The lesson materials include Trigonometric Problems and Equations including double angle formulas, fundamental trigonometric identities, and sum and difference formulas.
In this task students must investigate this conjecture to discover that it does not work in all cases: Pick any two integers. Look at the sum of their squares, the difference of their squares, and twice the product of the two integers you chose. Those three numbers are the sides of a right triangle.
These notes contain many sample calculations. It is important to do these yourself—type them in at your keyboard and see what happens on your screen—to get the feel of working in R.
Exercises in the middle of a section should be done immediately when you get to them, and make sure you have them right before moving on.
Many other similar introductions are scattered around the web; a partial list is in the “contributed documentation” section on the R web site (http://cran.r-project.org/other-docs.html). This particular version is limited (it has similar coverage to the standard Introduction to R manual and targets biologists who are neither programmers nor statisticians (yet).
This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4).
This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.
Understanding Linear Algebra is a freely available linear algebra textbook suitable for use in a first undergraduate linear algebra course. The text aims to support readers as they develop their ability to think about linear algebra conceptually, their computational fluency, and their understanding of the role that linear algebra plays in shaping our society. It is also designed to support an active learning classroom environment.
OLO: Students will be able to explain how changing the y-intercept and slope will change the appearance of the line.This lesson is to help students develop the relationship between slope(m) and y-intercept(b) by using an interactive coordinate plane to explore. This is a 2 part lesson with some teacher clarification in between the 2 parts. The teacher component in between the 2 parts is to make sure the students fully understand the relationships developed in part 1 before applying it to part 2.If you have a Face to Face class and just want a print out/editable version for Google Classroom Click Here
Using manipulatives, students figure out patterns in problems involving seating arrangements. They write algebraic equations to describe the relationships.
Students are introduced to algebraic expressions that use more than one variable and have multiple solutions. They figure out combinations of two items at different costs, with each combination adding up to 100.
In this seminar, you will learn about variables and how they are used in mathematical expressions. You will also learn what it means to be a like term and how to combine like terms. You will pull variables, expressions, and like terms together so that you can apply them moving forward.StandardsCC.2.2.HS.D.1Interpret the structure of expressions to represent a quantity in terms of its context.CC.2.2.HS.D.2Write expressions in equivalent forms to solve problems.
In this seminar you will learn how to solve problems where variables are on each side of the equation. You will apply techniques you have learned to this point involving inverse operations. You will use the techniques learned in this seminar to verify solutions to various other types of problems as you move forward, primarily when solving for any unknown variable. When solving equations with a variable on each side, you will first want to isolate the variable to one side and then use inverse operations to solve for it. StandardsCC.2.2.HS.D.8Apply inverse operations to solve equations or formulas for a given variable.
In this seminar you will learn about variation. There are two types - direct and inverse- and you will be able to identify and calculate each type. You will also learn how to solve real-world problems involving each type of variation, to better understand the importance of variation.StandardsCC.2.2.HS.D.8Apply inverse operations to solve equations or formulas for a given variable.CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities.MP.1. Make sense of problems and persevere in solving them.MP.2. Reason abstractly and quantitatively.MP.3. Construct viable arguments and critique the reasoning of others.MP.4. Model with mathematics.MP.5. Use appropriate tools strategically.MP.6. Attend to precision.MP.7. Look for and make use of structure.MP.8. Look for and express regularity in repeated reasoning.
This activity is a indoor lab where students gather data within their school to develop the concept of displacement in two dimensional motion.
Created for a first-year university course, this linear algebra textbook takes an unusual approach: it introduces vector spaces at the outset and deals with linear systems only after a thorough introduction to vector spaces. This approach is based on the authors' experience over the past 25 years that students often need more time to master vector spaces while traditional textbooks relegate the topic to the end of the course. In this way, these new notions at the heart of linear algebra that are often considered abstract and difficult in an introductory course can then be used in the rest of the course as well as in different contexts.
Students work in groups to find a "treasure" using a treasure map with directions written in vector format.
This collection of videos presents Professor Strang’s updated vision of how linear algebra could be taught.
It starts with six brief videos, recorded in 2020, containing many ideas and suggestions about the recommended order of topics in teaching and learning linear algebra. Topics include A New Way to Start Linear Algebra, The Column Space of a Matrix, The Big Picture of Linear Algebra, Orthogonal Vectors, Eigenvalues and Eigenvectors, and Singular Values and Singular Vectors.
An additional brief video, recorded in 2021, Finding the Nullspace: Solving Ax = 0 by Elimination, computes the nullspace of any matrix A.
In 2023, Professor Strang recorded a new one-hour video, Five Factorizations of a Matrix, providing an overall look at linear algebra by highlighting five different ways that a matrix gets factored.
Prepared with pre-algebra or algebra 1 classes in mind, this module leads students through the process of graphing data and finding a line of best fit while exploring the characteristics of linear equations in algebraic and graphic formats. Then, these topics are connected to real-world experiences in which people use linear functions. During the module, students use these scientific concepts to solve the following hypothetical challenge: You are a new researcher in a lab, and your boss has just given you your first task to analyze a set of data. It being your first assignment, you ask an undergraduate student working in your lab to help you figure it out. She responds that you must determine what the data represents and then find an equation that models the data. You believe that you will be able to determine what the data represents on your own, but you ask for further help modeling the data. In response, she says she is not completely sure how to do it, but gives a list of equations that may fit the data. This module is built around the legacy cycle, a format that incorporates educational research feindings on how people best learn.
Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat’s pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.
This set of problems for Linear Algebra in the open-source WeBWorK mathematics platform was created under a Round Eleven Mini-Grant for Ancillary Materials Creation. The problems were created for an implementation of the CC-BY Lyrix open textbook A First Course in Linear Algebra. Also included as an additional file are the selected and modified Lyryx Class Notes for the textbook.
Topics covered include:
Linear Independence
Linear Transformations
Matrix of a Transformation
Isomorphisms
Eigenvalues and Eigenvectors
Diagonalization
Orthogonality