Unit 3: Functions   Have you ever had to convert from a Celsius temperature to a Fahrenheit temperature? If you know it is 25 degrees Celsius outside, there is a function that you could use to figure out the Fahrenheit temperature. A function is a rule that assigns to each input exactly one output. So, when you input 25°C, there is only one possible Fahrenheit temperature, 77°F, that reflects how warm it is outside. In this unit, you will work with comparing functions, graphing functions, and using functions to model relationships between quantities. Through the unit you will work with linear and non-linear functions.

Completing this unit should take approximately 11 hours and 45 minutes.

☐    Subunit 3.1: 3 hours and 45 minutes

☐    Subunit 3.2: 2 hours and 15 minutes

☐    Subunit 3.3: 3 hours and 30 minutes

☐    Subunit 3.4: 2 hours and 15 minutes

Unit3 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Evaluate functions using the rule that each input has exactly one output. - Recognize that the graph of a function is the set of ordered pairs consisting of an input and output. - Compare properties of two functions represented in different ways. - Interpret and give examples of linear and non-linear equations. - Construct a function to model a linear relationship between two quantities. - Use tables or graphs to interpret rate of change of a linear function. - Sketch a graph that exhibits the features of a described function. - Describe what a graph looks like using qualitative features.

Standards Addressed (Common Core): - CCSS.Math.Content.8.F.A.1 - CCSS.Math.Content.8.F.A.2 - CCSS.Math.Content.8.F.A.3 - CCSS.Math.Content.8.F.B.4 - CCSS.Math.Content.8.F.B.5 - CCSS.ELA-Literacy.W.8.2

3.1 What Is a Function?   A function is a relation between variables. Picture a candy factory with all the machines inputting ingredients (sugar!) and outputting your favorite sweet treat. If you input a different variety of ingredients, then the output will be a different sweet treat. A function is the math version of a candy factory. For every input there is only one output. If you input a different value, it will lead to the output of a different value. During this subunit you will be looking for patterns to help you figure out the function, and you will begin to look at how functions show up in both tables and graphs.

• Explanation: Monterey Institute for Technology and Education: HippoCampus: “Inductive Patterns” Link: Monterey Institute for Technology and Education: HippoCampus: “Inductive Patterns” (Flash)

Instructions: On the left side of the screen, select Algebra I -An Open Course (2011) under the “Presentations” heading. On the pop-out menu, scroll down to the “Functions and Patterns” heading and select “Inductive Patterns.” While you watch the video, sketch the triangle example, including the picture, sequence, terms, and eventually the mathematical representation. Notice how you don’t want to just think “add two” from one term to the next, because that would not be beneficial to you as you look at a larger number of struts.

Watching the video and taking notes should take approximately 15 minutes.

• Explanation: Monterey Institute for Technology and Education: HippoCampus: “Representing Functions and Relations” Link: Monterey Institute for Technology and Education: HippoCampus: “Representing Functions and Relations” (Flash)

Instructions: On the left side of the screen, select Algebra I - An Open Course (2011) under the “Presentations” heading. On the pop-out menu, scroll down to the “Functions and Patterns” heading and select “Representing Functions and Relations.” While you watch, listen closely to why the Just Bicycles shop represents a function, and the Multi-Cycles shop does not. The learning objective for this video is that you understand what makes a situation a function. The video represents this in words, equations, tables, and graphs. Take notes of each of these situations as well as the definition for a function. With about 1:48 left in the video, the instructor shows you important information on a graph and then follows it up with notes on variables, relations, and functions. Pay close attention and take notes on this portion of the video.

Watching the video and taking notes should take approximately 15 minutes.

• Explanation: Monterey Institute for Technology and Education: HippoCampus: “Domain and Range” Link: Monterey Institute for Technology and Education: HippoCampus: “Domain and Range” (Flash)

Instructions: On the left side of the screen, select Algebra I - An Open Course (2011) under the “Presentations” heading. On the pop-out menu, scroll down to the “Functions and Patterns” heading and select “Domain and Range.” The focus of this video is for you to learn about domain and range. While watching the video, make sure you are able to define domain and range, but also take notes on how to write each in the various ways mentioned throughout the video: words, sets, tables, and ordered pairs.

Watching the video and taking notes should take approximately 15 minutes.

• Did I Get This? Activity: Southern Nevada Regional Professional Development Program: “Domain and Range” Link: Southern Nevada Regional Professional Development Program: “Domain and Range” (PDF)

Instructions: Complete practice questions 1-10 involving domain and range. Check your answers here.

Completing the practice questions and checking solutions should take approximately 30 minutes.

• Explanation: CK-12: “Input-Output Tables for Function Rules” Link: CK-12: “Input-Output Tables for Function Rules” (HTML)

Instructions: Most of the information in this resource will be review of the videos you just watched. Skim the reading while focusing on the bold sentences. Complete Examples A, B, and C and be sure you understand the solutions. Write down any of the vocabulary words that you don’t already have in your notes. Complete the “Guided Practice” section. Do not watch the video.

Taking notes and completing practice problems should take approximately 30 minutes.

• Explanation: Monterey Institute for Technology and Education: HippoCampus: “Representing Patterns” Link: Monterey Institute for Technology and Education: HippoCampus: “Representing Patterns” (Flash)

Instructions: On the left side of the screen, select Algebra I - An Open Course (2011) under the “Presentations” heading. On the pop-out menu, scroll down to the “Functions and Patterns” heading and select “Representing Patterns.” While you watch, write down and understand the definition of independent and dependent variable. Take note where you would find each on a table and a graph. Notice that for each graphing input (year) there is one graphing output (sun spot activity).

Watching the video and taking notes should take approximately 15 minutes.

• Explanation: James Sousa’s Mathispower4u: “Introduction to Functions – Part 1” Link: James Sousa’s Mathispower4u: “Introduction to Functions – Part 1” (YouTube)

Instructions: This video will give you another glimpse at some of the information you have already learned in this subunit, and you will get to hear it explained by another instructor. There are examples throughout the video, as well. When a specific slide poses a question, pause the video and see if you can answer the question yourself before the instructor gives the detailed explanation about how to solve it. In your notebook, you should add the example problems and any additional notes that you find helpful.

Watching the video and taking notes should take approximately 15 minutes.

• Did I Get This? Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “My Summer Job Options: Investigations” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “My Summer Job Options: Investigations” (PDF)

Instructions: For each investigation, write an equation, complete the table, and graph the situation. You will be able to check your equations and tables on page 4. Then, use your equation, table, and graph to answer the questions found here. There are 9 questions total. Answer all 9 of the questions and check your solutions on pages 2-3. As you work, consider how you could use each equation, table, or graph to answer the questions.

Completing this activity should take approximately 45 minutes.

• Checkpoint: Southern Nevada Regional Professional Development Program: “Relations and Functions” Link: Southern Nevada Regional Professional Development Program: “Relations and Functions” (PDF)

Instructions: Read through pages 2-5. While you read, make sure you have all the vocabulary words written in your notes. Add to your notes, as necessary. Complete the example problems on each page and be sure you understand each solution. Page 4 talks about the vertical line test. Make sure you understand how the vertical line test can help you determine if a graph is a function. This will be an important strategy to use in the future.

Pages 6-8 have 10 assessment questions. Complete these questions, then check your answers here and make sure you understand the solution.

Taking notes and completing practice problems should take approximately 45 minutes.

3.2 Proportional, Linear, and Non-Linear Functions   There are different forms of functions. In this subunit, you will learn about the difference between proportional, linear, and non-linear functions and start to develop strategies you can use to recognize the differences between these types of functions by just looking at a table, at a graph, or at the equation. You will build on these ideas as you continue in your math career. At this point, you will see that both proportional and linear functions are the most familiar to you.

• Explanation: Monterey Institute for Technology and Education: HippoCampus: “Proportional Functions” Link: Monterey Institute for Technology and Education: HippoCampus: “Proportional Functions” (Flash)

Instructions: On the left side of the screen, select Algebra I - An Open Course (2011) under the “Presentations” heading. On the pop-out menu, scroll down to the “Functions and Patters” heading and select “Proportional Functions.” While you watch, write down the definition of proportional functions and be sure you understand how the functions are written using the output, input, and constant. With about 55 seconds left, the instructor gives two pieces of information that lead to a proportional function. Make sure you take notes and sketch the graph of this information.

Watching the video and taking notes should take approximately 15 minutes.

• Explanation: Monterey Institute for Technology and Education: HippoCampus: “Linear Functions” Link: Monterey Institute for Technology and Education: HippoCampus: “Linear Functions” (Flash)

Instructions: On the left side of the screen, select Algebra I - An Open Course (2011) under the “Presentations” heading. On the pop-out menu, scroll down to the “Functions and Patterns” heading and select “Linear Functions.” While you watch, write down the definition of linear functions and make sure you understand what they are. Take notes on the difference between a proportional function and a linear function. Look for how the difference shows up in the equation, table, and graph. When the instructor shows the graph for the gym membership option with about 1:55 left in the video, consider why a line does not connect the points. She quickly answers the question at the 1:40 mark. Rewind the video and think about this, and watch it again if you need more time to process this idea. Watch the remainder of the video and take any additional notes about linear and proportional functions.

Watching the video and taking notes should take approximately 15 minutes.

• Explanation: Monterey Institute for Technology and Education: HippoCampus: “Non-Linear Functions” Link: Monterey Institute for Technology and Education: HippoCampus: “Non-Linear Functions” (Flash)

Instructions: On the left side of the screen, select Algebra I - An Open Course (2011) under the “Presentations” heading. On the pop-out menu, scroll down to the “Functions and Patterns” heading and select “Non-Linear Functions.” While you watch, write down the definition of non-linear functions and make sure you understand what they are. Take notes on the difference between linear and non-linear functions. Throughout the video the instructor gives some examples of non-linear functions: inverse variation, quadratic formula, and exponential. Take notes and do a quick sketch of each graph in your notebook.

Watching the video and taking notes should take approximately 15 minutes.

• Did I Get This? Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Function Families” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Function Families” (PDF)

Instructions: Look at Function Family Sets 1, 2, and 3. Just by looking at the functions (don’t make a table or a graph yet), consider: does one of the equations look different than the others? Predict what you think each graph will look like. Using your graphing calculator, graph each equation in the specific set. Decide which equation does not “fit” with the rest of the Function Family.

Using the information from the previous videos, define each family as “linear,” “quadratic,” or “exponential.” Decide which family the outcast equation should be moved to from each specific set.

Completing this activity should take approximately 30 minutes.

• Did I Get This? Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Function Families Summary Resource Sheet” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Function Families Summary Resource Sheet” (PDF)

Instructions: Complete the table and graph for each of the three types of functions. When you write the “Key Features” section, consider this an opportunity to explain to a younger student about each of these types of functions. Write 3-5 sentences about each type of function using information from the table, graph, and equation. Explain the difference between each type of function. Explain some commonalities that would be found for each specific types of graph (example: all linear graphs look like ___). Use appropriate mathematical vocabulary words.

Completing this activity should take approximately 30 minutes.

• Checkpoint: Illustrative Mathematics: “Introduction to Linear Functions” Link: Illustrative Mathematics: “Introduction to Linear Functions” (PDF)

Completing this activity should take approximately 30 minutes.

3.3 Modeling Relationships with Functions   When a situation can be described with a function, people can make predictions for scenarios. For example, once you know a relation between speed and time, you can predict how much longer you will have to sit in the car before arriving at your destination. Perhaps you are saving your allowance for a new pair of jeans. You could use a function to model how many weeks until you have saved the necessary amount of money.

• Explanation: Monterey Institute for Technology and Education: HippoCampus: “Rate of Change and Slope” Link: Monterey Institute for Technology and Education: HippoCampus: “Rate of Change and Slope” (Flash)

Instructions: On the left side of the screen, select Algebra I - An Open Course (2011) under the “Presentations” heading. On the pop-out menu, scroll down to the “Analyze and Graph Linear Equations, Functions, and Relations” heading and select “Rate of Change and Slope.” You already have experience with slope. This video will add to that understanding as it uses the terminology rate of change. Take notes as you watch the video about how to find slope.

Watching the video and taking notes should take approximately 15 minutes.

• Explanation: James Sousa’s Mathispower4u: “Ex: Determine Which Tables Represent a Linear Function or Linear Relationship” Link: James Sousa’s Mathispower4u: “Ex: Determine Which Tables Represent a Linear Function or Linear Relationship” (YouTube)

Instructions: This video will give you an opportunity to use a table to determine if it represents a linear function. As the video starts, pause the video and try to figure out which of the three tables represents a linear function. Continue watching and listen to how the instructor solved the problems.

Watching the video and solving the example problem should take approximately 15 minutes.

• Explanation: James Sousa’s Mathispower4u: “Ex: Given Linear Function, Find the Rate of Change and Initial Value” Link: James Sousa’s Mathispower4u: “Ex: Given Linear Function, Find the Rate of Change and Initial Value” (YouTube)

Instructions: This video will give you an opportunity to use a real-world situation to answer questions about linear functions. As the video starts, pause the video and try to figure out the answers to the two questions on the first slide. Continue watching and listen to how the instructor solves the problems.

Watching the video and taking notes should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Modeling with a Linear Function” Link: Illustrative Mathematics: “Modeling with a Linear Function” (PDF)

Instructions: Read each of the scenarios and decide if they could be modeled by the function provided. It might be helpful to consider a few different inputs and interpret what the outputs would mean. Answer questions a-f. Check your solutions on page 2.

Completing this activity and checking solutions should take approximately 15 minutes.

• Explanation: James Sousa’s Mathispower4u: “Ex. Determine If Statements Represent Functions” Link: James Sousa’s Mathispower4u: “Ex. Determine If Statements Represent Functions” (YouTube)

Instructions: Begin watching the video and take any notes from the second slide that you don’t already have in your notebook. Pause the video at the 1:05 mark. Decide which of the four statements represent a function. Watch the remainder of the video and check your solutions with the instructor.

Watching the video and solving the examples should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Battery Charging” Link: Illustrative Mathematics: “Battery Charging” (PDF)

Instructions: Read about Sam and his need to charge the battery on a few of his devices. As you read, consider the rate of change for both the MP3 player and the video game. For this problem we will assume the rate of change stays consistent through the completion of the battery charge. Solve questions a and b. Read the commentary and check your solutions on pages 2-4.

Completing this activity and checking solutions should take approximately 15 minutes.

• Did I Get This? Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Filling the Pool” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Filling the Pool” (PDF)

Instructions: Read about Max and Joey in the section titled “The Task.” Consider the scenario and decide how you want to solve the problem. Solve the problem. On pages 2-5, a few solution strategies are explained. Compare how you solved the task to the other solutions. How are the strategies alike? How are the strategies unique?

Completing this activity and checking your solution should take approximately 30 minutes.

• Checkpoint: Illustrative Mathematics: “Distance Across the Channel” Link: Illustrative Mathematics: “Distance Across the Channel” (PDF)

Instructions: Read about the situation of water flow in the channel. Answer questions a-e. You can check the solutions on pages 2-3.

If you find that getting started on the problem is a challenge, consider the extremes of the channel with no water (0 depth) versus the channel with water at the highest level. Use this information to help assign two input-output pairs. Once you have two points on a line, you can use this information to find the rate of change (slope).

Completing this activity should take approximately 45 minutes.

• Checkpoint: Southern Nevada Regional Professional Development Program: “Unit 2 Practice Test” Link: Southern Nevada Regional Professional Development Program: “Unit 2 Practice Test” (PDF)

Completing this activity should take approximately 45 minutes.

3.4 Qualitatively Describing a Function   Graphs tell stories about what is happening in specific scenarios. When you look at a graph, you should be able to get a snapshot of what is happening even without looking at details such as numbers on the x-axis and y-axis. For example, if you were looking at a time (x-axis) and distance (y-axis) graph for a hike that you recently took, how would your lunch stop show up on the graph? How would a steep uphill climb show up on the graph? If you raced your younger sibling back to the car, how would that rate of change show up on the graph? All these examples are qualitative situations for describing a function. In this subunit, you will write qualitative interpretations of graphs and sketch graphs using qualitative descriptions.

• Explanation: CK-12: “Rate of Change” Link: CK-12: “Rate of Change” (HTML)

Instructions: Read the introduction about rate of change and Example A. Make sure you understand the solution. Skip down to Example C. This shows a graph that tells a story of a delivery truck throughout one workday. Try to figure out what is happening at each of segments A-E before looking at the solutions. Use the labels of the x-axis and y-axis to make interpretations about the segments. Read and understand the solution for Example C. Skip to the “Practice” section. Answer the question that asks you to identify each section of the graph about Mark’s cycle ride.

Completing this activity should take approximately 30 minutes.

• Did I Get This? Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Matching Graphs Activity” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Matching Graphs Activity” (PDF)

Instructions: Look at each of the seven graphs on the first page. As you look at each graph, think about the rate of change that is occurring. Scroll down to see only the top part of page 2. Match each of the descriptors with the graphs. Check your solutions on the bottom of page 2.

Completing this activity should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Velocity vs. Distance” Link: Illustrative Mathematics: “Velocity vs. Distance” (PDF)

Instructions: Write a paragraph to describe each graph shown in the task. Use detail about what might be happening at each rate of change on the two graphs. Read the commentary and solution on page

1.
Completing this activity should take approximately 15 minutes.

2. CCSS.Math.Content.8.F.B.5
3. CCSS.ELA-Literacy.W.8.2

• Did I Get This? Activity: Illustrative Mathematics: “Riding by the Library” Link: Illustrative Mathematics: “Riding by the Library” (PDF)

Instructions: Read about Nina and her bike ride from home to school. Sketch the graphs for questions a-b. If you are uncertain about what to do, scroll down part of the way on page 2 and read the list of bulleted points about the first graph. This should help you organize your thought on how to sketch the first graph. Another bulleted list on the bottom of page 2 can help you understand how to sketch the second graph. Check the graphed solutions on pages 2-3.

Completing this activity should take approximately 15 minutes.

• Checkpoint: Illustrative Mathematics: “Bike Race” Link: Illustrative Mathematics: “Bike Race” (PDF)

Completing this activity should take approximately 15 minutes.