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Quadratics

Learning Module

Target Audience and Expected Outcomes

Target Audience:

This learning module is intended for middle or high school students in an Algebra course. They should have background knowledge in solving linear equations, graphing lines and some knowledge of factoring.

By the end of this learning module, students will be able to graph quadratic equations from vertex form, standard form, and factored form. Students will also be able to complete the square to convert between standard and vertex forms.

As a culminating project, students will create a graphing quiz for a peer.

Common Core

  • CCSS.MATH.CONTENT.HSA.SSE.B.3.A: Factor a quadratic expression to reveal the zeros of the function it defines.  
  • CCSS.MATH.CONTENT.HSA.SSE.B.3.B: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines
  • CCSS.MATH.CONTENT.HSA.REI.B.4.A: Use the method of completing the square to transform any quadratic equation in xinto an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.
  • CCSS.MATH.CONTENT.HSF.IF.C.7.A: Graph linear and quadratic functions and show intercepts, maxima, and minima.
  • CCSS.MATH.CONTENT.HSF.IF.C.8.A: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
  • CCSS.MATH.CONTENT.HSF.IF.C.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Mathematical Practices

  • CCSS.MATH.PRACTICE.MP1: Make sense of problems and persevere in solving them.
  • CCSS.MATH.PRACTICE.MP2: Reason abstractly and quantitatively.
  • CCSS.MATH.PRACTICE.MP3: Construct viable arguments and critique the reasoning of others
  • CCSS.MATH.PRACTICE.MP4: Model with mathematics.
  • CCSS.MATH.PRACTICE.MP5: Use appropriate tools strategically.
  • CCSS.MATH.PRACTICE.MP6: Attend to precision.
  • CCSS.MATH.PRACTICE.MP7: Look for and make use of structure.
  • CCSS.MATH.PRACTICE.MP8: Look for and express regularity in repeated reasoning.

Update 1: Pre-test/prior knowledge

For the Student

Welcome to our first update! 

Please complete the survey that was sent out.

After each update, you will be requested to make a comment and/or write an update to be shared with the learning community. You can respond to your classmates' questions by typing @ followed by their name.

Make a comment: Introduce yourself by sharing your name and 2 fun facts about yourself. Also include 2 things that you wonder about how to use Scholar or what we will be doing.

For the Teacher

This survey will give a baseline picture of each student's understanding of the vocabulary used in this unit.

Update 2: Solving Quadratics

For the Student

This should be review. Use any resources you like to refresh your memory on solving quadratics using the following three methods:

  1. Taking square roots: Practice
  2. Factoring: Practice
  3. Using the Quadratic Formula: Practice

Use Khan Academy to practice. You should keep trying until you can get 4 problems in a row correct.

Here is a preview of some of the vocabulary we will be using for the rest of this module.

Make a comment: What is your favorite method of solving quadratic equations? Why do you like it better than the others?

For the Teacher

As a follow-up to the survey, this should be a week of reviewing background knowledge of solving quadratics before the graphing and conceptualization/application starts.

Update 3: Will it Hit the Hoop?

For the Student

Go to http://student.desmos.com and create an account or sign in with Google. Then type in the course code provided by the teacher (example: hpxbb). Follow the instructions on each page. When you finish, come back here and follow the directions at the end of the post:

Make a Comment: Was it easier to predict or analyze? Why? Comment below with your strategies for making the best predictions and for creating the best parabolas when you were analyzing. What made the extension questions more difficult than the others?

Write an Update: Post an update to our community including a description of another scenario where lines of best fit would not be successful in making predictions. The scenario does not need to necessarily have the same pattern of movement as the basketball one, but your update should contain a description (and potentially a picture or video) supporting your claim that something more than \(y=mx+b\) is required.

For the Teacher

You will need to create a free account at teacher.desmos.com and create a class code for the "Will it Hit the Hoop?" Activity. Share that code with the students in the update.

Students will complete the assignment in Desmos Activity Builder. The goal is to develop the need for parabolic curves of best fit in order to make predictions.

Here is a teacher's guide: 

Will_20It_20Hit_20the_20Hoop_3F_20_E2_80_A2_20Teacher_20Guide.pdf

You can pause the class at any time and have them all look at the same screen. This would be best done in a synchronous setting.

In an asynchronous setting, students can login later and see their classmates' responses and you can moderate some of the discussion to target specific ideas of how closely their predictions matched what actually happened, etc.

For each Desmos activity, you can see individual student screens and responses, as well as a summary of responses. The user interface is fairly intuitive, and all student work is saved in the teacher screen.

Update 4: Graphing Practice

For the Student

Watch the following Khan Academy videos on graphing quadratic equations. While you watch, work through the examples on paper. When finished, you should be able to define the following vocabulary terms:

  • axis of symmetry
  • zeros
  • x-intercepts
  • roots
  • vertex
  • parabola
  • factored form
  • vertex form
Media embedded October 1, 2017
Media embedded October 1, 2017
Media embedded October 1, 2017
Media embedded October 1, 2017

 After you have worked through the examples in those videos, practice your graphing skills using the following Desmos Activity: Match My Parabola (Go to http://student.desmos.com, sign in with your account, and use the code: QSCPD).

Make a comment: When you are finished with the Match My Parabola activity, write a comment below with either a question you have about graphing quadratics or a statement about which form of equation you find is the hardest to graph (and why). Please answer each other's questions if you know the answer!

For the Teacher

The students will watch the Khan Academy videos highlighting axis of symmetry, zeros/x-intercepts, vertex, and parabola vocabulary. Students may struggle with assumed prior knowledge of factoring, function notation, or substitution. If that happens, direct students to review these skills using alternate Khan Academy videos: factoring, function notation

For the Match My Parabola activity, go to http://teacher.desmos.com and head to Match My Parabola. You should create a new course code and edit the "for the student" page with the new code so you can see their progress. 

Here is the teacher guide from Desmos: 

Match_20My_20Parabola_20_E2_80_A2_20Teacher_20Guide.pdf

Update 5: Domain restrictions; Marbleslides

For the Student

Now that you have had some practice with graphing quadratic equations, it's time for a challenge. Your goal will be to collect all the stars on each challenge page. Pay careful attention to which numbers you are changing and how the changes affect the graph of the parabola.

Go to http://student.desmos.com, login and use class code GZMGW

There are different types of screens:

FIx it:

On this screen you should follow the directions carefully and pay attention to what happens when you change a number or a sign. Make sure to reset or change the number back before you change a different one!

Predict/verify:

On these screens you will be asked to write a hypothesis about what you think will happen. You will then have a chance to test your prediction on the verify screen.

Challenge:

On these screens you can create your own parabolas, with or without domain restrictions to collect all the stars. Bragging rights awarded to students who can collect all the stars with one parabola per challenge!

Here's an example of how to add domain restrictions (the {x<5} at the end means Desmos will only graph the function for values of x that are less than 5):

Final Challenge and Update:

On screen #23, at the end of the challenge, you will see a blank screen where you can create your own Marbleslide Challenge.

Post a comment: Starting with the quadratic equation \(y=x^2\), in the comment below, make one change to the equation and describe how that change would affect the graph. (ex. Changingthe equation to \(y=(x-2)^2\) would move the vertex of the parabola 2 units to the right).

Make an Update: On screen #23, create a marbleslide with two or more parabolas, with some domain restrictions (like the ones used in the Fix It and Challenge pages). When you are finished creating, take a screen shot (instructions for Mac, PC, Chromebook, iPad) of your marbleslide. Include that screenshot, along with a list of ordered pairs of where you would place the stars to be collected in this marbleslide in an update.

Also, comment on at least two of your classmates' updates with what you notice or wonder about how they created their marbleslide.

For the Teacher

Students will continue practicing their graphing skills using Desmos Marbleslides: Parabolas. The goal will be to catch all the stars using the fewest parabolas for each page. Again, you may want to navigate to the teacher dashboard with your own account and create a new class code to see their progress.

At the end, students will create their own marbleslide using two or more quadratic functions and some domain restrictions. They will create an update with a screenshot of their marbleslide, a list of ordered pairs where they would place the stars, and a description of their thought process.

Look for creative or unusual graphs in the updates as ones that go above and beyond the requirements. Highlight them in a synchronous session or link to them in the comments of the student update.

Update 6: Completing the Square

For the Student

To solve a quadratic equation is to find the values of the variable that make an equation true. For quadratic equations, finding solutions is synonomous with finding the x-intercepts, roots, or zeros of the graph of the quadratic function.

One method of solving is to factor. This is something you have done before. Another method is called completing the square, and in addition to being a useful strategy for solving quadratics, it is also a convenient method of converting a quadratic equation from standard form (\(y=ax^2+bx+c\)) to vertex form (\(y=a(x-h)^2+k\)) which makes it easier to graph the function.

Watch the following videos on a quadratics solving strategy called Completing the Square. While watching, work through the examples along with the instructor and make note of any questions you have along the way.

Media embedded October 1, 2017
Media embedded October 1, 2017
Media embedded October 1, 2017

 Complete the odd problems on this .pdf as practice:

Vertex_20Form_20of_20Parabolas.pdf

Make a comment: Comment on this post with a question about completing the square or solving quadratic equations.

Write an update: Research an application of quadratic equations that you would like to explore more. We have already seen an application involving gravity and falling objects. What other applications can you find? 

Comment on at least two of your classmates' updates.

For the Teacher

For a more thorough understanding of the graphs of quadratic equations in standard form, completing the square is introduced to convert between standard and vertex forms. Here are the solutions to the .pdf.

Vertex_20Form_20of_20Parabolas_20with_20answers.pdf

 As you read student updates, consider whether they are actually applications of quadratic functions and compile a list of options for further exploration.

Update 7: Project

For the Student

It is now time to create a quiz on Graphing Quadratics for someone who has gone through this learning module. Pay close attention to the rubric as you are creating. You will be asked to give feedback on a peer's quiz and thenr revise your own.

For the Teacher

Update 8: Wrapping it up

For the Student

For the Teacher