Lesson Plan Overview

Title: Lessons on Quadratic Functions: UDL Integrated Approach
Author: Ulysses Lalunio
Subject: Mathematics
Grade Level: 9th
Unit: Patterns & Transformations, Relations & Functions


Unit Description

This unit is designed with UDL principles in mind and a humble attempt to provide a lesson plan model for teachers interested in integrating UDL in their classroom repertoire. Those principles provide ideas such as leveraging existing technologies to engage teachers in adopting instructional strategies that enhance learning experiences of their students (Annot 01, All). The unit assumes students' prerequisite general understanding of linear functions and relations. The following support for this prerequisite or background knowledge is provided here: (Video Lecture, Online Resource,  Printable Text Reference, Interractive Software

While the content or topic focus is on quadratic functions, the overall unit goal is for students to demonstrate the transformation of a quadratic function and analyze it for various solutions through multiple representations.  The unit can be divided into one 90-minute lessons or two 45-minute lessons. This time allocation may be "recalibrated" as needed to support appropriate attention for various activities to accommodate differentiated instructions. 

Lesson Description

Teacher will engage students in both formal and informal approach to introduce the topic of 2nd degree polynomial functions or quadratic.  He will provide brief background knowledge by engaging the students in exploration activities that allow them to make connections and arrive at an understanding of quadratic functions (Annot 02, All). To enhance and sustain engagement, students will be immersed in various modalities such as multimedia, online manipulatives and group activities to achieve the learning goals.(Annot 03, ALL)

State Standards

CA Standards Addressed:

8.0 Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system. 

9.0 Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b) 2+ c.


Unit Goals:
Understand patterns, functional representations, relationships and functions.
Analyze and create multiple representations of mathematical situations and structures using algebraic symbols.
3.  Specify and describe locations and spatial relationships using coordinate geometry and other representational systems.

Lesson Goals: (Annot. 04, All)

Share and discuss briefly the following goals to the class and invite students to comment or pitch in other goals they think are relevant to the lesson either in writing or verbal.  Suggestions will remain anonymous.

1. Students will identify the defining characteristics of a quadratic function.

• Distinguish how the characteristics are represented symbolically and graphically

• Recognize patterns and relationships among certain characteristics of the function (for example, deducing the y-intercept by noticing patterns in the roots.)

• Describe the meaning of the vertex, y-intercept, and x-intercept(s) by analyzing the behavior of a quadratic function at and around those points

2. Students investigate the coefficients in symbolic expressions of quadratic functions, and how varying coefficients are reflected in graphs.

3. Students identify significant points on a parabola and how these are reflected in symbolic expressions.

4. Students make connections between graphical and symbolic representations by examining and manipulating the three symbolic forms of a quadratic function in order to predict shape, orientation (opens upward or downward), and location.

               Three forms of quadratic equation:

               polynomial:  y = ax2 + bx + c,
vertex:  y = a (x-h)2 + k,
               root:  y = a (x-r1)(x-r2)

5.      Students will gain deeper understanding of quadratic functions as they become more aware and motivated to observe and reflect on its application in their environment (Annot 05, All).


Resource Preparations:

  • Arrange the learning environment:
    -  reserve computer lab if necessary
    -  set up and test technologies (internet connections, website, java downloadable)
  • If the students are not familiar with SeeingMath or Graph Sketcher have 2-3 minute session to demonstrate the software.  Both software are very intuitive and have readily accessible online help tabs.
  • If necessary, prepare electronic or paper handouts containing a quick guide with step-by-step instructions on some common input routines and command buttons.
  • The activity is recommended to be completed in pairs or small groups.  It can be accomplished individually but may require more time.

Priming & Introduction:
(15-20 minutes)
Have students complete engagement and exploration activities (a & b) that allow them to get motivated and acquainted with the subject matter. (Annot. 06, All)

a) Watch a 2-minute video on Quadratic Functions & Formula. Click on the image to play.

b) Have students break into groups of 3 and have them build a table of values for a parent function y = x2.  Tell students to include negative and positive values of x.

Allow each group to assign different roles (in rotation) of who would investigate the function using online manipulatives (SeeingMath, Graph Sketcher, Quadratic Function Explorer) to generate graphs or by manual sketching using graph paper and pencil.

Encourage students to share and discuss their generated visual output among their group mates. (Annot. 07, All) Ask them to compare and contrast to previous knowledge of linear function's graphs and table of values.  Pose guide questions that help students to focus and monitor their understanding: What is the graphical representation of a quadratic function?  How are linear patterns different from quadratic patterns?

Guided Learning (Practice):(Annot. 08, All)

1) This is a multimedia activity for students who have better facility with a combination of sound, text and animated visual displays, click on the voicethread link to play the video. Students are encouraged to post questions and share comments using VT to extend the discussions and understanding of the quadratic function using this platform. This activity can also be done as a homework. VT questions and comments can be accommodated in text, voice record, webcam, phone message, webcam, or file uploads such as powerpoint illustrations, etc. (If the students are not familiar with posting comments, have 2-3 minute session to demonstrate site.  This could be assigned prior to class.  VT has brief interactive instructional videos available).

(Click this icon to watch how to graphs quadratic functions.)

2) If the student is more comfortable with text or print equivalent format of the lesson, click here. This can be done independently or by grouping together students of similar preference through this medium.

3) Leave the default function y = x2 (SeeingMath) from the previous activity.  Using SeeingMath, ask the students to click on New command button and have them plot a quadratic functions f(x) = x2 - 6x + 8 by plugging the numeric coefficients.

4) Have students share and describe the new function f(x) by comparing (vertex, x and y intercepts, shape, location) with the original function y = x2. Use the vertex form and the root form tabs/buttons to visualize or check the values of the coordinates of the vertex and roots.

5) If the students are not comfortable with using SeeingMath and typing the coefficients, they may use Quadratic Explorer.  This uses slider to change the values of the coefficients in a given function and has less sophistications on display. (Annot. 09, All)

6) Explore more relevant examples (Annot 10, All) by asking students to create two more functions g(x) and h(x). Have them write the algebraic expressions of the transformations of f(x).  To illustrate: g(x) = -f(x): that is g(x) becomes -x2 + 6x - 8,  and h(x) = f(-x): that is h(x) becomes x+ 6x + 8. Have them to graph both g(x) and h(x). 

Independent Learning (Practice):
1) Have students work in small groups. H
ave each group come up with its own quadratic function f(x) and ask them to plot this new function and then generate both g(x) = -f(x) and h(x) = f(-x). 

2) Have them describe the behavior of the transformed equations by asking them to create two-column notes with critical attributes of the quadratic functions such as vertex, intercepts, and line of symmetry. (Annot. 10All)

3) Since the students have been using technology to plot and visualize the graphs, they are able to go through more examples and immediately see what happens to the graph display when they change different parts of the equations. Have them check their own understanding with their group partners.

4) The teacher can step back from the spotlight and allow the students to discover and learn from each other (Annot. 07All)Pose open-ended questions occasionally while moving around ready to lend a helping hand: "How de we know that parabola opens upward or downward?  How do we know if the function has x-intercepts or y-intercepts?"  

5) Students then summarize ideas to help them remember the attributes (characteristics) and how to find those attributes in the graph and in the equation.  This can be done as a homework or groupwork in the next class session.  Inform the students that they can submit the summary (homework) in one of the following options: essay format, PowerPoint slides, post a voice comment on VT, a list of educational online games about quadratic functions with brief description and personal rating on engagement. (Annot. 11. All)


Review & Summary Activity - Click to play the video.



Othe Clarifying Summary Exhibits:
Present other forms of representing the concepts and approach to problem-solving.  Concept maps or charts, procedure roadmaps, and other mindtools  that illustrate key constructs reinforce students' understanding and promote critical thinking.

Last lesson:

1) Elaborate with student understanding of quadratic solutions by using real-world situations such as maximizing area or volume (if one dimension is a constant value), projectile
motions, or building archways. (Annot. 05, All


2) Pose questions such as: Why is it important to real life?  In what careers might they use quadratic equations?  Make it exploratory.  Have students visit Living Mathematics website: http://plus.maths.org/issue30/features/quadratic/index-gifd.html

Video is also available to elicit ideas (Annot. 08, All):

3) Break students into small groups by career choice and have them solve real world problems for that career that use the quadratic function.  This could even be a couple of sessions - solving their problem one day & then the groups presenting to the class what they did the next, so all could learn about the different career options. This activity could also be done as a homework.  (Annot. 12, ALL)

Assessment & Evaluation

Formative/Ongoing: (Annot. 10, All)
Students' understanding was monitored throughout several activities and were mentioned explicitly in the methods section.  In addition, we can also design a test posing open-ended questions that help students focus on explanations and assumptions, questions such as:

1)     Explain how points on the function f(x) pre-image are mapped onto the image.
2)     Explain what components are critical in describing the shape, location, and the opening or orientation of the graph (parabola).
3)     Present a graphical image of a parabola without its symbolic or algebraic representation, then introduce three choices of the polynomial form (y = ax2 + bx + c) with varying coefficients.  Ask the students which function best describes the parabola and have them explain.
4)     Explain what condition(s) will make quadratic a linear function, or what’s the effect of making the coefficient of x2 less than 0, equal to 0, greater than 0?

Again, these open-ended questions can also be assigned as individual homework or small group activities and students are free to choose their mode of responding to the questions such as by voicethread, oral explanations, essay format, sample problems, slides, graphical solutions, algebraic solutions, etc.