Lesson 11 | Functions, Graphs and Features | 9th Grade Mathematics

Objective

Analyze functions and identify parent functions of graphs. Identify variables of a situation and the scale of the associated graph. Represent a situation in a graph, table, and description.

Common Core Standards

Core Standards

The core standards covered in this lesson

F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Interpreting Functions

F.IF.B.4 — For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

The foundational standards covered in this lesson

8.F.B.5

Functions

8.F.B.5 — Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Define variables presented in a problem in context.
Identify an appropriate domain and range based on the context presented.
Model a contextual situation graphically, representing the intervals where a function is increasing/decreasing/constant, whether a function is linear, intercepts, and other key points.
Interpret a situation represented graphically for intervals of interest, points of interest, and comparison in rates of change.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In this unit, students have worked to define variables in context, name the assumptions/levels of precision that are necessary for a model, and interpret parts and features of a model presented graphically and in a table. We will continue to work within parts of the modeling cycle (although not linearly) throughout the year, but this is the first lesson where students will be given a context and asked to apply their knowledge from the unit.
The target task for this lesson is Anchor Problem #2.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Consider the story:
Darryl lives on the third floor of his apartment building. His bike is locked up outside on the ground floor. At 3:00 p.m., he leaves to go run errands, but as he is walking down the stairs, he realizes he forgot his wallet. He goes back up the stairs to get it and then leaves again. As he tries to unlock his bike, he realizes that he forgot his keys. One last time, he goes back up the stairs to get his keys. He then unlocks his bike, and he is on his way out at 3:10 p.m.

Sketch a graph that depicts Darryl’s change in elevation over time.

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 1 > Topic A > Lesson 3 — Example 1

Algebra I > Module 1 > Topic A > Lesson 3 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem 2

Sam, John, and Bill all went to the basketball game at the gym and left from the high school. Sam’s mom picked him up, John took public transportation because he had to stop at the Copley library to pick up a reserved book, and Bill biked from the high school to the elementary school.

Below are the modes of transportation necessary to make their full trip, and some information about the times and distances associated with each travel method.

Guiding Questions

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Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Choose a few of the matching cards from here to use—not all of them are appropriate given what the students have learned. Since we have not focused on writing an equation for any situation, do not include this in the matching graph.
Include problems that review major concepts from the unit.

Illustrative Mathematics Tides
Inside Mathematics Performance Assessment Tasks Grades 3-High School Vincent's Graphs

Lesson 10

Lesson Map

Topic A: Features of Functions

Model a contextual situation graphically using appropriate scales and features.

F.IF.B.4 F.IF.B.6 N.Q.A.1 N.Q.A.2 N.Q.A.3

Define functions and evaluate points in tables, graphs, and contextual situations using function notation.

F.IF.A.1 F.IF.A.2

Identify features of functions, including x-intercept and y-intercept, in context. Evaluate function notation in context.

F.IF.A.2 F.IF.C.9

Identify the domain and range through contextual situations, and explore domain and range on a graph. Represent domain and range with inequalities.

F.IF.A.1 F.IF.B.5

Calculate and interpret the rate of change from two points on a graph, in a situation, or in function notation.

F.IF.A.2 F.IF.B.4 F.IF.B.6

Describe and sketch functions using the features of domain and range, intercepts, function behavior, and the value of the function.

F.IF.B.4

Analyze the key features of a contextual situation and model these graphically.

F.IF.B.4 F.IF.B.5 F.IF.B.6 N.Q.A.2

Topic B: Nonlinear Functions

Draw quadratic functions represented contextually. Identify key features of the graph and relate to context.

A.CED.A.2 F.IF.B.5 F.IF.C.7.A F.LE.A.3

Sketch an exponential function that represents a situation. Identify key features of the graph and relate to context.

A.CED.A.2 F.IF.B.5 F.IF.C.7.E F.LE.A.3

Draw a graph to represent a system of functions. Identify the solution to a system represented graphically and in context.

A.REI.D.11 F.IF.A.2 F.IF.B.5

Analyze functions and identify parent functions of graphs. Identify variables of a situation and the scale of the associated graph. Represent a situation in a graph, table, and description.

F.IF.B.4

Functions, Graphs and Features

Objective

Common Core Standards

Core Standards

Interpreting Functions

Foundational Standards

Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem Set

Additional Practice

Lesson Map

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