New Colorado P-12 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:

2. Different models and representations can be used to compare fractional parts

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Use ideas of fraction equivalence and ordering to: (CCSS: 4.NF)
    • Explain equivalence of fractions using drawings and models.4
    • Use the principle of fraction equivalence to recognize and generate equivalent fractions. (CCSS: 4.NF.1)
    • Compare two fractions with different numerators and different denominators,5 and justify the conclusions.6 (CCSS: 4.NF.2)
  2. Build fractions from unit fractions by applying understandings of operations on whole numbers. (CCSS: 4.NF)
    • Apply previous understandings of addition and subtraction to add and subtract fractions.7
      • Compose and decompose fractions as sums and differences of fractions with the same denominator in more than one way and justify with visual models.
      • Add and subtract mixed numbers with like denominators.8 (CCSS: 4.NF.3c)
      • Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators.9 (CCSS: 4.NF.3d)
    • Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. (CCSS: 4.NF.4)
      • Express a fraction a/b as a multiple of 1/b.10 (CCSS: 4.NF.4a)
      • Use a visual fraction model to express a/b as a multiple of 1/b, and apply to multiplication of whole number by a fraction.11 (CCSS: 4.NF.4b)
      • Solve word problems involving multiplication of a fraction by a whole number.12 (CCSS: 4.NF.4c)

Inquiry Questions:

  1. How can different fractions represent the same quantity?
  2. How are fractions used as models?
  3. Why are fractions so useful?
  4. What would the world be like without fractions?

Relevance & Application:

  1. Fractions and decimals are used any time there is a need to apportion such as sharing food, cooking, making savings plans, creating art projects, timing in music, or portioning supplies.
  2. Fractions are used to represent the chance that an event will occur such as randomly selecting a certain color of shirt or the probability of a certain player scoring a soccer goal.
  3. Fractions are used to measure quantities between whole units such as number of meters between houses, the height of a student, or the diameter of the moon.

Nature Of:

  1. 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.
  2. Mathematicians construct viable arguments and critique the reasoning of others. (MP)
  3. Mathematicians model with mathematics. (MP)

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

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

2. Parts of a whole can be modeled and represented in different ways

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Develop understanding of fractions as numbers. (CCSS: 3.NF)
    • Describe a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; describe a fraction a/b as the quantity formed by a parts of size 1/b. (CCSS: 3.NF.1)
    • Describe a fraction as a number on the number line; represent fractions on a number line diagram.2 (CCSS: 3.NF.2)
    • Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. (CCSS: 3.NF.3)
      • Identify two fractions as equivalent (equal) if they are the same size, or the same point on a number line. (CCSS: 3.NF.3a)
      • Identify and generate simple equivalent fractions. Explain3 why the fractions are equivalent.4 (CCSS: 3.NF.3b)
      • Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.5 (CCSS: 3.NF.3c)
      • Compare two fractions with the same numerator or the same denominator by reasoning about their size. (CCSS: 3.NF.3d)
      • Explain why comparisons are valid only when the two fractions refer to the same whole. (CCSS: 3.NF.3d)
      • Record the results of comparisons with the symbols >, =, or <, and justify the conclusions.6 (CCSS: 3.NF.3d)

Inquiry Questions:

  1. How many ways can a whole number be represented?
  2. How can a fraction be represented in different, equivalent forms?
  3. How do we show part of unit?

Relevance & Application:

  1. Fractions are used to share fairly with friends and family such as sharing an apple with a sibling, and splitting the cost of lunch.
  2. Equivalent fractions demonstrate equal quantities even when they are presented differently such as knowing that 1/2 of a box of crayons is the same as 2/4, or that 2/6 of the class is the same as 1/3.

Nature Of:

  1. Mathematicians use visual models to solve problems.
  2. Mathematicians make sense of problems and persevere in solving them. (MP)
  3. Mathematicians reason abstractly and quantitatively. (MP)

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

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

3. Expressions can be represented in multiple, equivalent forms

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Interpret the structure of expressions.(CCSS: A-SSE)
    • Interpret expressions that represent a quantity in terms of its context.* (CCSS: A-SSE.1)
      • Interpret parts of an expression, such as terms, factors, and coefficients. (CCSS: A-SSE.1a)
      • Interpret complicated expressions by viewing one or more of their parts as a single entity.13 (CCSS: A-SSE.1b)
    • Use the structure of an expression to identify ways to rewrite it.14 (CCSS: A-SSE.2)
  2. Write expressions in equivalent forms to solve problems. (CCSS: A-SSE)
    • Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* (CCSS: A-SSE.3)
      • Factor a quadratic expression to reveal the zeros of the function it defines. (CCSS: A-SSE.3a)
      • Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (CCSS: A-SSE.3b)
      • Use the properties of exponents to transform expressions for exponential functions.15 (CCSS: A-SSE.3c)
    • Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. * (CCSS: A-SSE.4)
  3. Perform arithmetic operations on polynomials. (CCSS: A-APR)
    • Explain that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. (CCSS: A-APR.1)
  4. Understand the relationship between zeros and factors of polynomials. (CCSS: A-APR)
    • State and apply the Remainder Theorem.17 (CCSS: A-APR.2)
    • Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. (CCSS: A-APR.3)
  5. Use polynomial identities to solve problems. (CCSS: A-APR)
    • Prove polynomial identities18 and use them to describe numerical relationships. (CCSS: A-APR.4)
  6. Rewrite rational expressions. (CCSS: A-APR)
  7. Rewrite simple rational expressions in different forms.19 (CCSS: A-APR.6)

Inquiry Questions:

  1. When is it appropriate to simplify expressions?
  2. The ancient Greeks multiplied binomials and found the roots of quadratic equations without algebraic notation. How can this be done?

Relevance & Application:

  1. The simplification of algebraic expressions and solving equations are tools used to solve problems in science. Scientists represent relationships between variables by developing a formula and using values obtained from experimental measurements and algebraic manipulation to determine values of quantities that are difficult or impossible to measure directly such as acceleration due to gravity, speed of light, and mass of the earth.
  2. The manipulation of expressions and solving formulas are techniques used to solve problems in geometry such as finding the area of a circle, determining the volume of a sphere, calculating the surface area of a prism, and applying the Pythagorean Theorem.

Nature Of:

  1. Mathematicians abstract a problem by representing it as an equation. They travel between the concrete problem and the abstraction to gain insights and find solutions.
  2. Mathematicians construct viable arguments and critique the reasoning of others. (MP)
  3. Mathematicians model with mathematics. (MP)
  4. Mathematicians look for and express regularity in repeated reasoning. (MP)

13 For example, interpret \(P(1+r)^n\) as the product of P and a factor not depending on P. (CCSS: A-SSE.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: A-SSE.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: A-SSE.3c)

16 For example, calculate mortgage payments. (CCSS: A-SSE.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: A-APR.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: A-APR.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: A-APR.6)

Content Area: Mathematics
Grade Level Expectations: Eighth Grade
Standard: 2. Patterns, Functions, and Algebraic Structures

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

1. Linear functions model situations with a constant rate of change and can be represented numerically, algebraically, and graphically

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Describe the connections between proportional relationships, lines, and linear equations. (CCSS: 8.EE)
  2. Graph proportional relationships, interpreting the unit rate as the slope of the graph. (CCSS: 8.EE.5)
  3. Compare two different proportional relationships represented in different ways.1 (CCSS: 8.EE.5)
  4. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane. (CCSS: 8.EE.6)
  5. Derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. (CCSS: 8.EE.6)

Inquiry Questions:

  1. How can different representations of linear patterns present different perspectives of situations?
  2. How can a relationship be analyzed with tables, graphs, and equations?
  3. Why is one variable dependent upon the other in relationships?

Relevance & Application:

  1. Fluency with different representations of linear patterns allows comparison and contrast of linear situations such as service billing rates from competing companies or simple interest on savings or credit.
  2. Understanding slope as rate of change allows individuals to develop and use a line of best fit for data that appears to be linearly related.
  3. The ability to recognize slope and y-intercept of a linear function facilitates graphing the function or writing an equation that describes the function.

Nature Of:

  1. Mathematicians represent functions in multiple ways to gain insights into the relationships they model.
  2. Mathematicians model with mathematics. (MP)

1 For example, compare a distance-time graph to a distance-time 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

Prepared Graduates: (Click on a Prepared Graduate Competency to View Articulated Expectations) - (Remove PGC Filter)

Concepts and skills students master:

1. Properties of arithmetic can be used to generate equivalent expressions

Evidence Outcomes 21st Century Skill and Readiness Competencies

Students Can:

  1. Use properties of operations to generate equivalent expressions. (CCSS: 7.EE)
    • Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. (CCSS: 7.EE.1)
    • Demonstrate that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.1 (CCSS: 7.EE.2)

Inquiry Questions:

  1. How do symbolic transformations affect an equation or expression?
  2. How is it determined that two algebraic expressions are equivalent?

Relevance & Application:

  1. The ability to recognize and find equivalent forms of an equation allows the transformation of equations into the most useful form such as adjusting the density formula to calculate for volume or mass.

Nature Of:

  1. Mathematicians abstract a problem by representing it as an equation. They travel between the concrete problem and the abstraction to gain insights and find solutions.
  2. Mathematicians reason abstractly and quantitatively. (MP)
  3. Mathematicians look for and express regularity in repeated reasoning. (MP)

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)