This work is the introductory section to s unit on quadratic equations, parabolic models, and the related skills. The work is designed to be semi-independent and peer group driven while in the classroom.
Quadratics, Modeling, Parabola, Function
This course is intended for freshman or sohpomore level students in a standards base math emtics classroom. This learning module is designed for investigation into quadratic functions, graphs, and equations as the first part of the unit. In the previous Math 2 curriculum used, students developed the skills to understand and create quadratic tasks from investigation and discussion within small groups. Some of the conceptual understanding gets lost in the new format of the course compared to more didactic models of teaching mathematics. In this module, the benefits of discovery and discussion will be present and used for students to learn from as well as other multi media elements to round out some of the content. Math 2 is also a course that students can test into from middle school and skip the Math 1 curriculum, with this they also lose some of the building blocks to how the tasks work and some assumed knowledge is not present because of skipping the Math 1 year. By including other sources for content from Math 1, students have the opportunity to skip, review, or have just-in-time support for the Math 2 topic they are pursuing.
The learning module utilizes a variety of approaches from the EPOL 481 course. The learning module here follows the new learning/transformative method of teaching for students giving them choice in time, investigation, and autonomy within some structures that also provide concrete examples for the mathematics tasks. This module is meant to be discussion and inquiry based, following sudent lead.
Learning Sequence:
| Prior Year 1 related topics: | Following in Year 2: | Following in Year 3: |
|---|---|---|
|
Variables and Notation |
Rate of Change |
Formal unction Notation |
Prior Knowledge:
Progression
Beginning with graphical quadratic functions, the unit is split into two parts. The first part working through transformations and the area model of multiplication for a groundwork understanding to build on manipulative skills. The first part of this unit also looks at sumbolic representations for quadratics.
Module Duration:
You will have 6 regulaly scheduled 90 minute blocks to complete this module, as well as a 7th block for the assessment portion. The Peer Reviewed Project will be done outside of class time.
The expectation is that for the first module, you come with the Pre-Lesson Tasks done to engage in lesson with your peers or indepedently during the class. There are three main parts of each module, the Pre-Lesson, Lesson, and Assignments. Depending on how you choose to space out your work, either the Pre-Lesson or the Assignments can be completed in class and the other will be done outside of class. The choice of which to prioritize in class is yours, and thus you will be placed in a group that fits with the timing your selected. The main lesson will always be completed in class, splitting time between independent work and group work.
Learning Objectives:
Required Materials:
Module Duration:
90 minutes blocks, 6-7 blocks
The work in this module is intended to take place during the scheduled class time where the students have access to each other and the teacher. Though, students may not finish all "assignment" tasks before the end of the class period, these tasks are meant to be more indepedent and able to be worked on alone or with online communicaiton.
Purpose:
This unit uses a variety of contexts—projectile motion, areas and volumes, the Pythagorean theorem, and economics—to develop students’ understanding of quadratic functions and their representations, as well as methods for solving quadratic equations. The central problem involves a rocket used to launch a fireworks display. The height of the rocket is described by a quadratic function, and the questions involve vertices and x-intercepts, which are fundamental features of the graphs of quadratic functions. Over the course of the unit, students strengthen their abilities to work with algebraic symbols and to relate algebraic representations to problem situations. Specifically, they see that rewriting quadratic expressions in special ways, either in factored form or in vertex form, provides insight into the graphs of the corresponding functions. Establishing this connection between algebra and geometry is a primary goal of the unit. (2015 Interactive Mathematics Programs)
Method/Delivery:
Students can do selected tasks and watch modules assignments at home, coming to class prepared to debrief on their findings and create new conclusions before continuing to next module. Some tasks may be done in class for better small group collaboration and discourse. The schedule is time flexible as some discussions may need more time for understanding and conclusions. The facilitation of the discussions by the teacher is limited, only using guiding questions when then discussion is slowing down. Most frequently though, larger class sizes have a few that continue to ask questions until everyone is settled with the work they have done.
Supports for students who struggle with the flipped classroom are provided during synchornous learning time as the "flipped" part of the classroom is done in class to also have discussion intermittently with peers and support of the teacher resource.
Common Core State Standards:
Essential Question: How can a rocket’s path be modeled by a quadratic function?
Learning Objectives:
Pre-Lesson Task: Textbook: A Victory Celebration. In this task you will sketch a model of the rocket's motion and paraphrase the situation to look for specific heights and times within the fireworks path.
Discussion: Share findings in small groups and create a presentation of your work. Share your steps and findings. Answer the following questions as a summary:
Parabolas Intro Summary:
Assignments:
Classroom Task: Pair up and work through this Desmos Activity: Poly Graph: Parabolas, Part 2
Independent Update Task: Textbook - A Corral Variation. 1. Create a visual model then answer the questions: What dimensions give the maximum area? Why does this give you the maximum area? What is the maximum area?
Comment: What method did you use to work through this task, did it work? Are you convinced this is THE method to solve the task? Then comment on at least 2 other peer methods, at least 1 should have used a different method for you to compare.
Purpose: The Victory Celebration task is a projectile-motion problem that looks at the essential question in an informal way. Students should explore the input and output of data given by the standard form equation. New key vocabulary may arise during this task.
Method/Delivery: Students will come into class having previewed and attempted the task. Have a group of volunteers read the task aloud and dissect the important information. Group students in small groups based on similar methods for solving the task. Have each group create their own presentation so the whole class can see the different methods for approach.
Notes: The video and assignment tasks can be done together, but in pairs for discussion. In the online discussio, students should engage in the questions about A Corral Variation showing different equations and methods. Use those to highlight their differences in the next class after the assignment.
Essential Question: How do parabolas move and change shape?
Learning Objectives:
Pre-Lesson Task: Independent Task: Match My Parabola
Debrief: Facilitated discussion on what was discovered (observations, struggles, etc.) via the Desmos task. Use the Parabolas and Equations I, II, and III tasks to summarize and graph in large group via Desmos Graphing Calculator projected on board.
Transforming Quadratic Functions Summary:
Assignments:
Independent Task: Textbook - Vertex Form for Parabolas & Using Vertex Form
Create Update: Use the Demsos Graphing Calculator to use the parent function for a quadratic and create 5 transformed graphs. 3 of which should have 1 or more transformations. Use the space below each graphs equation to describe how the graph changes according to the specific parts of the equation. Share with a partner and review each other's graphs.
Comment: What method did you use to work through this task, did it work? Are you convinced this is THE method to solve the task? Then comment on at least 2 other peer methods, at least 1 should have used a different method for you to compare.
Purpose: These activities are meant for students to seperately discover the three types of transformations for polynomial functions. After discovering these individually, they then combine all of the equations into one in the following lesson.
Method/Delivery: Demonstrate the general process in the graphic calculator with the equation from the rocket. Ask students to formalize the form that equation is in, drawing out the important values of the a,b,c coefficients. After students have concluded their discussion, make sure the idea of concavity was mentioned. If not introduce this idea and have them create their own graphs with different concavities to test their work starting in different quadrants.
Essential Question: What does vertex form provide?
Learning Objectives:
Pre-Lesson Task: Independent Video:
Application: Partner Task: Textbook - Crossing the Axis
What are the parameters from the equation doing to the graph?
Assignment:
Independent Practice: Desmos Will It Hit the Hoop?
Independent Task: Textbook - Is It a Homer?
Create Update: Take a real world analogy of motion that creates a parabolic curve and create a visual model with the critical information, but no equation. Post the task to your class page and ask your peers create the equation that represents the motion of the situation.
Comment: What method did you use to work through this task, did it work? Are you convinced this is THE method to solve the task? Then comment on at least 2 other peer methods, at least 1 should have used a different method for you to compare.
Purpose: Students are combining the dilation and translations from the previous lesson here to see how vertext form can do all of these changes at once. Inquire for students to try changing random numbers one at a time to see what happens to the graphs.
Method/Delivery: Have students summarize what the videos told them in groups as they start to work on the application task. This task should presnent a chart for the x-intercepts determined. This is great to formalize for better student organization.
Notes: The absolute value of a will determing how vertically stretched the graph is regardless of concavity. If this comes up, it can be discussed, but is not lost if not mentioned. The assignment task is for students to show the flight path of a ball is symmetrical, creating a parabola, and a predictable path of motion. There are two paths they can take this information, both should be presented in the comments of the task.
Essential Question: How can you use algebra to change a standard form equation into vertex form?
Learning Objectives:
Pre-Lesson Task: Partner Task: Textbook - A Lot of Changing Sides. Draw a model for each of the questions, use different colors or submols to connect your algebraic expression to the diagram.
Discussion: Discuss: With another pair:
Choose one of the problems from the task to create a situation for on a poster for the whole class.
Apply: Small Group: Textbook - Distributing the Area I
Answer these questions on the whole class discussion board (you can provide an example, but try to explain the concept in mathematical discourse first):
Reading: Independent Reference: Textbook - Views of the Distributive Property. In your own words, describe what the vertical multiplication is with binomials. Connect that concept to a vidual model.
Area Model Summary:
Assignment:
Independent Task: Textbook - Distributing the Area II
Update Post: Use the results of your work from this lesson to describe what the process means for quadratic functions. What applications beyond plots of land does this model support?
Comment: What method did you use to work through this task, did it work? Are you convinced this is THE method to solve the task? Then comment on at least 2 other peer methods, at least 1 should have used a different method for you to compare.
Extra Practice: Desmos Build Bigger Field
Purpose: The following activities focus on the symbolic representation for quadratic functions and the algebraic process for factoring and multiplying. The area model is used as a tool to multiply polynomials and complete the square for changing equation form. Students will be moving back and forth between graphical and symbolic representations in this lesson and the following lesson. As we get more into the unit, think about the main ideas of creating the classroom to be student centered and student led. Any and all of these steps can be guided by students for better comprehension in the classroom:
So far we have begun to discover the quadratic function, it's applications, and features of the equation and graph. From what you have learned so far you will create a Discovery Task. This will be similar to the tasks you have completed to find new properties and rules for quadratic functions (and other topics from year 1).
Goal: Your task is to create your own Discovery MathematicsTask. This should include at least 3 parts that build to 3 or more concept discoveries. As this will be a peer reviewed work, your classmates will look at your task, possibly attempt it, and critique what is great about it and what needs to be worked on. From this critique you will more thoroughly develop your task and submit it for final review. In the final stage, a peer from a different class will attempt and reflect on the task as part of the accessibility to many learners aspect of the task
Elements: Make sure to include the following peices of work in your task.
Sample Task:
Purpose:
Student's are to show that they understand the meaningful parts of mathematics creation adn the content that they are currently studying. As part of the meaningful mathematics within this curriculum students have been practicing and working on understanding deeper the 8 Mathematical Practices. This is part of an ongoing portfolio of task creations throughout the year to show their level of understanding within these practices. This is the first of two for the Quadratics unit where they will show how deeply they understand the parts of a quadratic function and models.
Notes:
Reviewer's Rubric
Essential Question: How does multiplying binomials, specifically squares, rewrite into a standard form equation?
Learning Objectives:
Pre-Lesson Tasks:
Reading: Khan Academy Article - Factoring Quadratics: Perfect Squares
Activity in Small Groups: Textbook - Square It!
Debrief: Draw the area model for one of the square tasks. Determine the three essential steps for the visual model of the distributive property described in Square It!
Videos:
_C3_83_C2_82_C3_82_C2_B2_20_20_20Algebra_20I_20_20_20High_20School_20Math_20_20_20Khan_20Academy__.jpg?Expires=1716091970&Signature=L0ktX6R9~cawf2d-xwFb90d6MkHetpCSG7MmqSDrAbT8xUjK4ZX6msG0ywna4B~kZZQ3Icz-8r3gl8RGq8mLmfI~-AwFP7~U-TOBlSBFjyUqYcwOLnyrpTH7o5BdcePcnLeSlMElEkPVWntjL4BmSk990EVgDcvIIa~bz6U1Zg8X3DoyfOcQxk-Uimt6iURWYiBfYDoGXwyDh7yOSX5QJ~2tZGR5vm7XMJtPfHanvx~olfcEO6EiKCSKEGVMW90GvltnlqlNqStNmiZHiQuNEKo6iJwcHSj6NE-WnlttLzmLdOlmlKA6OABpbzCvj9BHVDKFvEQWmowiQrS5r~GKYg__&Key-Pair-Id=APKAJELYXGUCCDL7FUQA)
Assignment:
Independent Task: Textbook - Squares and Expansion
Watch Videos: Comment Update: Using the videos below, comment on an essential step that you think will have a common error or misconception of the task when applied.
Comment: What method did you use to work through the tasks in this section, did it work? Are you convinced this is THE method to solve the task? Then comment on at least 2 other peer methods, at least 1 should have used a different method for you to compare.
Purpose: This will be an application of the multiplication skills developed in rewriting expressions.
Notes: This section should go very smoothly, if students understoof the previous lesson. Students seem to like this lesson more becuase of the consistency that perfect squares provide them with in every form the equations.
Essential Question: How can everything from this module be applied into a real world context?
Learning Objectives:
Collaboration Task: Small Groups: Textbook - Vertex Form to Standard Form. Create a group presentation (whiteboard or slides) to share. This should include:
Discussion: How do we know what the length of the cardboard support should be? Why can't it be anything else?
Assignment:
Independent Practice Task: Textbook - How Much Can They Drink?
Update Post: Make a model and answer one of the extension questions in the discussion board (include a picture of your model):
Comment: What method did you use to work through the tasks in this section, did it work? Are you convinced this is THE method to solve the task? Then comment on at least 2 other peer methods, at least 1 should have used a different method for you to compare.
Purpose: As a summary of Quadratics Part 1, students will practice with the algebraic manipulation and parabolas in a real science context.
Method/Delivery: Students have not yet inquired about the directrix and the focus of a parabolas, this is a good palce to bring this up for them to take notes on. As a summary lesson, give students plenty of time to work together and collaborate before summarizing their own ideas. This space is used for final questions and comments before the assessments.
Notes: One important property of parabolas is that they concentrate incoming engery, light, at the focus. This property is the main focus of the real world situation of the last problem in colaboration. Students may inquire about building a model to use outside, they shoudl create one and try it outside the school!
Method/Delivery: Have students respond to this before the end of the course, to be included in Analytics
Notes: Could add more questions
To demonstrate your level of understanding from this unit, you will complete an in class assessment as well as the following take home assesment. These two parts will account for 1/3 of your standard assessment grade each. The final third will come from the Peer Reviewed Task that you began after Lesson 4.
Take Home Assessment
In Class Assessment
"Completing the square" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:completing-square-quadratics/v/solving-quadratic-equations-by-completing-the-square
Desmos. (n.d.). Build a Bigger Field. Retrieved September, 2020, from https://teacher.desmos.com/activitybuilder/custom/56e19b4183ba3908118725dd
Desmos. (n.d.). Will It Hit the Hoop? Retrieved September, 2020, from https://teacher.desmos.com/activitybuilder/custom/56e0b6af0133822106a0bed1
Desmos. (n.d.). Match My Parabola. Retrieved September, 2020, from https://teacher.desmos.com/activitybuilder/custom/5605bb6200701ed10fb0931a
Desmos. (n.d.). Graphing Calculator. Retrieved September, 2020, from http://www.desmos.com/calculator
Desmos. (n.d.). Polygraph: Parabolas, Part 2. Retrieved September, 2020, from https://teacher.desmos.com/activitybuilder/custom/574f12421390db611564fa32
Fendel, D., Resek, D., Alper, L., & Fraser, S. (2015). Interactive Mathematics Program, Integrate High School Mathematics (2nd ed.). Mount Kisco, NY: Interactive Mathematics Program and It's About Time.
"Graphing quadratics: Vertex form" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:vertex-form/v/graphing-a-parabola-in-vertex-form
"Intro to parabola transformations" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:transform-quadratic-functions/v/shifting-and-scaling-parabolas
Jones, M. (n.d.). Keys to Making Thinking Visible [Digital image]. Retrieved October, 2020, from https://www.pinterest.com/pin/398005685789291943/?nic_v2=1a5JbVIes
Self Made Inforgraphic
Mathematics Standards. (n.d.). Retrieved September, 2020, from http://www.corestandards.org/Math
Morgan, J. (2020). Algebra, Awesome Inc. Retrieved October, 2020, from https://www.resourceaholic.com/p/resource-library-key-stage-34-algebra.html
"Multiplying binomials: Area model" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratics-multiplying-factoring/x2f8bb11595b61c86:multiply-binomial/v/area-model-for-multiplying-binomials
"Parabolas intro" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:intro-parabolas/v/parabolas-intro
Quintanilla, J. (2016). Engaging students: Fitting data to a quadratic function. Retrieved October, 2020, from https://meangreenmath.com/tag/parabola/
"Scaling & reflecting parabolas" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:transform-quadratic-functions/v/example-translating-parabola
"Shifting parabolas" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:transform-quadratic-functions/v/example-translating-parabola
Splung. (2020). Projectile Motion. Retrieved October, 2020, from http://www.splung.com/content/sid/2/page/projectiles
"Squaring a binomial (old)" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra-home/alg-polynomials/alg-special-products-of-polynomials/v/special-products-of-polynomials-1
"Squaring binomials of the form (x a)²" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra-home/alg-polynomials/alg-special-products-of-polynomials/v/pattern-for-squaring-simple-binomials
"Vertex form introduction" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:vertex-form/v/vertex-form-intro
"Worked example: Complete the square intro" (video). Khan Academy. Retrieved September, 2020, from https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:completing-square-quadratics/v/ex1-completing-the-square