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## Content Area: MathematicsGrade Level Expectations: High SchoolStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. The complex number system includes real numbers and imaginary numbers Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Extend the properties of exponents to rational exponents. (CCSS: N-RN)Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.1 (CCSS: N-RN.1)Rewrite expressions involving radicals and rational exponents using the properties of exponents. (CCSS: N-RN.2) Use properties of rational and irrational numbers. (CCSS: N-RN)Explain why the sum or product of two rational numbers is rational. (CCSS: N-RN.3)Explain why the sum of a rational number and an irrational number is irrational. (CCSS: N-RN.3)Explain why the product of a nonzero rational number and an irrational number is irrational. (CCSS: N-RN.3) Perform arithmetic operations with complex numbers. (CCSS: N-CN)Define the complex number i such that $i^2 = –1$, and show that every complex number has the form a + bi where a and b are real numbers. (CCSS: N-CN.1)Use the relation $i^2 = –1$ and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. (CCSS: N-CN.2) Use complex numbers in polynomial identities and equations. (CCSS: N-CN)Solve quadratic equations with real coefficients that have complex solutions. (CCSS: N-CN.7) Inquiry Questions: When you extend to a new number systems (e.g., from integers to rational numbers and from rational numbers to real numbers), what properties apply to the extended number system? Are there more complex numbers than real numbers? What is a number system? Why are complex numbers important? Relevance & Application: Complex numbers have applications in fields such as chaos theory and fractals. The familiar image of the Mandelbrot fractal is the Mandelbrot set graphed on the complex plane. Nature Of: Mathematicians build a deep understanding of quantity, ways of representing numbers, and relationships among numbers and number systems. Mathematics involves making and testing conjectures, generalizing results, and making connections among ideas, strategies, and solutions. Mathematicians look for and make use of structure. (MP) Mathematicians look for and express regularity in repeated reasoning. (MP)

1 For example, we define $5^\frac{1}{3}$ to be the cube root of 5 because we want $(5^\frac{1}{3})^3=5^{(\frac{1}{3})3}$ to hold, so $(5^\frac{1}{3})^3$ must equal 5. (CCSS: N-RN.1)

## Content Area: MathematicsGrade Level Expectations: Eighth GradeStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. In the real number system, rational and irrational numbers are in one to one correspondence to points on the number line Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Define irrational numbers.1 Demonstrate informally that every number has a decimal expansion. (CCSS: 8.NS.1)For rational numbers show that the decimal expansion repeats eventually. (CCSS: 8.NS.1)Convert a decimal expansion which repeats eventually into a rational number. (CCSS: 8.NS.1) Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.2 (CCSS: 8.NS.2) Apply the properties of integer exponents to generate equivalent numerical expressions.3 (CCSS: 8.EE.1) Use square root and cube root symbols to represent solutions to equations of the form $x^2=p$ and $x^3=p$, where $p$ is a positive rational number. (CCSS: 8.EE.2) Evaluate square roots of small perfect squares and cube roots of small perfect cubes.4 (CCSS: 8.EE.2) Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.5 (CCSS: 8.EE.3) Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. (CCSS: 8.EE.4)Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.6 (CCSS: 8.EE.4)Interpret scientific notation that has been generated by technology. (CCSS: 8.EE.4) Inquiry Questions: Why are real numbers represented by a number line and why are the integers represented by points on the number line? Why is there no real number closest to zero? What is the difference between rational and irrational numbers? Relevance & Application: Irrational numbers have applications in geometry such as the length of a diagonal of a one by one square, the height of an equilateral triangle, or the area of a circle. Different representations of real numbers are used in contexts such as measurement (metric and customary units), business (profits, network down time, productivity), and community (voting rates, population density). Technologies such as calculators and computers enable people to order and convert easily among fractions, decimals, and percents. Nature Of: Mathematics provides a precise language to describe objects and events and the relationships among them. Mathematicians reason abstractly and quantitatively. (MP) Mathematicians use appropriate tools strategically. (MP) Mathematicians attend to precision. (MP)

1 Know that numbers that are not rational are called irrational. (CCSS: 8.NS.1)

2 e.g., $\pi^2$. (CCSS: 8.NS.2)
For example, by truncating the decimal expansion of $\sqrt{2}$, show that $\sqrt{2}$ is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. (CCSS: 8.NS.2)

3 For example, $3^2\times3^{–5}=3^{–3}=\frac{1}{3}^3=\frac{1}{27}$. (CCSS: 8.EE.1)

4 Know that $\sqrt{2}$ is irrational. (CCSS: 8.EE.2)

5 For example, estimate the population of the United States as 3 times $10^8$ and the population of the world as 7 times $10^9$, and determine that the world population is more than 20 times larger. (CCSS: 8.EE.3)

6 e.g., use millimeters per year for seafloor spreading. (CCSS: 8.EE.4)

## Content Area: MathematicsGrade Level Expectations: Sixth GradeStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 3. In the real number system, rational numbers have a unique location on the number line and in space Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Explain why positive and negative numbers are used together to describe quantities having opposite directions or values.11 (CCSS: 6.NS.5)Use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. (CCSS: 6.NS.5) Use number line diagrams and coordinate axes to represent points on the line and in the plane with negative number coordinates.12 (CCSS: 6.NS.6)Describe a rational number as a point on the number line. (CCSS: 6.NS.6)Use opposite signs of numbers to indicate locations on opposite sides of 0 on the number line. (CCSS: 6.NS.6a)Identify that the opposite of the opposite of a number is the number itself.13 (CCSS: 6.NS.6a)Explain when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. (CCSS: 6.NS.6b)Find and position integers and other rational numbers on a horizontal or vertical number line diagram. (CCSS: 6.NS.6c)Find and position pairs of integers and other rational numbers on a coordinate plane. (CCSS: 6.NS.6c) Order and find absolute value of rational numbers. (CCSS: 6.NS.7)Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.14 (CCSS: 6.NS.7a)Write, interpret, and explain statements of order for rational numbers in real-world contexts.15 (CCSS: 6.NS.7b)Define the absolute value of a rational number as its distance from 0 on the number line and interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.16 (CCSS: 6.NS.7c)Distinguish comparisons of absolute value from statements about order.17 (CCSS: 6.NS.7d) Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane including the use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. (CCSS: 6.NS.8) Inquiry Questions: Why are there negative numbers? How do we compare and contrast numbers? Are there more rational numbers than integers? Relevance & Application: Communication and collaboration with others is more efficient and accurate using rational numbers. For example, negotiating the price of an automobile, sharing results of a scientific experiment with the public, and planning a party with friends. Negative numbers can be used to represent quantities less than zero or quantities with an associated direction such as debt, elevations below sea level, low temperatures, moving backward in time, or an object slowing down Nature Of: Mathematicians use their understanding of relationships among numbers and the rules of number systems to create models of a wide variety of situations. Mathematicians construct viable arguments and critique the reasoning of others. (MP) Mathematicians attend to precision. (MP)

11 e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge). (CCSS: 6.NS.5)

12 Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane. (CCSS: 6.NS.6)

13 e.g., –(–3) = 3, and that 0 is its own opposite. (CCSS: 6.NS.6a)

14 For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right. (CCSS: 6.NS.7a)

15 For example, write –3$^o$C > –7$^o$ C to express the fact that –3$^o$ C is warmer than –7$^o$ C. (CCSS: 6.NS.7b)

16 For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars. (CCSS: 6.NS.7c)

17 For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars. (CCSS: 6.NS.7d)

## Content Area: MathematicsGrade Level Expectations: Fifth GradeStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. The decimal number system describes place value patterns and relationships that are repeated in large and small numbers and forms the foundation for efficient algorithms Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Explain that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. (CCSS: 5.NBT.1)Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. (CCSS: 5.NBT.2)Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. (CCSS: 5.NBT.2)Use whole-number exponents to denote powers of 10. (CCSS: 5.NBT.2) Read, write, and compare decimals to thousandths. (CCSS: 5.NBT.3)Read and write decimals to thousandths using base-ten numerals, number names, and expanded form.1 (CCSS: 5.NBT.3a)Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. (CCSS: 5.NBT.3b) Use place value understanding to round decimals to any place. (CCSS: 5.NBT.4) Convert like measurement units within a given measurement system. (CCSS: 5.MD)Convert among different-sized standard measurement units within a given measurement system.2 (CCSS: 5.MD.1)Use measurement conversions in solving multi-step, real world problems. (CCSS: 5.MD.1) Inquiry Questions: What is the benefit of place value system? What would it mean if we did not have a place value system? What is the purpose of a place value system? What is the purpose of zero in a place value system? Relevance & Application: Place value is applied to represent a myriad of numbers using only ten symbols. Nature Of: Mathematicians use numbers like writers use letters to express ideas. Mathematicians look closely and make use of structure by discerning patterns. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians reason abstractly and quantitatively. (MP) Mathematicians construct viable arguments and critique the reasoning of others. (MP)

1 e.g., 347.392 = 3 x 100 + 4 x 10 + 7 x 1 + 3 x 1/10 + 9 x 1/100 + 2 x 1/1000. (CCSS: 5.NBT.3a)

2 e.g., convert 5 cm to 0.05 m. (CCSS: 5.MD.1)

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 4. The concepts of multiplication and division can be applied to multiply and divide fractions (CCSS: 5.NF) Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). (CCSS: 5.NF.3) Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.8 (CCSS: 5.NF.3) Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. 9 In general, (a/b) × (c/d) = ac/bd. (CCSS: 5.NF.4a) Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. (CCSS: 5.NF.4b)Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. (CCSS: 5.NF.4b) Interpret multiplication as scaling (resizing). (CCSS: 5.NF.5)Compare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.10 (CCSS: 5.NF.5a)Apply the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1. (CCSS: 5.NF.5b) Solve real world problems involving multiplication of fractions and mixed numbers.11 (CCSS: 5.NF.6) Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.12 (CCSS: 5.NF.7a) Interpret division of a whole number by a unit fraction, and compute such quotients.13 (CCSS: 5.NF.7b) Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions.14 (CCSS: 5.NF.7c) Inquiry Questions: Do adding and multiplying always result in an increase? Why? Do subtracting and dividing always result in a decrease? Why? How do operations with fractional numbers compare to operations with whole numbers? Relevance & Application: Rational numbers are used extensively in measurement tasks such as home remodeling, clothes alteration, graphic design, and engineering. Situations from daily life can be modeled using operations with fractions, decimals, and percents such as determining the quantity of paint to buy or the number of pizzas to order for a large group. Rational numbers are used to represent data and probability such as getting a certain color of gumball out of a machine, the probability that a batter will hit a home run, or the percent of a mountain covered in forest. Nature Of: Mathematicians explore number properties and relationships because they enjoy discovering beautiful new and unexpected aspects of number systems. They use their knowledge of number systems to create appropriate models for all kinds of real-world systems. Mathematicians make sense of problems and persevere in solving them. (MP) Mathematicians model with mathematics. (MP) Mathematicians look for and express regularity in repeated reasoning. (MP)

8 e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? (CCSS: 5.NF.3)

9 For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (CCSS: 5.NF.4a)

10 Explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number. (CCSS: 5.NF.5b)
Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number (CCSS: 5.NF.5b)

11 e.g., by using visual fraction models or equations to represent the problem. (CCSS: 5.NF.6)

12 For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. (CCSS: 5.NF.7a)

13 For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. (CCSS: 5.NF.7b)

14 e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? (CCSS: 5.NF.7c)

## Content Area: MathematicsGrade Level Expectations: Fourth GradeStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. The decimal number system to the hundredths place describes place value patterns and relationships that are repeated in large and small numbers and forms the foundation for efficient algorithms Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Generalize place value understanding for multi-digit whole numbers (CCSS: 4.NBT)Explain that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. (CCSS: 4.NBT.1)Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. (CCSS: 4.NBT.2)Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. (CCSS: 4.NBT.2)Use place value understanding to round multi-digit whole numbers to any place. (CCSS: 4.NBT.3) Use decimal notation to express fractions, and compare decimal fractions (CCSS: 4.NF)Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.1 (CCSS: 4.NF.5)Use decimal notation for fractions with denominators 10 or 100.2 (CCSS: 4.NF.6)Compare two decimals to hundredths by reasoning about their size.3 (CCSS: 4.NF.7) Inquiry Questions: Why isn’t there a “oneths” place in decimal fractions? How can a number with greater decimal digits be less than one with fewer decimal digits? Is there a decimal closest to one? Why? Relevance & Application: Decimal place value is the basis of the monetary system and provides information about how much items cost, how much change should be returned, or the amount of savings that has accumulated. Knowledge and use of place value for large numbers provides context for population, distance between cities or landmarks, and attendance at events. Nature Of: Mathematicians explore number properties and relationships because they enjoy discovering beautiful new and unexpected aspects of number systems. They use their knowledge of number systems to create appropriate models for all kinds of real-world systems. Mathematicians reason abstractly and quantitatively. (MP) Mathematicians look for and make use of structure. (MP)

1 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. (CCSS: 4.NF.6)

2 For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. (CCSS: 4.NF.6)

3 Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. (CCSS: 4.NF.7)

## Content Area: MathematicsGrade Level Expectations: Third GradeStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. The whole number system describes place value relationships and forms the foundation for efficient algorithms Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Use place value and properties of operations to perform multi-digit arithmetic. (CCSS: 3.NBT)Use place value to round whole numbers to the nearest 10 or 100. (CCSS: 3.NBT.1)Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. (CCSS: 3.NBT.2)Multiply one-digit whole numbers by multiples of 10 in the range 10–90 using strategies based on place value and properties of operations.1 (CCSS: 3.NBT.3) Inquiry Questions: How do patterns in our place value system assist in comparing whole numbers? How might the most commonly used number system be different if humans had twenty fingers instead of ten? Relevance & Application: Knowledge and use of place value for large numbers provides context for distance in outer space, prehistoric timelines, and ants in a colony. The building and taking apart of numbers provide a deep understanding of the base 10 number system. Nature Of: Mathematicians use numbers like writers use letters to express ideas. Mathematicians look for and make use of structure. (MP) Mathematicians look for and express regularity in repeated reasoning. (MP)

1 e.g., 9 × 80, 5 × 60. (CCSS: 3.NBT.3)

## Content Area: MathematicsGrade Level Expectations: Second GradeStandard: 1. Number Sense, Properties, and Operations

1 e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: (CCSS: 2.NBT.1)
100 can be thought of as a bundle of ten tens — called a "hundred." (CCSS: 2.NBT.1a)
The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). (CCSS: 2.NBT.1b)

2 Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. (CCSS: 2.NBT.7)

## Content Area: MathematicsGrade Level Expectations: First GradeStandard: 1. Number Sense, Properties, and Operations

1 10 can be thought of as a bundle of ten ones — called a "ten." (CCSS: 1.NBT.2a)
The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. (CCSS: 1.NBT.2b)
The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones). (CCSS: 1.NBT.2c)

## Content Area: MathematicsGrade Level Expectations: KindergartenStandard: 1. Number Sense, Properties, and Operations

 Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter) Concepts and skills students master: 1. Whole numbers can be used to name, count, represent, and order quantity Evidence Outcomes 21st Century Skill and Readiness Competencies Students Can: Use number names and the count sequence. (CCSS: K.CC)Count to 100 by ones and by tens. (CCSS: K.CC.1)Count forward beginning from a given number within the known sequence.1 (CCSS: K.CC.2)Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20.2 (CCSS: K.CC.3) Count to determine the number of objects. (CCSS: K.CC)Apply the relationship between numbers and quantities and connect counting to cardinality.3 (CCSS: K.CC.4)Count and represent objects to 20.4 (CCSS: K.CC.5) Compare and instantly recognize numbers. (CCSS: K.CC)Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group.5 (CCSS: K.CC.6)Compare two numbers between 1 and 10 presented as written numerals. (CCSS: K.CC.7)Identify small groups of objects fewer than five without counting Inquiry Questions: Why do we count things? Is there a wrong way to count? Why? How do you know when you have more or less? What does it mean to be second and how is it different than two? Relevance & Application: Counting is used constantly in everyday life such as counting plates for the dinner table, people on a team, pets in the home, or trees in a yard. Numerals are used to represent quantities. People use numbers to communicate with others such as two more forks for the dinner table, one less sister than my friend, or six more dollars for a new toy. Nature Of: Mathematics involves visualization and representation of ideas. Numbers are used to count and order both real and imaginary objects. Mathematicians attend to precision. (MP) Mathematicians look for and make use of structure. (MP)

1 instead of having to begin at 1. (CCSS: K.CC.2)

2 with 0 representing a count of no objects. (CCSS: K.CC.3)

3 When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. (CCSS: K.CC.4a)
Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. (CCSS: K.CC.4b)
Understand that each successive number name refers to a quantity that is one larger. (CCSS: K.CC.4c)

4 Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration. (CCSS: K.CC.5)
Given a number from 1–20, count out that many objects. (CCSS: K.CC.5)

5 e.g., by using matching and counting strategies. (CCSS: K.CC.6)