In this activity, students will watch a short video clip that displays how quickly past viruses have spread and how deadly they are. This will activate students' prior knowledge about rates of growth and provide the context for the simulation activity they will engage in during the next activity.
This activity results from the ALEX Resource Development Summit.
This exploration provides students the opportunity to actively engage in creating their own knowledge about exponential models. They are given the support to conduct their own simulation and record the information, make predictions using the data that they have collected, and compare their predictions to the technology generated models. Additionally, the technology element is necessary to make sense of the data in a more efficient manner as compared to hand calculations of procedures. The emphasis is to produce a deep conceptual understanding of rates of change of exponential functions in multiple representations and use that information to build up procedural fluency.
This activity allows teachers to formatively assess students' understanding of multiplicative rates of change from different representations. The activity presents four different viruses of different spread rates, and students are to determine which virus is the most virulent and justify their responses.
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.
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.
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.
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).
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.
Students will apply their critical thinking skills to learn about multiplication and division of exponents. This interactive exercise focuses on positive and negative exponents and combining exponents in an effort to help students recognize patterns and determine a rule.
This resource is part of the Math at the Core: Middle School collection.