# Courses of Study : Mathematics

Proportional Reasoning
Analyze proportional relationships and use them to solve real-world and mathematical problems.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 4 Learning Activities: 3 Classroom Resources: 1
1. Calculate unit rates of length, area, and other quantities measured in like or different units that include ratios or fractions. [Grade 7, 1]
Unpacked Content Evidence Of Student Attainment:
Students:
• Compute a unit rate for ratios that compare quantities with different units.
• Determine the unit rate for a given ratio, including unit rates expressed as a complex fraction.
• Example, if a runner runs mile every hour, a student should be able to write the ratio as a complex fraction.)
Teacher Vocabulary:
• Unit rate
• Ratio
• Unit
• Complex fractions
Knowledge:
Students know:
• What and how to calculate a unit rate to represent a given relationship between quantities.
• Quantities compared in ratios are not always whole numbers but can be represented by fractions or decimals.
• A fraction can be used to represent division.
Skills:
Students are able to:
Compute unit rates associated with ratios of fractional:
• Lengths.
• Areas.
• Quantities measured in like or different units.
Understanding:
Students understand that:
• the two measurements that create a unit rate are always different (miles per gallon, dollars per hour).
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 4 Classroom Resources: 4
2. Represent a relationship between two quantities and determine whether the two quantities are related proportionally.

a. Use equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional.

b. Identify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions.

c. Explain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate. [Grade 7, 2]
Unpacked Content Evidence Of Student Attainment:
Students:
• Decide whether a relationship between two quantities is proportional.
• Recognize that not all relationships are proportional.
• Use equivalent ratios in a table or a coordinate graph to demonstrate a proportional relationship.
• Identify the constant of proportionality when a proportional relationship exists between two quantities.
• Interpret a variety of models (tables, graphs, equations, diagrams and verbal descriptions) to identify the constant of proportionality.
• Explain the meaning of a point (x, y) in the context of a real-world problem.

• Example: If a boy charges \$6 per hour to mow lawns, this relationship can be graphed on the coordinate plane. The point (1, 6) contains the unit rate or constant of proportionality, 6.
Teacher Vocabulary:
• Equivalent ratios proportional
• Coordinate plane
• Ratio table
• Unit rate
• Constant of proportionality
• Equation
• Ordered pair
Knowledge:
Students know:
• (2a) how to explain whether a relationship is proportional.
• (2b) that the constant of proportionality is the same as a unit rate.
• (2b) where the constant of proportionality can be found in a table, graph, equation or diagram.
• (2c) that the constant of proportionality or unit rate can be found on a graph of a proportional relationship where the input value or x-coordinate is 1.
Skills:
Students are able to:
• (2a) model a proportional relationship using a table of equivalent ratios.
• Use a coordinate graph to decide whether a relationship is proportional by plotting ordered pairs and observing whether the graph is a straight line through the origin.
• (2b) translate a written description of a proportional relationship into a table, graph, equation or diagram.
• Read and interpret these to find the constant of proportionality.
• (2c) model a proportional relationship using coordinate graphing.
• Explain the meaning of the point (1, r), where r is the unit rate or constant of proportionality.
Understanding:
Students understand that:
• (2a) a proportional relationship requires equivalent ratios between quantities. Students understand how to decide whether two quantities are proportional.
• (2b) the constant of proportionality is the unit rate. Students are able to identify the constant of proportionality for a proportional relationship and explain its meaning in a real-world context.
• (2c) the context of a problem can help them interpret a point on a graph of a proportional relationship.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 8 Learning Activities: 1 Classroom Resources: 7
3. Solve multi-step percent problems in context using proportional reasoning, including simple interest, tax, gratuities, commissions, fees, markups and markdowns, percent increase, and percent decrease. [Grade 7, 3]
Unpacked Content Evidence Of Student Attainment:
Students:
• Use proportional reasoning strategies including setting up and solving proportions to solve problems involving simple interest, tax, gratuities, commissions, fees, markups, percent increase, markdowns or percent decrease.

• Example: Students might be asked to "order" from a menu for lunch then calculate the tax and gratuities to determine the total cost.
Teacher Vocabulary:
• Proportion
• Simple interest
• Tax
• Gratuities
• Commissions
• Fees
• Markups and markdowns
• Percent increase
• Percent decrease
Knowledge:
Students know:
• how to interpret a real-world problem to determine what is being asked.
• how to interpret the solution in the context of the problem.
Skills:
Students are able to:
• Write and solve proportions to help them solve real-world problems involving percent.
• Solve problems that require them to calculate: simple interest, tax, gratuities, commission, fees, mark ups, markdowns, percent increase and percent decrease.
Understanding:
Students understand that:
• proportional reasoning requires interpretation or making sense of percent problems.
• Solving problems and determining their calculated answers may require further computation.
Diverse Learning Needs:
Analyze the relationship between proportional and non-proportional situations.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 3 Learning Activities: 3
4. Determine whether a relationship between two variables is proportional or non-proportional. [Grade 8, 7]
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:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
5. 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. [Grade 8, 8]
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
• Constant of Proportionality
• Proportionality
• Dependent variable
• Independent 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.
• The 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:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 2 Learning Activities: 1 Classroom Resources: 1
6. 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. [Grade 8, 9]
Unpacked Content Evidence Of Student Attainment:
Students:
• can 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.
• Can 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.
• Can explain why any two points on a given line will have the same slope.
• Can 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 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 you will have lines with varying y-intercepts. even when the slope is the same.
• you can use 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:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
7. 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. [Grade 8, 10]
Unpacked Content Evidence Of Student Attainment:
Students:
• Analyze and explain the difference between proportional and non-proportional linear 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:
Number Systems and Operations
Apply and extend prior knowledge of addition, subtraction, multiplication, and division to operations with rational numbers.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 11 Learning Activities: 5 Classroom Resources: 6
8. Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals.

a. Identify and explain situations where the sum of opposite quantities is 0 and opposite quantities are defined as additive inverses.

b. Interpret the sum of two or more rational numbers, by using a number line and in real-world contexts.

c. Explain subtraction of rational numbers as addition of additive inverses.

d. Use a number line to demonstrate that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

e. Extend strategies of multiplication to rational numbers to develop rules for multiplying signed numbers, showing that the properties of the operations are preserved.

f. Divide integers and explain that division by zero is undefined. Interpret the quotient of integers (with a non-zero divisor) as a rational number.

g. Convert a rational number to a decimal using long division, explaining that the decimal form of a rational number terminates or eventually repeats. [Grade 7, 4]
Unpacked Content Evidence Of Student Attainment:
Students:
• Apply their knowledge of addition and subtraction of rational numbers to describe real-world contexts.
• Use physical and visual models to add and subtract integers.
• Add and subtract rational numbers.
• Model multiplication and division of rational numbers.
• Apply the distributive property to rational numbers.
• Convert rational numbers to a decimal using long division to determine if the result is terminating or repeating.
Teacher Vocabulary:
• Integers
• Rational numbers
• opposite quantities
• Absolute value
• Terminating decimals
• Repeating decimals
Knowledge:
Students know:
• a number and its opposite have a sum of 0.
• A number and its opposite are called additive inverses.
• properties of operations.
• Absolute value represents distance on a number line, therefore it is always non-negative.
• Every quotient of integers (with non-zero divisor) is a rational number.
• If p and q are integers, then -(p/q) = (-p)/q = p/(-q).
• The decimal form of a rational number terminates in 0s or eventually repeats.
Skills:
Students are able to:
• add rational numbers.
• Subtract rational numbers.
• Represent addition and subtraction on a number line diagram.
• Describe situations in which opposite quantities combine to make 0.
• Find the opposite of a number.
• Interpret sums of rational numbers by describing real-world contexts.
• Show that the distance between two rational numbers on the number line is the absolute value of their difference.
• Use absolute value in real-world contexts involving distances.
• Multiply and divide rational numbers.
• Convert a rational number to a decimal using long division.
Understanding:
Students understand that:
• p + q is the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative.
• Subtraction of rational numbers is the same as adding the additive inverse, p - q = p + (-q).
• If a factor is multiplied by a number greater than one, the answer is larger than that factor.
• If a factor is multiplied by a number between 0 and 1, the answer is smaller than that factor.
• Multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers.
• Integers can be divided, provided that the divisor is not zero.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 4 Learning Activities: 3 Classroom Resources: 1
9. Solve real-world and mathematical problems involving the four operations of rational numbers, including complex fractions. Apply properties of operations as strategies where applicable. [Grade 7, 5]
Unpacked Content Evidence Of Student Attainment:
Students:
• Apply knowledge of addition and subtraction of rational numbers to describe real-world contexts.
• Solve multi-step problems using numerical expressions that involve addition, subtraction, multiplication, and/or division of rational numbers, including problems that involve complex fractions.
Teacher Vocabulary:
• Rational numbers
• Complex fractions
• properties of operations
Knowledge:
Students know:
• Include situations involving elevation, temperature changes, debits and credits, and proportional relationships with negative rates of change.
• Develop greater fluency with evaluating numerical expressions, using the properties of operations to increase their flexibility in approach.
Skills:
Students are able to:
• Interpret products and quotients of rational numbers by describing real-world contexts.
• Solve real-world and mathematical problems involving the four operations with rational numbers.
Understanding:
Students understand that:
• it important to be able to write numeric expressions in multiple ways.
• Rational numbers can represent values in real-world situations.
• properties of operations learned with whole numbers in elementary apply to rational numbers
Diverse Learning Needs:
Understand that the real number system is composed of rational and irrational numbers.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
10. 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 in a pattern or terminates.

b. Convert a decimal expansion that repeats in a pattern into a rational number. [Grade 8, 1]
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:
• 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.
• Every real number has a decimal expansion that repeats, terminates, or is both non-repeating and non-terminating.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
11. Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of irrational numbers. [Grade 8, 2]
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:
Algebra and Functions
Create equivalent expressions using the properties of operations.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 3 Learning Activities: 3
12. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. [Grade 7, 6]
Unpacked Content Evidence Of Student Attainment:
Students:
• Factor a linear expression with an integer as the greatest common factor.
• Interpret the parts of an expression, such as the coefficient, constant, term, and variable, based on the context of problem.
Teacher Vocabulary:
• Term
• like terms
• Constant
• Factor
• Expression
• Rational coefficient
Knowledge:
Students know:
• How to add, subtract, multiply, and divide rational numbers.
• A(b + c) = ab + ac.
• how to find the greatest common factor of two or more terms.
Skills:
Students are able to:
• Apply properties of operations as strategies to add and subtract linear expressions with rational coefficients.
• Apply properties of operations as strategies to factor linear expressions with rational coefficients.
• Apply properties of operations as strategies to expand linear expressions with rational coefficients.
Understanding:
Students understand that:
• Only like terms can be combined, e.g., x + y = x + y but x + x = 2x.
• To factor an expression, one must factor out the greatest common factor.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 1 Learning Activities: 1
13. Generate expressions in equivalent forms based on context and explain how the quantities are related. [Grade 7, 7]
Unpacked Content Evidence Of Student Attainment:
Students:
Write an expression for a situation and determine equivalent expressions for the same situation.
• Interpret and apply problem solving strategies in solving multi-step real-world word problems.
Teacher Vocabulary:
• Like terms
• Equivalent expressions
• Distributive property
• Factor
Knowledge:
Students know:
• Properties of operations can be used to identify or create equivalent linear expressions.
• Equivalent expressions can reveal real-world and mathematical relationships, and some forms of equivalent expressions can provide more insight than others.
Skills:
Students are able to:
• Determine whether two expressions are equivalent.
• Rewrite expressions into equivalent forms by combining like terms, using the distributive property, and factoring.
Understanding:
Students understand that:
• Rewriting an expression in different forms in a problem context can clarify the problem.
• Rewriting an expression can clarify how the quantities in the problem are related.
Diverse Learning Needs:
Apply concepts of rational and integer exponents.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 8 Learning Activities: 3 Classroom Resources: 5
14. Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions. [Grade 8, 3]
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. i.e. 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:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
15. 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. [Grade 8, 4]
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:
• 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:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 7 Learning Activities: 6 Classroom Resources: 1
16. Express and compare very large or very small numbers in scientific notation. [Grade 8, 5]

a. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used, expressing answers in scientific notation. [Grade 8, 6]

b. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. [Grade 8, 6a]

c. Interpret scientific notation that has been generated by technology. [Grade 8, 6b]
Unpacked Content Evidence Of Student Attainment:
Students:
• Rewrite numbers using scientific notation.
• Use use numbers in scientific notation to estimate measurements and values.
• 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 base ten system and is the reason a 10 is used as the base number.
• Raising or lowering an exponent is has an effect on the place value of the decimal expansion.
• That scientific notation is formed using a base ten system.
• how to apply laws for multiplying and dividing exponents
Skills:
Students are able to:
• Write numbers in standard notation 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.
• 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:
• 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).
• 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:
Solve real-world and mathematical problems using numerical and algebraic expressions, equations, and inequalities.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 3 Learning Activities: 3
17. Solve multi-step real-world and mathematical problems involving rational numbers (integers, signed fractions, and decimals), converting between forms as needed. Assess the reasonableness of answers using mental computation and estimation strategies. [Grade 7, 8]
Unpacked Content Evidence Of Student Attainment:
Students:
• Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form.
• Interpret solutions of problems with rational numbers in the context of the problem.
• Assess the reasonableness of answers using mental computation and estimation strategies.
• Use variables to represent quantities in a real-world or mathematical problem.
Teacher Vocabulary:
• Rational numbers
• Integers
• Estimation
Knowledge:
Students know:
• Multiplying by 0.1 is the same as multiplying by 1/10 , the value of 10 percent can also be found by simply dividing by 10.
• 1% = 0.01 = 1 100 .
• Since multiplying by 0.01 is the same as multiplying by 1/100 , the value of 1 percent can also be found by simply dividing by 100.
• Adding a percent of a number onto the original number is the same thing as adding that percent to 100 and then finding that new percent of the number.
• Finding more than 100% of a number must yield an answer that is larger than the original number.
Skills:
Students are able to:
• Convert between different forms of a rational number.
• Add, subtract, multiply and divide rational numbers.
Understanding:
Students understand that:
• One form of a number may be more advantageous to use in a problem context than another form.
• Using estimation strategies helps to determine the reasonableness of answers.
• Finding one percent or ten percent of a number can facilitate solving percent problems.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 3 Classroom Resources: 3
18. Use variables to represent quantities in a real-world or mathematical problem and construct algebraic expressions, equations, and inequalities to solve problems by reasoning about the quantities.

a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem.
[Grade 7, 9, and linear portion of Algebra I with Probability, 11]
Unpacked Content Evidence Of Student Attainment:
Students:
• Write and solve mathematical equations (or inequalities) to model real-world problems.
• Interpret the solution to an equation in the context of a problem.
• Interpret the solution set of an inequality in the context of a problem.
Teacher Vocabulary:
• Algebraic expressions
• Equations
• Inequalities
• Greater than
• Greater than or equal to
• less than
• less than or equal to
Knowledge:
Students know:
• p(x + q) = px + pq, where p and q are specific rational numbers.
• When multiplying or dividing both sides of an inequality by a negative number, every term must change signs and the inequality symbol reversed.
• In the graph of an inequality, the endpoint will be a closed circle indicating the number is included in the solution set (≤ or ≥) or an open circle indicating the number is not included in the solution set ( < or >).
Skills:
Students are able to:
• Use variables to represent quantities in a real-world or mathematical problem.
• Construct equations (px + q = r and p(x + q) = r) to solve problems by reasoning about the quantities.
• Construct simple inequalities (px + q > r or px + q < r) to solve problems by reasoning about the quantities.
• Graph the solution set of an inequality.
Understanding:
Students understand that:
• Real-world problems can be represented through algebraic expressions, equations, and inequalities.
• The inequality symbol reverses when multiplying or dividing both sides of an inequality by a negative number, and why.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
19. Create equations in two variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear functions. [Algebra I with Probability, 12 partial]
Unpacked Content Evidence Of Student Attainment:
Students:
• Create two variable equations. graph them based on various representations, such as verbal descriptions, and use them to solve problems.
• Graph equations on coordinate axes with labels and scales clearly labeling the axes defining what the values on the axes represent and the unit of measure.
Teacher Vocabulary:
• System of equations
• Scale
• Linear Function
Knowledge:
Students know:
• How to construct a linear function that models the relationship between two quantities.
• Graph linear equations.
• That the graph of a function is the set of ordered pairs consisting of input and a corresponding output.
• That the graph of a two-variable equation represents the set of all solutions to the equation.
Skills:
Students are able to:
• Create equations in two variables from tables or verbal descriptions.
• Graph the relationship between two variable equations. and use graph to recognize key features of the graph.
Understanding:
Students understand that:
• Why their equations were created and how they relate to the given real-world context.
• Scaling of coordinate axes needs to appropriate to the context given.
• Graphs can be used to make predictions about possible solutions to a two variable equation or system.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 1 Learning Activities: 1
20. Represent constraints by equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear. [Algebra I with Probability, 13 partial]
Unpacked Content Evidence Of Student Attainment:
Students:
• Create a system of linear equations or a system of linear inequalities that model a real-world situation. The system can include inequalities that limit the domain and range.
• Describe the origins of created equations and demonstrate its relation to the context given.
Teacher Vocabulary:
• System of equations
• Scale
• Linear Function
• Constraint
Knowledge:
Students know:
• How to construct a linear function that models the relationship between two quantities.
• how to graph two variable equations using appropriate scale.
• how to interpret a graph of two variable equations in context.
Skills:
Students are able to:
• Write and graph a system of linear equations or inequalities based on real-world context.
• Interpret the solutions to equations or inequalities as a viable or nonviable answer based on models created from the equations or inequalities.
Understanding:
Students understand that:
• The solutions to linear systems of equations/inequalities, while sometimes infinite in theory, can be limited to have more realistic meaning in real-world contexts.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 5 Learning Activities: 1 Classroom Resources: 4
21. 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. [Grade 8, 11]
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 equations 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:
Explain, evaluate, and compare functions.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 1 Learning Activities: 1
22. Identify the effect on the graph of replacing f(x) by f(x) + k, k · f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and explain the effects on the graph using technology, where appropriate. Limit to linear functions. [Algebra I with Probability, 23]
Unpacked Content Evidence Of Student Attainment:
Students:
• Find k values when given the graphs of linear functions or when expressed in the forms f(x) + k. k(f(x)). f (kx). and f (x +k).
• Use appropriate technology to experiment with these cases to verify the effects of manipulating k values for linear functions.
Teacher Vocabulary:
• Linear function
• Slope
• y-intercept
Knowledge:
Students know:
• Linear relationships have input and output values that have an associated graph, including a y-intercept.
• parallel lines have the same slope but different y-intercepts.
Skills:
Students are able to:
• Compare functions with the same slopes graphically while manipulating k values.
• Explore functions with a calculator or graphing software to develop a relationship between the coefficient on x and the slope.
Understanding:
Students understand that:
• Linear functions can shift based on factors other than the independent variable.
• The shift of a function is not the same as the stretch of a function.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 1 Classroom Resources: 1
23. Construct a function to model the 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 from two points in a table or graph. [Grade 8, 16]
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:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 1 Classroom Resources: 1
24. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). Limit to linear equations. [Algebra I with Probability, 19]
Unpacked Content Evidence Of Student Attainment:
Students:
• Use mathematical language to tell why the x-coordinates for the point of intersection of the equations y=f(x) and y=g(x) are the solutions to f(x) = g(x).
• Justify their reasoning based on previous experiences with systems of linear equations.
Teacher Vocabulary:
• x-intercepts
• y-intercepts
• Point of intersection
• One solution
Knowledge:
Students know:
• That a point of intersection between two linear functions represents one solution to those functions.
Skills:
Students are able to:
• Use mathematical language to explain why the x-coordinates are the same at intersection for y = f(x) and y = g(x).
Understanding:
Students understand that:
• That in cases of a system of linear equations, there is sometimes only one, common point for each one that yields a solution. This is different from previous experiences with single linear equations where every point on its line is a solution set.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
25. Find approximate solutions by graphing the functions, making tables of values, or finding successive approximations, using technology where appropriate.
Note: Include cases where f(x) is linear and g(x) is constant or linear. [Algebra I with Probability, 19 edited]
Unpacked Content Evidence Of Student Attainment:
Students:
• Use graphs and tables to find appropriate solutions for y = f(x) and y = g(x).
Teacher Vocabulary:
• X-intercepts
• y-intercepts
• point of intersection
• one solution
• Approximation
Knowledge:
Students know:
• how to use a table to graph and analyze a function.
• Estimate values between points on a table and graph.
Skills:
Students are able to:
• find and make use of successive approximation as method to solve the system y = f(x) and y = g(x)
Understanding:
Students understand that:
• to be more precise with solutions to a system of linear equations, it is best to make use of successive approximation (even using adequate technology) rather than one estimate based on a single representation.
Diverse Learning Needs:
Data Analysis, Statistics, and Probability
Make inferences about a population using random sampling.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 3 Classroom Resources: 3
26. Examine a sample of a population to generalize information about the population.

a. Differentiate between a sample and a population.

b. Compare sampling techniques to determine whether a sample is random and thus representative of a population, explaining that random sampling tends to produce representative samples and support valid inferences.

c. Determine whether conclusions and generalizations can be made about a population based on a sample.

d. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest, generating multiple samples to gauge variation and make predictions or conclusions about the population.

e. Informally explain situations in which statistical bias may exist. [Grade 7, 10]
Unpacked Content Evidence Of Student Attainment:
Students:
• Distinguish between a population and a sample population, and identify both for statistical questions.
• Understand that a population characteristic is determined using data from the entire population, whereas a sample statistic is determined using data from a sample of the population.
• Describe different ways that data can be collected to answer a statistical question.
• Understand why a sample of a population may be useful or necessary to answer a statistical question.
Teacher Vocabulary:
• Population
• Sample
• biased
• Unbiased
• Sampling techniques
• Random sampling
• Representative samples
• Inferences
Knowledge:
Students know:
• a random sample can be found by various methods, including simulations or a random number generator.
• Samples should be the same size in order to compare the variation in estimates or predictions.
Skills:
Students are able to:
• determine whether a sample is random or not and justify their reasoning.
• Use the center and variability of data collected from multiple same-size samples to estimate parameters of a population.
• Make inferences about a population from random sampling of that population.
• Informally assess the difference between two data sets by examining the overlap and separation between the graphical representations of two data sets.
Understanding:
Students understand that:
• statistics can be used to gain information about a population by examining a sample of the populations.
• Generalizations about a population from a sample are valid only if the sample is representative of that population.
• Random sampling tends to produce representative samples and support valid inferences
• The way that data is collected, organized and displayed influences interpretation.
Diverse Learning Needs:
Make inferences from an informal comparison of two populations.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
27. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. [Grade 7, 11]
Unpacked Content Evidence Of Student Attainment:
Students:
• Determine which measure of center best represents the typical value in the data set.
• Calculate measures of variability (range, interquartile range, and mean absolute deviation), noting how larger values indicate that values are more spread out from the center of the distribution.
Teacher Vocabulary:
• Visual overlap
• Measure of variability
• Data distribution
Knowledge:
Students know:
• Populations can be compared using measures of center and measures of variability
Skills:
Students are able to:
• informally assess the degree of visual overlap of two numerical data distributions with similar variabilities.
• Measure the difference between the centers by expressing it as a multiple of a measure of variability.
Understanding:
Students understand that:
• outliers skew data, which in turn affects the display.
• Measures of center give information about the location of mean, median and mode, whereas measures of variability give information about how spread out the data is.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 1 Classroom Resources: 1
28. Make informal comparative inferences about two populations using measures of center and variability and/or mean absolute deviation in context. [Grade 7, 12]
Unpacked Content Evidence Of Student Attainment:
Students:
• Use measures of center for numerical data from random samples to draw informal comparative inferences about two populations.
• Use measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.
Teacher Vocabulary:
• Mean
• Median
• Mode
• Mean absolute deviation
Knowledge:
Students know:
• measures of center are insufficient to compare populations. measures of variability are necessary to assess if data sets are significantly different or not.
• Mean is the sum of the numerical values divided by the number of values.
• Median is the number that is the midpoint of an ordered set of numerical data.
• Mode is the data value or category occurring with the greatest frequency (there can be no mode, one mode, or several modes).
• Mean absolute deviation of a data set is found by the following steps: 1) calculate the mean 2) determine the deviation of each variable from the mean 3) divide the sum of the absolute value of each deviation by the number of data points.
• Range is a number found by subtracting the minimum value from the maximum value.
Skills:
Students are able to:
• find the measures of center of a data set.
• Find the interquartile range of a data set and use to compare variability between data sets.
Understanding:
Students understand that:
• outliers skew data, which in turn affects the display.
• Measures of center give information about the location of mean, median and mode, whereas measures of variability give information about how spread out the data is.
• The mean absolute deviation of a data set describes the average distance that points within a data set are from the mean of the data set.
Diverse Learning Needs:
Investigate probability models.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 2 Classroom Resources: 2
29. Use a number between 0 and 1 to represent the probability of a chance event occurring, explaining that larger numbers indicate greater likelihood of the event occurring, while a number near zero indicates an unlikely event. [Grade 7, 13]
Unpacked Content Evidence Of Student Attainment:
Students:
• Accurately describe the likelihood of an event occurring.
• Describe the probability of an event occurring on a scale of 0 to 1 and using appropriate vocabulary based on the scale.
• Categorize and order the probabilities of events by their likelihood.
• Use words like impossible, very unlikely, unlikely, equally likely/unlikely, likely, very likely, and certain to describe the probabilities of events.
Teacher Vocabulary:
• probability
• Event
• Chance
• likely
• Unlikely
• very unlikely
• very likely
• Equally likely
• Impossible
• Certain
Knowledge:
Students know:
• probability is equal to the ratio of favorable number of outcomes to total possible number of outcomes.
• As a number for probability increases, so does the likelihood of the event occurring.
• A probability near 0 indicates an unlikely event.
• A probability around 1/2 indicates an event that is neither unlikely nor likely.
• A probability near 1 indicates a likely event.
• probability is equal to the ratio of favorable number of outcomes to total possible number of outcomes.
Skills:
Students are able to:
• approximate the probability of a chance event.
• Use words like impossible, very unlikely, unlikely, equally likely/unlikely, likely, very likely, and certain to describe the probabilities of events.
Understanding:
Students understand that:
• the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.
• An event that is equally likely or equally unlikely has a probability of about 0.5 or �.
• The sum of the probabilities of an event and its complement must be 1.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 3 Learning Activities: 2 Classroom Resources: 1
30. Define and develop a probability model, including models that may or may not be uniform, where uniform models assign equal probability to all outcomes and non-uniform models involve events that are not equally likely.

a. Collect and use data to predict probabilities of events.

b. Compare probabilities from a model to observe frequencies, explaining possible sources of discrepancy. [Grade 7, 14]
Unpacked Content Evidence Of Student Attainment:
Students:
• Develop uniform (all outcomes have the same probability) and non-uniform (outcomes with different probabilities) probability models and use them to find probabilities of simple events.
• Explain possible sources of discrepancy if the agreement between the probability model and observed frequencies is not good.
• Estimate the probability of an event happening in an experiment.
• Compare the accuracy of estimated probabilities from different experiments to the actual probability.
Teacher Vocabulary:
• Probability model
• Uniform model
• non-uniform model
• observed frequencies
Knowledge:
Students know:
• the probability of any single event can be expressed using terminology like impossible, unlikely, likely, or certain or as a number between 0 and 1, inclusive, with numbers closer to 1 indicating greater likelihood.
• A probability model is a visual display of the sample space and each corresponding probability.
• probability models can be used to find the probability of events.
• A uniform probability model has equally likely probabilities.
• Sample space and related probabilities should be used to determine an appropriate probability model for a random circumstance.
Skills:
Students are able to:
• make predictions before conducting probability experiments, run trials of the experiment, and refine their conjectures as they run additional trials.
• Collect data on the chance process that produces an event.
• Use a developed probability model to find probabilities of events.
• Compare probabilities from a model to observed frequencies
• Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
Understanding:
Students understand that:
• long-run frequencies tend to approximate theoretical probability.
• predictions are reasonable estimates and not exact measures.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
31. Approximate the probability of an event by using data generated by a simulation (experimental probability) and compare it to theoretical probability.

a. Observe the relative frequency of an event over the long run, using simulation or technology, and use those results to predict approximate relative frequency. [Grade 7, 15]
Unpacked Content Evidence Of Student Attainment:
Students:
• Predict the approximate relative frequency of an event given the probability.
• Compare the accuracy of estimated probabilities from different experiments to the actual probability.
• Describe how a single event can be simulated using an experiment.
Teacher Vocabulary:
• Experimental probability
• simulation
• Theoretical probability
• Relative frequency
Knowledge:
Students know:
• relative frequencies for experimental probabilities become closer to the theoretical probabilities over a large number of trials.
• Theoretical probability is the likelihood of an event happening based on all possible outcomes.
• long-run relative frequencies allow one to approximate the probability of a chance event and vice versa.
Skills:
Students are able to:
• approximate the probability of a chance event.
• observe an event's long-run relative frequency.
Understanding:
Students understand that:
• real-world outcomes can be simulated using probability models and tools.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 4 Classroom Resources: 4
32. Find probabilities of simple and compound events through experimentation or simulation and by analyzing the sample space, representing the probabilities as percents, decimals, or fractions.

a. Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams, and determine the probability of an event by finding the fraction of outcomes in the sample space for which the compound event occurred.

b. Design and use a simulation to generate frequencies for compound events.

c. Represent events described in everyday language in terms of outcomes in the sample space which composed the event. [Grade 7, 16]
Unpacked Content Evidence Of Student Attainment:
Students:
• Conduct probability experiments to quantify and interpret likeliness of an event occurring.
• Design and use a simulation to generate frequencies for compound events
• Analyze the results from a simulation of a compound event to estimate the probability of the compound event.
Teacher Vocabulary:
• Tree diagram
• Compound probability
• Simulation
• Sample space
• Event
Knowledge:
Students know:
• how the sample space is used to find the probability of compound events.
• A compound event consists of two or more simple events.
• A sample space is a list of all possible outcomes of an experiment.
• how to make an organized list.
• how to create a tree diagram.
Skills:
Students are able to:
• find probabilities of compound events using organized lists, tables, tree diagrams and simulations
• Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams.
• For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.
• Design a simulation to generate frequencies for compound events.
• Use a designed simulation to generate frequencies for compound events.
Understanding:
Students understand that:
• the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
• A compound event can be simulated using an experiment.
Diverse Learning Needs:
Geometry and Measurement
Construct and describe geometrical figures, analyzing relationships among them.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 1 Classroom Resources: 1
33. Solve problems involving scale drawings of geometric figures including computation of actual lengths and areas from a scale drawing and reproduction of a scale drawing at a different scale. [Grade 7, 17]
Unpacked Content Evidence Of Student Attainment:
Students:
• Solve problems involving scale drawings.
• Use a scale factor to reproduce a scale drawing at a different scale.
• Determine the scale factor for a scale drawing.
Teacher Vocabulary:
• Scale drawing
• Reproduction
• Scale factor
Knowledge:
Students know:
• how to calculate actual measures such as area and perimeter from a scale drawing.
• Scale factor impacts the length of line segments, but it does not change the angle measurements.
• There is a proportional relationship between the corresponding sides of similar figures.
• A proportion can be set up using the appropriate corresponding side lengths of two similar figures.
• If a side length is unknown, a proportion can be solved to determine the measure of it.
Skills:
Students are able to:
• find missing lengths on a scale drawing.
• Use scale factors to compute actual lengths, perimeters, and areas in scale drawings.
• Use a scale factor to reproduce a scale drawing at a different scale.
Understanding:
Students understand that:
• scale factor can enlarge or reduce the size of a figure.
• Scale drawings are proportional relationships.
• Applying a scale factor less than one will shrink a figure.
• Applying a scale factors greater than one will enlarge a figure.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
34. Construct geometric shapes (freehand, using a ruler and a protractor, and using technology) given measurement constraints with an emphasis on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. [Grade 7, 18]
Unpacked Content Evidence Of Student Attainment:
Students:
• Determine if a unique triangle can be made when given three specific conditions of a triangle.
• Explain why three given conditions about a triangle may result in more than one or no triangles.
Teacher Vocabulary:
• Construct
• protractor
• Angle measures
• Constraints
• Acute triangle
• Right triangle
• obtuse triangle
• Isosceles triangle
• Scalene triangle
• Equilateral triangle
Knowledge:
Students know:
• if three side lengths will create a unique triangle or no triangle.
Skills:
Students are able to:
• freehand, draw geometric shapes with given conditions.
• Using a ruler and protractor, draw geometric shapes with given conditions.
• Using technology, draw geometric shapes with given conditions.
• Construct triangles from three measures of angles or sides.
• Identify the conditions that determine a unique triangle, more than one triangle, or no triangle.
Understanding:
Students understand that:
• from their experiences with constructions what conditions are necessary to construct a triangle.
• only certain combinations of angle and side measures will create triangles.
• Constructing a triangle requires a specific relationship between the legs of the triangle and a specific sum between the angles of the triangle.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 2 Learning Activities: 1 Classroom Resources: 1
35. Describe the two-dimensional figures created by slicing three-dimensional figures into plane sections. [Grade 7, 19]
Unpacked Content Evidence Of Student Attainment:
Students:
• Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Teacher Vocabulary:
• Two-dimensional figures
• Three-dimensional solids
• plane sections
Knowledge:
Students know:
• the difference between a two-dimensional and three-dimensional figure.
• The names and properties of two-dimensional shapes.
• The names and properties of three-dimensional solids.
Skills:
Students are able to:
• discover two-dimensional shapes from slicing three-dimensional figures. For example, students might slice a clay rectangular prism from different perspectives to see what two-dimensional shapes occur from each slice.
Understanding:
Students understand that:
• slices the prism from different planes may provide a different two-dimensional shape.
• There are specific two-dimensional shapes result from slicing a three-dimensional figure.
Diverse Learning Needs:
Solve real-world and mathematical problems involving angle measure, area, surface area, and volume.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 6 Classroom Resources: 6
36. Explain the relationships among circumference, diameter, area, and radius of a circle to demonstrate understanding of formulas for the area and circumference of a circle.

a. Informally derive the formula for area of a circle.

b. Solve area and circumference problems in real-world and mathematical situations involving circles. [Grade 7, 20]
Unpacked Content Evidence Of Student Attainment:
Students:
• Solve problems with the circumference and area of a circle.
Teacher Vocabulary:
• Diameter
• Circle
• Area
• Circumference
• π
Knowledge:
Students know:
• the ratio of the circumference of a circle and its diameter is always π.
• The formulas for area and circumference of a circle.
Skills:
Students are able to:
• use the formula for area of a circle to solve problems.
• Use the formula(s) for circumference of a circle to solve problems.
• Give an informal derivation of the relationship between the circumference and area of a circle.
Understanding:
Students understand that:
• area is the number of square units needed to cover a two-dimensional figure.
• Circumference is the number of linear units needed to surround a circle.
• The circumference of a circle is related to its diameter (and also its radius).
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 1 Classroom Resources: 1
37. Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle in a figure. [Grade 7, 21]
Unpacked Content Evidence Of Student Attainment:
Students:
• Find the values of angles using complementary and supplementary angle relationships and equations.
• Identify angle relationships in angle diagrams involving vertical, supplementary, and complementary angles.
• Write equations to represent relationships between known and unknown angle measurements.
• Determine the measures of unknown angles and judge the reasonableness of the measures.
Teacher Vocabulary:
• Supplementary angles
• Complementary angles
• vertical angles
Knowledge:
Students know:
• supplementary angles are angles whose measures add to 180 degrees.
• Complementary angles are angles whose measures add to 90 degrees.
• vertical angles are opposite angles formed when two lines intersect.
• Adjacent angles are non-overlapping angles which share a common vertex and side.
Skills:
Students are able to:
• write a simple equation to find an unknown angle.
• Identify and determine values of angles in complementary and supplementary relationships.
• Identify pairs of vertical angles in angle diagrams.
• Identify pairs of complementary and supplementary angles in angle diagrams.
• Use vertical, complementary, and supplementary angle relationships to find missing angles.
Understanding:
Students understand that:
• vertical angles are the pair of angles formed across from one another when two lines intersect, and that the measurements of vertical angles are congruent.
• Complementary angles are angles whose measures add up to 90o, and supplementary angles are angles whose measures add up to 180o.
• Relationships between angles depends on where the angles are located.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
38. 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. [Grade 8, 25]
Unpacked Content Evidence Of Student Attainment:
Students:
• Can find missing angles when presented angles formed by a transversal cutting through parallel lines.
• Can 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
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).
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 6 Classroom Resources: 6
39. Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right rectangular prisms. [Grade 7, 22]
Unpacked Content Evidence Of Student Attainment:
Students:
• Find efficient ways to determine surface area of right prisms and right pyramids by analyzing the structure of the shapes and their nets.
• Use the formulas for volume of prisms and pyramids to solve multi-step real-world problems.
• Use the formula for volume to find missing measurements of a prism.
Teacher Vocabulary:
• Area
• volume
• Surface area
• Two-dimensional figures
• Three-dimensional solids
• Triangles
• polygons
• Cubs
• Right rectangular prisms
Knowledge:
Students know:
• That volume of any right prism is the product of the height and area of the base.
• The volume relationship between pyramids and prisms with the same base and height.
• The surface area of prisms and pyramids can be found using the areas of triangular and rectangular faces.
Skills:
Students are able to:
• Find the area and perimeter of two-dimensional objects composed of triangles, quadrilaterals, and polygons.
• Use a net of a three-dimensional figure to determine the surface area.
• Find the volume and surface area of pyramids, prisms, or three-dimensional objects composed of cubes, pyramids, and right prisms.
Understanding:
Students understand that:
• Two-dimensional and three-dimensional figures can be decomposed into smaller shapes to find the area, surface area, and volume of those figures.
• The area of the base of a prism multiplied by the height of the prism gives the volume of the prism.
• The volume of a pyramid is 1/3 the volume of a prism with the same base.
Diverse Learning Needs:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 0
40. 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. [Grade 8, 29]
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:
• Pi
• Volume
• Cylinder
• Cone
• Sphere
Knowledge:
Students 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:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 11 Classroom Resources: 11
41. Use formulas to calculate the volumes of three-dimensional figures to solve real-world problems. [Grade 8, 30]
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:
• 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:
Understand congruence and similarity using physical models or technology.
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 3 Classroom Resources: 3
42. 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. [Grade 8, 22]
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:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 3 Classroom Resources: 3
43. Use coordinates to describe the effect of transformations (dilations, translations, rotations, and reflections) on two-dimensional figures. [Grade 8, 23]
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 congruency/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:
 Mathematics (2019) Grade(s): 7 Accelerated All Resources: 7 Classroom Resources: 7
44. 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. [Grade 8, 24]
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: