congruent shapes, match concrete examples of similar shapes).

Representation: Understand the following concepts and vocabulary: figures,

congruent, similar. Describe circles, squares, rectangles, and triangles by telling

about their shape, sides, lines, and angles.

Resources

Content Module Coordinate Plane: Click hereContent Module Perimeter, Area and Volume: Click here Curriculum Resource Guide Measurement and Geometry: Click hereElement Card 8th: Click here

MAFS.8.G.1.AP.4b: Identify two-dimensional figures as similar or congruent given coordinate plane representations.

EUs

Concrete: Recognize how the space inside a figure increases when the

sides are lengthened. Use virtual manipulatives to create dilations, reflections,

rotations, and translations. Use the virtual manipulatives to identify coordinate points,

similarity, or congruent. (E.g., lay the shapes on top of each other to determine congruence. Stretch or shrink shapes to determine similarity.)

Representation: Understand the following concepts and vocabulary: similar,

area, length, width, volume, square, rectangle, prism. Given a picture, identify dimensions needed to calculate area,

and volume. Compare greater than, less than, equal/same figures in two

and three dimensions.

Resources Content Module Perimeter, Area and Volume: Click here Element Card 8th: Click here

MAFS.8.G.1.AP.4c: Compare area and volume of similar figures.

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Concrete: Recognize how the space inside a figure increases when the

sides are lengthened. Multiply whole numbers, fractions, and decimals in order to

compare area, and volume. Use graph paper to count the area inside of a figure and use

the area to compare the sizes of the figures. Use cubes to count the volume of a figure and use the volume

to compare the sizes of the figures.Representation: Understand the following concepts and vocabulary: similar,

area, length, width, volume, square, rectangle, prism. Compare: greater than, less than, equal/same, figures in two

and three dimensions. Calculate the area and/or volume of two figures. Use the

area/volume to compare their sizes.

Resources

Content Module Perimeter, Area and Volume: Click here Curriculum Resource Guide Measurement and Geometry: Click hereMASSI: Click hereElement Card 8th: Click here

MAFS.8.G.1.5: 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.Related Access Points

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MAFS.8.G.1.AP.5a: Use angle relationships to find the value of a missing angle.EUs Concrete:

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Given an angle measure, draw an angle. Recognize that a triangle consists of three angles that total

180 degrees. Recognize that the angle measure of a straight line is 180

degrees. Use a protractor to measure the missing angle.Representation: Understand the following concepts and vocabulary: acute,

obtuse, right, straight line, transversal, vertical angles, corresponding angles, alternate interior angles, supplementary angles.

Match or identify angle measurements. Describe triangles and parallel lines by telling about their

shape, sides, lines, and angles. Use appropriate tools as needed. Use addition or subtraction to determine the missing angle

measurement in triangles. (E.g., Angle A = 60 degrees, Angle B = 40 degrees, Angle A + Angle B = 100 degrees, therefore Angle C = 180 – 100 = 80 degrees).

Identify vertical angles, corresponding angles, alternate interior angles. Understand that the angles are congruent.

Resources Curriculum Resource Guide Measurement and Geometry: Click here

MAFS.8.G.2.6: Explain a proof of the Pythagorean Theorem and its converse.Related Access Points

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MAFS.8.G.2.AP.6a: Measure the lengths of the sides of multiple right triangles to determine a relationship.

EUs Concrete: Identify the formula for the Pythagorean Theorem. Identify what each variable in the Pythagorean Theorem

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represents. Label the legs and the hypotenuse of the given figure. Measure the lengths of the legs and the hypotenuse of the

given figure.Representation: Understand the following concepts and vocabulary:

Pythagorean Theorem, length, right triangle, hypotenuse, leg, and angle.

Use a graphic organizer to organize the measurements of the legs and hypotenuse, using appropriate tools as needed. Click here

Resources Curriculum Resource Guide Measurement and Geometry: Click here

MAFS.8.G.2.7: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Remarks/Examples:Examples of Opportunities for In-Depth Focus The Pythagorean theorem is useful in practical problems, relates to grade-level work in irrational numbers and plays an important role mathematically in coordinate geometry in high school.

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MAFS.8.G.2.AP.7a: Find the hypotenuse of a two-dimensional right triangle using the Pythagorean theorem.

EUs

Concrete: Identify the formula for the Pythagorean Theorem. Identify what each variable in the Pythagorean Theorem

represents. Label the legs and the hypotenuse of the given figure. Measure the lengths of the legs and the hypotenuse of the

given figure.

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Representation: Understand the following concepts and vocabulary:

Pythagorean Theorem, length, right triangle, legs, hypotenuse and angle.

Enter information into the formula for the Pythagorean Theorem to solve problems. Click here

Use a graphic organizer to calculate a missing side using the Pythagorean Theorem, using appropriate tools as needed.

Resources Curriculum Resource Guide Measurement and Geometry: Click here

MAFS.8.G.2.8: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Related Access Points

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MAFS.8.G.2.AP.8a:Apply the Pythagorean Theorem to determine lengths/distances between two points in a coordinate system by forming right triangles.

EUs

Concrete: Identify the formula for the Pythagorean Theorem. Identify what each variable in the Pythagorean Theorem

represents. Label the legs and the hypotenuse of the given figure Measure the lengths of the legs and the hypotenuse of the

given figure.Representation: Understand the following concepts and vocabulary:

Pythagorean Theorem, length, right triangle, legs, hypotenuse, and angle.

Use a graphic organizer to calculate a missing side using the Pythagorean Theorem, using appropriate tools as needed.

Enter information into the formula for the Pythagorean Theorem to solve problems. Click here

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Resources Curriculum Resource Guide Measurement and Geometry: Click here

MAFS.8.G.3.9: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Remarks/Examples:Fluency Expectations or Examples of Culminating Standards When students learn to solve problems involving volumes of cones, cylinders, and spheres — together with their previous grade 7 work in angle measure, area, surface area and volume (7.G.2.4–2.6) — they will have acquired a well-developed set of geometric measurement skills. These skills, along with proportional reasoning (7.RP) and multistep numerical problem solving (7.EE.2.3), can be combined and used in flexible ways as part of modeling during high school — not to mention after high school for college and careers.

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MAFS.8.G.3.AP.9a: Using a calculator, apply the formula to find the volume of three-dimensional shapes (i.e., cubes, spheres and cylinders).

EUs

Concrete: Identify attributes (length, width, height, diameter, radius,

circumference) of a three-dimensional shape. Using appropriate tools, multiply whole numbers, fractions,

and decimals. Identify the formulas for cones, spheres and cylinders using

appropriate tools.Representation: Understand the following concepts and vocabulary: volume,

cylinder, cone, height, length, width, diameter, radius, circumference, cube, sphere, side, and pi.

Recognize that the volume of three-dimensional shapes can be found by finding the area of the base and multiplying that by the height.

Understand the parts of the formulas for cones, spheres, and cylinders.

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Apply the formulas for cones, spheres, and cylinders using appropriate tools.

ResourcesCurriculum Resource Guide Measurement and Geometry: Click hereElement Card 8th: Click here

MAFS.8.NS.1.1: 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.Related Access Points

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MAFS.8.NS.1.AP.1a:Distinguish between rational and irrational numbers. Show that any number that can be expressed as a fraction is a rational number.

EUs

Concrete: Use manipulatives to represent whole numbers as a fraction

(e.g., 3 whole circles each divided in half is equal to 6/2) Use manipulatives to represent a fraction. Understand that the use of 3.14 for π is a rounded,

approximated number (e.g., use 22/7 in a calculator to approximate π).

Representation: Identify the symbol for π in writing and on a calculator. Identify 3.14 as π. Understand the following concepts, symbols, and

vocabulary: irrational numbers, rational numbers, fraction, decimal, π.

ResourcesCurriculum Resource Guide Fractions and Decimals:Click here Element Card 8th: Click here

MAFS.8.NS.1.AP.1b: Using whole number numerators from 8 to 20 and odd whole

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number denominators from 3 to 7, identify rational decimal expansions.  

EUs

Concrete: Use base ten manipulatives to divide numbers. Understand that the use of 3.14 for π is a rounded,

approximated number (e.g., use 22/7 in a calculator to approximate π).

Match characteristics of irrational and rational numbers.Representation: Use a calculator to divide numbers from 8 to 20 and odd

whole number divisors from 3 to 7 Identify the characteristics of an irrational number. Identify irrational decimal quotients

ResourcesCurriculum Resource Guide Fractions and Decimals:Click here Element Card 8th: Click here

MAFS.8.NS.1.AP.1c: Round or truncate rational decimal expansions to the hundredths place.

EUs

Concrete: Identify place value to the tenths, hundredths and

thousandths place. Use a number line to determine which number is closer to

the given value (i.e., given 30.433, place the number on the number line between 30.43 and 30.44. Use the distance between the numbers, to determine that 30.433 is closer to 30.43 than it is to 30.44).

Representation: Understand the following concepts, symbols, and

vocabulary: place value, ones, decimal, tenths, hundredths, thousandths.

Apply the rule for rounding (e.g., find number on a number line—if five or greater, round up, if less than five, round down).

Identify the nearest tenth, nearest hundredth.

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Resources Element Card 8th: Click here

MAFS.8.NS.1.2: 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.Related Access Points

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MAFS.8.NS.1.AP.2a: Locate approximations of irrational numbers on a number line.

EUs

Concrete: Locate whole numbers on a number line. Locate decimal numbers on a number line. Locate fractions on a number line. Use a calculator to find the square root of a number. Use the square root of a number to place a value on the

number line.Representation: Round an irrational number to the nearest whole number. Round an irrational number to the nearest tenths place. Round an irrational number to the hundredths place. Round an irrational number to the thousandths place.

Resources Element Card 8th: Click here

MAFS.8.SP.1.1: 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.

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MAFS.8.SP.1.AP.1a: Graph data using line graphs, histograms or box plots.

EUs

Concrete: Enter data into a graph or histogram using manipulatives, as

needed. Identify a graph histogram and box plot and the kind of

information each displays (the use for each display). Identify a representation of a histogram as having greater or

less frequency of members/events related to a single variable (e.g., compare number of boys in soccer to girls in a graph).

Representation: Understand the following concepts and vocabulary: best fit

line, variable, outliers, histogram, clustering, box plot, frequency.

Understand basic information from simple graphs (e.g., interpret a bar graph using the understanding that the taller column on a graph has a higher frequency, the shorter column on a graph has a lower frequency).

Graph a series of data points on a coordinate grid.

MAFS.8.SP.1.AP.1b: Graph bivariate data using scatter plots and identify possible associations between the variables.

EUs

Concrete: Locate points on the x-axis and y-axis on an adapted grid

(not necessarily numeric). Identify a similar distribution when given a choice of three

(e.g., when shown a normal distribution, can select a second example of a normal distribution from three choices).

Representation: Understand the following concepts and vocabulary: best fit

line, variable, outliers. Graph a series of data points on a coordinate grid. Identify the associations between the variables using

supports (E.g., use a template to determine the correlation,

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use a pre-made scatter plot transparency and place on top of a given scatter plot to determine associations).

Resources Curriculum Resource Guide Data Analysis: Click here Element Card 8th: Click here

MAFS.8.SP.1.AP.1c: Using box plots and scatter plots, identify data points that appear to be outliers.

EUs

Concrete: Identify an outlier on a line graph. Identify a similar distribution when given a choice of three

(e.g., when shown a normal distribution, can select a second example of a normal distribution from three choices).

Representation: Understand the following concepts and vocabulary: best fit

line, variable, outliers, box plot, scatter plots, data points. Order data in numerical order. Identify the median of a data set. Identify the associations between the variables using

supports.

MAFS.8.SP.1.2: 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.Related Access Points

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MAFS.8.SP.1.AP.2a: Draw the line of best fit on a scatter plot.EUs Concrete:

Draw a line between two points on a graph. Identify when data points are close together or spread out.

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Use appropriate manipulatives (uncooked spaghetti noodle, clear ruler, popsicle stick, etc.) to approximate the line of best fit.

Representation: Understand the following concepts and vocabulary: best fit

line, variable, outliers, box plot, scatter plots, data points. Given three choices, select the line of best fit.

MAFS.8.SP.1.AP.2b: Identify outliers on a scatter plot given the line of best fit.

EUsConcrete: Draw a line between two points on a graph. Identify when data points are close together or spread out.Representation: Understand the following concepts and vocabulary: best fit

line, variable, outliers, box plot, scatter plots, data points. Identify linear and non-linear data in various scatter plots.

MAFS.8.SP.1.3: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.Related Access Points

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MAFS.8.SP.1.AP.3a: Interpret the slope and the y-intercept of a line in the context of data plotted from a real-world situation.

EUs Concrete:

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Identify the slope (m) in an equation using a formula (i.e., y = mx + b).

Identify the y-intercept (b) using a formula (i.e., y = mx + b).Representation: Understand the following concepts and vocabulary: slope

(m), y-intercept (b). Match or select an appropriate meaning of the slope (m) and

y-intercept (b) for the context of the problem. Identify the slope and y-intercept within the context of a

real-world situation. For example: In 1998, Linda purchased a house for $144,000 (y-intercept). In 2009 the house was worth $245,000. Find the average annual rate of change (slope) in dollars per year in the value of the house. Round your answer to the nearest dollar.

MAFS.8.SP.1.4: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?Related Access Points

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MAFS.8.SP.1.AP.4a: Analyze displays of bivariate data to develop or select appropriate claims about those data.

EUs

Concrete: Identify a similar distribution when given a choice of three

(e.g., when shown a normal distribution, can select a second example of a normal distribution from three choices).

Identify the appropriate statement when given a relationship between two variables (may use graphic supports such as highlighted transparency of an association).

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Representation: Understand the following concepts and vocabulary: variable,

claim, association, relative frequency, frequency, two-way table, categorical variable, bivariate

Explain the associations between the variables using supports (e.g., the selection of the highlighted transparency and make a statement).

ResourcesCurriculum Resource Guide Data Analysis: Click hereElement Card 8th: Click hereMASSI: Click here

MAFS.K12.MP.1.1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

MAFS.K12.MP.2.1: Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

MAFS.K12.MP.3.1: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

MAFS.K12.MP.4.1: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their 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.

MAFS.K12.MP.5.1: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

MAFS.K12.MP.6.1: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

MAFS.K12.MP.7.1: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can 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.

MAFS.K12.MP.8.1:Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x² + x + 1), and (x – 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

LAFS.68.RST.1.3: Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks.LAFS.68.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 6–8 texts and topics.LAFS.68.RST.3.7: Integrate quantitative or technical information expressed in words in a text with a version of that information expressed visually (e.g., in a flowchart, diagram, model, graph, or table).

LAFS.68.WHST.1.1: Write arguments focused on discipline-specific content. a. Introduce claim(s) about a topic or issue, acknowledge and distinguish the claim(s) from alternate or opposing claims, and

organize the reasons and evidence logically. b. Support claim(s) with logical reasoning and relevant, accurate data and evidence that demonstrate an understanding of

the topic or text, using credible sources. c. Use words, phrases, and clauses to create cohesion and clarify the relationships among claim(s), counterclaims, reasons,

and evidence. d. Establish and maintain a formal style. e. Provide a concluding statement or section that follows from and supports the argument presented.

LAFS.68.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.

LAFS.8.SL.1.1: Engage effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grade 8 topics, texts, and issues, building on others’ ideas and expressing their own clearly.

a. Come to discussions prepared, having read or researched material under study; explicitly draw on that preparation by referring to evidence on the topic, text, or issue to probe and reflect on ideas under discussion.

b. Follow rules for collegial discussions and decision-making, track progress toward specific goals and deadlines, and define individual roles as needed.

c. Pose questions that connect the ideas of several speakers and respond to others’ questions and comments with relevant evidence, observations, and ideas.

d. Acknowledge new information expressed by others, and, when warranted, qualify or justify their own views in light of the evidence presented.

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LAFS.8.SL.1.AP.1a: Use information and feedback to refine understanding.

LAFS.8.SL.1.AP.1b: Use information and feedback to clarify meaning for readers.

LAFS.8.SL.1.AP.1c: Discuss how own view or opinion changes using new information provided by others.

LAFS.8.SL.1.2: Analyze the purpose of information presented in diverse media and formats (e.g., visually, quantitatively, orally) and evaluate the motives (e.g., social, commercial, political) behind its presentation.

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LAFS.8.SL.1.AP.2a:Analyze the purpose of information presented in diverse media (e.g., visually, personal communication, periodicals, social media).

LAFS.8.SL.1.AP.2b:Identify the motives behind information presented in diverse media and formats (e.g., visually, personal communication, periodicals, social media).

LAFS.8.SL.1.AP.2c: Evaluate the motives and purpose behind information presented in diverse media and formats for persuasive reasons.

LAFS.8.SL.1.3: Delineate a speaker’s argument and specific claims, evaluating the soundness of the reasoning and relevance and sufficiency of the evidence and identifying when irrelevant evidence is introduced.Related Access Points

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LAFS.8.SL.1.AP.3a: Evaluate the soundness of reasoning and the relevance and sufficiency of evidence provided in an argument.

LAFS.8.SL.1.AP.3b: Identify when irrelevant evidence is introduced within an argument.

LAFS.8.SL.1.AP.3c:Evaluate the soundness or accuracy (e.g., Does the author have multiple sources to validate information?) of reasons presented to support a claim.

ELD.K12.ELL.MA.1: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.

ELD.K12.ELL.SI.1: English language learners communicate for social and instructional purposes within the school setting.