Courses of Study : Mathematics

Number Systems and Operations
Understand that the real number system is composed of rational and irrational numbers.
Mathematics (2019)
Grade(s): 8
All Resources: 2
Classroom Resources: 2
1. Define the real number system as composed of rational and irrational numbers.

a. Explain that every number has a decimal expansion; for rational numbers, the decimal expansion repeats or terminates.

b. Convert a decimal expansion that repeats into a rational number.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Provide an example of both rational and irrational numbers in ratio form as well as the decimal expansion taken from the quotient of that ratio.
  • Convert a repeating decimal into a rational number.
Teacher Vocabulary:
  • Real Number System
  • Ratio
  • Rational Number
  • Irrational Number
Knowledge:
Students: know that any ratio a/b, where b is not equal to zero, has a quotient attained by dividing a by b.
  • know that the real number system contains natural numbers, whole numbers, integers, rational, and irrational numbers.
  • know that every real number has a decimal expansion that is repeating, terminating, or is non-repeating and non-terminating.
  • Skills:
    Students are able to:
    • define the real number system by giving its components.
    • Explain the difference between rational and irrational numbers. specifically how their decimal expansions differ.
    • Convert a ratio into its decimal expansion and take a decimal expansion back to ratio form.
    Understanding:
    Students understand that:
    • all real numbers are either rational or irrational and
    • Every real number has a decimal expansion that repeats, terminates, or is both non-repeating and non-terminating.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.1.1: Define expanding decimals, terminating decimals, rational number, and irrational number.
    M.8.1.2: Identify and give examples of rational numbers.
    M.8.1.3: Demonstrate how to convert fractions to decimals.
    M.8.1.4: Recall steps for division of fractions.

    Prior Knowledge Skills:
    • Define rational number.
    • Plot pairs of integers and/or rational numbers on a coordinate plane.
    • Arrange integers and /or rational numbers on a horizontal or vertical number line.
    • Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
    • Identify a rational number as a point on the number line.
    • Recognize place value of whole numbers and decimals.
    • Give examples of rational numbers.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.1 Add and subtract fractions with like denominators (e.g. halves, thirds, fourths, tenths).
    M.AAS.8.1a Add and subtract decimals to the hundredths place.
    M.AAS.8.1b Convert a fraction with a denominator of 100 to a decimal.


    Mathematics (2019)
    Grade(s): 8
    All Resources: 2
    Classroom Resources: 2
    2. Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of the irrational numbers.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Estimate the value of an irrational number and use that estimate to compare an irrational number to other numbers and to place irrational numbers on a number line alongside or between rational numbers.
    Teacher Vocabulary:
    • Rational
    • Irrational
    Knowledge:
    Students know:
    • the difference between a rational and an irrational number.
    • That real numbers and their decimal expansions can be approximated using a common place value to compare those expansions.
    Skills:
    Students know: the difference between a rational and an irrational number.
  • That real numbers and their decimal expansions can be approximated using a common place value to compare those expansions.
  • Understanding:
    Students understand that:
    • an estimation of the value of an irrational number can be used to compare an irrational number to other numbers and to place them on a number line.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.2.1: Define square root, expressions, and approximations.
    M.8.2.2: Identify properties of exponents.
    M.8.2.3: Recall how to compare numbers.
    M.8.2.4: Identify perfect squares and square roots.
    M.8.2.5: Demonstrate how to locate points on a vertical or horizontal number line.
    M.8.2.6: Recall how to estimate.

    Prior Knowledge Skills:
    • Define equivalent, simplify, term, distributive property, associative property of addition and multiplication, and the commutative property of addition and multiplication.
    • Simplify expressions with parentheses (Ex. 5(4 + x) = 20 + 5x).
    • Combine terms that are alike of a given expression.
    • Recognize the property demonstrated in a given expression.
    • Discuss various strategies for solving real-world and mathematical problems.
    • Recall steps for solving fractional problems.
    • Identify properties of operations for addition and multiplication.
    • Recall the rules for multiplication and division of rational numbers.
    • Recall the rules for addition and subtraction of rational numbers.
    • Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.2 Compare quantities represented as decimals in real-world examples to the hundredths place.


    Algebra and Functions
    Apply concepts of integer exponents and radicals.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 7
    Learning Activities: 3
    Classroom Resources: 4
    3. Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Use their understanding of exponents as repeated multiplication to create equivalent expressions and justify integer exponent properties.
    Teacher Vocabulary:
    • Integer Exponent
    Knowledge:
    Students know:
    • that whole number exponents indicate repeated multiplication of the base number and that these exponents indicate the actual number of factors being produced.
    Skills:
    Students are able to:
    • Develop integer exponent operations in order to generate equivalent expressions through addition, multiplication, division and raising a power by another power with expressions containing integer exponents.
    Understanding:
    Students understand that:
    • just as whole number exponents represent repeated multiplication, negative integer exponents represent repeated division by the base number.
    • The exponent can be translated (visually, listing out the factors) to represent the exact number of factors being repeated so that the use of integer exponent operations ("rules") can be proven/make sense.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.3.1: Define exponent, power, coefficient, integers, equivalent, and numerical expression.
    M.8.3.2: Restate negative exponents as positive exponents in the form 1/xy .
    M.8.3.3: Restate zero exponents as 1 (x0 = 1).
    M.8.3.4: Recognize to add exponents when multiplying terms with like bases (Property of product of powers).
    M.8.3.5: Recognize to subtract exponents when dividing terms with like bases (Property of quotient of powers).
    M.8.3.6: Compute a numerical expression with positive exponents.
    M.8.3.7: Restate exponential numbers as repeated multiplication.
    M.8.3.8: Compute problems with adding and subtracting integers.

    Prior Knowledge Skills:
    • Define exponent, numerical expression, algebraic expression, variable, base, power, square of a number, and cube of a number.
    • Compute a numerical expression with exponents, with or without a calculator.
    • Restate exponential numbers as repeated multiplication.
    • Choose the correct value to replace each variable in the expression (Substitution).
    • Calculate the multiplication of single or multi-digit whole numbers, with or without a calculator.
    • Define integers, positive and negative numbers.
    • Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.
    • Give examples of positive and negative numbers to represent quantities having opposite directions in real-world contexts.
    • Discuss the measure of centering of 0 in relationship to positive and negative numbers.
    • Discover that the opposite of the opposite of a number is the number itself.
    • Show on a number line that numbers that are equal distance from 0 and on opposite sides of 0 have opposite signs.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 3
    Lesson Plans: 1
    Classroom Resources: 2
    4. Use square root and cube root symbols to represent solutions to equations.

    a. Evaluate square roots of perfect squares (less than or equal to 225) and cube roots of perfect cubes (less than or equal to 1000).

    b. Explain that the square root of a non-perfect square is irrational.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Evaluate expressions involving squared and cubed numbers.
    • Solve equations with radicals with a square or cube root solution.
    Teacher Vocabulary:
    • Radical
    • Square Root
    • Cube Root
    Knowledge:
    Students know:
    • that the square root of a non-perfect square is an irrational number.
    • Equations can potentially have two solutions.
    • how to identify a perfect square/cube.
    Skills:
    Students are able to:
    • define a perfect square/cube.
    • Evaluate radical expressions representing square and cube roots.
    • Solve equations with a squared or cubed variable.
    Understanding:
    Students understand that:
    • there is an inverse relationship between squares and cubes and their roots.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.4.1: Define square root, cube root, inverse, perfect square, perfect cube, and irrational number.
    M.8.4.2: Recognize the inverse operation of squaring a number is square root and the inverse of cubing a number is cube root.
    M.8.4.3: Restate exponential numbers as repeated multiplication.
    M.8.4.4: Calculate the multiplication of single or multi-digit whole numbers.
    M.8.4.5: Recognize rational and irrational numbers.

    Prior Knowledge Skills:
    • Restate exponential numbers as repeated multiplication.
    • Define rational number.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.4 Calculate the square of numbers 1 through 10.


    Mathematics (2019)
    Grade(s): 8
    All Resources: 1
    Classroom Resources: 1
    5. Estimate and compare very large or very small numbers in scientific notation.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Rewrite numbers using scientific notation.
    • Use numbers in scientific notation to estimate measurements and values.
    Teacher Vocabulary:
    • Scientific Notation
    Knowledge:
    Students know:
    • that scientific notation is formed using the base ten system and is the reason a 10 is used as the base number.
    • Raising or lowering an exponent has an effect on the place value of the decimal expansion.
    Skills:
    Students are able to:
    • write numbers in standard form in scientific notation.
    • Convert numbers from scientific notation back to standard form.
    • Use information given in scientific notation to estimate very large or small quantities given in real world contexts.
    Understanding:
    Students understand that:
    • the movement of decimals in converting between scientific and standard notation is a function of an exponent.
    • Every decimal place represents a power of ten (this is a connection many students have not made yet when thinking about place value).
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.5.1: Recognize a fraction as division of the denominator into the numerator.
    M.8.5.2: Demonstrate that when multiplying powers of like bases; add the exponents (Property of products of powers).
    M.8.5.3: Demonstrate that when dividing powers of like bases; subtract the exponents (Property of quotient of powers).
    M.8.5.4 : Demonstrate how to convert fractions to a decimal, with or without a calculator.
    M.8.5.5: Recall how to write numbers in scientific notation.
    M.8.5.6: Recall estimation strategies.

    Prior Knowledge Skills:
    • Define the parts of a division problem including divisor, dividend, and quotient.
    • Write a division equation.
    • Apply the signs ÷ and = to the action of separating sets.
    • Recognize division as either repeated subtraction, parts of a set, parts of a whole, or the inverse of multiplication.
    • Model grouping with basic division facts partitioned equally (e.g. 8/2).
    • Apply properties of operations as strategies to subtract.
    • Subtract within 20.
    • Represent equal groups using manipulatives.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.5 Find the square root of the perfect squares up to 100.


    Mathematics (2019)
    Grade(s): 8
    All Resources: 1
    Classroom Resources: 1
    6. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

    a. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.

    b. Interpret scientific notation that has been generated by technology.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Use the laws of exponents to multiply and divide expressions containing numbers written in scientific and decimal notation to solve real-world problems.
    • Compare numbers written in scientific notation and express the multiplicative relationship between the numbers.
    Teacher Vocabulary:
    • Multiplicative Relationship
    • Scientific Notation
    Knowledge:
    Students know:
    • that scientific notation is formed using a base ten system.
    • how to apply laws for multiplying and dividing exponents.
    Skills:
    Students are able to:
    • perform multiplication and division with numbers expressed in scientific notation to solve real-world problems, including problems where both scientific and decimal notation are used.
    • Choose between appropriate units of measure when determining solutions or estimating
    Understanding:
    Students understand that:
    • scientific notation has real-world applications for very large and very small quantities found in many disciplines.
    • performing scientific notation operations are another application of integer exponent operations.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.6.1: Define scientific notation.
    M.8.6.2: Calculate multiplication and division of scientific notation, with or without a calculator.
    M.8.6.3: Recall properties of exponents.
    M.8.6.4: Recall how to write a number using scientific notation.
    M.8.6.5: Restate exponents as repeated multiplication.
    M.8.6.6: Discuss the real world application of scientific notation (very large or very small quantities).
    M.8.6.7: Demonstrate difference of scientific notation symbol between paper and calculator.

    Prior Knowledge Skills:
    • Recall that exponents are repeated multiplication.
    • Demonstrate the ability to multiply and divide a number by a power of ten.
    • Recognize the place value changes when multiplying/dividing by powers of ten.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.6 Identify irrational numbers as non perfect squares (e.g. discriminate between perfect and non perfect squares).


    Analyze the relationship between proportional and non-proportional situations.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 0
    7. Determine whether a relationship between two variables is proportional or non-proportional.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Describe a given relationship as proportional or non-proportional when given in various contexts.
    Teacher Vocabulary:
    • Ratio
    • Proportion
    • Proportional
    • Independent variable
    • Dependent variable
    Knowledge:
    Students know:
    • how to use rates and scale factors to find equivalent ratios.
    • What a unit rate is and how to find it when needed.
    Skills:
    Students are able to:
    • Recognize whether ratios are in a proportional relationship using tables and verbal descriptions.
    Understanding:
    Students understand that:
    • a proportion is a relationship of equality between quantities.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.7.1: Define proportional, independent variable, dependent variable, unit rate.
    M.8.7.2: Recall equivalent ratios and origin on a coordinate (Cartesian) plane.
    M.8.7.3: Recall how to write a ratio of two quantities as a fraction.
    M.8.7.4: Identify the unit rate of two quantities.
    M.8.7.5: Recall that for a relationship to be proportional, both variables must start at zero.

    Prior Knowledge Skills:
    • Define unit rate, proportion, and rate.
    • Create a ratio or proportion from a given word problem.
    • Calculate unit rate by using ratios or proportions.
    • Write a ratio as a fraction.
    • Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
    • Create a ratio or proportion from a given word problem, diagram, table, or equation.
    • Calculate unit rate or rate by using ratios or proportions with or without a calculator.
    • Restate real world problems or mathematical problems.
    • Construct a graph from a set of ordered pairs given in the table of equivalent ratios.
    • Calculate missing input and/or output values in a table with or without a calculator.
    • Draw and label a table of equivalent ratios from given information.
    • Identify the parts of a table of equivalent ratios (input, output, etc.).
    • Compute the unit rate, unit price, and constant speed with or without a calculator.
    • Create a proportion or ratio from a given word problem.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 4
    Classroom Resources: 4
    8. Graph proportional relationships.

    a. Interpret the unit rate of a proportional relationship, describing the constant of proportionality as the slope of the graph which goes through the origin and has the equation y = mx where m is the slope.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Represent given proportional relationships with graphs.
    • Determine the characteristics that remain consistent in proportional relationships, such as the unit rate and inclusion of the origin.
    • Use a graphical representation of a proportional relationship in context to: explain the meaning of any point (x, y). explain the meaning of (0, 0). and why it is included.
    Teacher Vocabulary:
    • Ratio
    • Proportion
    • Proportional
    • Independent variable
    • Dependent variable
    • y-intercept
    • origin
    Knowledge:
    Students know:
    • what a proportion is and how it is represented on a table or verbally.
    • how to graph coordinates and identify the origin and quadrants on the coordinate plane.
    Skills:
    Students are able to:
    • create graphs to visually verify a constant rate as a straight line through the corresponding coordinates and the origin.
    • Identify the unit rate (constant of proportionality) within two quantities in a proportional relationship shown on a graph and in the form y =mx.
    Understanding:
    Students understand that:
    • unit rate is sometimes referred to as the constant of proportionality.
    • proportional relationships are represented by a straight line that runs through the origin.
    • y=mx is the equation form that represents all proportions, where m is the rate of change/constant of proportionality which can now be called the slope.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.8.1: Define proportional relationships, unit rate, and slope.
    M.8.8.2: Demonstrate how to write ratios.
    M.8.8.3: Recall how to solve proportions using cross products.
    M.8.8.4: Recall how to find the unit rate.
    M.8.8.5: Demonstrate how to graph on a Cartesian plane.
    M.8.8.6: Recall that for a relationship to be proportional, the graph must pass through the origin.
    M.8.8.7: Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.

    Prior Knowledge Skills:
    • Define unit rate, proportion, and rate.
    • Create a ratio or proportion from a given word problem.
    • Calculate unit rate by using ratios or proportions.
    • Write a ratio as a fraction.
    • Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
    • Create a ratio or proportion from a given word problem, diagram, table, or equation.
    • Calculate unit rate or rate by using ratios or proportions with or without a calculator.
    • Restate real world problems or mathematical problems.
    • Construct a graph from a set of ordered pairs given in the table of equivalent ratios.
    • Calculate missing input and/or output values in a table with or without a calculator.
    • Draw and label a table of equivalent ratios from given information.
    • Identify the parts of a table of equivalent ratios (input, output, etc.).

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.8 Using a real-world scenario, match a table with its graph. Identify proportional or nonproportional relationships.


    Mathematics (2019)
    Grade(s): 8
    All Resources: 3
    Learning Activities: 1
    Classroom Resources: 2
    9. Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y-intercept.

    a. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in a coordinate plane.

    b. Given two distinct points in a coordinate plane, find the slope of the line containing the two points and explain why it will be the same for any two distinct points on the line.

    c. Graph linear relationships, interpreting the slope as the rate of change of the graph and the y-intercept as the initial value.

    d. Given that the slopes for two different sets of points are equal, demonstrate that the linear equations that include those two sets of points may have different y-intercepts.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Analyze linear equations in the form y=mx + b as representing a line where m represents the rate of change, called the slope of the line when graphed. and b is the initial value, called the y-intercept when graphed.
    • Create similar right triangles by connecting the "rise over run" between any two points on a given line and use them to show why their slopes are the same.
    • Explain why any two points on a given line will have the same slope.
    • Graph linear relationships on a coordinate plane when given in multiple contexts.
    Teacher Vocabulary:
    • Slope
    • Rate of change
    • Initial Value
    • Y-intercept
    Knowledge:
    Students know:
    • how to graph points on a coordinate plane.
    • Where to graph the initial value/y-intercept.
    • Understand how/why triangles are similar.
    • how to interpret y=mx equations.
    Skills:
    Students are able to:
    • create a graph of linear equations in the form y = mx + b and recognize m as the slope and b as the y-intercept.
    • point out similar triangles formed between pairs of points and know that they have the same slope between any pairs of those points.
    • Show that lines may share the same slope but can have different y-intercepts.
    • Interpret a rate of change as the slope and the initial value as the y-intercept.
    Understanding:
    Students understand that:
    • Slope is a graphic representation of the rate of change in linear relationships and the y-intercept is a graphic representation of an initial value in a linear relationship.
    • When given an equation in the form y = mx + b it generally symbolizes that there will be lines with varying y-intercepts. even when the slope is the same.
    • Use of the visual of right triangles created between points on a line to explain why the slope is a constant rate of change.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.9.1: Define linear functions, nonlinear functions, slope, and y-intercept.
    M.8.9.2: Recall how to solve problems using the distributive property.
    M.8.9.3: Recognize linear equations.
    M.8.9.4: Identify ordered pairs.
    M.8.9.5: Recognize ordered pairs.
    M.8.9.6: Define similar triangles, intercept, slope, vertical, horizontal, and origin.
    M.8.9.7: Recognize similar triangles.
    M.8.9.8: Generate the slope of a line using given ordered pairs.
    M.8.9.9: Analyze the graph to determine the rate of change.
    M.8.9.10: Demonstrate how to plot points on a coordinate plane using ordered pairs from table.
    M.8.9.11: Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
    M.8.9.12: Graph a function given the slope-intercept form of an equation.
    M.8.9.13: Recognize that two sets of points with the same slope may have different y-intercepts.
    M.8.9.14: Graph a linear equation given the slope-intercept form of an equation.

    Prior Knowledge Skills:
    • Define ordered pairs.
    • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
    • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
    • Identify which signs indicate the location of a point in a coordinate plane.
    • Recall how to plot ordered pairs on a coordinate plane.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 0
    10. Compare proportional and non-proportional linear relationships represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions) to solve real-world problems.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Analyze and explain the difference between a proportional and non-proportional linear relationship given in various contexts.
    • Use evidence gathered from data given by linear relationships to make sense of and solve real-world problems.
    Teacher Vocabulary:
    • Proportional
    Knowledge:
    Students know:
    • the difference between proportional and non-proportional linear relationships.
    • What rate of change/slope represents as well as the meaning of initial value/y-intercepts when given in a variety of contexts.
    Skills:
    Students are able to:
    • qualitatively and quantitatively compare linear relationships in different ways when those relationships are presented within real-world problems.
    Understanding:
    Students understand that:
    • real-world linear relationships can be compared using any representation they choose. based on their understanding of proportions and functions.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.10.1: Define proportional and nonproportional.
    M.8.10.2: Recall that for two relationships to be proportional they must have the same unit rate and pass through the origin on a coordinate plane.
    M.8.10.3: Apply the rule of proportional relationship to real world context.

    Prior Knowledge Skills:
    • Define unit rate, proportion, and rate.
    • Calculate unit rate by using ratios or proportions.
    • Write a ratio as a fraction.
    • Define proportions and proportional relationships.
    Analyze and solve linear equations and systems of two linear equations.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 6
    Learning Activities: 1
    Classroom Resources: 5
    11. Solve multi-step linear equations in one variable, including rational number coefficients, and equations that require using the distributive property and combining like terms.

    a. Determine whether linear equations in one variable have one solution, no solution, or infinitely many solutions of the form x = a, a = a, or a = b (where a and b are different numbers).

    b. Represent and solve real-world and mathematical problems with equations and interpret each solution in the context of the problem.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Recognize and explain when linear equations have one solution, infinitely many solutions, or no solution with and without completing the solving process.
    • Solve one variable equation with the same variable on both sides and require use of the distributive property.
    • Analyze and explain solutions in the context of a real-world problem.
    Teacher Vocabulary:
    • one solution
    • no solution
    • Infinitely many solutions
    • like terms
    • Distributive property
    Knowledge:
    Students know:
    • how to solve one and two step equations with one variable.
    • Write linear equations given real-world contexts.
    • That a solution to an equation can represent a real-world quantity.
    Skills:
    Students are able to:
    • apply the distributive property and combine like terms to simplify an equation.
    • Recognize a solution as representing one solution, no solution, or infinite solutions.
    • Analyze and solve a real-world problem and write an appropriate equation for it that leads to a solution that can be explained within the context of the problem.
    Understanding:
    Students understand that:
    • equations can now have more than one solution in given real-world scenarios.
    • The distributive property and combining like terms are essential to simplifying an equation. therefore making it easier to solve.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.11.1: Define linear equation, coefficient, distributive property and variable.
    M.8.11.2: Recall how to solve equations for a missing variable.
    M.8.11.3: Recall properties of operation for addition and multiplication.
    M.8.11.4: Solve multi-step equations.
    M.8.11.5: Identify properties of operations.

    a.
    M.8.11.6: Identify how many solutions the linear equation may or may not have.
    M.8.11.7: Recall how to solve equations by using substitution.

    b.
    M.8.11.8: Create an equation to represent a real world situation or mathematical problem.
    M.8.11.9: Analyze the solution in context of a real world problem.

    Prior Knowledge Skills:
    • Recognize properties of numbers (Distributive, Associative, Commutative).
    • Define equation, inequality, and variable.
    • Set up equations and inequalities to represent the given situation, using correct mathematical operations and variables.
    • Calculate a solution or solution set by combining like terms, isolating the variable, and/or using inverse operations.
    • Test the found number or number set for accuracy by substitution.
    • Recall solving one step equations and inequalities.
    • Recognize properties of numbers (Distributive, Associative, Commutative).
    • Define equation and variable.
    • Set up an equation to represent the given situation, using correct mathematical operations and variables.
    • Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations.
    • Test the found number for accuracy by substitution.

    • Example: Is 5 an accurate solution of 2(x + 5)=12 .
    • Identify the unknown, in a given situation, as the variable.
    • List given information from the problem.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 9
    Learning Activities: 2
    Lesson Plans: 3
    Classroom Resources: 4
    12. Solve systems of two linear equations in two variables by graphing and substitution.

    a. Explain that the solution(s) of systems of two linear equations in two variables corresponds to points of intersection on their graphs because points of intersection satisfy both equations simultaneously.

    b. Interpret and justify the results of systems of two linear equations in two variables (one solution, no solution, or infinitely many solutions) when applied to real-world and mathematical problems.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will generate the given y-value for both equations.
    • Recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have infinitely many solutions.
    • Use substitution to solve a system, given two linear equations in slope-intercept form or one equation in standard form and one in slope-intercept form.
    • Make sense of their solutions by making connections between algebraic and graphical solutions and the context of the system of linear equations.
    Teacher Vocabulary:
    • System of linear equations
    • Point of intersection
    • One solution
    • No solution
    • Infinitely many solutions
    • Parallel lines
    • Slope-intercept form of a linear equation
    • Standard form of a linear equation
    Knowledge:
    Students know:
    • The properties of operations and equality and their appropriate application.
    • Graphing techniques for linear equations (using points, using slope-intercept form, using technology).
    • Substitution techniques for algebraically finding the solution to a system of linear equations.
    Skills:
    Students are able to:
    • generate a table from an equation.
    • Graph linear equations.
    • Identify the ordered pair for the point of intersection.
    • Explain the meaning of the point of intersection (or lack of intersection point) in context.
    • Solve a system algebraically using substitution when both equations are written in slope-intercept form or one is written in standard form and the other in slope-intercept form.
    Understanding:
    Students understand that:
    • any point on a line when substituted into the equation of the line, makes the equation true and therefore, the intersection point of two lines must make both equations true.
    • Graphs and equations of linear relationships are different representations of the same relationships, but reveal different information useful in solving problems, and allow different solution strategies leading to the same solutions.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.12.1: Define variables.
    M.8.12.2: Recall how to estimate.
    M.8.12.3: Recall how to solve linear equations.
    M.8.12.4: Demonstrate how to graph solutions to linear equations.
    M.8.12.5: Recall how to graph ordered pairs on a Cartesian plane.
    M.8.12.6: Recall that linear equations can have one solution (intersecting), no solution (parallel lines), or infinitely many solutions (graph is simultaneous).
    M.8.12.7: Define simultaneous.
    M.8.12.8: Recall how to solve linear equations.
    M.8.12.9: Recall properties of operations for addition and multiplication.
    M.8.12.10: Discover that the intersection of two lines on a coordinate plane is the solution to both equations.
    M.8.12.11: Define point of intersection.
    M.8.12.12: Recall how to solve linear equations.
    M.8.12.13: Demonstrate how to graph on the Cartesian plane.
    M.8.12.14: Identify ordered pairs.
    M.8.12.15: Recall how to solve linear equations in two variables by using substitution.
    M.8.12.16: Create a word problem from given information.
    M.8.12.17: Recall how to solve linear equations.
    M.8.12.18: Explain how to write an equation to solve real-world mathematical problems.

    Prior Knowledge Skills:
    • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
    • Demonstrate an understanding of an extended coordinate plane.
    • Draw a four-quadrant coordinate plane.
    • Draw and extend vertical and horizontal number lines.
    • Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
    • Recall how to graph points in all four quadrants of the coordinate plane.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.12 Solve two-step linear equations where coefficients are less than 10 and answers are integers.


    Explain, evaluate, and compare functions.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 2
    Learning Activities: 1
    Classroom Resources: 1
    13. Determine whether a relation is a function, defining a function as a rule that assigns to each input (independent value) exactly one output (dependent value), and given a graph, table, mapping, or set of ordered pairs.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Define a function as a rule assigning each input exactly one output.
    • Identify functions when given a relation as graph, table of values, mapping, or set of ordered pairs.
    Teacher Vocabulary:
    • Relation
    • Function
    • Input
    • Output
    Knowledge:
    Students know:
    • how to interpret a graph, table, mapping, and ordered pairs.
    Skills:
    Students are able to:
    • give an accurate definition of a function.
    • Analyze graphs, tables, mappings, and sets of ordered pairs to determine if a relation is a function.
    Understanding:
    Students understand that:
    • Functions assign every input one output, but they may see outputs repeat.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.13.1: Define function, ordered pairs, input, output.
    M.8.13.2: Demonstrate how to plot points on a Cartesian plane using ordered pairs.
    M.8.13.3: Recall how to complete input/output tables.
    M.8.13.4: Recognize numeric patterns.
    M.8.13.5: Given a function, create a rule.

    Prior Knowledge Skills:
    • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
    • Demonstrate an understanding of an extended coordinate plane.
    • Draw a four-quadrant coordinate plane.
    • Draw and extend vertical and horizontal number lines.
    • Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
    • Recall how to graph points in all four quadrants of the coordinate plane.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.13 Determine whether a relation is a function given a graph or a table.


    Mathematics (2019)
    Grade(s): 8
    All Resources: 0
    14. Evaluate functions defined by a rule or an equation, given values for the independent variable.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Find the value of outputs when given specific inputs for a function.
    Teacher Vocabulary:
    • Evaluate
    • Input
    • Output
    • Function
    Knowledge:
    Students know:
    • how to apply order of operations.
    • That every input will produce one output for a given function.
    Skills:
    Students are able to:
    • analyze a rule or an equation
    • Substitute given values for the input to produce a desired output.
    Understanding:
    Students should understand that:
    • An output for any function is controlled by the input for that function. This is important to help reinforce/establish the concept of inputs being the independent variable and outputs representing the dependent variable.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.14.1: Define functions, independent variables, and dependent variables.
    M.8.14.2: Evaluate a function rule given the independent variable.

    Prior Knowledge Skills:
    • Define equation and variable.
    • Set up an equation to represent the given situation, using correct mathematical operations and variables.
    • Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations.
    • Test the found number for accuracy by substitution.

    • Example: Is 5 an accurate solution of 2(x + 5)=12.
    • Identify the unknown, in a given situation, as the variable.
    • List given information from the problem.
    • Recalling one-step equations.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.15 Identify linear and nonlinear functions graphically.


    Mathematics (2019)
    Grade(s): 8
    All Resources: 1
    Classroom Resources: 1
    15. Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.

    a. Distinguish between linear and non-linear functions.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Describe the comparison of linear functions qualitatively and quantitatively by discussing and analyzing the rates of change (slopes), initial values (y-intercepts), and any points of intersection.
    • Tell the difference between functions that are linear and those that are non-linear by analyzing information in a variety of contexts.
    Teacher Vocabulary:
    • Function
    • Linear
    • Non-linear
    • Slope
    Knowledge:
    Students know:
    • how to find rates of change and initial values for function represented multiple ways.
    • how to graph functions when given an equation, table, or verbal description.
    Skills:
    Students are able to:
    • identify the differences between functions represented in multiple contexts.
    • Tell the differences between linear and nonlinear functions.
    Understanding:
    Students understand that:
    • Converting to different representations of functions can assist in their comparisons of linear functions qualitatively and quantitatively.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.15.1: Define rate of change.
    M.8.15.2: Recognize linear and nonlinear functions.
    M.8.15.3: Recall how to read/interpret information from a table.
    M.8.15.4: Identify algebraic expressions.
    M.8.15.5: Recall how to name points on a Cartesian plane using ordered pairs.
    M.8.15.6: Compare and contrast the differences between linear and nonlinear functions.

    Prior Knowledge Skills:
    • Define expression, equivalent, and equivalent expressions.
    • Recall mathematical terms such as sum, difference, etc.
    • Recognize that a variable without a written coefficient is understood to have a coefficient of one.
    • Recall how to convert mathematical terms to mathematical symbols and numbers and vice versa.
    • Restate numerical expressions with words.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.15 Identify linear and nonlinear functions graphically.


    Use functions to model relationships between quantities.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 2
    Classroom Resources: 2
    16. Construct a function to model a linear relationship between two variables.

    a. Interpret the rate of change (slope) and initial value of the linear function from a description of a relationship or from two points in a table or graph.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Create the graphical representation of linear function, given a linear equation in slope-intercept form, using the initial value and rate of change.
    • give meaning rates of change and the initial values of linear functions in different contexts.
    Teacher Vocabulary:
    • Function
    • Linear
    • Non-linear
    • Slope
    • y-intercept
    Knowledge:
    Students know:
    • that the rate of change of a function is the ratio of change in the output to the change in the input.
    • how to find the rate of change/slope as well as the initial value/y-intercept.
    Skills:
    Students are able to:
    • construct the graph of a linear function.
    • Identify the slope and y-intercept of functions in different contexts.
    Understanding:
    Students understand that:
    • terms such as slope and y-intercept describe a graphical representation of a linear function and correlate their meaning to the rate of change and initial value, where the input is 0.
    • Using the units from a context appropriately is needed to make their description of rate of change and initial value accurate.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.16.1: Define function, rate of change, and initial value.
    M.8.16.2: Recall how to complete an input/output function table.
    M.8.16.3: Recall how to find the rate of change (slope) in a linear equation.
    M.8.16.4: Recall how to name points from a graph (ordered pairs).
    M.8.16.5: Analyze real-world situations to identify the rate of change and initial value from a table, graph, or description.

    Prior Knowledge Skills:
    • Solve an equation by substituting a value to find an output.
    • Find the coordinates of an ordered pair.
    • Recognize how the steepness of a graphed line changes vertically and horizontally.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 1
    Classroom Resources: 1
    17. Analyze the relationship (increasing or decreasing, linear or non-linear) between two quantities represented in a graph.
    Unpacked Content
    Evidence Of Student Attainment:
    Students: Given graphical representations of functions,
    • Use mathematical reasoning to analyze the graphs and describe the functional relationships between two quantities.
    Teacher Vocabulary:
    • Increasing
    • Decreasing
    • Linear
    • Non-linear
    Knowledge:
    Students know:
    • Characteristics of representations for functions in graphic form.
    Skills:
    Students are able to:
    • Use mathematical vocabulary and understanding of functions to describe relationships between two quantities.
    Understanding:
    Students understand that:
    • functions can be represented in a variety of ways, each of which provides unique perspectives of the relationship between the variables.
    • Graphs of functions are useful to compare characteristics of different relationships.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.17.1: Define qualitative, increase, and decrease.
    M.8.17.2: Distinguish the difference between linear and nonlinear functions.
    M.8.17.3: Recall how to plot points on a Cartesian plane.
    M.8.17.4: Identify parts of the Cartesian plane.
    M.8.17.5: Recognize ordered pairs.
    M.8.17.6: Compare and contrast the relationship between two quantities in a graph.

    Prior Knowledge Skills:
    • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
    • Demonstrate an understanding of an extended coordinate plane.
    • Draw a four-quadrant coordinate plane.
    • Draw and extend vertical and horizontal number lines.
    • Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
    • Recall how to graph points in all four quadrants of the coordinate plane.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.17 Given a simple scatter plot of points in a straight line, describe the relationship between the two quantities.


    Data Analysis, Statistics, and Probability
    Investigate patterns of association in bivariate data.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 0
    18. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities, describing patterns in terms of positive, negative, or no association, linear and non-linear association, clustering, and outliers.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given sets of bivariate measurement data or contextual situations in which bivariate measurement data must be collected,
    • Construct and interpret scatter plots.
    • Describe visual patterns observed, (e.g., clustering, outliers, positive or negative association, linear, and nonlinear association).
    Teacher Vocabulary:
    • Scatter plots
    • Bivariate measurement data
    • Clustering
    • Outliers
    • Positive and negative association
    • No association
    • Linear and nonlinear association
    Knowledge:
    Students know:
    • representations for bivariate data and techniques for constructing each (tables, scatter plots).
    Skills:
    Students are able to:
    • Construct a scatter plot to represent a set of bivariate data.
    • Use mathematical vocabulary to describe and interpret patterns in bivariate data.
    Understanding:
    Students understand that:
    • Using different representations and descriptors of a data set can be useful in seeing important features of the situation being investigated.
    • Negative association in bivariate data can be a very strong association but is an inverse relationship.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.18.1: Define bivariate scatter plot, outlier, cluster, linear, nonlinear, and positive and negative association.
    M.8.18.2: Describe patterns found in a scatter plot.
    M.8.18.3: Demonstrate how to label and plot information on a scatter plot (dot plot).
    M.8.18.4: Distinguish the difference between positive and negative correlation.
    M.8.18.5: Recall how to describe the spread of the scatter plot (dot plot).

    Prior Knowledge Skills:
    • Define numerical data set, measure of variation, and measure of center.
    • Relate the measure of variation, of a data set, with the concept of range.
    • Relate the measure of the center for a numerical data set with the concept of measure of center.
    • Define numerical data set, quantitative, measure of center, median, frequency distribution, and attribute.
    • Compare and contrast the center and variation.
    • Collect the data.
    • Organize the data.
    • Describe how attribute was measured including units of measurement.
    • Identify the attribute used to create the numerical set.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 0
    19. Given a scatter plot that suggests a linear association, informally draw a line to fit the data, and assess the model fit by judging the closeness of the data points to the line.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a variety of scatterplots representing bivariate data,
    • Determine if the plots suggest a linear relationship.
    • Informally fit a straight line to the data.
    • Assess the model fit by judging the closeness of the data points to the line.
    Teacher Vocabulary:
    • Scatter plot
    • Linear association
    • Quantitative variable
    Knowledge:
    Students know:
    • Patterns found on scatter plots of bivariate data, (linear/non-linear, positive/negative).
    • Strategies for informally fitting straight lines to bivariate data with a linear relationship.
    • Methods for finding the distance between two points on a coordinate plane and between a point and a line.
    Skills:
    Students are able to:
    • Construct a scatter plot to represent a set of bivariate data.
    • Use mathematical vocabulary to describe and interpret patterns in bivariate data.
    • Use logical reasoning and appropriate strategies to draw a straight line to fit data that suggest a linear association.
    • Use mathematical vocabulary, logical reasoning, and closeness of data points to a line to judge the fit of the line to the data.
    Understanding:
    Students understand that:
    • Using different representations and descriptors of a data set can be useful in seeing important features of the situation being investigated.
    • When visual examination of a scatter plot suggests a linear association in the data, fitting a straight line to the data can aid in interpretation and prediction.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.19.1: Define scatter plot, outlier, linear, quantitative, line of best fit, and variable.
    M.8.19.2: Analyze scatter plots to determine line of best fit.
    M.8.19.3: Explain how to draw informal inferences from data distributions.
    M.8.19.4: Recall how to summarize numerical data sets in relation to their context.
    M.8.19.5: Recognize the concept of outlier and its relationship to the data distribution.
    M.8.19.6: Draw an estimate for a line of best fit.

    Prior Knowledge Skills:
    • Define numerical data set, measure of variation, and measure of center.
    • Relate the measure of variation, of a data set, with the concept of range.
    • Relate the measure of the center for a numerical data set with the concept of measure of center.
    • Define numerical data set, quantitative, measure of center, median, frequency distribution, and attribute.
    • Compare and contrast the center and variation.
    • Collect the data.
    • Organize the data.
    • Describe how attribute was measured including units of measurement.
    • Identify the attribute used to create the numerical set.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 0
    20. Use a linear model of a real-world situation to solve problems and make predictions.

    a. Describe the rate of change and y-intercept in the context of a problem using a linear model of a real-world situation.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a contextual or mathematical situation involving bivariate measurement data,
    • Represent the situation graphically and algebraically, describe the relationship between the two models, and interpret the slope and the y-intercept of the line in order to find answers to questions.
    • Use the equation and graph to make predictions about unobserved data in context.
    Teacher Vocabulary:
    • Linear model
    • Bivariate measurement data
    • Slope
    • y-intercept
    Knowledge:
    Students know:
    • strategies for determining slope and y-intercept of a linear model.
    Skills:
    Students are able to:
    • Represent contextual and mathematical situations involving bivariate measurement data with a linear relationship algebraically and graphically.
    • Use mathematical vocabulary to describe and interpret slopes and y-intercepts of lines which represent contextual situations involving bivariate data.
    • Make predictions about unobserved data using the equation and graph.
    Understanding:
    Students understand that:
    • Modeling bivariate data with scatter plots and fitting a straight line to the data can aid in interpretation of the data and predictions about unobserved data.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.20.1: Define slope, intercept, linear, equation, and bivariate.
    M.8.20.2: Recall how to determine the rate of change (slope) from a graph.
    M.8.20.3: Identify the parts of the slope-intercept form of an equation.
    M.8.20.4: Recognize how to read a graph.
    M.8.20.5: Recall how to write an equation in slope-intercept form.
    M.8.20.6: Apply the identification of the slope and the y-intercept to a real-world situation.
    M.8.20.7: Create a graph to model a real-word situation.

    Prior Knowledge Skills:
    • Define equation and variable.
    • Set up an equation to represent the given situation, using correct mathematical operations and variables.
    • Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations.
    • Test the found number for accuracy by substitution.
      Example: Is 5 an accurate solution of 2(x + 5)=12?.
    • Identify the unknown, in a given situation, as the variable.
    • List given information from the problem.
    • Recalling one-step equations.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 0
    21. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects, using relative frequencies calculated for rows or columns to describe possible associations between the two variables.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    Given a contextual or mathematical situation involving bivariate categorical data,
  • 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.
  • Teacher Vocabulary:
    • Two-way table
    • Rows
    • Columns
    • Bivariate categorical data
    • Frequencies
    • Relative frequencies
    • Categorical variables
    Knowledge:
    Students know:
    • Characteristics of data sets that distinguish categorical data from measurement data.
    Skills:
    Students are able to:
    • Construct two-way tables for categorical data.
    • Find relative frequencies for cells in the two-way tables.
    • Conjecture about patterns of association in the two-way tables and explain the reasoning that leads to the conjecture.
    Understanding:
    Students understand that:
    • organizing categorical data in two-way tables can aid in identifying patterns of association in the data.
    • Relative frequencies, rather than just absolute frequencies, need to be calculated from two-way tables to identify patterns of association.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.21.1: Define relative frequency and frequency.
    M.8.21.2: Design a two-way table.
    M.8.21.3: Analyze a two-way table containing categorical variables.
    M.8.21.4: Calculate relative frequency.
    M.8.21.5: Discuss relative frequency.
    M.8.21.6: Design a table.
    M.8.21.7: Recall how to calculate frequency.
    M.8.21.8: Recall how to collect data.

    Prior Knowledge Skills:
    • Define numerical data set, quantitative, measure of center, median, frequency distribution, and attribute.
    • Compare and contrast the center and variation.
    • Collect the data.
    • Organize the data.
    • Describe how attribute was measured including units of measurement.
    • Identify the attribute used to create the numerical set.
    Geometry and Measurement
    Understand congruence and similarity using physical models or technology.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 2
    Classroom Resources: 2
    22. Verify experimentally the properties of rigid motions (rotations, reflections, and translations): lines are taken to lines, and line segments are taken to line segments of the same length; angles are taken to angles of the same measure; and parallel lines are taken to parallel lines.

    a. Given a pair of two-dimensional figures, determine if a series of rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are congruent; describe the transformation sequence that verifies a congruence relationship.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Confirm characteristics of the figures. such as lengths of line segments, angle measures, and parallel lines as they develop a definition for congruent figures.
    • Use mathematical vocabulary to distinguish between a pair of congruent figures, noting that the figure prior to the transformation is called the preimage and the post-transformation figure is called the image.
    • Examine two figures to identify the rigid transformation(s) that produced the image from the pre-image. they can recognize the symbol for congruency (≅) and write statements of congruence.
    Teacher Vocabulary:
    • Congruent
    • Rotation
    • Reflection
    • Translation
    Knowledge:
    Students know:
    • How to measure line segments and angles.
    • That similar figures have congruent angles.
    • The definition/concept of what a figure does when it undergoes a rotation, reflection, and translation.
    • How to perform a translation, reflection, and rotation.
    Skills:
    Students are able to:
    • verify by measuring and comparing lengths of a figure and its image that after a figure has been translated, reflected, or rotated its corresponding lines and line segments remain the same length.
    Understanding:
    Students understand that:
    • congruent figures have the same shape and size.
    • Two figures in the plane are said to be congruent if there is a sequence of rigid motions that takes one figure onto the other.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.22.1: Define rotation, reflection, and translation.
    M.8.22.2: Recognize translations (slides), rotations (turns), and reflections (flips).
    M.8.22.3: Distinguish between lines and line segments.
    M.8.22.4: Demonstrate how to measure length.
    M.8.22.5: Demonstrate how to use a protractor to measure angles.
    M.8.22.6: Identify parallel lines.
    M.8.22.7: Define congruent and sequence.
    M.8.22.8: Compare translations to reflections.
    M.8.22.9: Compare reflections to rotations.
    M.8.22.10: Compare rotations to translations.
    M.8.22.11: Identify attributes of two-dimensional figures.
    M.8.22.12: Identify congruent figures.

    Prior Knowledge Skills:
    • Define ordered pairs.
    • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
    • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
    • Identify which signs indicate the location of a point in a coordinate plane.
    • Recall how to plot ordered pairs on a coordinate plane.
    • Define reflections.
    • Define reflections.
    • Calculate the distances between points having the same first or second coordinate using absolute value.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.22 Identify 3 different transformations (e.g., reflection, rotation, translation).


    Mathematics (2019)
    Grade(s): 8
    All Resources: 1
    Classroom Resources: 1
    23. Use coordinates to describe the effect of transformations (dilations, translations, rotations, and reflections) on two-dimensional figures.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Describe the changes occurring to the x-and y-coordinates of a figure after a transformation.
    Teacher Vocabulary:
    • Coordinates
    • Congruent
    • Rotation
    • Reflection
    • Translation
    • Dilation
    • Scale factor
    Knowledge:
    Students know:
    • What it means to translate, reflect, rotate, and dilate a figure.
    • How to perform a translation, reflection, rotation, and dilation of a figure.
    • How to apply (x, y) notation to describe the effects of a transformation.
    Skills:
    Students are able to:
    • Select and apply the proper coordinate notation/rule when given a specific transformation for a figure.
    • Graph a pre-image/image for a figure on a coordinate plane when given a specific transformation or sequence of transformations.
    Understanding:
    Students understand that:
    • the use of coordinates is also helpful in proving the congruence/proportionality between figures.
    • The relationships between coordinates of a preimage and its image for dilations represent scale factors learned in previous grade levels.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.23.1: Define dilation.
    M.8.23.2: Recall how to find scale factor.
    M.8.23.3: Give examples of scale drawings.
    M.8.23.4: Recognize translations.
    M.8.23.5: Recognize reflections.
    M.8.23.6: Recognize rotations.

    Prior Knowledge Skills:
    • Define scale, scale drawings, length, area, and geometric figures.
    • Locate/use scale on a map.
    • Identify proportional relationships.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.23 Recognize the reflection (across the x or y axis) and translation (across quadrants) of a two dimensional figure on a coordinate plane (limited to non-equilateral rectangles and triangles).


    Mathematics (2019)
    Grade(s): 8
    All Resources: 5
    Classroom Resources: 5
    24. Given a pair of two-dimensional figures, determine if a series of dilations and rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are similar; describe the transformation sequence that exhibits the similarity between them.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Explain how transformations can be used to prove that two figures are similar.
    • Describe a sequence of transformations to prove or disprove that two figures are similar or congruent.
    Teacher Vocabulary:
    • Translation
    • Reflection
    • Rotation
    • Dilation
    • Scale factor
    Knowledge:
    Students know:
    • How to perform rigid transformations and dilations graphically and algebraically (applying coordinate rules).
    • What makes figures similar and congruent.
    Skills:
    Students are able to:
    • Use mathematical language to explain how transformations can be used to prove that two figures are similar or congruent.
    • Demonstrate/perform a series of transformations to prove or disprove that two figures are similar or congruent.
    Understanding:
    Students understand that:
    • There is a proportional relationship between corresponding characteristics of the figures, such as lengths of line segments, and angle measures as they develop a definition for similarity between figures.
    • The coordinate plane can be used as tool because it gives a visual image of the relationship between the two figures.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.24.1: Define similar.
    M.8.24.2: Recognize dilations.
    M.8.24.3: Recognize translations.
    M.8.24.4: Recognize rotations.
    M.8.24.5: Recognize reflections.
    M.8.24.6: Identify similar figures.
    M.8.24.7: Analyze an image and its dilation to determine if the two figures are similar.

    Prior Knowledge Skills:
    • Define ordered pairs.
    • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
    • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
    • Identify which signs indicate the location of a point in a coordinate plane.
    • Recall how to plot ordered pairs on a coordinate plane.
    • Define reflections.
    • Calculate the distances between points having the same first or second coordinate using absolute value.
    Analyze parallel lines cut by a transversal.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 4
    Lesson Plans: 1
    Classroom Resources: 3
    25. Analyze and apply properties of parallel lines cut by a transversal to determine missing angle measures.

    a. Use informal arguments to establish that the sum of the interior angles of a triangle is 180 degrees.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Find missing angles when presented angles formed by a transversal cutting through parallel lines.
    • Write equations to find missing angles when an angle is represented by a variable expression.
    • Prove that all triangles have an interior angle sum of 180 degrees by using given angle relationships that form a triangle.
    Teacher Vocabulary:
    • Transversal
    • Corresponding Angles
    • Vertical Angles
    • Alternate Interior Angles
    • Alternate Interior Angles
    • Supplementary
    • Adjacent
    Knowledge:
    Students know:
    • That a straight angle is 180 degrees
    • That a triangle has three interior angles whose sum is 180 degrees.
    • The definition of transversal.
    • How to write and solve two-step equations.
    Skills:
    Students are able to:
    • Make conjectures about the relationships and measurements of the angles created when two parallel lines are cut by a transversal.
    • Informally prove that the sum of any triangle's interior angles will have the same measure as a straight angle.
    Understanding:
    Students understand that:
    • Missing angle measurements can be found when given just one angle measurement along a transversal cutting through parallel lines.
    • Every exterior angle is supplementary to its adjacent interior angle.
    • Parallel lines cut by a transversal will yield specific angle relationships that are connected to the concepts of rigid transformations (i.e. vertical angles are reflections over a point. corresponding angles can be viewed as translations).
    • The sum of the interior angles of a triangle is 180 degrees.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.25.1: Define exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, corresponding angles, and transversal.
    M.8.25.2: Identify attributes of triangles.
    M.8.25.3: Identify exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, and corresponding angles.
    M.8.25.4: Identify a transversal.
    M.8.25.5: Apply properties to find missing angle measures.
    M.8.25.6: Discover the Angle Sum Theorem (sum of the interior angles of a triangle equal 180 degrees).

    Prior Knowledge Skills:
    • Define supplementary angles, complementary angles, vertical angles, adjacent angles, parallel lines, perpendicular lines, and intersecting lines.
    • Discuss strategies for solving mulit-step problems and equations.
    • Identify all types of angles.
    • Identify right angles and straight angles.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.25 Compare any angle to a right angle using greater than, less than, or congruent to the right angle.


    Understand and apply the Pythagorean Theorem.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 5
    Classroom Resources: 5
    26. Informally justify the Pythagorean Theorem and its converse.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Use mathematical reasoning and vocabulary to verbally explain the Pythagorean Theorem and its converse.
    Teacher Vocabulary:
    • Pythagorean Theorem
    • Converse
    Knowledge:
    Students know:
    • the Pythagorean Theorem
    Skills:
    Students are able to:
    • Use mathematical reasoning and vocabulary to verbally explain a proof of the Pythagorean Theorem and its converse.
    Understanding:
    Students understand that:
    • theorems represent generalizations about geometric relationships that are used to solve problems.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.26.1: Define a right angle, Pythagorean Theorem, converse, and proof.
    M.8.26.2: Recognize examples of right triangles.
    M.8.26.3: Demonstrate how to find square roots.
    M.8.26.4: Solve problems with exponents.

    Prior Knowledge Skills:
    • Define supplementary angles, complementary angles, vertical angles, adjacent angles, parallel lines, perpendicular lines, and intersecting lines.
    • Discuss strategies for solving mulit-step problems and equations.
    • Identify all types of angles.
    • Identify right angles and straight angles.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.26 Identify vertical angles given two parallel lines cut by a transversal.


    Mathematics (2019)
    Grade(s): 8
    All Resources: 3
    Learning Activities: 1
    Lesson Plans: 1
    Classroom Resources: 1
    27. Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Given real-world and mathematical problems that can be represented on a coordinate plane, apply the Pythagorean Theorem in order to solve problems and justify solutions and solution paths for finding side lengths (distances between points) in right triangles within the problem contexts.
    Teacher Vocabulary:
    • Pythagorean Theorem
    Knowledge:
    Students know:
    • The Pythagorean Theorem.
    • The operations and labeling within a coordinate system.
    Skills:
    Students are able to:
    • Solve equations involving one variable and square root.
    • Represent real-world and mathematical contexts involving right triangles in a variety of formats (drawings on coordinate planes, equations).
    • Justify solutions and solution paths using conceptual understandings and vocabulary related to the Pythagorean Theorem (right angle, hypotenuse).
    Understanding:
    Students understand that:
    • The properties of right triangles can be used to solve problems.
    • Theorems represent general relationships that are true for all shapes that fit certain criteria.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.27.1: Recall how to name points on a Cartesian plane using ordered pairs.
    M.8.27.2: Recognize ordered pairs (x, y).
    M.8.27.3: Solve problems using the Pythagorean Theorem, with or without a calculator.
    M.8.27.4: Identify right triangles.
    M.8.27.5: Demonstrate how to find square roots, with or without a calculator.
    M.8.27.6: Solve problems with exponents, with or without a calculator.

    Prior Knowledge Skills:
    • Define area, special quadrilaterals, right triangles, and polygons.
    • Analyze the area of other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
    • Apply area formulas to solve real-world mathematical problems.
    • Demonstrate how the area of a rectangle is equal to the sum of the area of two equal right triangles.
    • Explain how to find the area for rectangles.
    • Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
    • Recognize and demonstrate that two right triangles make a rectangle.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.27 Use the pythagorean theorem to find the hypotenuse when given the measures of two legs in a real-world context. Limit to Pythagorean triples.


    Mathematics (2019)
    Grade(s): 8
    All Resources: 7
    Learning Activities: 1
    Classroom Resources: 6
    28. Apply the Pythagorean Theorem to determine unknown side lengths of right triangles, including real-world applications
    Unpacked Content
    Evidence Of Student Attainment:
    Students: Given real-world and mathematical problems in two and three dimensions,
    • Apply the Pythagorean Theorem in order to solve problems and justify solutions and solution paths for finding side lengths in right triangles within the problem contexts.
    Teacher Vocabulary:
    • Pythagorean Theorem
    Knowledge:
    Students know:
    • The Pythagorean Theorem.
    • Appropriate labeling of a right triangle, leg and hypotenuse.
    Skills:
    Students are able to:
    • Solve equations involving one variable and square root.
    • Represent real-world and mathematical contexts involving right triangles in a variety of formats (drawings, equations).
    • Justify solutions and solution paths using conceptual understandings and vocabulary related to the Pythagorean Theorem (right angle, hypotenuse).
    Understanding:
    Students understand that:
    • the properties of right triangles can be used to solve problems.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.28.1: Discuss strategies for solving real-world and mathematical problems.
    M.8.28.2: Solve problems using the Pythagorean Theorem, with or without a calculator.
    M.8.28.3: Identify right triangles.
    M.8.28.4: Demonstrate how to find square roots, with or without a calculator.
    M.8.28.5: Solve problems with exponents, with or without a calculator.

    Prior Knowledge Skills:
    • Define exponent, numerical expression, algebraic expression, variable, base, power, square of a number, and cube of a number.
    • Compute a numerical expression with exponents, with or without a calculator.
    • Restate exponential numbers as repeated multiplication.
    • Choose the correct value to replace each variable in the expression (Substitution).

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.27 Use the pythagorean theorem to find the hypotenuse when given the measures of two legs in a real-world context. Limit to Pythagorean triples.


    Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
    Note: Students must select and use the appropriate unit for the attribute being measured when determining length, area, angle, time, or volume.
    Mathematics (2019)
    Grade(s): 8
    All Resources: 0
    29. Informally derive the formulas for the volume of cones and spheres by experimentally comparing the volumes of cones and spheres with the same radius and height to a cylinder with the same dimensions.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Verify the relationship between the volume of cones and spheres in comparison to the volume of a cylinder with the same radius and height using experimental evidence.
    Teacher Vocabulary:
    • Radius
    • Pi
    • Volume
    • Cylinder
    • Cone
    • Sphere
    Knowledge:
    Student know:
    • The difference between volume and surface area.
    • That volume is defined as the number of unit cubes needed to create or fill the 3-dimensional figure.
    Skills:
    Students are able to:
    • Find the volume of cones, cylinders, and spheres.
    • Show the relationship between the volume of a cone, a cylinder, and a sphere with the same radius.
    Understanding:
    Students understand that:
    • Volume can be seen as layers of the base for a cylinder, but not for the cone or sphere.
    • When radius and height are equal, one sphere will fill 2/3 of a cylinder and a cone will only take up 1/3 of a cylinder's volume. as a result this is reflected in their formulas for volume.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.29.1: Define volume.
    M.8.29.2: Identify cone, sphere, and cylinder.
    M.8.29.3: Recall the meaning of a radius and diameter.
    M.8.29.4: Compare and contrast cone, sphere, and cylinder.
    M.8.29.5: Derive the formulas for the volume of a cone, cylinder, and sphere.

    Prior Knowledge Skills:
    • Define volume, surface area, triangles, quadrilaterals, polygons, cubes, and right prisms.
    • Discuss strategies for solving real-world mathematical problems.
    • Recall formulas for calculating volume and surface area.
    • Identify the attributes of triangles, quadrilaterals, polygons, cubes, and right prisms.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.27 Use the pythagorean theorem to find the hypotenuse when given the measures of two legs in a real-world context. Limit to Pythagorean triples.


    Mathematics (2019)
    Grade(s): 8
    All Resources: 10
    Classroom Resources: 10
    30. Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems.
    Unpacked Content
    Evidence Of Student Attainment:
    Students:
    • Understand that the application of volume formulas and the relationship between these three formulas can be used in combinations when determining solutions involving real-world cylinders, cones, and spheres.
    Teacher Vocabulary:
    • Radius
    • Pi
    • Volume
    • Cylinder
    • Cone
    • Sphere
    Knowledge:
    Students know:
    • The volume formulas for cylinders, cones, and spheres.
    • That 3.14 is an approximation of pi commonly used in these volume formulas.
    • That composite three dimensional objects in the real world can be created by combining cylinders, cones, and spheres in part or whole.
    Skills:
    Students are able to:
    • Calculate the volume of cones, cylinders, and spheres given in real-world contexts. often times approximating solutions to a specified decimal place.
    • Identify the components of a composite figure as being portions of or whole cylinders, cones, and spheres.
    • Combine the results of calculations to find volume for real-world composite figures.
    Understanding:
    Students understand that:
    • the application of volume formulas and the relationship between these three formulas can be used in combinations when determining solutions involving real-world cylinders, cones, and spheres.
    Diverse Learning Needs:
    Essential Skills:
    Learning Objectives:
    M.8.30.1: Define formula, volume, cone, cylinders, spheres, and height.
    M.8.30.2: Discuss the measure of volume and give examples.
    M.8.30.3: Solve problems with exponents, with or without a calculator.
    M.8.30.4: Recall how to find circumference of a circle, with or without a calculator.
    M.8.30.5: Identify parts of a circle.
    M.8.30.6: Calculate the volume of three-dimensional figures.
    M.8.30.7: Solve real-world problems using the volume formulas for three-dimensional figures, with or without a calculator.

    Prior Knowledge Skills:
    • Define volume, surface area, triangles, quadrilaterals, polygons, cubes, and right prisms.
    • Discuss strategies for solving real-world mathematical problems.
    • Recall formulas for calculating volume and surface area.
    • Identify the attributes of triangles, quadrilaterals, polygons, cubes, and right prisms.
    • Define diameter, radius, circumference, area of a circle, and formula.
    • Identify and label parts of a circle.
    • Recognize the attributes of a circle.
    • Apply the formula of area and circumference to real world mathematical situations.

    Alabama Alternate Achievement Standards
    AAS Standard:
    M.AAS.8.30 Use the formulas for perimeter, area, and volume to solve real-world and mathematical problems (where volume problems are limited to finding the volume of cylinders and rectangular prisms).