New Colorado P12 Academic Standards
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Content Area: Mathematics
Grade Level Expectations: High School
Standard: 1. Number Sense, Properties, and Operations
Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations)  (Remove PGC Filter) 

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Evidence Outcomes  21st Century Skill and Readiness Competencies 
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^{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: NRN.1)
Content Area: Mathematics
Grade Level Expectations: Eighth Grade
Standard: 1. Number Sense, Properties, and Operations
Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations)  (Remove PGC Filter) 

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Evidence Outcomes  21st Century Skill and Readiness Competencies 
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^{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: Mathematics
Grade Level Expectations: Sixth Grade
Standard: 1. Number Sense, Properties, and Operations
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Evidence Outcomes  21st Century Skill and Readiness Competencies 
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^{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: Mathematics
Grade Level Expectations: Fifth Grade
Standard: 1. Number Sense, Properties, and Operations
Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations)  (Remove PGC Filter) 

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Evidence Outcomes  21st Century Skill and Readiness Competencies 
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^{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) 

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Evidence Outcomes  21st Century Skill and Readiness Competencies 
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^{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 50pound 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) ๗ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ๗ 4 = 1/12 because (1/12) ื 4 = 1/3. (CCSS: 5.NF.7a)
^{13} For example, create a story context for 4 ๗ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ๗ (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 1/3cup servings are in 2 cups of raisins? (CCSS: 5.NF.7c)
Content Area: Mathematics
Grade Level Expectations: Fourth Grade
Standard: 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:


Evidence Outcomes  21st Century Skill and Readiness Competencies 
Students Can:

Inquiry Questions:
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^{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: Mathematics
Grade Level Expectations: Third Grade
Standard: 1. Number Sense, Properties, and Operations
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Evidence Outcomes  21st Century Skill and Readiness Competencies 
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^{1} e.g., 9 ื 80, 5 ื 60. (CCSS: 3.NBT.3)
Content Area: Mathematics
Grade Level Expectations: Second Grade
Standard: 1. Number Sense, Properties, and Operations
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Evidence Outcomes  21st Century Skill and Readiness Competencies 
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^{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 threedigit 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: Mathematics
Grade Level Expectations: First Grade
Standard: 1. Number Sense, Properties, and Operations
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Evidence Outcomes  21st Century Skill and Readiness Competencies 
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Inquiry Questions:
Relevance & Application:
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^{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: Mathematics
Grade Level Expectations: Kindergarten
Standard: 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:


Evidence Outcomes  21st Century Skill and Readiness Competencies 
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Inquiry Questions:
Relevance & Application:
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^{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 120, count out that many objects. (CCSS: K.CC.5)
^{5} e.g., by using matching and counting strategies. (CCSS: K.CC.6)