Sal solves a linear system with 3 variables by representing it with an augmented matrix and bringing the matrix to reduced row-echelon form. Gaussian Elimination is the method, reduced row echelon is just the final result.
This lesson unit is intended to help teachers assess how well students are able to: recognize the differences between equations and identities; substitute numbers into algebraic statements in order to test their validity in special cases; resist common errors when manipulating expressions such as 2(x Đ 3) = 2x Đ 3; (x + 3)_ = x_ + 3_; and carry out correct algebraic manipulations. It also aims to encourage discussion on some common misconceptions about algebra.
Determining whether 3 vectors are linearly independent and/or span R3. Created by Sal Khan.
Proficiency Scales
Math
Algebra
Algebra Essentials
Created by Special School District of St. Louis County
This course forms an introduction to a selection of mathematical topics that are not covered in traditional mechanical engineering curricula, such as differential geometry, integral geometry, discrete computational geometry, graph theory, optimization techniques, calculus of variations and linear algebra. The topics covered in any particular year depend on the interest of the students and instructor. Emphasis is on basic ideas and on applications in mechanical engineering. This year, the subject focuses on selected topics from linear algebra and the calculus of variations. It is aimed mainly (but not exclusively) at students aiming to study mechanics (solid mechanics, fluid mechanics, energy methods etc.), and the course introduces some of the mathematical tools used in these subjects. Applications are related primarily (but not exclusively) to the microstructures of crystalline solids.
In this video segment from Cyberchase, Hacker and the CyberSquad race to reach the Good Vibration on staircases that grow at different rates and have steps of varying sizes.
Speedy Feedback for the teacher to quickly check for student understanding.
In this seminar you will explore square roots and see how they are used throughout mathematics. You will see how “squaring” a number and “taking the square root” relate to each other. Also presented will be how to estimate square roots when given a number. You will have plenty of opportunity to try practice problems, review, and even create a short presentation. You will apply techniques you have learned involving estimating. You will use the techniques learned in this seminar to solve problems involving square roots moving forward and ultimately be able to verify solutions. When working with square roots, you will identify the patterns in perfect square numbers, as well as in techniques such as squaring and cubing numbers. By the end, you should have a better understanding of what a square root is and how to handle expressions that involve square roots.StandardsCC.2.1.HS.F.1Apply and extend the properties of exponents to solve problems with rational exponents.CC.2.2.HS.D.6Extend the knowledge of rational functions to rewrite in equivalent forms.
In this seminar you will learn when quadratic equations can be solved using square root techniques, and when they can be solved by factoring and using the zero product property. You will apply techniques you have learned involving radical expressions and factoring. You will use the techniques learned in this seminar to verify solutions to various other types of problems as you move forward. When solving quadratic equations using this method, you will identify the patterns found in the solutions which will allow you to make graphic interpretations of the solutions, as well as allow you to identify when a quadratic function may (or may not) be solved using these techniques.StandardsCC.2.2.HS.C.5Construct and compare linear, quadratic, and exponential models to solve problems.
This unit is a multidisciplinary unit created for a high school math classroom, designed to combine statistics and hydrology. In this unit, students will learn about the water cycle and water budgets within the watershed. The unit starts with learning about basic budgeting in a watershed, comparable to financial budgeting, and expands to creating linear regressions based on the relationship between precipitation, discharge, and evapotranspiration in a watershed system. Students will be able to synthesize the information they learn about the watershed to learn about topics such as graphing points, lines, creating scatterplots, and creating linear regressions for the line of best fit. By teaching statistics through the lens of the watershed, the primary objective is to facilitate active, engaged learners who understand how math can be usefully applied to various contexts in the world around us while gaining a deeper appreciation for the water resources on Earth.
This was designed for a Geometry classroom, but could be modified for Pre-Algebra- Statistics based on student needs and interest level.
This lesson unit is intended to help teachers assess how well students are able to: form and solve linear equations involving factorizing and using the distributive law. In particular, this unit aims to help teachers identify and assist students who have difficulties in: using variables to represent quantities in a real-world or mathematical problem and solving word problems leading to equations of the form px + q = r and p(x + q) = r.
This seminar will explore how to solve a system of linear equations using substitution. It will also describe what it means for a system of equations to have no solution or infinitely many solutions. Applications of systems of equations will also be explored.StandardsCC.2.2.HS.D.10Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.
This problem provides students with an opportunity to discover algebraic structure in a geometric context. More specifically, the student will need to divide up the given polygons into triangles and then use the fact that the sum of the angles in each triangle is 180_.
Parts (d) and (e) of this task constitute a very advanced application of the skill of making use of structure: in (d) students are being asked to use the defining property of even and odd functions to manipulate expressions involving function notation. In (e) they are asked to see the structure in the system of two equations involving functions.
Playing the role of engineers in collaborations with the marketing and production teams in a chocolate factory, students design a container for a jumbo chocolate bar. The projects constraints mean the container has to be a regular trapezoidal prism. The design has to optimize the material used to construct the container; that is, students have to find the dimensions of the container with the maximum volume possible. After students come up with their design, teams present a final version of the product that includes creative branding and presentation. The problem-solving portion of this project requires students to find a mathematical process to express the multiple variables in the prism’s volume formula as a single variable cubic polynomial function. Students then use technology to determine the value for which this function has a maximum and, with this value, find the prism’s optimal dimensions.
Roller coasters projects are frequently used in middle and high school physics classes to illustrate the principle of conservation of mechanical energy. Potential energy transforms to kinetic energy and vice versa, with gravity being the driving force during the entire process. Even though friction force is mentioned, it is rarely considered in the velocity calculations along the coasters’ paths. In this high school lesson, the friction force is considered in the process. Using basic calculus and the work-energy theorem for non-conservative forces, the friction along a curved path is quantified, and the cart’s velocity along this path is predicted. This activity and its associated lesson are designed for AP Calculus. Practice problems/answers, a PowerPoint® presentation and student notes are provided.
This task is not about computing the final price of the shirt but about using the structure in the computation to make a general argument.
This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.
This course is an elementary introduction to number theory with no algebraic prerequisites. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions.
Discussion on explaining parabolas.