中學數學、邏輯與問題解決：函數與體積

This is the sixth 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, learners will be introduced to the concepts of functions as a way to make sense about real world relationships using terms like input, output, and function.  We will describe functions as increasing or decreasing and consider the relationships between independent and dependent variables using tables, equations, and graphs to represent what we discover and synthesize our understanding of relationships and proportionality developed in earlier units as well as rate of change and slope. Then, we will pivot to focus on three dimensional shapes and derive volume formulas as we analyze and describe cross sections and the relationships between shapes. We will also understand area, surface area, and volumes as functions. 
Week 1: We will begin with completing a readiness check that covers prerequisite skills as well as preview skills used in the unit. This will allow modification of pacing and instruction to support students throughout the unit with additional review or acceleration. Then we will introduce inputs, outputs, and functions. We will discuss the equation for functions and introduce them as tables and graphs. 
Week 2: We will continue graphing functions based on real world problems and contexts and connecting various representations of functions according to meaning. Then we will connect our new knowledge of functions to our previous work with linear relationships exploring linear functions and linear models. 
Week 3: We will solidify our work with linear relationships by exploring piecewise linear functions. Then, we will complete a mid-unit assessment to ensure understanding. Lastly, we will pivot to understanding three dimensional shapes by slicing solids and filling containers to build a conceptual framework for volume of right prisms. 
Week 4: We will expand our work through decomposing figures to understand area and then expand to review surface area of right prisms. Then we will work through applying volume and surface area to various shapes and introducing cylinders as a variation of a prism. 
Week 5: We will build off of our work with cylinders to work with cones. We will practice finding the volume of a cone in comparison to cylinders and working backwards to find cone dimensions. We will begin to tie our work together by thinking of scaling the dimensions of cones as a function and understanding how scaling one or two dimensions of a three dimension shape impacts area and volume and working to represent this visually. 
Week 6: We will begin to work with spheres starting by thinking of a hemisphere in relation to cones and cylinders. From this work, students will derive the volume of a sphere. Then we will complete a math-based art project using cylinders, cones, and spheres. Finally we will work towards building prisms using a variety of tools before taking our end of unit assessment. 
Week 6: We will