In Part 1 of this unit, students will learn about data collection, graphing skills (both by hand and computer aided [Desmos]), and the fundamental mathematical patterns of the course: horizontal line, proportional, linear, quadratic, and inverse. Students perform several experiments, each targeting a different pattern and build the mathematical models of physical phenomena. During each experiment, students start with an uninformed wild guess, then through inquiry and making sense through group consensus, can make an accurate data informed prediction.

For this task, students view images of containers and imagine them placed under a steady stream of water. Students determine what the graphs of the containers would look like when plotting the height of the water level against the volume of water as the containers fill up.

Given a set of points on a graph, students will find one linear function and one quadratic function which, between them, pass through all the points.

Adapted from mathematicsvisionproject.com’s Material Overview:
The Mathematics Vision Project (MVP) was created as a resource for teachers to implement the Common Core State Standards (CCSS) using a task-based approach that leads to skill and efficiency in mathematics by first developing understanding. The MVP approach develops the Standards of Mathematical Practice through experiential learning. Students engage in mathematical problem solving, guided by skilled teachers, in order to achieve mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. The MVP authors created a curriculum where students do not learn solely by either “internalizing what a teacher or book says or, on the other hand, solely by inventing mathematics on their own.”
The MVP classroom experience begins by confronting students with an engaging problem and allows them to grapple with solving it. As students’ ideas emerge, take form, and are shared, the teacher deliberately orchestrates the student discussions and explorations toward a focused math goal. Students justify their own thinking while clarifying, describing, comparing, and questioning the thinking of others leading to refined thinking and mathematical fluency. What begin as ideas become concepts that lead to formal, traditional math definitions and properties. Strategies become algorithms that lead to procedures supporting efficiency and consistency. Representations become tools of communication which are formalized as mathematical models. Students learn by doing mathematics.

This lesson uses the model of a 4 x 4 cube viewed from the top, front, and right side.The first task is to find the integer value representing the number of cubes that can be viewed as well as those that cannot be seen. The second task is to find a polynomial model that represents this solution.  Besides drawing a picture and counting the cubes, think of a method which requires multiplication, addition, and subtraction of the sides that will result in a quadratic solution.  Check the model and evaluating it when n=4 and compare answer to Task 1. The third task is to check to see if the model will hold true when the number of units in one side is greater than 4. The fourth task is an optional extension for students to predict a model for a rectangular prism with a square base using two variables.  An addtional extension can be used to review probability and odds.  

Students carry out an interactive, geometric "proof without words" for the algebraic technique of completing the square in this interactive. The page also includes directions and a link to the final solution.