Introduction to Quadratic Functions

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SWBAT interpret tables and sketch graphs of quadratic functions. SWBAT interpret the domain of a quadratic function in terms of context. SWBAT graph quadratic functions accurately and describe the graphs using mathematical terminology.

Big Idea

Students use modeling and technology as tools to come to a deeper understanding of quadratic functions using real life contexts!

Entry Ticket: Introduction to Quadratic Functions

15 minutes

The purpose of the Entry Ticket: Intro. to Quadratic Functions is to activate students’ prior knowledge about working with functions. I start by having students work on the Entry Ticket as soon as they enter the class – as the year has progressed it has become more and more automatic that students take out their binders and get to work on the Entry Ticket rather than milling around or socializing. This also frees up a couple of quick minutes for me to take care of housekeeping (attendance, etc.) and not waste valuable instructional time.

This entry ticket reviews work students completed earlier in the year with linear functions. The entry ticket accomplishes two goals:

  1. Providing a real-world context helps students understand and apply concepts (Bagel World is a local hot spot for high school students and pretty much every one else that lives in town!)
  2. Providing students with opportunities to practice skills that I am going to ask them to apply to a new situation. 

I tell students that the questions on the entry ticket review ideas we are going to build on in class today. The entry ticket focuses on a linear relationship, while we will focus on quadratic relationships in class today.

About 5 minutes into class, I ask students to turn-and-talk with a partner about the Entry Ticket. I prompt them to share how they solved the problem. I encourage students to identify the algebraic rules used to solve each problem. After a few minutes of partner work, we will review the Entry Ticket as a class. I plan to start by asking groups to share any discrepancies/errors and how to correct them.

After discussing the Entry Ticket, I turn my attention to the agenda for the day. We review the objectives as a class. I explain how this lesson’s objective fits into the bigger objectives of the unit: to support students who have difficulty seeing the big picture and/or shifting back and forth between the gestalt and the details of lessons and unitsI typically have students write down the homework assignment during this time and I hand out copies of the homework, asking students to file the homework in their binders. 

After the day’s agenda has been reviewed, the class shifts to the middle of the lesson.


Introductory Videos, Notetaking and Class Discussions

20 minutes

After reviewing the objectives for the day’s lesson, I cue students to open to a new piece of paper in their notebook. Many of my students continue to benefit from an explicit directive to take notes. Then, I let students know that I want them to take notes as they watch a 2-minute video. I write the following focus prompt on a whiteboard: Based on the video, describe different characteristics of contexts that involve quadratic functions.

I then show this as an introduction to contexts involving quadratic functions:

Source Url (April 7 2014)

After showing the video I ask students to take an additional two minutes to respond to the prompt. We then have a 3-5 minute class discussion where I give students a chance to share their ideas. In order to differentiate instruction, I lead the conversation patiently. It is important to allow students sufficient time to integrate their ideas. It is beneficial to provide multiple opportunities for students to express themselves (written and verbal). 

After the second video is complete, I ask students to complete the following Think-Pair-Share protocol:

Think – 2 minutes to write down your thoughts and update your notes from watching the videos

Pair – 3-5 minutes to compare and contrast your ideas with a partner

Share – 5-10 minute class discussion of ideas answering the prompt “Describe different characteristics of quadratic functions and their graphs”

Part of the reason for setting up this subsection in the way I did was to provide opportunities for students to engage in the four major domains of language (listening, reading, speaking and writing). 

Extension Video: I sometime also show the first couple of minutes on the following video that shows examples of parabolas in Super Mario Brothers. 

Vocabulary: Quadratic Functions and Equations

15 minutes

Since I believe students need to be taught skills of conversation and language use in school, I next have students work independently on the vocabulary for this unit. I explicitly teach students the difference between brick words and mortar words and how to use them in an academic conversation.

To accomplish these goals efficiently, I have students work on iPad Minis using a math dictionary to look up the definitions for terms. As an alternative, students could use an online math dictionary. I recommend Wolfram Alpha as it provides excellent knowledge, is accurate and also provides good visual examples for many terms.

Instructional Note: With some classes, I might choose to provide students with the definitions of the words (see: ______ to allow students to focus more on generating multiple representations of meaning for each word to develop a deeper understanding of the vocabulary terms. I have included two versions of the vocabulary for teachers - one with definitions and one without. 

Today, as the students wrap up, I have students file the work in a vocabulary section of their notebook and remind them vocabulary is part of the Notebook Check. We will be using the vocabulary later in the unit for a writing exercise on creating their own functions. 

Technology Note: The use of iPad Miinis is supported by a technology grant for Modeling with Mathematics and Universal Design for Learning in the Math Classroom by the Hardscrabble Education Fund.

Explicit Instruction and Active Note-Taking: Key Features of Quadratic Functions

20 minutes

I review the PowerPoint Slides: Intro. to Quadratic Functions to provide students with direct instruction and examples to practice. I start at a basic level, but I expect to move quickly. My goal is to deepen student understanding of the features of quadratic functions.

I begin an introduction to the basic form of a quadratic function. I show students a number of graphs and equations as examples. The next slide focuses on the value of the coefficient of the leading term, a, in a quadratic equation. I include the situation when a=0 because I want students to make the connection between quadratic functions and linear functions. A linear function is related to a quadratic in so far as it can be considered a quadratic with a value of a = 0.  

Then, I discuss two examples of graphing quadratic functions with students. I chose two examples that can factor without having to complete the square. I want to focus on the basic ideas necessary to graph a quadratic function. 

After graphing the two functions, the class then shifts to determining the domain and range of quadratic functions.

Relating the Domain of a Function to a Graph

20 minutes

As a means of introducing students to identifying the domain of a quadratic function, I ask students to watch a LearnZillion video that demonstrates how to determine the domain and range of a quadratic function from its graph. Before starting the video, I put the words domain and range on the whiteboard and ask students to brainstorm things they remember about the two terms.

I then show the video:

LearnZillion Video: Determining the Domain and Range of a Parabola from its Graph

URL: (April 7 2014)

After the video, I give students the opportunity to ask questions they have about the domain of a quadratic function. Then, I show students an example of a quadratic function on the board. As a class, we determine the domain and range of the function from its graph.

To give students the opportunity to practice on their own, I display graphs of quadratic functions (and their equations) on the SmartBoard. I may well use the examples from the PowerPoint Slides.  As we look at the functrions, I plan to complete the following discussion protocol with students:

Think: 3-5 minutes working in pairs (writing down their own responses)

Pair: 2-3 minutes talking with another pair (this makes students work in groups of 3-5) comparing their answers and asking clarifying questions of each other).

Share: 5 minutes or so reviewing the examples as a class. 

Exit Ticket and Homework

10 minutes

To conclude today's lesson, I will ask students to complete an Exit Ticket: Introduction to Quadratic Functions. The Exit Ticket uses a similar context as the entry ticket. I designed the task to help students reflect on the characteristics of linear and quadratic functions. I will ask my students to work on their own, so that I can use the Exit Ticket as an informal assessment. On the Exit Ticket I ask students to explain their reasoning. I want them to take time at the end of class to integrate their thinking. 

Tonight's Homework: Introduction to Quadratic Functions assignment asks students to find 3 examples of quadratic functions in real life. I ask students to identify examples that were not included in the class videos. I provide them with an idea organizer to complete. I provide this resource to help the students focus their ideas and choose supporting examples.  This homework supports a focus on using academic language.