This exploration can be done in class near the beginning of a …
This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is.
While not a full-blown modeling problem, this task does address some aspects …
While not a full-blown modeling problem, this task does address some aspects of modeling as described in Standard for Mathematical Practice 4. Also, students often think that time must always be the independent variable, and so may need some help understanding that one chooses the independent and dependent variable based on the way one wants to view a situation.
This task can be used as a quick assessment to see if …
This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match.
This task emphasizes the expectation that students know linear functions grow by …
This task emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.
The goal of this task is to get students to focus on …
The goal of this task is to get students to focus on the shape of the graph of the equation y=ex and how this changes depending on the sign of the exponent and on whether the exponential is in the numerator or denominator. It is also intended to develop familiarity, in the case of f and k, with the functions which are used in logistic growth models, further examined in ``Logistic Growth Model, Explicit Case'' and ``Logistic Growth Model, Abstract Verson.''
In this task, students use trigonometric functions to model the movement of …
In this task, students use trigonometric functions to model the movement of a point around a wheel and, in the case of part (c), through space (F-TF.5). Students also interpret features of graphs in terms of the given real-world context (F-IF.4).
This is a direct task suitable for the early stages of learning …
This is a direct task suitable for the early stages of learning about exponential functions. Students interpret the relevant parameters in terms of the real-world context and describe exponential growth.
The purpose of this task is to probe students' ability to correlate …
The purpose of this task is to probe students' ability to correlate symbolic statements about a function using function notation with a graph of the function, and to interpret their answers in terms of the quantities between which the function describes a relationship
The purpose of this task is to help students learn to read …
The purpose of this task is to help students learn to read information about a function from its graph, by asking them to show the part of the graph that exhibits a certain property of the function. The task could be used to further instruction on understanding functions, or as an assessment tool with the caveat that it requires some amount of creativity to decide how to best illustrate some of the statements.
This problem provides an opportunity to experiment with modeling real data. Populations …
This problem provides an opportunity to experiment with modeling real data. Populations are often modeled with exponential functions and in this particular case we see that, over the last 200 years, the rate of population growth accelerated rapidly, reaching a peak a little after the middle of the 20th century and now it is slowing down.
This is a fast-paced introductory course to the C++ programming language. It …
This is a fast-paced introductory course to the C++ programming language. It is intended for those with little programming background, though prior programming experience will make it easier, and those with previous experience will still learn C++-specific constructs and concepts. This course is offered during the Independent Activities Period (IAP), which is a special 4-week term at MIT that runs from the first week of January until the end of the month.
This is a undergraduate course. It will cover normed spaces, completeness, functionals, …
This is a undergraduate course. It will cover normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators; Lebesgue measure, measurable functions, integrability, completeness of L-p spaces; Hilbert space; compact, Hilbert-Schmidt and trace class operators; as well as spectral theorem.
This task lets students explore the differences between linear and non-linear functions. …
This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions. The task lends itself to an extended discussion comparing the differences that students have found and relating them back to the equation and the graph of the two functions.
This is a text for a two-term course in introductory real analysis …
This is a text for a two-term course in introductory real analysis for junior or senior mathematics majors and science students with a serious interest in mathematics. Prospective educators or mathematically gifted high school students can also benefit from the mathematical maturity that can be gained from an introductory real analysis course.
The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics. The standard elementary calculus sequence is the only specific prerequisite for Chapters 1–5, which deal with real-valued functions. (However, other analysis oriented courses, such as elementary differential equation, also provide useful preparatory experience.) Chapters 6 and 7 require a working knowledge of determinants, matrices and linear transformations, typically available from a first course in linear algebra. Chapter 8 is accessible after completion of Chapters 1–5.
In the middle grades, students have lots of experience analyzing and comparing …
In the middle grades, students have lots of experience analyzing and comparing linear functions using graphs, tables, symbolic expressions, and verbal descriptions. In this task, students may choose a representation that suits them and then reason from within that representation.
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