New Colorado P12 Academic Standards
Current Display Filter: Mathematics  High School
Content Area: Mathematics
Grade Level Expectations: High School
Standard: 1. Number Sense, Properties, and Operations
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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)
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Content Area: Mathematics
Grade Level Expectations: High School
Standard: 2. Patterns, Functions, and Algebraic Structures
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^{1} If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (CCSS: FIF.1)
^{2} For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n1) for n \(\geq\) 1. (CCSS: FIF.3)
^{3} Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (CCSS: FIF.4)
^{4} For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (CCSS: FIF.5)
^{5} presented symbolically or as a table. (CCSS: FIF.6)
^{6} For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10,. (CCSS: FIF.8b)
^{7} For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (CCSS: FIF.9)
^{8} For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (CCSS: FBF.1b)
^{9} both positive and negative. (CCSS: FBF.3)
^{10} Include recognizing even and odd functions from their graphs and algebraic expressions for them. (CCSS: FBF.3)
^{11} Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
For example, f(x) =2 \(x^3\) or f(x) = (x+1)/(x–1) for x \(\neq\) 1. (CCSS: FBF.4a)
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^{12} include reading these from a table. (CCSS: FLE.2)
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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)
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^{20} Include equations arising from linear and quadratic functions, and simple rational and exponential functions. (CCSS: ACED.1)
^{21} For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. (CCSS: ACED.3)
^{22} For example, rearrange Ohm's law V = IR to highlight resistance R. (CCSS: ACED.4)
^{23} e.g., for \(x^2 = 49\). (CCSS: AREI.4b)
^{24} e.g., with graphs. (CCSS: AREI.6)
^{25} For example, find the points of intersection between the line y = –3x and the circle \(x^2 + y^2 = 3\). (CCSS: AREI.7)
^{26} which could be a line. (CCSS: AREI.10)
^{27} Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. (CCSS: AREI.11)
^{28} e.g., using technology to graph the functions, make tables of values, or find successive approximations. (CCSS: AREI.11)
Content Area: Mathematics
Grade Level Expectations: High School
Standard: 3. Data Analysis, Statistics, and Probability
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^{1} including joint, marginal, and conditional relative frequencies.
^{2} rate of change. (CCSS: SID.7)
^{3} constant term. (CCSS: SID.7)
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^{4} e.g., using simulation. (CCSS: SIC.2)
For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? (CCSS: SIC.2)
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^{5} the set of outcomes. (CCSS: SCP.1)
^{6} "or," "and," "not". (CCSS: SCP.1)
^{7} For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. (CCSS: SCP.4)
^{8} For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. (CCSS: SCP.5)
Content Area: Mathematics
Grade Level Expectations: High School
Standard: 4. Shape, Dimension, and Geometric Relationships
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^{9} Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. (CCSS: GSRT.4)
^{10} Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. (CCSS: GC.2)
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^{11} For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, \(\sqrt{3}\)) lies on the circle centered at the origin and containing the point (0, 2). (CCSS: GGPE.4)
^{12} e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point. (CCSS: GGPE.5)
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^{13} Use dissection arguments, Cavalieri's principle, and informal limit arguments. (CCSS: GGMD.1)
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^{14} e.g., modeling a tree trunk or a human torso as a cylinder. (CCSS: GMG.1)
^{15} e.g., persons per square mile, BTUs per cubic foot. (CCSS: GMG.2)
^{16} e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios. (CCSS: GMG.3)