Matrix product associativity
(View Complete Item Description)Showing that matrix products are associative. Created by Sal Khan.
Material Type: Lecture
Showing that matrix products are associative. Created by Sal Khan.
Material Type: Lecture
The Mathematics Vision Project (MVP) curriculum has been developed to realize the vision and goals of the New Core Standards of Mathematics. The Comprehensive Mathematics Instruction (CMI) framework is an integral part of the materials. You can read more about the CMI framework in the Utah Mathematics Teacher Journal. (UCTM, 2009)
Material Type: Full Course, Lesson Plan, Student Guide, Textbook, Unit of Study
Table of Contents: Introduction to the Materials Module 1: Sequences Module 2: Linear & Exponential Functions Module 3: Features of Functions Module 4: Equations & Inequalities Module 5: Systems of Equations & Inequalities Module 6: Transformations & Symmetry Module 7: Congruence, Construction & Proof Module 8: Connecting Algebra & Geometry Module 9: Modeling Data Core Correlation
Material Type: Textbook
This course is designed for Senior to bridge the gap in mathematics between highschool and college.
Material Type: Lesson Plan, Textbook
This task focuses on a remarkable fact which comes out of the construction of the inscribed circle in a triangle: the angle bisectors of the three angles of triangle ABC all meet in a point.
Material Type: Activity/Lab
This task examines, in a graphical setting, the impact of adding a scalar, multiplying by a scalar, and making a linear substitution of variables on the graph of a function f. The setting here is abstract as there is no formula for the function f. The focus is therefore on understanding the geometric impact of these three operations.
Material Type: Activity/Lab
For these particular triangles, three reflections were necessary to express how to move from ABC to DEF. Sometimes, however, one reflection or two reflections will suffice. Since any rigid motion will take triangle ABC to a congruent triangle DEF, this shows the remarkable fact that any rigid motion of the plane can be expressed as one reflection, a composition of two reflections, or a composition of three reflections.
Material Type: Activity/Lab
This is a simple exercise in creating equations from a situation with many variables. By giving three different scenarios, the problem requires students to keep going back to the definitions of the variables, thus emphasizing the importance of defining variables when you write an equation. In order to reinforce this aspect of the problem, the variables have not been given names that remind the student of what they stand for. The emphasis here is on setting up equations, not solving them.
Material Type: Activity/Lab
The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence. In this problem, we considered SSA. Also insufficient is AAA, which determines a triangle up to similarity. Unlike SSA, AAS is sufficient because two pairs of congruent angles force the third pair of angles to also be congruent.
Material Type: Activity/Lab
This task is the last in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in HS algebra. Students write and solve inequalities, and represent the solutions graphically. The progression of the content standards is 6.EE.8 to 7.EE.4 to A-REI.12.
Material Type: Activity/Lab
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.
Material Type: Activity/Lab
This task shows that the three perpendicular bisectors of the sides of a triangle all meet in a point, using the characterization of the perpendicular bisector of a line segment as the set of points equidistant from the two ends of the segment. The point so constructed is called the circumcenter of the triangle.
Material Type: Activity/Lab
This task can be implemented in a variety of ways. For a class with previous exposure to the incenter or angle bisectors, part (a) could be a quick exercise in geometric constructions,. Alternatively, this could be part of a full introduction to angle bisectors, culminating in a full proof that the three angle bisectors are concurrent, an essentially complete proof of which is found in the solution below.
Material Type: Activity/Lab