This task is designed to develop the ideas of features of functions using a situation. Features of functions such as increasing/decreasing and maximum/minimum can be difficult for students to understand, even in a graphical representation if they are not used to reading a graph from left to right. A situation using the water level of a pool over a period of time can provide opportunities for students to make connections to these features. While some parts of the graph need to come before others (emptying the pool before filling the pool), other situations can be switched around (emptying the water with buckets and emptying the water with a hose).

Students relate the solutions of a quadratic equation in one variable to the zeros of the function it defines. They sketch graphs of quadratic functions from tables, expressions, and verbal descriptions of relationships in real-world contexts, identifying key features of the quadratic functions from their graphs. Students also graph and show the intercepts and minimum or maximum point.

Students have been using function notation in various forms and have become more comfortable with features of functions. In this task, the purpose is for students to practice their understanding of the following:
• Distinguish between input and output values when using notation
• Evaluate functions for inputs in their domains
• Determine the solution where the graphs of f(x) and g(x) intersect based on tables of values and by interpreting graphs
• Combine standard function types using arithmetic operations (finding values of f(x)+ g(x))
• Create graphs of functions given conditions.

Students interpret quadratic functions from graphs and tables: zeros (-intercepts), -intercept, the minimum or maximum value (vertex), the graph's axis of symmetry, positive and negative values for the function, increasing and decreasing intervals, and the graph's end behavior. Students determine an appropriate domain and range for a function's graph and when given a quadratic function in a context, recognize restrictions on the domain.

This task provides opportunities for students to show their understanding of functions in various representations by making matches (3 cards in a set).

Students will extend their study of functions to include function notation and the concepts of domain and range by exploring examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.

Students will learn how to apply knowledge of functions to mathematically describe real-life situations
and objects.

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.

The purpose of this task is for students to combine functions, make sense of function notation, and connect multiple representations (context, equations, and graphs). Students will also address features of functions as they solve problems that arise from this context.

Finding key features of quadratic equations based off graphs, quadratic functions and factored quadratic functions.

This video that solves a word problem about a ball being shot in the air. The equation for the height of the ball as a function of time is quadratic.

Students use linear functions and systems of linear functions to help create a space suit.

Students will synthesize what they have learned about functions to select the correct function type in a series of modeling problems. Students must also draw on their study of statistics, using graphs and functions to model a context presented with data and/or tables of values. In this module, the modeling cycle is used as the organizing structure, rather than function type.

This lesson requires students to explore quadratic functions by examining the family of functions described by y = a (x - h)^2 + k. This lesson plan is based on the activity Tremain Nelson used in the video for Part I of this workshop.

This lesson teaches students about inverse variation by exploring the relationship between the heights of a fixed amount of water poured into cylindrical containers of different sizes as compared to the area of the containers' bases.

The purpose of this task is for students to interpret and highlight features of functions using contexts. This task provides opportunities for students to practice skills they have already learned as well as solidifying their knowledge of features of functions. This task first asks students to make observations from a graph. There are several observations to make and by having students make these observations, they are accessing their background knowledge as a way to prepare for this task. In the following sections, students solidify their understanding of domain and distinguish between the domain of a function and the domain of a situation. They also use function notation to interpret the meaning of the situation.

The purpose of the task is to show that graphs can tell a story about the variables that are involved. Together with tables of values, they can paint a complete picture of a situation. These tasks all describe situations for which the given quantity is decreasing.