New Colorado P12 Academic Standards
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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:

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^{4} Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. (CCSS: 4.NF.1)
^{5} e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, (CCSS: 4.NF.2)
^{6} e.g., by using a visual fraction model. (CCSS: 4.NF.2)
^{7} Understand a fraction a/b with a > 1 as a sum of fractions 1/b. (CCSS: 4.NF.3)
Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. (CCSS: 4.NF.3a)
Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. (CCSS: 4.NF.3b)
^{8} e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. (CCSS: 4.NF.3c)
^{9} e.g., by using visual fraction models and equations to represent the problem. (CCSS: 4.NF.3d)
^{10} For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). (CCSS: 4.NF.4a)
^{11} For example, 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) (CCSS: 4.NF.4b)
^{12} e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? (CCSS: 4.NF.4c)
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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^{2} Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. (CCSS: 3.NF.2a)
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. (CCSS: 3.NF.2b)
^{3} e.g., 1/2 = 2/4, 4/6 = 2/3). (CCSS: 3.NF.3b)
^{4} e.g., by using a visual fraction model.(CCSS: 3.NF.3b)
^{5} Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. (CCSS: 3.NF.3c)
^{6} e.g., by using a visual fraction model. (CCSS: 3.NF.3d)
Content Area: Mathematics
Grade Level Expectations: High School
Standard: 2. Patterns, Functions, and Algebraic Structures
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Evidence Outcomes  21st Century Skill and Readiness Competencies 
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^{13} For example, interpret \(P(1+r)^n\) as the product of P and a factor not depending on P. (CCSS: ASSE.1b)
^{14} For example, see \(x^4  y^4\) as \((x^2)^2 – (y^2)^2\), thus recognizing it as a difference of squares that can be factored as \((x^2 – y^2)(x^2 + y^2)\). (CCSS: ASSE.2)
^{15} For example the expression \(1.15^t\) can be rewritten as \((1.15^\frac{1}{12})^{12t}\) \(\approx\) \(1.012^{12t}\) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. (CCSS: ASSE.3c)
^{16} For example, calculate mortgage payments. (CCSS: ASSE.4)
^{17} For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). (CCSS: AAPR.2)
^{18} For example, the polynomial identity \((x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2\) can be used to generate Pythagorean triples. (CCSS: AAPR.4)
^{19} write \(\frac{a(x)}{b(x)}\) in the form \(q(x) + \frac{r(x)}{b(x)}\), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. (CCSS: AAPR.6)
Content Area: Mathematics
Grade Level Expectations: Eighth Grade
Standard: 2. Patterns, Functions, and Algebraic Structures
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^{1} For example, compare a distancetime graph to a distancetime equation to determine which of two moving objects has greater speed. (CCSS: 8.EE.5)
Content Area: Mathematics
Grade Level Expectations: Seventh Grade
Standard: 2. Patterns, Functions, and Algebraic Structures
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Concepts and skills students master:


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

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Relevance & Application:
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^{1} For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05." (CCSS: 7.EE.2)