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The coaches have developed numerous video lessons that cover key LLAMA concepts. These videos can be played for your students as stand‑alone lessons and contain suggested pauses for students to engage, practice skills, and practice developing arguments. Alternatively, you can the videos to plan your own lessons or as review of the arguments and content your students have already learned.
The coaches have developed numerous video lessons that cover key LLAMA concepts. These videos can be played for your students as stand‑alone lessons and contain suggested pauses for students to engage, practice skills, and practice developing arguments. Alternatively, you can the videos to plan your own lessons or as review of the arguments and content your students have already learned.
The Converse of the Pythagorean Theorem
This lesson helps students develop a “proposed” counterexample to the converse of the Pythagorean theorem with its accessible approach. Students enjoy applying and practicing the Pythagorean theorem as they discover that the counterexample was not a real example at all. The Grade 8 concept of congruent triangles is emphasized through the Grade 7 notion of the SSS congruence theorem. The concept of “eliminating the possibility of counterexamples” through indirect argumentation is made accessible to all students.
Irrational Numbers
Irrational numbers can be a tricky concept for students to understand. This lesson asks the key question, “Can you find integers that solve the equation ?” Not only does this question pave the way to understanding irrational numbers, this question develops rich understanding of the meaning of the Grade 8 word “solve.” Students look for structure in the prime structure of and to learn why this key equation is not solvable. What an awesome way to teach students that “irrational” means “not expressible as a quotient of integers” and to teach students about indirect argument!
Systems of Equations: Arguments Involving Slope
Solving systems of equations takes a great deal of practice and reflection on a variety of strategies. This lesson has several practice problems that apply understanding of what it means to solve systems of equations and understanding of the structure of equations that define a system with one, two, or no solution. The lesson helps students develop rich connections between slope, parallel versus intersection lines, and types of results from solving a system of equations (no solution, one solution, infinitely many solutions). Emphasis on techniques such as graphing and substitution are also present.
Proving Pythagorean Using Area
Common Core calls for all Grade 8 students to be able to explain a proof of the Pythagorean theorem. Students not only need to see the proof—students need to memorize key steps and their meanings. This lesson meets this standard and makes this process more accessible.
Students build a proof with the help of the teacher, while reviewing numerous Grade 8 skills and concepts. Students use rigid motions to construct copies of a right triangle, prove that objects created are squares, find the area of squares and the area of triangles, and apply equation‑solving steps to derive the Pythagorean theorem, while learning to use a referent drawing in an argument as the Common Core mathematical practices recommend.
Generic Example Arguments: Odd Number Sums and Exponent Properties
Generic example argument is critical to meeting Common Core standards at any grade level. The literature has shown that even typical Grade 3 students can create these arguments. This lesson solidifies this important idea by addressing a familiar case, the sum of odd numbers, then reviewing for Smarter Balance by addressing exponent properties in a meaningful way. Even if you have already taught many of the LLAMA conceptual pillars, going back to this lesson prior to the Smarter Balance assessments can solidify students’ understanding of mathematical structure!
Skepticism and Functions: A Claim About Prime Outputs
Cases of students’ lack of skepticism about empirical arguments are well documented in the education literature as a problem that can be fixed. This lesson teaches about functions, while attending to notions of skepticism and general argument in the same lesson! The focus on prime outputs discusses the notion of input and output and domains of functions, while encouraging students to be cautious about arguments that do not show the structure of “why.”
Lines, Transversals, and Angles
Common Core calls for students to make observations and arguments about lines cut by transversals and the angles created. This student pursues that goal with a vigor and rigor! (In an accessible way, of course.) Students review several critical Grade 8 concepts and skills as they develop arguments for lines, transversals, and the angles created. Prior results reviewed and used include results about triangles and the measures of their interior angles, supplemental angles, solving systems of equations, and parallel and intersecting lines—all the while learning about contrapositive arguments.
Depending on where you are in your teaching of viable argumentation, one or more of these videos is right for you. The first link reviews indirect reasoning by complimenting the LLAMA copies lesson called “CP11b Sudoku Logic.” The second link reviews reasoning with the 2-by-2 mental models table (e.g., “All mammals live on land,” etc.), which many of you have called a “Punnett square.” The third link goes straight to the content argument. Many of you will want to go straight to Part 3, but others will wish to start with Parts 1 or 2.
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