Name 
Description 
MA.6.AR.1.1:  Given a mathematical or realworld context, translate written descriptions into algebraic expressions and translate algebraic expressions into written descriptions.
Examples: The algebraic expression 7.2x20 can be used to describe the daily profit of a company who makes $7.20 per product sold with daily expenses of $20.  
MA.6.AR.1.2:  Translate a realworld written description into an algebraic inequality in the form of x > a, x < a, x ≥ a or x ≤ a. Represent the inequality on a number line. Clarifications: Clarification 1: Variables may be on the left or right side of the inequality symbol. 
Examples:
Mrs. Anna told her class that they will get a pizza if the class has an average of at least 83 out of 100 correct questions on the semester exam. The inequality g ≥ 83 can be used to represent the situation where students receive a pizza and the inequality g < 83 can be used to represent the situation where students do not receive a pizza.  
MA.6.AR.1.3:  Evaluate algebraic expressions using substitution and order of operations.
Examples: Evaluate the expression , where a=1 and b=15.  
MA.6.AR.1.4:  Apply the properties of operations to generate equivalent algebraic expressions with integer coefficients.
Examples: Example: The expression 5(3x+1) can be rewritten equivalently as 15x+5.
Example: If the expression 2x+3x represents the profit the cheerleading team can make when selling the same number of cupcakes, sold for $2 each, and brownies, sold for $3 each. The expression 5x can express the total profit.
 
MA.6.AR.2.1:  Given an equation or inequality and a specified set of integer values, determine which values make the equation or inequality true or false.Clarifications: Clarification 1: Problems include the variable in multiple terms or on either side of the equal sign or inequality symbol. 
Examples: Determine which of the following values make the inequality x+1<2 true: 4,2,0,1.  
MA.6.AR.2.2:  Write and solve onestep equations in one variable within a mathematical or realworld context using addition and subtraction, where all terms and solutions are integers.Clarifications: Clarification 1: Instruction includes using manipulatives, drawings, number lines and inverse operations.
Clarification 2: Instruction includes equations in the forms x+p=q and p+x=q, where x,p and q are any integer.
Clarification 3: Problems include equations where the variable may be on either side of the equal sign.

Examples: The equations 35+x=17, 17=35+x and 17x=35 can represent the question “How many units to the right is 17 from 35 on the number line?”  
MA.6.AR.2.3:  Write and solve onestep equations in one variable within a mathematical or realworld context using multiplication and division, where all terms and solutions are integers.Clarifications: Clarification 1: Instruction includes using manipulatives, drawings, number lines and inverse operations.
Clarification 2: Instruction includes equations in the forms , where p≠0, and px=q.
Clarification 3: Problems include equations where the variable may be on either side of the equal sign.
 
MA.6.AR.2.4:  Determine the unknown decimal or fraction in an equation involving any of the four operations, relating three numbers, with the unknown in any position.Clarifications: Clarification 1: Instruction focuses on using algebraic reasoning, drawings, and mental math to determine unknowns.
Clarification 2: Problems include the unknown and different operations on either side of the equal sign. All terms and solutions are limited to positive rational numbers.

Examples: Given the equation , x can be determined to be because is more than .  
MA.6.AR.3.1:  Given a realworld context, write and interpret ratios to show the relative sizes of two quantities using appropriate notation: , a to b, or a:b where b ≠ 0.Clarifications: Clarification 1: Instruction focuses on the understanding that a ratio can be described as a comparison of two quantities in either the same or different units.
Clarification 2: Instruction includes using manipulatives, drawings, models and words to interpret parttopart ratios and parttowhole ratios.
Clarification 3: The values of a and b are limited to whole numbers.
 
MA.6.AR.3.2:  Given a realworld context, determine a rate for a ratio of quantities with different units. Calculate and interpret the corresponding unit rate.Clarifications: Clarification 1: Instruction includes using manipulatives, drawings, models and words and making connections between ratios, rates and unit rates.
Clarification 2: Problems will not include conversions between customary and metric systems.

Examples: Tamika can read 500 words in 3 minutes. Her reading rate can be described as which is equivalent to the unit rate of words per minute.  
MA.6.AR.3.3:  Extend previous understanding of fractions and numerical patterns to generate or complete a two or threecolumn table to display equivalent parttopart ratios and parttoparttowhole ratios.Clarifications: Clarification 1: Instruction includes using twocolumn tables (e.g., a relationship between two variables) and threecolumn tables (e.g., parttoparttowhole relationship) to generate conversion charts and mixture charts. 
Examples: The table below expresses the relationship between the number of ounces of yellow and blue paints used to create a new color. Determine the ratios and complete the table.
Yellow (part)  1.5  3   9  Blue (part)  2  4    New Color (whole)    12  21 
 
MA.6.AR.3.4:  Apply ratio relationships to solve mathematical and realworld problems involving percentages using the relationship between two quantities.Clarifications: Clarification 1: Instruction includes the comparison of to in order to determine the percent, the part or the whole. 
Examples: Gerald is trying to gain muscle and needs to consume more protein every day. If he has a protein shake that contain 32 grams and the entire shake is 340 grams, what percentage of the entire shake is protein? What is the ratio between grams of protein and grams of nonprotein?  
MA.6.AR.3.5:  Solve mathematical and realworld problems involving ratios, rates and unit rates, including comparisons, mixtures, ratios of lengths and conversions within the same measurement system.Clarifications: Clarification 1: Instruction includes the use of tables, tape diagrams and number lines.  
MA.6.DP.1.1:  Recognize and formulate a statistical question that would generate numerical data.
Examples: The question “How many minutes did you spend on mathematics homework last night?” can be used to generate numerical data in one variable.  
MA.6.DP.1.2:  Given a numerical data set within a realworld context, find and interpret mean, median, mode and range.Clarifications: Clarification 1: Numerical data is limited to positive rational numbers. 
Examples: The data set {15,0,32,24,0,17,42,0,29,120,0,20}, collected based on minutes spent on homework, has a mode of 0.  
MA.6.DP.1.3:  Given a box plot within a realworld context, determine the minimum, the lower quartile, the median, the upper quartile and the maximum. Use this summary of the data to describe the spread and distribution of the data.Clarifications: Clarification 1: Instruction includes describing range, interquartile range, halves and quarters of the data. 
Examples: The middle 50% of the population can be determined by finding the interval between the upper quartile and the lower quartile.  
MA.6.DP.1.4:  Given a histogram or line plot within a realworld context, qualitatively describe and interpret the spread and distribution of the data, including any symmetry, skewness, gaps, clusters, outliers and the range. 
MA.6.DP.1.5:  Create box plots and histograms to represent sets of numerical data within realworld contexts.Clarifications: Clarification 1: Instruction includes collecting data and discussing ways to collect truthful data to construct graphical representations.
Clarification 2: Within this benchmark, it is the expectation to use appropriate titles, labels, scales and units when constructing graphical representations.
Clarification 3: Numerical data is limited to positive rational numbers.

Examples: The numerical data set {15,0,32,24,0,17,42,0,29,120,0,20}, collected based on minutes spent on homework, can be represented graphically using a box plot.  
MA.6.DP.1.6:  Given a realworld scenario, determine and describe how changes in data values impact measures of center and variation.Clarifications: Clarification 1: Instruction includes choosing the measure of center or measure of variation depending on the scenario.
Clarification 2: The measures of center are limited to mean and median. The measures of variation are limited to range and interquartile range.
Clarification 3: Numerical data is limited to positive rational numbers.
 
MA.6.GR.1.1:  Extend previous understanding of the coordinate plane to plot rational number ordered pairs in all four quadrants and on both axes. Identify the x or yaxis as the line of reflection when two ordered pairs have an opposite x or ycoordinate. 
MA.6.GR.1.2:  Find distances between ordered pairs, limited to the same xcoordinate or the same ycoordinate, represented on the coordinate plane. 
MA.6.GR.1.3:  Solve mathematical and realworld problems by plotting points on a coordinate plane, including finding the perimeter or area of a rectangle.Clarifications: Clarification 1: Instruction includes finding distances between points, computing dimensions of a rectangle or determining a fourth vertex of a rectangle.
Clarification 2: Problems involving rectangles are limited to cases where the sides are parallel to the axes.
 
MA.6.GR.2.1:  Derive a formula for the area of a right triangle using a rectangle. Apply a formula to find the area of a triangle.Clarifications: Clarification 1: Instruction focuses on the relationship between the area of a rectangle and the area of a right triangle.
Clarification 2: Within this benchmark, the expectation is to know from memory a formula for the area of a triangle.
 
MA.6.GR.2.2:  Solve mathematical and realworld problems involving the area of quadrilaterals and composite figures by decomposing them into triangles or rectangles.Clarifications: Clarification 1: Problem types include finding area of composite shapes and determining missing dimensions.
Clarification 2: Within this benchmark, the expectation is to know from memory a formula for the area of a rectangle and triangle.
Clarification 3: Dimensions are limited to positive rational numbers.
 
MA.6.GR.2.3:  Solve mathematical and realworld problems involving the volume of right rectangular prisms with positive rational number edge lengths using a visual model and a formula.Clarifications: Clarification 1: Problem types include finding the volume or a missing dimension of a rectangular prism.  
MA.6.GR.2.4:  Given a mathematical or realworld context, find the surface area of right rectangular prisms and right rectangular pyramids using the figure’s net.Clarifications: Clarification 1: Instruction focuses on representing a right rectangular prism and right rectangular pyramid with its net and on the connection between the surface area of a figure and its net.
Clarification 2: Within this benchmark, the expectation is to find the surface area when given a net or when given a threedimensional figure.
Clarification 3: Problems involving right rectangular pyramids are limited to cases where the heights of triangles are given.
Clarification 4: Dimensions are limited to positive rational numbers.
 
MA.6.NSO.1.1:  Extend previous understanding of numbers to define rational numbers. Plot, order and compare rational numbers.Clarifications: Clarification 1: Within this benchmark, the expectation is to plot, order and compare positive and negative rational numbers when given in the same form and to plot, order and compare positive rational numbers when given in different forms (fraction, decimal, percentage).
Clarification 2: Within this benchmark, the expectation is to use symbols (<, > or =).
 
MA.6.NSO.1.2:  Given a mathematical or realworld context, represent quantities that have opposite direction using rational numbers. Compare them on a number line and explain the meaning of zero within its context.Clarifications: Clarification 1: Instruction includes vertical and horizontal number lines, context referring to distances, temperatures and finances and using informal verbal comparisons, such as, lower, warmer or more in debt.
Clarification 2: Within this benchmark, the expectation is to compare positive and negative rational numbers when given in the same form.

Examples: Jasmine is on a cruise and is going on a scuba diving excursion. Her elevations of 10 feet above sea level and 8 feet below sea level can be compared on a number line, where 0 represents sea level.  
MA.6.NSO.1.3:  Given a mathematical or realworld context, interpret the absolute value of a number as the distance from zero on a number line. Find the absolute value of rational numbers.Clarifications: Clarification 1: Instruction includes the connection of absolute value to mirror images about zero and to opposites.
Clarification 2: Instruction includes vertical and horizontal number lines and context referring to distances, temperature and finances.
 
MA.6.NSO.1.4:  Solve mathematical and realworld problems involving absolute value, including the comparison of absolute value.Clarifications: Clarification 1: Absolute value situations include distances, temperatures and finances.
Clarification 2: Problems involving calculations with absolute value are limited to two or fewer operations.
Clarification 3: Within this benchmark, the expectation is to use integers only.

Examples: Michael has a lemonade stand which costs $10 to start up. If he makes $5 the first day, he can determine whether he made a profit so far by comparing 10 and 5.  
MA.6.NSO.2.1:  Multiply and divide positive multidigit numbers with decimals to the thousandths, including using a standard algorithm with procedural fluency.Clarifications: Clarification 1: Multidigit decimals are limited to no more than 5 total digits.  
MA.6.NSO.2.2:  Extend previous understanding of multiplication and division to compute products and quotients of positive fractions by positive fractions, including mixed numbers, with procedural fluency.Clarifications: Clarification 1: Instruction focuses on making connections between visual models, the relationship between multiplication and division, reciprocals and algorithms.  
MA.6.NSO.2.3:  Solve multistep realworld problems involving any of the four operations with positive multidigit decimals or positive fractions, including mixed numbers.Clarifications: Clarification 1: Within this benchmark, it is not the expectation to include both decimals and fractions within a single problem.  
MA.6.NSO.3.1:  Given a mathematical or realworld context, find the greatest common factor and least common multiple of two whole numbers.Clarifications: Clarification 1: Within this benchmark, expectations include finding greatest common factor within 1,000 and least common multiple with factors to 25.Clarification 2: Instruction includes finding the greatest common factor of the numerator and denominator of a fraction to simplify a fraction. 
Examples: Example: Middleton Middle School’s band has an upcoming winter concert which will have several performances. The bandleader would like to divide the students into concert groups with the same number of flute players, the same number of clarinet players and the same number of violin players in each group. There are a total of 15 students who play the flute, 27 students who play the clarinet and 12 students who play the violin. How many separate groups can be formed?
Example: Adam works out every 8 days and Susan works out every 12 days. If both Adam and Susan work out today, how many days until they work out on the same day again?
 
MA.6.NSO.3.2:  Rewrite the sum of two composite whole numbers having a common factor, as a common factor multiplied by the sum of two whole numbers.Clarifications: Clarification 1: Instruction includes using the distributive property to generate equivalent expressions.  
MA.6.NSO.3.3:  Evaluate positive rational numbers and integers with natural number exponents.Clarifications: Clarification 1: Within this benchmark, expectations include using natural number exponents up to 5.  
MA.6.NSO.3.4:  Express composite whole numbers as a product of prime factors with natural number exponents. 
MA.6.NSO.3.5:  Rewrite positive rational numbers in different but equivalent forms including fractions, terminating decimals and percentages.Clarifications: Clarification 1: Rational numbers include decimal equivalence up to the thousandths place. 
Examples: The number can be written equivalently as 1.625 or 162.5%  
MA.6.NSO.4.1:  Apply and extend previous understandings of operations with whole numbers to add and subtract integers with procedural fluency.Clarifications: Clarification 1: Instruction begins with the use of manipulatives, models and number lines working towards becoming procedurally fluent by the end of grade 6. Clarification 2: Instruction focuses on the inverse relationship between the operations of addition and subtraction. If p and q are integers, then pq=p+(q) and p+q=p(q).  
MA.6.NSO.4.2:  Apply and extend previous understandings of operations with whole numbers to multiply and divide integers with procedural fluency. 
MA.7.AR.1.1:  Apply properties of operations to add and subtract linear expressions with rational coefficients.
Examples: is equivalent to .  
MA.7.AR.1.2:  Determine whether two linear expressions are equivalent.
Examples: Are the expressions and equivalent?  
MA.7.AR.2.1:  Write and solve onestep inequalities in one variable within a mathematical context and represent solutions algebraically or graphically.Clarifications: Clarification 1: Instruction focuses on the properties of inequality. Refer to Properties of Operations, Equality and Inequality (Appendix D).
Clarification 2: Instruction includes inequalities in the forms ;; x±p>q and p±x>q, where p and q are specific rational numbers and any inequality symbol can be represented.
Clarification 3: Problems include inequalities where the variable may be on either side of the inequality symbol.
 
MA.7.AR.2.2:  Write and solve twostep equations in one variable within a mathematical or realworld context, where all terms are rational numbers.Clarifications: Clarification 1: Instruction focuses the application of the properties of equality. Refer to Properties of Operations, Equality and Inequality (Appendix D).
Clarification 2: Instruction includes equations in the forms px±q=r and p(x±q)=r, where p, q and r are specific rational numbers.
Clarification 3: Problems include linear equations where the variable may be on either side of the equal sign.
 
MA.7.AR.3.1:  Apply previous understanding of percentages and ratios to solve multistep realworld percent problems.Clarifications: Clarification 1: Instruction includes discounts, markups, simple interest, tax, tips, fees, percent increase, percent decrease and percent error. 
Examples: Example: 23% of the junior population are taking an art class this year. What is the ratio of juniors taking an art class to juniors not taking an art class?
Example: The ratio of boys to girls in a class is 3:2. What percentage of the students are boys in the class?
 
MA.7.AR.3.2:  Apply previous understanding of ratios to solve realworld problems involving proportions.
Examples: Example: Scott is mowing lawns to earn money to buy a new gaming system and knows he needs to mow 35 lawns to earn enough money. If he can mow 4 lawns in 3 hours and 45 minutes, how long will it take him to mow 35 lawns? Assume that he can mow each lawn in the same amount of time.
Example: Ashley normally runs 10kilometer races which is about 6.2 miles. She wants to start training for a halfmarathon which is 13.1 miles. How many kilometers will she run in the halfmarathon? How does that compare to her normal 10K race distance?
 
MA.7.AR.3.3:  Solve mathematical and realworld problems involving the conversion of units across different measurement systems.
Examples: Clarification 1: Problem types are limited to length, area, weight, mass, volume and money.  
MA.7.AR.4.1:  Determine whether two quantities have a proportional relationship by examining a table, graph or written description.Clarifications: Clarification 1: Instruction focuses on the connection to ratios and on the constant of proportionality, which is the ratio between two quantities in a proportional relationship.  
MA.7.AR.4.2:  Determine the constant of proportionality within a mathematical or realworld context given a table, graph or written description of a proportional relationship.
Examples: Example: A graph has a line that goes through the origin and the point (5,2). This represents a proportional relationship and the constant of proportionality is .
Example: Gina works as a babysitter and earns $9 per hour. She can only work 6 hours this week. Gina wants to know how much money she will make. Gina can use the equation e=9h, where e is the amount of money earned, h is the number of hours worked and 9 is the constant of proportionality.
 
MA.7.AR.4.3:  Given a mathematical or realworld context, graph proportional relationships from a table, equation or a written description.Clarifications: Clarification 1: Instruction includes equations of proportional relationships in the form of y=px, where p is the constant of proportionality.  
MA.7.AR.4.4:  Given any representation of a proportional relationship, translate the representation to a written description, table or equation.Clarifications: Clarification 1: Given representations are limited to a written description, graph, table or equation.
Clarification 2: Instruction includes equations of proportional relationships in the form of y=px, where p is the constant of proportionality.

Examples: Example: The written description, there are 60 minutes in 1 hour, can be represented as the equation m=60h.
Example: Gina works as a babysitter and earns $9 per hour. She would like to earn $100 to buy a new tennis racket. Gina wants to know how many hours she needs to work. She can use the equation , where e is the amount of money earned, h is the number of hours worked and is the constant of proportionality.
 
MA.7.AR.4.5:  Solve realworld problems involving proportional relationships.
Examples: Gordy is taking a trip from Tallahassee, FL to Portland, Maine which is about 1,407 miles. On average his SUV gets 23.1 miles per gallon on the highway and his gas tanks holds 17.5 gallons. If Gordy starts with a full tank of gas, how many times will he be required to fill the gas tank?  
MA.7.DP.1.1:  Determine an appropriate measure of center or measure of variation to summarize numerical data, represented numerically or graphically, taking into consideration the context and any outliers.Clarifications: Clarification 1: Instruction includes recognizing whether a measure of center or measure of variation is appropriate and can be justified based on the given context or the statistical purpose.
Clarification 2: Graphical representations are limited to histograms, line plots, box plots and stemandleaf plots.
Clarification 3: The measure of center is limited to mean and median. The measure of variation is limited to range and interquartile range.
 
MA.7.DP.1.2:  Given two numerical or graphical representations of data, use the measure(s) of center and measure(s) of variability to make comparisons, interpret results and draw conclusions about the two populations.Clarifications: Clarification 1: Graphical representations are limited to histograms, line plots, box plots and stemandleaf plots.
Clarification 2: The measure of center is limited to mean and median. The measure of variation is limited to range and interquartile range.
 
MA.7.DP.1.3:  Given categorical data from a random sample, use proportional relationships to make predictions about a population.
Examples: Example: O’Neill’s Pillow Store made 600 pillows yesterday and found that 6 were defective. If they plan to make 4,300 pillows this week, predict approximately how many pillows will be defective.
Example: A school district polled 400 people to determine if it was a good idea to not have school on Friday. 30% of people responded that it was not a good idea to have school on Friday. Predict the approximate percentage of people who think it would be a good idea to have school on Friday from a population of 6,228 people.
 
MA.7.DP.1.4:  Use proportional reasoning to construct, display and interpret data in circle graphs.Clarifications: Clarification 1: Data is limited to no more than 6 categories.  
MA.7.DP.1.5:  Given a realworld numerical or categorical data set, choose and create an appropriate graphical representation.Clarifications: Clarification 1: Graphical representations are limited to histograms, bar charts, circle graphs, line plots, box plots and stemandleaf plots.  
MA.7.DP.2.1:  Determine the sample space for a simple experiment.Clarifications: Clarification 1: Simple experiments include tossing a fair coin, rolling a fair die, picking a card randomly from a deck, picking marbles randomly from a bag and spinning a fair spinner.  
MA.7.DP.2.2:  Given the probability of a chance event, interpret the likelihood of it occurring. Compare the probabilities of chance events.Clarifications: Clarification 1: Instruction includes representing probability as a fraction, percentage or decimal between 0 and 1 with probabilities close to 1 corresponding to highly likely events and probabilities close to 0 corresponding to highly unlikely events.
Clarification 2: Instruction includes P(event) notation. Clarification 3: Instruction includes representing probability as a fraction, percentage or decimal.
 
MA.7.DP.2.3:  Find the theoretical probability of an event related to a simple experiment.Clarifications: Clarification 1: Instruction includes representing probability as a fraction, percentage or decimal.
Clarification 2: Simple experiments include tossing a fair coin, rolling a fair die, picking a card randomly from a deck, picking marbles randomly from a bag and spinning a fair spinner.
 
MA.7.DP.2.4:  Use a simulation of a simple experiment to find experimental probabilities and compare them to theoretical probabilities.Clarifications: Clarification 1: Instruction includes representing probability as a fraction, percentage or decimal.
Clarification 2: Instruction includes recognizing that experimental probabilities may differ from theoretical probabilities due to random variation. As the number of repetitions increases experimental probabilities will typically better approximate the theoretical probabilities.
Clarification 3: Experiments include tossing a fair coin, rolling a fair die, picking a card randomly from a deck, picking marbles randomly from a bag and spinning a fair spinner.

Examples: Investigate whether a coin is fair by tossing it 1,000 times and comparing the percentage of heads to the theoretical probability 0.5.  
MA.7.GR.1.1:  Apply formulas to find the areas of trapezoids, parallelograms and rhombi.Clarifications: Clarification 1: Instruction focuses on the connection from the areas of trapezoids, parallelograms and rhombi to the areas of rectangles or triangles.
Clarification 2: Within this benchmark, the expectation is not to memorize area formulas for trapezoids, parallelograms and rhombi.
 
MA.7.GR.1.2:  Solve mathematical or realworld problems involving the area of polygons or composite figures by decomposing them into triangles or quadrilaterals.Clarifications: Clarification 1: Within this benchmark, the expectation is not to find areas of figures on the coordinate plane or to find missing dimensions.  
MA.7.GR.1.3:  Explore the proportional relationship between circumferences and diameters of circles. Apply a formula for the circumference of a circle to solve mathematical and realworld problems.Clarifications: Clarification 1: Instruction includes the exploration and analysis of circular objects to examine the proportional relationship between circumference and diameter and arrive at an approximation of pi (π) as the constant of proportionality.
Clarification 2: Solutions may be represented in terms of pi (π) or approximately.
 
MA.7.GR.1.4:  Explore and apply a formula to find the area of a circle to solve mathematical and realworld problems.Clarifications: Clarification 1: Instruction focuses on the connection between formulas for the area of a rectangle and the area of a circle.
Clarification 2: Problem types include finding areas of fractional parts of a circle.
Clarification 3: Solutions may be represented in terms of pi (π) or approximately.

Examples: If a 12inch pizza is cut into 6 equal slices and Mikel ate 2 slices, how many square inches of pizza did he eat?  
MA.7.GR.1.5:  Solve mathematical and realworld problems involving dimensions and areas of geometric figures, including scale drawings and scale factors.Clarifications: Clarification 1: Instruction focuses on seeing the scale factor as a constant of proportionality between corresponding lengths in the scale drawing and the original object.
Clarification 2: Instruction includes the understanding that if the scaling factor is k, then the constant of proportionality between corresponding areas is k² .
Clarification 3: Problem types include finding the scale factor given a set of dimensions as well as finding dimensions when given a scale factor.
 
MA.7.GR.2.1:  Given a mathematical or realworld context, find the surface area of a right circular cylinder using the figure’s net.Clarifications: Clarification 1: Instruction focuses on representing a right circular cylinder with its net and on the connection between surface area of a figure and its net.
Clarification 2: Within this benchmark, the expectation is to find the surface area when given a net or when given a threedimensional figure.
Clarification 3: Within this benchmark, the expectation is not to memorize the surface area formula for a right circular cylinder.
Clarification 4: Solutions may be represented in terms of pi (π) or approximately.
 
MA.7.GR.2.2:  Solve realworld problems involving surface area of right circular cylinders.Clarifications: Clarification 1: Within this benchmark, the expectation is not to memorize the surface area formula for a right circular cylinder or to find radius as a missing dimension.
Clarification 2: Solutions may be represented in terms of pi (π) or approximately.
 
MA.7.GR.2.3:  Solve mathematical and realworld problems involving volume of right circular cylinders.Clarifications: Clarification 1: Within this benchmark, the expectation is not to memorize the volume formula for a right circular cylinder or to find radius as a missing dimension.
Clarification 2: Solutions may be represented in terms of pi (π) or approximately.
 
MA.7.NSO.1.1:  Know and apply the Laws of Exponents to evaluate numerical expressions and generate equivalent numerical expressions, limited to wholenumber exponents and rational number bases.Clarifications: Clarification 1: Instruction focuses on building the Laws of Exponents from specific examples. Refer to the K12 Formulas (Appendix E) for the Laws of Exponents.
Clarification 2: Problems in the form must result in a wholenumber value for p.  
MA.7.NSO.1.2:  Rewrite rational numbers in different but equivalent forms including fractions, mixed numbers, repeating decimals and percentages to solve mathematical and realworld problems.
Examples: Justin is solving a problem where he computes and his calculator gives him the answer 5.6666666667. Justin makes the statement that ; is he correct?  
MA.7.NSO.2.1:  Solve mathematical problems using multistep order of operations with rational numbers including grouping symbols, wholenumber exponents and absolute value.Clarifications: Clarification 1: Multistep expressions are limited to 6 or fewer steps.  
MA.7.NSO.2.2:  Add, subtract, multiply and divide rational numbers with procedural fluency. 
MA.7.NSO.2.3:  Solve realworld problems involving any of the four operations with rational numbers.Clarifications: Clarification 1: Instruction includes using one or more operations to solve problems.  
MA.8.AR.1.1:  Apply the Laws of Exponents to generate equivalent algebraic expressions, limited to integer exponents and monomial bases.
Examples: The expression is equivalent to .  
MA.8.AR.1.2:  Apply properties of operations to multiply two linear expressions with rational coefficients.
Examples: The product of (1.1+x) and (2.3x) can be expressed as 2.53x2.3x² or 2.3x²2.53x.  
MA.8.AR.1.3:  Rewrite the sum of two algebraic expressions having a common monomial factor as a common factor multiplied by the sum of two algebraic expressions.
Examples: The expression 99x11x³ can be rewritten as 11x(9x²) or as 11x(9+x² ).  
MA.8.AR.2.1:  Solve multistep linear equations in one variable, with rational number coefficients. Include equations with variables on both sides.Clarifications: Clarification 1: Problem types include examples of onevariable linear equations that generate one solution, infinitely many solutions or no solution.  
MA.8.AR.2.2:  Solve twostep linear inequalities in one variable and represent solutions algebraically and graphically.Clarifications: Clarification 1: Instruction includes inequalities in the forms px±q>r and p(x±q)>r, where p, q and r are specific rational numbers and where any inequality symbol can be represented.
Clarification 2: Problems include inequalities where the variable may be on either side of the inequality.
 
MA.8.AR.2.3:  Given an equation in the form of x²=p and x³=q, where p is a whole number and q is an integer, determine the real solutions.Clarifications: Clarification 1: Instruction focuses on understanding that when solving x²=p, there is both a positive and negative solution.
Clarification 2: Within this benchmark, the expectation is to calculate square roots of perfect squares up to 225 and cube roots of perfect cubes from 125 to 125.
 
MA.8.AR.3.1:  Determine if a linear relationship is also a proportional relationship.Clarifications: Clarification 1: Instruction focuses on the understanding that proportional relationships are linear relationships whose graph passes through the origin.
Clarification 2: Instruction includes the representation of relationships using tables, graphs, equations and written descriptions.
 
MA.8.AR.3.2:  Given a table, graph or written description of a linear relationship, determine the slope.Clarifications: Clarification 1: Problem types include cases where two points are given to determine the slope.
Clarification 2: Instruction includes making connections of slope to the constant of proportionality and to similar triangles represented on the coordinate plane.
 
MA.8.AR.3.3:  Given a table, graph or written description of a linear relationship, write an equation in slopeintercept form. 
MA.8.AR.3.4:  Given a mathematical or realworld context, graph a twovariable linear equation from a written description, a table or an equation in slopeintercept form. 
MA.8.AR.3.5:  Given a realworld context, determine and interpret the slope and yintercept of a twovariable linear equation from a written description, a table, a graph or an equation in slopeintercept form.Clarifications: Clarification 1: Problems include conversions with temperature and equations of lines of fit in scatter plots. 
Examples: Raul bought a palm tree to plant at his house. He records the growth over many months and creates the equation h=0.21m+4.9, where h is the height of the palm tree in feet and m is the number of months. Interpret the slope and yintercept from his equation.  
MA.8.AR.4.1:  Given a system of two linear equations and a specified set of possible solutions, determine which ordered pairs satisfy the system of linear equations.Clarifications: Clarification 1: Instruction focuses on the understanding that a solution to a system of equations satisfies both linear equations simultaneously.  
MA.8.AR.4.2:  Given a system of two linear equations represented graphically on the same coordinate plane, determine whether there is one solution, no solution or infinitely many solutions. 
MA.8.AR.4.3:  Given a mathematical or realworld context, solve systems of two linear equations by graphing.Clarifications: Clarification 1: Instruction includes approximating noninteger solutions.
Clarification 2: Within this benchmark, it is the expectation to represent systems of linear equations in slopeintercept form only.
Clarification 3: Instruction includes recognizing that parallel lines have the same slope.
 
MA.8.DP.1.1:  Given a set of realworld bivariate numerical data, construct a scatter plot or a line graph as appropriate for the context.Clarifications: Clarification 1: Instruction includes recognizing similarities and differences between scatter plots and line graphs, and on determining which is more appropriate as a representation of the data based on the context.
Clarification 2: Sets of data are limited to 20 points.

Examples: Example: Jaylyn is collecting data about the relationship between grades in English and grades in mathematics. He represents the data using a scatter plot because he is interested if there is an association between the two variables without thinking of either one as an independent or dependent variable.
Example: Samantha is collecting data on her weekly quiz grade in her social studies class. She represents the data using a line graph with time as the independent variable.
 
MA.8.DP.1.2:  Given a scatter plot within a realworld context, describe patterns of association.Clarifications: Clarification 1: Descriptions include outliers; positive or negative association; linear or nonlinear association; strong or weak association.  
MA.8.DP.1.3:  Given a scatter plot with a linear association, informally fit a straight line.Clarifications: Clarification 1: Instruction focuses on the connection to linear functions.
Clarification 2: Instruction includes using a variety of tools, including a ruler, to draw a line with approximately the same number of points above and below the line.
 
MA.8.DP.2.1:  Determine the sample space for a repeated experiment.Clarifications: Clarification 1: Instruction includes recording sample spaces for repeated experiments using organized lists, tables or tree diagrams.
Clarification 2: Experiments to be repeated are limited to tossing a fair coin, rolling a fair die, picking a card randomly from a deck with replacement, picking marbles randomly from a bag with replacement and spinning a fair spinner.
Clarification 3: Repetition of experiments is limited to two times except for tossing a coin.
 
MA.8.DP.2.2:  Find the theoretical probability of an event related to a repeated experiment.Clarifications: Clarification 1: Instruction includes representing probability as a fraction, percentage or decimal.
Clarification 2: Experiments to be repeated are limited to tossing a fair coin, rolling a fair die, picking a card randomly from a deck with replacement, picking marbles randomly from a bag with replacement and spinning a fair spinner.
Clarification 3: Repetition of experiments is limited to two times except for tossing a coin.
 
MA.8.DP.2.3:  Solve realworld problems involving probabilities related to single or repeated experiments, including making predictions based on theoretical probability.Clarifications: Clarification 1: Instruction includes making connections to proportional relationships and representing probability as a fraction, percentage or decimal.
Clarification 2: Experiments to be repeated are limited to tossing a fair coin, rolling a fair die, picking a card randomly from a deck with replacement, picking marbles randomly from a bag with replacement and spinning a fair spinner.
Clarification 3: Repetition of experiments is limited to two times except for tossing a coin.

Examples: Example: If Gabriella rolls a fair die 300 times, she can predict that she will roll a 3 approximately 50 times since the theoretical probability is .
Example: Sandra performs an experiment where she flips a coin three times. She finds the theoretical probability of landing on exactly one head as . If she performs this experiment 50 times (for a total of 150 flips), predict the number of repetitions of the experiment that will result in exactly one of the three flips landing on heads.
 
MA.8.F.1.1:  Given a set of ordered pairs, a table, a graph or mapping diagram, determine whether the relationship is a function. Identify the domain and range of the relation.Clarifications: Clarification 1: Instruction includes referring to the input as the independent variable and the output as the dependent variable.
Clarification 2: Within this benchmark, it is the expectation to represent domain and range as a list of numbers or as an inequality.
 
MA.8.F.1.2:  Given a function defined by a graph or an equation, determine whether the function is a linear function. Given an inputoutput table, determine whether it could represent a linear function.Clarifications: Clarification 1: Instruction includes recognizing that a table may not determine a function.  
MA.8.F.1.3:  Analyze a realworld written description or graphical representation of a functional relationship between two quantities and identify where the function is increasing, decreasing or constant.Clarifications: Clarification 1: Problem types are limited to continuous functions.
Clarification 2: Analysis includes writing a description of a graphical representation or sketching a graph from a written description.
 
MA.8.GR.1.1:  Apply the Pythagorean Theorem to solve mathematical and realworld problems involving unknown side lengths in right triangles.Clarifications: Clarification 1: Instruction includes exploring right triangles with naturalnumber side lengths to illustrate the Pythagorean Theorem.
Clarification 2: Within this benchmark, the expectation is to memorize the Pythagorean Theorem. Clarification 3: Radicands are limited to whole numbers up to 225.
 
MA.8.GR.1.2:  Apply the Pythagorean Theorem to solve mathematical and realworld problems involving the distance between two points in a coordinate plane.Clarifications: Clarification 1: Instruction includes making connections between distance on the coordinate plane and right triangles.
Clarification 2: Within this benchmark, the expectation is to memorize the Pythagorean Theorem. It is not the expectation to use the distance formula.
Clarification 3: Radicands are limited to whole numbers up to 225.

Examples: The distance between (2,7) and (0,6) can be found by creating a right triangle with the vertex of the right angle at the point (2,6). This gives a height of the right triangle as 1 unit and a base of 2 units. Then using the Pythagorean Theorem the distance can be determined from the equation 1²+2²=c², which is equivalent to 5=c². So, the distance is units.  
MA.8.GR.1.3:  Use the Triangle Inequality Theorem to determine if a triangle can be formed from a given set of sides. Use the converse of the Pythagorean Theorem to determine if a right triangle can be formed from a given set of sides. 
MA.8.GR.1.4:  Solve mathematical problems involving the relationships between supplementary, complementary, vertical or adjacent angles. 
MA.8.GR.1.5:  Solve problems involving the relationships of interior and exterior angles of a triangle.Clarifications: Clarification 1: Problems include using the Triangle Sum Theorem and representing angle measures as algebraic expressions.  
MA.8.GR.1.6:  Develop and use formulas for the sums of the interior angles of regular polygons by decomposing them into triangles.Clarifications: Clarification 1: Problems include representing angle measures as algebraic expressions.  
MA.8.GR.2.1:  Given a preimage and image generated by a single transformation, identify the transformation that describes the relationship.Clarifications: Clarification 1: Within this benchmark, transformations are limited to reflections, translations or rotations of images.
Clarification 2: Instruction focuses on the preservation of congruence so that a figure maps onto a copy of itself.
 
MA.8.GR.2.2:  Given a preimage and image generated by a single dilation, identify the scale factor that describes the relationship.Clarifications: Clarification 1: Instruction includes the connection to scale drawings and proportions.
Clarification 2: Instruction focuses on the preservation of similarity and the lack of preservation of congruence when a figure maps onto a scaled copy of itself, unless the scaling factor is 1.
 
MA.8.GR.2.3:  Describe and apply the effect of a single transformation on twodimensional figures using coordinates and the coordinate plane.Clarifications:
Clarification 1: Within this benchmark, transformations are limited to reflections, translations, rotations or dilations of images. Clarification 2: Lines of reflection are limited to the xaxis, yaxis or lines parallel to the axes. Clarification 3: Rotations must be about the origin and are limited to 90°, 180°, 270° or 360°.
Clarification 4: Dilations must be centered at the origin.
 
MA.8.GR.2.4:  Solve mathematical and realworld problems involving proportional relationships between similar triangles.
Examples: During a Tampa Bay Lightning game one player, Johnson, passes the puck to his teammate, Stamkos, by bouncing the puck off the wall of the rink. The path of the puck creates two line segments that form hypotenuses for each of two similar right triangles, with the height of each triangle the distance from one of the players to the wall of the rink. If Johnson is 12 feet from the wall and Stamkos is 3 feet from the wall. How far did the puck travel from the wall of the rink to Stamkos if the distance traveled from Johnson to the wall was 16 feet?  
MA.8.NSO.1.1:  Extend previous understanding of rational numbers to define irrational numbers within the real number system. Locate an approximate value of a numerical expression involving irrational numbers on a number line.Clarifications: Clarification 1: Instruction includes the use of number line and rational number approximations, and recognizing pi (π) as an irrational number.
Clarification 2: Within this benchmark, the expectation is to approximate numerical expressions involving one arithmetic operation and estimating square roots or pi (π).

Examples: Within the expression , the irrational number can be estimated to be between 5 and 6 because 30 is between 25 and 36. By considering and , a closer approximation for is 5.5. So, the expression is equivalent to about 6.5.  
MA.8.NSO.1.2:  Plot, order and compare rational and irrational numbers, represented in various forms.Clarifications: Clarification 1: Within this benchmark, it is not the expectation to work with the number e.
Clarification 2: Within this benchmark, the expectation is to plot, order and compare square roots and cube roots.
Clarification 3: Within this benchmark, the expectation is to use symbols (<, > or =).
 
MA.8.NSO.1.3:  Extend previous understanding of the Laws of Exponents to include integer exponents. Apply the Laws of Exponents to evaluate numerical expressions and generate equivalent numerical expressions, limited to integer exponents and rational number bases, with procedural fluency. 
MA.8.NSO.1.4:  Express numbers in scientific notation to represent and approximate very large or very small quantities. Determine how many times larger or smaller one number is compared to a second number.
Examples: Roderick is comparing two numbers shown in scientific notation on his calculator. The first number was displayed as 2.3147E27 and the second number was displayed as 3.5982E5. Roderick determines that the first number is about 10³² times bigger than the second number.  
MA.8.NSO.1.5:  Add, subtract, multiply and divide numbers expressed in scientific notation with procedural fluency.Clarifications: Clarification 1: Within this benchmark, for addition and subtraction with numbers expressed in scientific notation, exponents are limited to within 2 of each other. 
Examples: The sum of .  
MA.8.NSO.1.6:  Solve realworld problems involving operations with numbers expressed in scientific notation.Clarifications: Clarification 1: Instruction includes recognizing the importance of significant digits when physical measurements are involved.
Clarification 2: Within this benchmark, for addition and subtraction with numbers expressed in scientific notation, exponents are limited to within 2 of each other.
 
MA.8.NSO.1.7:  Solve multistep mathematical and realworld problems involving the order of operations with rational numbers including exponents and radicals.Clarifications: Clarification 1: Multistep expressions are limited to 6 or fewer steps.
Clarification 2: Within this benchmark, the expectation is to simplify radicals by factoring square roots of perfect squares up to 225 and cube roots of perfect cubes from 125 to 125.

Examples: The expression is equivalent to which is equivalent towhich is equivalent to .  
MA.K12.MTR.1.1:  Actively participate in effortful learning both individually and collectively. Mathematicians who participate in effortful learning both individually and with others:
 Analyze the problem in a way that makes sense given the task.
 Ask questions that will help with solving the task.
 Build perseverance by modifying methods as needed while solving a challenging task.
 Stay engaged and maintain a positive mindset when working to solve tasks.
 Help and support each other when attempting a new method or approach.
Clarifications: Teachers who encourage students to participate actively in effortful learning both individually and with others:
 Cultivate a community of growth mindset learners.
 Foster perseverance in students by choosing tasks that are challenging.
 Develop students’ ability to analyze and problem solve.
 Recognize students’ effort when solving challenging problems.
 
MA.K12.MTR.2.1:  Demonstrate understanding by representing problems in multiple ways. Mathematicians who demonstrate understanding by representing problems in multiple ways:  Build understanding through modeling and using manipulatives.
 Represent solutions to problems in multiple ways using objects, drawings, tables, graphs and equations.
 Progress from modeling problems with objects and drawings to using algorithms and equations.
 Express connections between concepts and representations.
 Choose a representation based on the given context or purpose.
Clarifications: Teachers who encourage students to demonstrate understanding by representing problems in multiple ways:  Help students make connections between concepts and representations.
 Provide opportunities for students to use manipulatives when investigating concepts.
 Guide students from concrete to pictorial to abstract representations as understanding progresses.
 Show students that various representations can have different purposes and can be useful in different situations.
 
MA.K12.MTR.3.1:  Complete tasks with mathematical fluency. Mathematicians who complete tasks with mathematical fluency:  Select efficient and appropriate methods for solving problems within the given context.
 Maintain flexibility and accuracy while performing procedures and mental calculations.
 Complete tasks accurately and with confidence.
 Adapt procedures to apply them to a new context.
 Use feedback to improve efficiency when performing calculations.
Clarifications: Teachers who encourage students to complete tasks with mathematical fluency: Provide students with the flexibility to solve problems by selecting a procedure that allows them to solve efficiently and accurately.
 Offer multiple opportunities for students to practice efficient and generalizable methods.
 Provide opportunities for students to reflect on the method they used and determine if a more efficient method could have been used.
 
MA.K12.MTR.4.1:  Engage in discussions that reflect on the mathematical thinking of self and others. Mathematicians who engage in discussions that reflect on the mathematical thinking of self and others:  Communicate mathematical ideas, vocabulary and methods effectively.
 Analyze the mathematical thinking of others.
 Compare the efficiency of a method to those expressed by others.
 Recognize errors and suggest how to correctly solve the task.
 Justify results by explaining methods and processes.
 Construct possible arguments based on evidence.
Clarifications: Teachers who encourage students to engage in discussions that reflect on the mathematical thinking of self and others: Establish a culture in which students ask questions of the teacher and their peers, and error is an opportunity for learning.
 Create opportunities for students to discuss their thinking with peers.
 Select, sequence and present student work to advance and deepen understanding of correct and increasingly efficient methods.
 Develop students’ ability to justify methods and compare their responses to the responses of their peers.
 
MA.K12.MTR.5.1:  Use patterns and structure to help understand and connect mathematical concepts. Mathematicians who use patterns and structure to help understand and connect mathematical concepts:  Focus on relevant details within a problem.
 Create plans and procedures to logically order events, steps or ideas to solve problems.
 Decompose a complex problem into manageable parts.
 Relate previously learned concepts to new concepts.
 Look for similarities among problems.
 Connect solutions of problems to more complicated largescale situations.
Clarifications: Teachers who encourage students to use patterns and structure to help understand and connect mathematical concepts: Help students recognize the patterns in the world around them and connect these patterns to mathematical concepts.
 Support students to develop generalizations based on the similarities found among problems.
 Provide opportunities for students to create plans and procedures to solve problems.
 Develop students’ ability to construct relationships between their current understanding and more sophisticated ways of thinking.
 
MA.K12.MTR.6.1:  Assess the reasonableness of solutions. Mathematicians who assess the reasonableness of solutions:  Estimate to discover possible solutions.
 Use benchmark quantities to determine if a solution makes sense.
 Check calculations when solving problems.
 Verify possible solutions by explaining the methods used.
 Evaluate results based on the given context.
Clarifications: Teachers who encourage students to assess the reasonableness of solutions: Have students estimate or predict solutions prior to solving.
 Prompt students to continually ask, “Does this solution make sense? How do you know?”
 Reinforce that students check their work as they progress within and after a task.
 Strengthen students’ ability to verify solutions through justifications.
 
MA.K12.MTR.7.1:  Apply mathematics to realworld contexts. Mathematicians who apply mathematics to realworld contexts:  Connect mathematical concepts to everyday experiences.
 Use models and methods to understand, represent and solve problems.
 Perform investigations to gather data or determine if a method is appropriate.
• Redesign models and methods to improve accuracy or efficiency.
Clarifications: Teachers who encourage students to apply mathematics to realworld contexts: Provide opportunities for students to create models, both concrete and abstract, and perform investigations.
 Challenge students to question the accuracy of their models and methods.
 Support students as they validate conclusions by comparing them to the given situation.
 Indicate how various concepts can be applied to other disciplines.
 
ELA.K12.EE.1.1:  Cite evidence to explain and justify reasoning.Clarifications: K1 Students include textual evidence in their oral communication with guidance and support from adults. The evidence can consist of details from the text without naming the text. During 1st grade, students learn how to incorporate the evidence in their writing.23 Students include relevant textual evidence in their written and oral communication. Students should name the text when they refer to it. In 3rd grade, students should use a combination of direct and indirect citations. 45 Students continue with previous skills and reference comments made by speakers and peers. Students cite texts that they’ve directly quoted, paraphrased, or used for information. When writing, students will use the form of citation dictated by the instructor or the style guide referenced by the instructor. 68 Students continue with previous skills and use a style guide to create a proper citation. 912 Students continue with previous skills and should be aware of existing style guides and the ways in which they differ.
 
ELA.K12.EE.2.1:  Read and comprehend gradelevel complex texts proficiently.Clarifications: See Text Complexity for gradelevel complexity bands and a text complexity rubric.  
ELA.K12.EE.3.1:  Make inferences to support comprehension.Clarifications: Students will make inferences before the words infer or inference are introduced. Kindergarten students will answer questions like “Why is the girl smiling?” or make predictions about what will happen based on the title page.
Students will use the terms and apply them in 2nd grade and beyond.  
ELA.K12.EE.4.1:  Use appropriate collaborative techniques and active listening skills when engaging in discussions in a variety of situations.Clarifications: In kindergarten, students learn to listen to one another respectfully.In grades 12, students build upon these skills by justifying what they are thinking. For example: “I think ________ because _______.” The collaborative conversations are becoming academic conversations. In grades 312, students engage in academic conversations discussing claims and justifying their reasoning, refining and applying skills. Students build on ideas, propel the conversation, and support claims and counterclaims with evidence.
 
ELA.K12.EE.5.1:  Use the accepted rules governing a specific format to create quality work.Clarifications: Students will incorporate skills learned into work products to produce quality work. For students to incorporate these skills appropriately, they must receive instruction. A 3rd grade student creating a poster board display must have instruction in how to effectively present information to do quality work.  
ELA.K12.EE.6.1:  Use appropriate voice and tone when speaking or writing.Clarifications: In kindergarten and 1st grade, students learn the difference between formal and informal language. For example, the way we talk to our friends differs from the way we speak to adults. In 2nd grade and beyond, students practice appropriate social and academic language to discuss texts.  
ELD.K12.ELL.MA.1:  English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. 