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Week of: May 11 to May 17 , 2020
Grade: 8 Content: Math
Learning Objective Blurb:
Greetings 8th^ graders! We hope you are safe and well with your families! This week, students learn about relations and functions. Students analyze mappings, sets of ordered pairs, sequences, tables, graphs and equations and determine which are functions.
We’ve included some video links to help you if you get stuck! This work will not be graded, just do your best and have fun!
Carnegie Learning: Use with Carnegie Resources provided below: Video 1: Derby Days- Slope Intercept Form of a line: https://vimeo.com/
Video 2: Derby Days- Slope Intercept Form of a line: https://vimeo.com/
Printable Resources: Skills Practice: Module 2 , Topic 2 , Lesson 4 Derby Days see below
Family Guide below
Practice Activities:
On-Line All students now have access to an on-line program called Mathia! Mathia- If you are already in Mathia, please continue to work in the program. If you are new to Mathia: Please see the log-in information attached.
Video Links:
Khan Academy: Refresh your memory with any or all of the following: What is a Function? https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th- functions-and-function-notation/v/what-is-a-function
TOPIC 3: Family Guide • M2-
Carnegie Learning Family Guide Grade 8
Module 2: Developing Function Foundations
TOPIC 3: INTRODUCTION TO
FUNCTIONS
In this topic, students explore functions
in terms of sequences, mappings, sets
of ordered pairs, graphs, tables, verbal
descriptions, and equations. Because
students have a strong foundation in
writing equations of lines, they can
construct equations for linear functions.
Students learn the formal definition of
a function and analyze functions and
relations represented in a wide variety of
ways. Finally, students further investigate
the focus function: the linear function.
Where have we been? Throughout elementary school, students described patterns and explained features of the pattern. They have also formed ordered pairs with terms of two sequences and compared the terms. Therefore, sequences are used as the entry point for this topic.
Where are we going? The study of functions is a predominant topic in high school mathematics. As students move into high school, they will develop and use formal notation (e.g., f ( x )) to denote and operate with functions. In high school, students will use sequences as a launching point for linear and exponential functions.
Using the Vertical Line Test to Determine
if a Relation Is a Function
A standard test to determine whether a graphed relation is a function is called the vertical
line test. If you draw a vertical line anywhere on the graph and cross more than one point, the
relation is not a function. The graph shown illustrates a relation that is not a function.
8 x
6
8
4 9
4
2 6
2
0 1 3 5 7
y
9
5
7
3
1 0
LESSON 3: One or More Xs to One Y • M2-
LEARNING GOALS
- Describe a functional relationship in terms of a rule which assigns to each input exactly one output.
- Determine whether a relation (represented as a mapping, set of ordered pairs, table, sequence, graph, equation, or context) is a function.
Throughout middle school, you have investigated different types of relationships between variable quantities: additive, multiplicative, proportional, and non-proportional. What are functional relationships?
WARM UP Evaluate each expression given the set of values {1, 6, 12, 25}.
- 5 x
- __^12 x 1 1
- x 2 8
One or More
Xs to One Y
Defining Functional Relationships
KEY TERMS
- mapping
- set
- relation
- input
- output
- function
- domain
- range
- scatter plot
- vertical line test
M2-206 • TOPIC 3: Introduction to Functions
Getting Started
What’s My Rule?
Rules can be used to generate sequences of numbers. They can also be used to generate ( x , y ) ordered pairs.
- Write an equation to describe the relationship between each independent variable x and the dependent variable y. Explain your reasoning.
a. (^) x y
26 212 23 0 0 12 3 24
b. (^) x y
1 22 5 210 21 2 210 20
c. (^) x y
210 9 22 1 0 21 5 4
d. (^) x y
0 2 4 4 5 4. 20 12
- Create your own table and have a partner determine the equation you used to build it.
You can
sketch the
graph to help
determine the
equation.
WORKED EXAMPLE
Each mapping represents a function because no input, or domain value, is mapped to more than one output, or range value.
M2-208 • TOPIC 3: Introduction to Functions
- Write the set of ordered pairs to represent each table.
a. Input Output
210 220
25 210
0 0
5 10
10 20
b. x y
20 210
10 25
0 0
10 5
20 10
The mappings and ordered pairs shown in Questions 1 through 3 form relations. A relation is any set of ordered pairs or the mapping between a set of inputs and a set of outputs. The first coordinate of an ordered pair in a relation is the input, and the second coordinate is the output. A function maps each input to one and only one output. In other words, a function has no input with more than one output. The domain of a function is the set of all inputs of the function. The range of a function is the set of all outputs of the function.
Notice the
use of set
notation when
writing the
domain and
range.
The range is {1, 3, 5, 7}. The range is {1, 3, 7}.
1 2 3 4
1 3 5 7
1 2 3 4
1 3
7
In each mapping shown, the domain is {1, 2, 3, 4}.
WORKED EXAMPLE
In the mapping shown, the domain is {1, 2, 3, 4, 5} and the range is {1, 3, 5, 7}.
This mapping does not represent a function.
NOTES
LESSON 3: One or More Xs to One Y • M2-
- State why the relation in the worked example shown is not a function.
- State the domain and range for each relation in Questions 2 and 3. Then, determine which relations represent functions. If the relation is not a function, explain why not.
1 2 3
5
4
1 3 5 7
LESSON 3: One or More Xs to One Y • M2-
Functions as Mapping Inputs to Outputs
AC T I V I T Y
You have determined if sets of ordered pairs represent functions. In this activity you will examine different situations and determine whether they represent functional relationships.
Read each context and decide whether it fits the definition of a function. Explain your reasoning.
- Input: Sue writes a thank-you note to her best friend. Output: Her best friend receives the thank-you note in the mail.
- Input: A football game is being telecast.
Output: It appears on televisions in millions of homes.
- Input: There are four puppies in a litter.
Output: One puppy was adopted by the Smiths, another by the Jacksons, and the remaining two by the Fullers.
- Input: The basketball team has numbered uniforms.
Output: Each player wears a uniform with her assigned number.
- Input: Beverly Hills, California, has the zip code 90210.
Output: There are 34,675 people living in Beverly Hills.
- Input: A sneak preview of a new movie is being shown in a local theater. Output: 65 people are in the audience.
M2-212 • TOPIC 3: Introduction to Functions
- Input: Tara works at a fast food restaurant on weekdays and a card store on weekends. Output: Tara’s job on any one day.
- Input: Janelle sends a text message to everyone in her contact list on her cell phone. Output: There are 41 friends and family on Janelle’s contact list.
Determining Whether a Relation Is a Function
AC T I V I T Y
Analyze the relations in each pair. Determine which relations are functions and which are not functions. Explain how you know.
- Mapping A Mapping B
10 11 12 13
1000 2000 3000
10 11 12 13
1000 2000 3000
WORKED EXAMPLE
In this scatter plot, the relation is not a function. The input value 4 can be mapped to two different outputs, 1 and 4. Those two outputs are shown as intersections to the vertical line drawn at x 5 4.
Consider the scatter plot shown.
A relation can be represented as a graph.
M2-214 • TOPIC 3: Introduction to Functions
Functions as Graphs
AC T I V I T Y
A scatter plot is a graph of a collection of ordered pairs that allows an exploration of the relationship between the points.
- Determine if each scatter plot represents a function. Explain your reasoning.
a.
x
6
4
4
2 6
2
0 1 3 5
y
5
3
1 0
Output
Input
b.
x
6
4
4
2 6
2
0 1 3 5
y
5
3
1 0
Output
Input
The vertical line test is a visual method used to determine whether a relation represented as a graph is a function. To apply the vertical line test, consider all of the vertical lines that could be drawn on the graph of a relation. If any of the vertical lines intersect the graph of the relation at more than one point, then the relation is not a function.
8 x
6
8
4 9
4
2 6
2
0 1 3 5 7
y 9
5
7
3
1 0
NOTES
LESSON 3: One or More Xs to One Y • M2-
- Use the definition of function to explain why the vertical line test works.
- Use the vertical line test to determine if each graph represents a function. Explain your reasoning.
a.
x
6
4
4
2 6
2
0 1 3 5
y
5
3
1 0
b.
x
6
4
4
2 6
2
0 1 3 5
y
5
3
1 0
- Use the 12 cards that you sorted in the previous lesson. Sort the graphs into two groups: functions and non-functions. Use the letter of each graph to record your findings.
Functions Non-functions
LESSON 3: One or More Xs to One Y • M2-
If you do not recognize the graph of the equation, use a graphing calculator to see the pattern.
- Determine whether each equation is a function. List three ordered pairs that are solutions to each. Explain your reasoning.
a. y 5 5 x 1 3 b. y 5 x^2
c. y 5 | x | d. x^2 1 y^2 5
e. y 5 4 f. x 5 2
- Explain what is wrong with Taylor's reasoning.
If two different
inputs go to
the same
output, it
can still be a
function.
Taylor The equationy^2 =x represents a function.
x y 4 2 9 3 25 5
NOTES
M2-218 • TOPIC 3: Introduction to Functions
TALK the TALK
Function Organizer
- Complete the graphic organizer for the concept of function. Write a definition for function in your own words. Then, create a problem situation that can be represented using a function. Finally, create a table of ordered pairs and sketch a graph to represent the function.
Function
Definition Problem Situation
Graph
Table/ Ordered Pairs
M2-220 • TOPIC 3: Introduction to Functions
Stretch Describe how you can tell from an equation whether a function is increasing, decreasing, or constant.
- The science teacher created the set of ordered pairs {(100, 6), (90, 5), (80, 3), (70, 1), (90, 4), (80, 2)} to represent six students' grades on the midterm to the number of hours that they had studied. Create a mapping from this set of ordered pairs. a. Is the relation a function? If the relation is not a function, explain why not. b. List all the inputs of the relation. c. List all the outputs of the relation. d. Instead of mapping grades to hours studied, the teacher decides to create a new diagram. This diagram maps hours studied to grades. Show the mapping that would result. e. Write the set of ordered pairs to represent the mapping in part (d). f. Is the relation in part (d) a function? If the relation is not a function, explain why not. g. Create a scatter plot. Does the graph agree with your conclusion from part (f)? Explain your reasoning.
- At the end of the year, a principal decides to create the given mapping. Input: the 82 total students in the history class Ouput: the final grades they received for the class Does this mapping fit the definition of a function? Explain your reasoning.
- Use the vertical line test to determine if each graph represents a function. Explain your reasoning. a. b.
x
y
2
4
6
8
–8 –6 –4 –2 0 2 4 6 8 x
y
2
4
6
8
–8 –6 –4 –2 0 2 4 6 8
LESSON 3: One or More Xs to One Y • M2-
Review Tell whether each graph is discrete or continuous. Also, tell whether each graph is increasing, decreasing, both, or neither.
3 4^ x
1
2
3
4
1 2 5 6 7 8 9 10
y
5
6
7
8
9
10
3 4^ x
1
2
3
4
1 2 5 6 7 8 9 10
y
5
6
7
8
9
10
Determine the slope and y -intercept of the linear relationship described by each equation.
- y 5 x __ 2 1 5 4. y 5 x __ 4
Calculate the slope of the line represented by each table.
x y
2 21
3 1.
4 4
5 6.
x y
2 8
4 2
6 24
9 213
