Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.*

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Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Write a function that describes a relationship between two quantities.*

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Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: 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 ‰äĘ 1 (x not equal to 1).

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Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

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Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Combine standard function types using arithmetic operations. 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.

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Learning Domain: Functions: Building Functions

Standard: Build a function that models a relationship between two quantities

Indicator: Determine an explicit expression, a recursive process, or steps for calculation from a context.

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Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x -1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x^2 + x + 1), and (x - 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

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Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

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Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Produce an invertible function from a non-invertible function by restricting the domain.

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Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

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Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: Find inverse functions.

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Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

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Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.

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Learning Domain: Functions: Building Functions

Standard: Build new functions from existing functions

Indicator: (+) Verify by composition that one function is the inverse of another.

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