HS3+Activities+and+Resources

__Activities __
1. What are some ways that the students can learn determinants of 2x2 and 3x3 matrices on their own and teach it to or share with their peers? calculating 2x2 determinants (interactive): [] calculating 3x3 determinants "basketweave" (interactive): [] calculating 2x2 determinants (plug and find):[] calculating 3x3 determinants (plug and find): []

2. Are there collaborative activities that students can do to connect the concept of 2x2 determinants with 3x3 determinants? playing cards activity: [] CLASS/TEAM activity: Teams/groups create a presentation/"lesson" on 2x2 matrices and other teams/groups work on 3x3 matrices. Once each team finishes the work, each member of the teams/groups will collaborate with another person from the other team/groups and create a new presentation that combines and connects the concept of determinants of 2x2 and 3x3 matrices.

Basket-weave: • Will not work for NxN matrix greater than N=3 • Very simple implementation Interactive basket-weave example: []
 * 3. What are the advantages of the "cofactor" method vs. the "basket-weave" method for a 3x3 determinant?**

Cofactor: • Works for all NxN matrices for N>= 3 • Helps students practice matrix notation by identifying and dealing with i-j elements Cofactor expansion: http://comp.uark.edu/~jjrencis/femur/Learning-Modules/Linear-Algebra/solving/determinant/cofactor_expansion.html


 * 4. How can teachers explain why Cramer's Rule is usually a better method of solving systems than the "traditional" three (substitution, graphing, elimination)?**


 * Graphing three variables on 2-D graph paper is difficult (even with special 'Avatar' glasses)
 * Have the students work through substitution and elimination and count the steps. They will see that there can be as many as 12 steps and these steps are 'non-repetitive' (i.e., the steps for one problem will probably differ from the steps for other problems).
 * Have the students work through Cramer's Rule for 3x3 matrices and count the **different** steps. They will notice there are no more than 8 steps and these will be the same from one problem to the next. This similarity of steps will allow for strong reinforcement.
 * Cramer's Rule for 3x3: http://www.analyzemath.com/Tutorial-System-Equations/cramers_rule.html

Ask students to come up with situations that would require solving a system of equations, either 2x2 or 3x3. Perhaps they can present their problem to the class, and the class can "rate" the various problems in terms of creativity, clarity, and difficulty.
 * 5. Which real-world applications are most effective for getting students to practice and learn the methods?**

The definition, vocabulary, and notation of basic matrix operations How to translate an application problem into the appropriate system of equations Knowledge of various linear applications, such as mixture problems, velocity, problems, etc. - students will practice throughout the year on translating words into equations and expressions - students can solve various linear applications through collaboration before producing their own application problem __For example:__ //Explain something like "36 gallons of a 25% alcohol solution" means: 25%, or one quarter, of the solution is pure alcohol. One quarter of 36 is 9. So that 36-gallon solution contains 9 gallons of pure alcohol.
 * 6. Which other standards/knowledge need to be included in a project application to allow us enough time to present it properly?**

Here is the problem: How many gallons of 30% alcohol solution and how many of 60% alcohol solution must be mixed to produce 18 gallons of 50% solution? "18 gallons of 50% solution" means: 50%, or half, is pure alcohol. The final solution, then, will have 9 gallons of pure alcohol. Let x be the number of gallons of 30% solution. Let y be the number of gallons of 60% solution. 1) Total number of gallons x+y=18 2) Gallons of pure alcohol .3x+.6y=9 2') 3x+6y=90 Equations 1) and 2') are the two equations in the two unknowns. The solutions are: x = 6 gallons, y = 12 gallons.//

__ Resources __
We can look at old textbooks and websites for examples of application problems. We can //ask students// what kinds of scenarios are interesting to them and which they would be interested in exploring more deeply.
 * 5. Which real-world applications are most effective for getting students to practice and learn the methods?**

Textbooks and websites such as practice the lower-level skills needed to solve application problems A "bottomless worksheet"of solving 3x3 systems for practice. Practice identifying keywordswhen translating word problems into equations Examples of translating word problems into equations. Another explanation with exercises of how to translate words into symbols.
 * 6. Which other standards/knowledge need to be included in a project application to allow us enough time to present it properly?**