Unit 1

REPRESENTING FUNCTIONAL RELATIONSHIPS

Unit 1 Overview

In this unit, students build on their understanding of functions and representations as they begin to explore functions represented in function notation. While students are comfortable with having the range defined as y, they now will use notation to show the range is f(x) = y. Function notation will be used to define a given relationship between two variables. Students will evaluate a function for a given input within the domain of the function. When given a real-world application, context will be used to interpret the meaning of a statement given in function notation.

Students will model problem situations using expressions focusing on linear, simple exponential, and simple quadratic models. In order to do this, students must be able to identify and apply appropriate units and quantities when modeling real-world situations and reasoning through possible solutions within context. Units should be chosen carefully in order to achieve a suitable level of accuracy and precision in the context of the problem. Students will represent functions symbolically in two variables, create graphs, and build a table of values. Students will interpret graphs in terms of the relationship between the quantities and in order to arrive at solutions to a real-world problem. Students will recognize the relationship between the contextual, graphical, numeric, and symbolic representations of a problem situation.

New Academic Vocabulary

  • Function
  • Function Notation (𝑓(𝑥) notation for 𝑦)
  • Relation

SCCCR Mathematical Content Standards:

A1.FIF.1a Understand that a function from one set (called the domain) to another set (called the “range”) assigns to each element of the domain exactly one element of the range.

A1.FIF.1b Represent a function using function notation and explain that 𝑓(𝑥) denotes the output of function 𝑓 that corresponds to the input 𝑥.

A1.FIF.1c Understand that the graph of a function labeled as 𝑓 is the set of all ordered pairs (𝑥, 𝑦) that satisfy the equation 𝑦 = 𝑓(𝑥).

A1.FIF.2 Evaluate functions and interpret the meaning of expressions involving function notation from a mathematical perspective and in terms of the context when the function describes a real-world situation.

A1.ACE.2 Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales.

A1.AREI.10 Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane.

    1. Use appropriate units, scales, and labels.

* Overarching standards - these standards should be emphasized throughout all units of instruction

A1.NQ.1 Use units of measurement to guide the solution of multi-step tasks. Choose and interpret appropriate labels, units, and scales when constructing graphs and other data displays.

A1.NQ.2 Label and define appropriate quantities in descriptive modeling contexts.

A1.NQ.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities in context.

A1.ASE.1 Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.)

A1.ACE.1 Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.)

A1.REI.1 Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original.

Common Misconceptions

  • The notation f(x) does not mean “f multiplied by x.”
  • The notation f(x) represents the range, or y-values.
  • A vertical, not horizontal, line test can help to verify graphically that a relationship is a function of x. The vertical line test should not be the only justification for determining whether a relation is a function.
  • Students may interchange domain and range.
  • Students find it difficult to choose a scale that displays all of the data accurately.
  • Many students do not fully understand that a line is made up of infinite points and have difficulty understanding that solutions are not limited to integers.
  • Students often confuse the x- and y-axes when graphing.
  • It may be difficult for students to conceptualize that an equation, a table of values, and a graph can all represent the same problem situation.
  • Students find it difficult to choose a scale that displays all of the data accurately.
  • Many students do not fully understand that a line is made up of infinite points and have difficulty understanding that solutions are not limited to integers.
  • Students often confuse the x- and y-axes when graphing.
  • It may be difficult for students to conceptualize that an equation, a table of values, and a graph can all represent the same problem situation.

(Common Core Companion: The Standards Decoded, High School: What They Say, What They Mean, How to Teach Them, Miles and Williams, 2016)

‘I can’ Statements you should learn

A1.FIF.1

  • I can define relation, domain, and range (K).
  • I can define a function as a relation in which each input (domain) has exactly one output (range) (K).
  • I can determine if a graph, table, or set of ordered pairs represents a function (R).
  • I can determine if stated rules (both numeric and non-numeric) produce ordered pairs that represent a function.

For example:

    • Student’s name and student’s date of birth will represent a function since every input was born once;
    • A date and a famous person born on that date may not represent a function since a date could be matched to more than one person;
    • A number and twice the input will represent a function since every number has only one double;
    • A number and a number less than two units from the input will not represent a function since there are many outputs that match each input (R).
  • I can explain that when x is an element of the input of a function, f(x); other letters (e.g., g(x) and h(x)) can also be used so we can distinguish one function from another (K).
  • I can explain that the graph of f is the graph of the equation y=f(x) (K).
  • I can define a function and understand the concept of single-valuedness.
  • I can understand function notation as a method to relate each input, x, to the corresponding output value f(x).
  • I can use function notation to describe functional relationships and differentiate between functions.
  • I can represent in different ways functions that are presented in function notation.
  • I can use function notation to evaluate functions for different inputs and outputs.

A1.FIF.2

  • I can decode function notation and explain how the output of a function is matched to its input (e.g., the function f(x)=3x2+4 squares the input, triples the square, and adds four to produce the output (R).
  • I can convert a table, graph, set of ordered pairs, or description into function notation by identifying the rule used to turn inputs into outputs and writing the rule (S).
  • I can use order of operations to evaluate a function for a given domain (input) value (S).
  • I can identify the numbers that are not in the domain of a function (e.g., 0 is not in the domain of g(x)=1xand negative numbers are not in the domain of h(x)=x (R).
  • I can choose inputs that make sense based on a problem situation (R).
  • I can analyze the input and output values of a function based on a problem situation (R).

A1.ACE.2

  • I can identify variables and quantities represented in a real-world problem (K).
  • I can determine the best model for the real-world problem (e.g., linear, exponential, quadratic) (R).
  • I can write the equation that best models the problem (S).
  • I can set up coordinate axes using an appropriate scale and label the axes (S).
  • I can graph equations on coordinate axes with appropriate labels and scales (S).

A1.AREI.10

  • I can explain that every point (x, y) on the graph of an equation represents values x and y that make the equation true (K).
  • I can verify that any point on a graph will result in a true equation when their coordinates are substituted into the equation (K).
  • I can define the set of solutions of an equation in two variables using its graph.

A1.NQ.1

A1.ACE.1

  • I can create linear equations to represent relationships between quantities.
  • I can graph equations using appropriate labels and scales.

Resources & Description

Shodor Interactivate - interactive digital tools on a variety of topics

Week of Inspirational Math - Jo Boaler (start the school year off for open, visual, creative mathematics all year long)

Clothesline Math - The greatest tool for teaching number sense is the number line. Unfortunately, this simple, powerful vehicle gets left behind in elementary grades. However, used properly throughout the secondary grades, the number line can develop deep, flexible number sense as well as conceptual understanding of variables, signed numbers, rules of exponents, algebraic expressions, geometric relationships, statistics and other key mathematical ideas.

Learn Desmos - your virtual calculator & graphing utility

Working with Venn Diagrams: https://www.math.tamu.edu/~kahlig/venn/venn.html

Chart of measurement conversions

Examples of dimensional analysis

Dimensional analysis student practice sheet

Fuel Efficiency problem from Illustrative Mathematics

Chart - general algorithm for solving linear equations

Article from the 2015 edition of Mathematical Teacher: Expression Polygon

Student activity for expression polygons

Geogebra Solving Basic Absolute Value Inequalities: https://www.geogebra.org/m/vgp7xR54

Additional Resources- Secondary Math 1