ALEX Resources

   View Standards     Standard(s): [MA2019] ACC-8 (8) 26 :

26. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [Algebra I with Probability, 24]

[MA2019] AL1-19 (9-12) 24 :

24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Learn about the systems literacy concept of exponential growth as a type of reinforcing feedback, using "Chess Wager, a video from Cyberchase in which Harry’s friend explains how putting a penny on one square of a chessboard and then doubling the amount on each subsequent square could generate a tremendous amount of money over time. This resource is part of the Systems Literacy Collection.

   View Standards     Standard(s): [MA2019] ACC-8 (8) 26 :

26. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [Algebra I with Probability, 24]

[MA2019] AL1-19 (9-12) 24 :

24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Following a profile of Elton Brand, an accomplished basketball player who uses math in his work, students are presented with a mathematical basketball challenge. In the challenge, students focus on understanding the Big Ideas of Algebra: patterns, relationships, equivalence, and linearity; learn to use a variety of representations, including modeling with variables; build connections between numeric and algebraic expressions; and use what they have learned previously about number and operations, measurement, proportionality, and discrete mathematics as applications of algebra.  This resource is part of the Math at the Core: Middle School Collection.

   View Standards     Standard(s): [MA2019] ACC-8 (8) 26 :

26. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [Algebra I with Probability, 24]

[MA2019] ACC-8 (8) 28 :

28. Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically. [Algebra I with Probability, 26]

[MA2019] AL1-19 (9-12) 24 :

24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

[MA2019] AL1-19 (9-12) 26 :

26. Use graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

In this video lesson, students investigate how quantities that grow quadratically compare to those that grow exponentially. They discover the reason that increasing exponential functions also eventually surpass increasing quadratic functions. By examining successive quotients for each type of function, students see that the outputs of quadratic functions are not multiplied by the same factor each time the input increases by one. In fact, these successive quotients get smaller as the inputs increase, while the outputs of the exponential function have the same multiplier. As they compare the two types of functions, they develop their understanding of quadratic expressions and how the shape of the graph differs between the two types of functions.

   View Standards     Standard(s): [MA2019] ACC-8 (8) 26 :

26. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [Algebra I with Probability, 24]

[MA2019] ACC-8 (8) 27 :

27. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). [Algebra I with Probability, 25]

[MA2019] AL1-19 (9-12) 24 :

24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

[MA2019] AL1-19 (9-12) 25 :

25. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

This is the first of several video lessons in which students construct quadratic functions to represent various situations. Here they investigate the movement of free-falling objects. Students analyze the vertical distances that falling objects travel over time and see that they can be described by quadratic functions. They use tables, graphs, and equations to represent and make sense of the functions. In subsequent lessons, students build on the functions developed here to represent projectile motions, providing a context to develop an understanding of the zeros, vertex, and domain of quadratic functions.

To express the relationship between distance and time, students need to see regularity in numerical values and express that regularity (MP8).

   View Standards     Standard(s): [MA2019] ACC-8 (8) 5 :

5. Use the structure of an expression to identify ways to rewrite it. [Algebra I with Probability, 5]

Example: See x⁴ - y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²).

[MA2019] ACC-8 (8) 26 :

26. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [Algebra I with Probability, 24]

[MA2019] AL1-19 (9-12) 5 :

5. Use the structure of an expression to identify ways to rewrite it.

Example: See x⁴ - y⁴ as (x²)² - (y²)², thus recognizing it as a difference of squares that can be factored as (x² - y²)(x² + y²).

[MA2019] AL1-19 (9-12) 24 :

24. Distinguish between situations that can be modeled with linear functions and those that can be modeled with exponential functions.

a. Show that linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals.

b. Define linear functions to represent situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Define exponential functions to represent situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

This video lesson transitions students from reasoning concretely and contextually about quadratic functions to reasoning about their representations in ways that are more abstract and formal (MP2).

In earlier grades, students reasoned about multiplication by thinking of the product as the area of a rectangle where the two factors being multiplied are the side lengths of the rectangle. In this lesson, students use this familiar reasoning to expand expressions such as (x + 4)(x + 7), where x + 4, and x + 7 are side lengths of a rectangle with each side length decomposed into x and a number. They use the structure in the diagrams to help them write equivalent expressions in expanded form, for example, x² + 11x + 28 (MP7). Students recognize that finding the sum of the partial areas in the rectangle is the same as applying the distributive property to multiply out the terms in each factor.

The terms “standard form” and “factored form” are not yet used and will be introduced in an upcoming lesson, after students have had some experience working with the expressions.