Matemáticas y lógica de secundaria: el teorema de Pitágoras y los números irracionales

This is the eighth course in year two of a series of middle school mathematics courses. These courses are taught in small-groups to provide individual instruction and social-learning opportunities aligned with a social constructionist or situated cognition view of learning. This series is based on an accelerated math curriculum that covers three years of content, aligned to Common Core Math Standards, over the course of two school years. It is perfect for students preparing to begin Algebra ahead of time, or those who need review and remedial support. 
The curriculum is problem-based this means instead of lectures or videos students work together as a small group to solve problems to discover principles and strategies with teacher guidance, as necessary. Therefore, we will spend approximately 90% of each class period working on problems and discussing them as a group. The use of discussion and problem-solving leads to generalized mathematics or proofs. This process prepares students well who may seek advanced mathematics in high-school or beyond. 
In this unit we build upon previous work from the past two years to understand rational and irrational numbers including geometric and symbolic representations of square and cube roots including common notation. We will work on cementing our understanding of fractions as division and the connection to decimals and plot rational and irrational numbers on the number line. We will use the Pythagorean theorem to solve problems in two and three dimensions based upon two distinct proofs. 
Week 1: We will begin with a unit readiness assessment which will allow the instructor to individualize and adjust the course. We will work to calculate the area of a tilted square using decomposition and work backwards from squares to the side length and formalize our understanding of square roots and square root notation and introduce the idea of irrational numbers by reviewing rational numbers. We will work to find the approximate location of irrational numbers on the number line. 
Week 2: We will investigate the relationship between the side lengths of right and non right triangles and introduce basic trigonometric vocabulary. Students will derive the geometric proof for the Pythagorean Theorem using a variety of representations and be introduced to the transformational proof. We will also proof the converse and learn how to identify whether or not a triangle is a right triangle by looking at the side lengths. We will use these to solve real world problems and find distances between points int he coordinate plane.
Week 3:  We will expand our work from square roots to cube roots and apply this to volume problems. We will begin moving to algebraic definitions of our work within the unit. We will explore decimal representations of rational and irrational numbers including understanding why infinite decimals are rational numbers. We will practice with a real-world lesson applying our knowledge to smart phone screen sizes to understand discussions of aspect ratio and value. Finally, we will take a unit assessment.