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Quadratics Part 2 Learning Module

Learning Module

Abstract

Part 2 of an intro quadratics unit for high school seniors. Inquiry and discussion based, student led.

Keywords

Quadratics, Modeling, Parabola, Functions, standard form, vertex form, solutions, factoring, zeros

Overview

This is the second half of a Math 2 unit based on investigation for quadratic functions, equations, and features of parabolic graphs. The first half of the unit's learning module is linked here. From the module 1 overview, the main practice and delivery for the module will be used in this portion of the work. Intended for high school Math 2 students, typically freshman or sophomores who have taken algebra or Math 1, students have the opportunity to skip, review, or have just-in-time support for the Math 2 topic they are pursuing.

This module aligns with practices from the EPOL 559 course. Specifically, in this part of the module, students have fluid differentiated groups, which can be adapted based on their understanding from lesson to lesson. The groups are self-determined by the students, based on their own reflections and confidence in each module's content students differentiate how much support or extension they need in the next lesson. There are also a variety of modalities to discover, understand, and compare ideas from the curriculum and with each other to make it more connected to their learning preferences. This module is student-lead based on inquiry and projects.

 

Math Review of Applications of Parabolas. (2014). [Cliff Diver Trajectory] [Photograph]. School Tutoring. https://schooltutoring.com/help/math-review-of-applications-of-parabolas/

The activities in this portion of the models work on the general assumption and emphasis of "doing and undoing" in algebraic thinking. Students will specifically discover how to do transforming equations, derive quadratic functions in real-world contents, interpret key points on a graph, and recontextualize their work.

Intended Learning Outcomes

For Students:

Learning Sequence:

Prior Year 1 Knowledge: Prior and Following Year 2 Knowledge: Following Year 3 Knowledge:

Variables and Notation

Basics of Graphing

Linear Equations, Graphs & Situations

Area and Perimeter of Rectangles

Prior:

Features of Quadratic Graphs

Completing the Square

Distributive Property

Quadratic Transformations

Following:

Rate of Change
Derivatives
Exponentials & Logrithmics
Number e

Formal Function Notation

Properties of Functions

Function Families

End Behavior & Asymptotes

Trigonometric Features and Transformations

Prior Knowledge:

  • The Pythagorean Theorem and features of triangles
  • What is a perfect square, how does that relate to the distributive property?

  • ​Developing and expressing algorithms for problem situations

  • Interpreting algebraic expressions in summary

  • Coordinate system, graphical to tabular, scaling

  • Multiple representations of descriptive situations

  • Points of intersection

Progression:

This is the second part of the quadratics unit. In this module, you will work through applications of quadratic functions, the features from the graphs, and how they directly connect to the equation's features. This is building upon the groundwork of the last module. There will also be supplemental work, either reinforcements or extensions to our current content at the end that is not required, but highly suggested (as this will prepare for World of Functions in Math 3).

Module Duration:

​You will have 6 regularly scheduled 90-minute blocks to complete this module, as well as a 7th block for the portfolio portion. The Peer Reviewed Project will be done outside of class time.

The expectation is that, similar to the previous module, you come with the Pre-Lesson Tasks done to engage in lessons with your peers or independently 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.

Within the assignments, there are three differentiated level groups. As you come into this lesson, you begin in the group you ended the last lesson on. Based on your level of understanding and achievement in that task from the lesson, you will move between the groups or stay the same. These groups are fluid and spaced intentionally for more personalized and structured work to connect your learning too. The amount of time each assignment will take you depends on the group you are in and if you need/want to work on another group's work as well.

Learning Objectives:

  • Mathematical Modeling: Expressing situations in terms of functions and equations, applying mathematical tools to models, interpreting mathematical results.
  • Graphs of Quadratic Functions: Understanding the role of the x-intercept in the graphs, recognizing the significance of the sign in the ax^2 term to determine the shape of the function.
  • Working with Algebraic Expressions: Using the different forms of the function to simplify the expression, identify certain quadratic expressions as perfect squares, differences of squares.
  • Solving Quadratic Equations: Finding the exact roots of an equation using the vertex form or factored form of the corresponding function, using the zero product rule of multiplication to solve equations via factoring.

Required Materials:

  • Graph paper (1 cm or 1 inch)

  • Graphing calculator (TI-84)

  • Computer with camera and internet access

  • Access to the classroom website and discussion board via the code provided in class

For Teachers:

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 independent and able to be worked on alone or with online communication. The class module is set up to have a discussion in person, with virtual learning, the discussion happens a bit less fluidly, have some guiding questions ready or facilitator students assigned to keep the conversation going.

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 module assignments at home, coming to class prepared to debrief on their findings and create new conclusions before continuing to the 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 the 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 synchronous 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:

  • A-SSE.3a. Factor a quadratic expression to reveal the zeros of the functions it defines.
  • A-APR.3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
  • A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • A-CED.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
  • A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
  • F-IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
  • F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
  • F-IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
  • F-BF.1. Write a function that describes a relationship between two quantities.
  • F-BF.3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

7. Pythagorean Theorem

For Students:

Essential Question:

How are the relationships within triangles connected to quadratic functions?

Learning Objectives:

  • Apply the Pythagorean Theorem to two right triangles within one scalene triangle
  • Derive a symbolic quadratic equation and solve for missing variables

Pre-Lesson Tasks:

Hypatia's Group - Review the Pythagorean Theorem videos from our previous lesson (in Do Bees Do It Best?) and answer the following questions. Then create a diagram showcasing the relationships of the sides with numbers and variables.

  • What do you feel confident about? What do you remember?
  • What do you need more support in understanding?

Euclid's Group - Create your own video explaining the Pythagorean Theorem with 2 or more numerical examples, one variable example, and 2 assignment questions for your peers.

Haynes' Group - Research 2 real-world applications for the Pythagorean Theorem, one should be two dimensional and three dimensional. Then create a video explaining your findings with questions that you have for your peers at the end. Begin the task Equilateral Efficiency, you will continue it after the lesson as well.

Activity and Discussion:

Small Groups: Leslie's Flowers (page 34)

Assignments:

​Update: Scale up the flower bed in Leslie's Flowers to some degree. Do all features of your graph still fit the model and function created during class? Create a diagram to go with your work.

Comment: What about this method/lesson can you connect to another concept in our geometry lessons? Describe the connection as well as its differences.

Hypatia's Task - Create your own video explaining the Pythagorean Theorem with 2 or more numerical examples, one variable example, and 2 assignment questions for your peers.

Euclid's Task - Emergency at Sea (page 35)

Hanes' Task - Emergency at Sea (page 35). Wrap up Equilateral Efficiency task and write up.

 

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.

 

For Teachers:

Purpose: Students will reinforce/solidify their understanding of the Pythagorean theorem and apply deductive reasoning skills to connect the side's relationships to the solution for missing variables. Students will need to break down the pieces of a quadratic expression to do this task.

Method/Delivery: Student-led, this lesson will focus on small group work to reason for solving two right triangles with 2 missing variables. The space at the end of class is used for final questions and comments before the assessments.

Notes: The extension task Equilateral Efficiency is used here for the students who need/want more opportunities to go beyond the text. This task develops Hero's formula, the relationship between sides of a triangle.

8. More Changing Forms

For Students:

Essential Question:

How can the process for changing equation forms be reversed to result in vertex form? What does vertex form tell us about our function?

Learning Objectives:

  • Completing the Square with a = 1, -1 from standard form
  • Interpret the results of vertex form
  • Develop a definition and process for finding the zeros of a function
  • Contextualizing the vertex and x-intercepts of functions

Pre-Lesson Tasks:

Hypatia's Group - ​Review the Vertex Form with this Desmos Activity: Vertex Form Exploration

Euclid's Group - ​Read this article on vertex form: Graphing Quadratic Functions: The Leading Coefficient/The Vertex

Haynes' Group - Create your own process for changing a function from standard to vertex form. Do not look up this process. This is not graded on accuracy, but development and discovery. Write out the process of your steps as you go along, formalizing your process in a concise way.

 

Activity and Discussion:

Via completing the square, we will complete #1 of Here Comes Vertex Form Together. After which, based on your initial comfort for this task you will move into your Assignment groups. 

Hypatia's Group will work on Here Comes Vertex Form as a whole group, in guided steps formalized together with the instructor. Euclid's Group will work on the task together, creating a comparison graph for each problem. Haynes' Group will complete the task and then create 2 of their own equations for other members to solve and present solutions upon.

At the end of the lesson, all groups will come back together to formatively assess with the task Finding Vertices and Intercepts. The results of this will determine assignment groups for this lesson.

Assignments:

​Update: Create a problem where you solve for the intercepts of a graph. When solving it provides two solutions, one with the wrong solution (due to at least 1 error in the work), and one solved correctly. Then explain where the error is and why it makes the problem incorrect. Also explain how this mistake might happen to many people, or what makes it an uncommon error.

Comment: One question you have about critical points on graphs and answer someone else's question.

Hypatia's Task - ​Describe the relationship between the features of standard form and vertex form equations, connecting the variables change throughout the steps.

Euclid's Task - Extension Task Check It Out! (page 68)

Haynes' Task - ​Extension Tasks Check It Out! (page 68) and Make Your Own Intercepts (page 75)

For Teachers:

Purpose: This is a new method for changing forms via two other methods we have seen. Here students are expected to bridge the two ideas in the task Here Comes Vertex Form.

Notes: ​Completing the square may need to be reviewed at this point, as well as the expansion step for balancing an equation on one side. This could be done by demonstrating #1 or having students try it on their own and presenting their ideas first. Whichever method would be more effective depending on how many students fall in each of the differentiated groups.

9. Finding Intercepts

For Students:

Essential Question:

How does the firework rocket travel? What are the important features defined in different parts of the function? How were they found?

Learning Objectives:

  • Use developing methods to solve the unit problem in a simpler context 
  • Complete the Square with an a coefficient great than 1
  • Find the intersection of a parabolic curve and a linear function

Pre-Lesson Tasks:

All Groups: Review page 4 (Victory Celebration) for context on the unit problem. Begin Another Rocket (page 38)

Discussion/Debrief:

Partnered Activity: Another Rocket followed by a whole-class debrief on the solution. 3 students will share their process and reasonings with their partners.

Independent Activity: Pens and Corrals in Vertex Form

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.

Assignments:

​Update: Your independent work for Pens and Corrals in Vertex Form along with 2 questions you have to extend the problem.

Comment: Find a video or article explaining the process of changing forms. Describe an example of the real-world application of parabolic features being relevant.

Hypatia's Task - Complete an Error Analysis for Peers Task from the other group based on the task Profiting from Widgets (page 39)

Euclid's Task - Pens and Corrals in Vertex Form Update and Comment.

Haynes' Task - Profiting from Widgets (page 39), but also create an incorrect response for others to find, describe, and correct the mistake on (Error Analysis for Peers)

For Teachers:

Purpose: Apply the process of transforming from standard form to vertex form in the context of two application problems, both of which have been previously addressed in one context but not yet formalized a process.

Method/Delivery: Students will work in groups on the task Another Rocket. The inital discussion will be based upon a leading coefficient being 16 rather than 1. This will begin the process described in the last lesson for students to work through the task. Once the task is complete, students should compare their answers to that from the original problem on page 4.

Notes: Pens and Corrals in Vertex Form should be done independently, then when the updates are posted, all will be held and then released once all submissions are in.

Peer Reviewed Task

For Students:

At this point in the unit, we have summarized our findings for quadratic equations in different forms, the features of those forms and conclusions we can make from them, factors, and intercepts. From what you have done, and the discovery task you did in the first portion of this unit, you will create a second discovery task. This can be an extension of the previous task you created (from new materials utilizing old concepts) or an original task based on the new content in this module. 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 pieces of work in your task.

  • The situation requires a visual model to be created.
  • Enough background information to have your students engaged in the material
  • A bridge/connecting statement that requires students to review ideas and extend their current thinking.
  • Multi-part element. This could be where you include your different concept discoveries or be the accessible component to your task.
  • At least 3 different concept discoveries. These should be sequentially connected but can take part in multiple lessons if needed.
  • At least 3 of the 8 Mathematical Practices that we have been studying in class should be underlying concepts within the task.
Wayne RESA. (2020). https://docs.google.com/file/d/0B19SejfvMU1rbW1sbEZ3Y1E1Qkk/edit

Sample Task: This task has one that we originally used to discover features of quadratic equations, the second page is what would be the third part of the task (I left out the second part, as you are doing that in this module).

Sample Extension Discovery Task

For Teachers:

Purpose: Students are to show that they understand the meaningful parts of mathematics creation and 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: This is the second discovery task creation for this unit.

Reviewer's Rubric

Task Creation Rubric

10. Factoring

For Students:

Essential Question:

What are factors and what do they mean?

Learning Objectives:

  • Define another symbolic method for solving quadratic equations
  • Connect factoring to the area model and FOIL process
  • Introduce the Zero Product Rule
  • Relate factors to x-intercept solutions

Pre-Lesson Tasks:

All Groups:​ Review factor trees and what factors of large numbers mean.

Discussion then Activity:

Independent Task: Factoring, this should be done with the steps written out to follow the pattern. In the next task, you will not need to write the steps out.

Partners: Let's Factor! and Solve That Quadratic!

Assignments:

​Update: Create an example in your life that could use the main idea and application of factoring.

Comment: Questions from Solve That Quadratic!​ or from Desmos Task: Factoring Practice

Hypatia's Group - Let's Factor Some More! (page 71) then check your work with the key posted after you submit.

Euclid's Group - Take one of the elements of your discovery-based project and connect it to factoring and x-intercepts.

Haynes' Group - Factor's of Research (page 74)

For Teachers:

Purpose: Factoring polynomials is a standard practice in mathematics. This factoring unit is included to connect the ideas in reverse to module 1 for Fireworks. Students will combine previous knowledge with the Zero Product Rule to connect these final features of functions to the graphs they have been looking at all along. 

Method/Delivery: The first task can be done completely independently with a partner think and share. The task Let's Factor! should evoke some discussion based on the equations that do not factor at all or do not factor into even portions. This discussion should happen as a whole class with observations for the graphs of many functions. The final task circles back a fifth time to the Corral problem and now looks at it through the lens of factoring and should show a simplified method for students. Make this point evident that while it may be simpler when an option, factoring is not always an option.

11. Quadratic Choices

For Students:

Essential Question:

How can you determine the most efficient and valuable way to solve a quadratic equation from any form?

Learning Objectives:

  • Synthesize the algebraic and geometric approaches for solving quadratic equations.

Pre-Lesson Tasks:

Hypatia's Group - ​Review notes on the different methods for simplifying equations, graphs, and functions. Write down at least two questions to ask your peers during class about the topic. Work through this Desmos Task: Intro to Quadratic Formula

Euclid's Group - Desmos Task: Intro the Quadratic Formula

Haynes' Group - Use the task The Quadratic Formula problem 2 to use the CTS process to solve for the quadratic formula from the standard form of an equation. Your goal is to isolate x.

 

Activity and Presentation:

In your groups you will work on the task Quadratic Choices, comparing the different ways to solve a quadratic function, then justifying why this was the best method for your work. When you are done, create a presentation based on the description for A Quadratic Summary, where you will give and explain the different algebraic processes you have discovered and their values. Each group will present to one other group, in preparation for the portfolio at the end of the unit.

Assignments:

​Update: Create your own video presentation on one of the methods for solving functions we have learned in this module, describe the great parts of that method and the shortfalls.

Comment: Watch 3 other peers' videos and comment on their thoughts of quadratic function solutions and methods.

For Teachers:

Purpose: Summarizing all of the methods and taking an individualized approach to what methods are superior at different times. Students should come to realize that we learn all methods because of the wide variety of expressions and equations that there are and how there is no one size fits all method.

Notes: The presentations could be done whole class depending on time but also depending on how many groups you have, it may become too repetitive to have every group share the same answer and process.

This video is just a fun song way to remember the Quadratic Formula, it is not necessary to teach or required memorization, but it is fun (especially if you speed it up a bit).

Media embedded November 24, 2020
Media embedded December 6, 2020

NikkiRose. (April 2020). Quadratic Formula Pop Goes the Weasel [video]. Youtube. https://www.youtube.com/watch?v=2lbABbfU6Zc

12. Supplemental Tasks

For Students:

Recommended Reading: A Summary of Quadratics and Other Polynomials:

Reference Reading from Textbook

These tasks provided are based on the topics you need or want more practice on. Tasks completed here should be included in your portfolio for the unit.

Task Guide:

  • Quadratic Symmetry (extension) Assign anytime after Vertex Form For Parabolas. Quadratic graphs are symmetrical about a vertical line through their vertex. In this activity, use the general vertex form of a parabola to explore the relationship between the coordinates of corresponding points on either side of the line of symmetry.
  • Subtracting Some Sums (reinforcement) or Subtracting Some Differences (reinforcement) Use this activity for those who are having difficulty simplifying with negatives.
  • Choosing Your Intercepts (reinforcement) This activity gives students practice with finding an equation in vertex form for a parabola given the vertex and x-intercepts.
  • A Lot of Symmetry (reinforcement) The special binomial product (x + n)(x – n) = x2 – n2 is introduced here in the context from A Lot of Changing Sides.
  • The Locker Problem (extension) This classic problem makes a good follow-up to Divisor Counting.
  • Let’s Factor Some More! (reinforcement) Assign after Let’s Factor! This activity encourages students to use an area model to factor quadratics in which the coefficient of the x2 term isn’t 1.
  • Vertex Form Challenge (extension) Assign after Solve That Quadratic! The activity gives students practice changing quadratic functions in standard form with leading coefficients other than 1 or –1 into vertex form. The problems involve fractions and decimals. This activity addresses a standard in the Common Core State Standards for Mathematics.
  • A Big Enough Corral (extension) Assign after Solve That Quadratic! This activity explores quadratic inequalities. To do Question 2, students must be able to factor quadratics.
  • Factors of Research (extension) Assign after Solve That Quadratic! The activity suggests further areas of exploration in the topic of factoring. Question 2 asks for a generalization of the difference of squares introduced in the supplemental activity A Lot of Symmetry. This activity addresses a standard in the Common Core State Standards for Mathematics.
  • Make Your Own Intercepts (extension) Assign after Quadratic Choices. The activity builds on the idea that students can now easily find a quadratic that has two given x-intercepts. For example, for intercepts x = 4 and x = 2, the quadratic y = (x – 4)(x – 2) will do. However, it isn’t the only quadratic with those intercepts. All quadratics y = a(x – 4)(x – 2) for any real number a will also work.
  • Quadratic Challenges (reinforcement) Assign after Quadratic Choices. The activity offers three more quadratic equations for students to solve. (A graphing calculator would make finding the requested decimal answers too easy.) This activity addresses a standard in the Common Core State Standards for Mathematics.
  • Standard Form, Factored Form, Vertex Form (reinforcement) Assign after Quadratic Choices. The activity pulls together the relationships among standard form, factored form, x-intercepts, vertex, and vertex form and makes a good group assignment.

For Teachers:

Purpose:​ These are the options of supplemental tasks given by the book for reinforcement and extension, students should practice based on the mutual reflection of tasks. Soon to come are videos/Desmos activities for support based on student understanding and reflection.

Knowledge Reflection Survey

For Students:

CG Scholar Survey

 

For Teachers:

Method/Delivery: Have students respond to this before the end of the course, to be included in Analytics

Notes: Could add more questions depending on what you want to learn from students.

Assessment & Evaluation

For Students:

Summary Assignments to include from the text:

Fireworks Summary

Portfolio Guidelines:

  • Include 2 peer-reviewed discovery tasks
  • Reflection
  • Questions remaining
  • Elements outlined in PDF
Fireworks Portfolio

 

For Teachers:

Purpose: The portfolio will include students' reflective work on their discovery tasks within the unit and other tasks that helped them to understand the value of changing forms, features of parabolas, solving real-world problems. They should also include parts of the unit that made them feel more comfortable with the mechanics of working with quadratic expressions and mathematical principles for understanding.

Notes: As in the previous module, there are two traditional assessment pieces to this unit. The third piece is the portfolio and reflection which is done at this point. The 2 peer-reviewed tasks are also included in this portfolio. Students could share their portfolios in class but this is optional. 

References

Barclay, M. (2020). Vertex Form Exploration. Desmos Activity Builder. https://teacher.desmos.com/activitybuilder/custom/5ce0094311d1e90d8e6cdcce 

Bentel, A. (October 2020). Quadratics Part 1 Learning Module. CGScholar EPOL 481

Donati, R. (n.d). Intro to Quadratic Formula. Desmos Activity Builder. https://teacher.desmos.com/activitybuilder/custom/5eb42c8922f9ab0cb89efc63

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.

Mathematics Standards. (n.d.). Retrieved November 2020, from http://www.corestandards.org/Math

Math Review of Applications of Parabolas. (2014). [Cliff Diver Trajectory] [Photograph]. School Tutoring. https://schooltutoring.com/help/math-review-of-applications-of-parabolas/

NikkiRose. (April 2010). Quadratic Formula Pop Goes the Weasel [Video]. Youtube. https://www.youtube.com/watch?v=2lbABbfU6Zc

Stapel, E. (n.d). Graphing Quadratic Functions: The Leading Coefficient/The Vertex. Purple Math.https://www.purplemath.com/modules/grphquad2.htm#:~:text=Parabolas%20always%20have%20a%20lowest,is%20called%20the%20%22vertex%22.&text=If%20the%20quadratic%20is%20written,point%20(h%2C%20k).

Veitch, M. (n.d). Factoring Practice. Desmos Activity Builder. https://teacher.desmos.com/activitybuilder/custom/5d2d25b179eab562f161c34c

Wayne RESA. (2020). https://docs.google.com/file/d/0B19SejfvMU1rbW1sbEZ3Y1E1Qkk/edit