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Unit 8 Overview

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SAMPLE Unit of Study: Algebra I

Quadratic Functions

Overview

Unit Description

This unit provides a firm foundation for students to work with a new function: the quadratic. The unit

starts with an introductory lesson, making connections back to students’ previous understanding of

functions, which began in grade 8 and gained sophistication earlier in this Algebra I course. It

culminates in the use of quadratic functions to solve problems and model mathematical and real-

world scenarios. Throughout the lessons in this unit, students explore the unique aspects of

quadratics and compare them to other functions (i.e., linear and exponential). Students familiarize

themselves with multiple representations—working with tables, graphs, and equations. Ample time

is spent exploring the standard, factored, and vertex forms of a quadratic function, along with the

key features that these forms allow students to easily identify. Additionally, students investigate

converting between forms, with a focus on completing the square.

Big Ideas

Any function that can be written in the form 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, where 𝑎 ≠ 0, is a quadtratic

function.

Relationships that can be modeled with quadratic functions will share the same characteristics,

such as:

o a maximum or minimum range value (vertex),

o symmetry (tabularly and graphically),

o axis of symmetry (graphically),

o intervals of increase/decrease (dependent on vertex),

o non-constant rate of change,

o and parabolic shape.

Writing and translating a quadratic function into different forms can help identify key features of

the function.

Many real-world situations that involve the squaring of an unknown can be modeled with a

quadratic function.

Unit 8 Overview


Essential Questions

What types of mathematical relationships can be modeled with quadratic functions?

What are the key features of quadratic functions?

Which key features of quadratic functions are revealed when moving between different forms?

What types of real-world phenomena can be modeled with quadratic functions?

Key Standards

The following focus standards are intended to guide teachers to be purposeful and strategic in both what

to include and what to exclude when teaching this unit. Although each unit emphasizes certain standards,

students are exposed to a number of key ideas in each unit, and as with every rich classroom learning

experience, these standards are revisited throughout the course to ensure that students master the

concepts with an ever-increasing level of rigor.

Factor a quadratic expression to reveal the zeros of the function it defines. A-SSE.3.a

Complete the square in a quadratic expression to reveal the maximum or minimum

value of the function it defines.

A-SSE.3.b

Solve quadratic equations by inspection (e.g., for 𝑥2 = 49), taking square roots,

completing the square, the quadratic formula and factoring, as appropriate to the initial

form of the equation. Recognize when the quadratic formula gives complex solutions

and write them as 𝑎 ± 𝑏𝑖 for real numbers 𝑎 and 𝑏.

A-REI.4.b

For a function that models a relationship between two quantities, interpret key features

of graphs and tables in terms of the quantities, and sketch graphs showing key features

given a verbal description of the relationship.

F-IF.4

Graph linear and quadratic functions and show intercepts, maxima, and minima. F-IF.7.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.

F-IF.8.a

Compare properties of two functions each represented in a different way (algebraically,

graphically, numerically in tables, or by verbal descriptions).

F-IF.9

Identify the effect on the graph of replacing 𝑓(𝑥) by 𝑓(𝑥) + 𝑘, 𝑘𝑓(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.

F-BF.3

Unit 8 Overview


Recommended Structures

The Unit Outline included in this document provides a framework for weekly instruction, practice, and

assessment. Each week of instruction includes digital lessons that students will complete independently,

as well as opportunities for whole-group and small-group teacher-led instruction.

The Unit Outline will use the following icons.

Preparation for Weekly Instruction Modifications for Special Populations

Learning Goals

Supporting English Learners

Edgenuity Digital Lessons

Work for Early Finishers

Additional Instructional Support

Developing Higher-Order Thinking

Common Misconceptions

& Reteaching Strategies

Supporting Foundational Math Skills

Social Emotional

Learning Connections

Unit 8 Outline


Week 1 – Standard & Vertex Form Unit 8: Quadratic Functions

Learning Goals This week, students will learn about the quadric function, its key features, and understand standard and factored

form.

Identify and evaluate quadratic functions in tables, graphs, and equations.

(F-IF.4, F-IF.7.a)

Graph a quadratic function given in standard form, identifying the key features of the graph.

(A-SSE.3.a, F-IF.7.a)

Graph a quadratic function given in factored form, identifying the key features of the graph.

(F-IF.4, F-IF.7.a, F-IF.9)

Edgenuity Digital Lessons Introduction to Quadratic Functions

Quadratic Functions: Standard Form

Quadratic Functions: Factored Form

Week at a Glance

Day 1 Students have done extensive work with linear functions in previous units. In the coming weeks they will learn

to use some of the same ideas—intercepts, rate of change, multiple representations of a function, modeling—

with a new type of function, the quadratic. Prepare students for this work by immersing them in comparisons of

applications of these two function types.

Use a linear model of a real-world scenario to review

key features of linear relationships. Use a function

that can be used to predict revenue when selling 10

notebooks at different prices in a school store.

Unit 8 Outline


Discuss how to determine that the function is linear, noting that the increase in revenue is the same for each

additional notebook sold. Make a table and graph of the model with the class to illustrate properties of linear

functions. Ask students to identify specific values of the function at different input values using both the table

and graph.

Ask students, “Do you think this is a good model?” If it does not come up in discussion, present the idea that this

model doesn’t allow for a decrease in sales when the price of a notebook becomes unrealistically high. The model

implies that ten notebooks will be sold even if the price is $1,000 per notebook!

After establishing a possible need for a better model, present students with the following function that takes

changes in sales based on selling price into consideration. Again, it shows that revenue equals the number of

items sold times the selling price.

Show that when x = 0, 10 notebooks are sold for $2 each

resulting in revenue of $20 and that g(1) can be used to

predict that when the price per notebook is $2.25, 9

notebooks are sold resulting in a revenue of $20.25.

Unit 8 Outline


Discuss how this function differs from the first model. While it is beneficial to talk about how this model is based

on the idea that ten notebooks will be sold when the selling price is $2 and that every increase in price of $0.25

leads to one fewer notebook sold, it isn’t necessary to do so at this point in the unit. If students in the class can

accept this detail without getting overwhelmed, talk with the class about the structure of the model. But

remember that the point of this introduction is to get the students to compare and contrast a linear model with

this new quadratic model that they will explore in the unit. You may also wish to show the function in standard

form to further illustrate the differences between linear and quadratic functions. The function can be rewritten

as 𝑔(𝑥) = −0.25𝑥2 + 0.5𝑥 + 20.

Split the class into groups of three to four students, and ask each group to create a table of values and a graph

for the second model. Groups should compare and contrast the two models and make three statements about

what the second model tells the students running the class store. Have the groups share their results. Groups

should note that the rate of change is not constant as it was with the linear model, there is a maximum possible

revenue, and that there are intervals in increasing and decreasing revenue.

Explain that the second function is a quadratic function. Students will be working with functions of this type in

many different forms over the next three weeks.

Unit 8 Outline


Day 2 Students will work independently on the digital lesson: “Introduction to Quadratic Functions.” Monitor students

who are struggling and provide individual attention as needed.

Day 3 Use data to identify students who did not pass the quiz from the previous digital lesson. These students will be

Group A. Students who passed the quiz will be Group B. During the first part of the class period, pull Group A

together for re-teaching while Group B students work on the second digital lesson (“Quadratic Functions:

Standard Form”). For the remaining time, work with students individually or in small groups as needed.

Common Misconceptions & Reteaching Strategies

Students will need three major skills from the first digitial lesson to be successful in the second

digital lesson:

Identifying a, b, and c in a quadratic function given in standard form

Understanding key features of a quadratic function (vertex, axis of symmetry)

Evaluating a quadratic function for a given input value

Reviewing the order of operations early in this unit can go a long way towards helping students evaluate

quadratic functions successfully. Even experienced students can attempt to evaluate a function such as

𝑓(𝑥) = 3𝑥2 + 2𝑥 − 1 for 𝑥 = 4 and carelessly multiply 3 by 4 before squaring. To review the necessity

of squaring before multiplying, present students with a series of related expressions such as (3•4)2 and

3•42 and ask them to simplify each. Note that without the parentheses, the exponent is only applied to

the base directly adjacent to it. In the given function, the term 3𝑥2 means 3•𝑥2, so squaring comes

before multiplication.

Day 4 Developing Higher Order Thinking

Open the class period with a discussion question.

What are the things that make quadratic functions unique?

Unit 8 Outline


How are they different from the other types of functions you’ve worked with before?

Encourage students to sketch graphs, create tables, and give equations—using examples and non-

examples—and evaluate the claims of others during the discussion.

After the discussion have students work on finishing “Quadratic Functions: Standard Form” and moving on to

“Quadratic Functions: Factored Form” for the remainder of the class period. Monitor students who are struggling

and provide individual attention as needed.

Day 5 Some students will need this day to finish the week’s required digital lessons. Other students will be finished.

Refer to the work for early finishers for those that have completed the required lessons.

Modifications for Special Populations

Supporting English Learners Low Proficiency High Proficiency

Front-load needed vocabulary before students begin

the lesson on Day 2. (Vocabulary needs will vary with

each student population, but consider including

quadratic function, linear function, vertex, maximum

range, minimum range, symmetry, axis of symmetry,

non-constant rate of change, mid-point, parabolic

shape [parabola], intervals of increase/decrease,

coefficient, and constant term).

Students will complete a compare/contrast graphic

organizer of quadratic functions vs. other algebraic

functions. This not only allows background knowledge

to be activated, but also allows for students to explore

similarities and differences of the new material being

presented with functions previously studied.

Work for Early Finishers Challenge students to write a quadratic function, if possible, given certain constraints. If an answer isn’t possible,

explain why. If it is possible, is there more than one way to do it?

Write a quadratic function with a y-intercept of 12 and x-intercepts that are 8 units apart.

Unit 8 Outline


Write a quadratic funtion that has no x-intercepts.

Write a quadratic function that has no y-intercept.

Give students time to create similar challenges for others.

Allow early finishers to begin the first digital lesson from Week 2, in which they will investigate the vertex form of

a quadratic function.

Supporting Foundational

Math Skills

For many students already familiar with domain and range, this will be the first time they have thought about a

range that isn’t all real numbers. Students often try to memorize a set of rules based on the location of the vertex

and the value of a without ever looking at what is going on with the graph. Get students thinking past

memorization and towards understanding by sketching graphs when thinking about range. Ask students to

consider what they can determine if they know the location of the vertex and the direction in which it opens.

Are there any limits on what x can be? How do you know?

Are there any limits on what f(x) can be? How does the graph support your answer?

Additionally, for success in this unit, students will need to activate their prior knowledge of exponents, solving

quadratic equations, input and output tables, and using coordinate pairs on the coordinate plane. Provide

students with examples of quadratic equations. Ask students what makes an equation quadratic. A useful visual

aid can be found at https://www.mathsisfun.com/algebra/quadratic-equation.html.

Social Emotional


The group work on day 1 and the discussion question on day 4 both give students an opportunity to use social-

awareness and interpersonal skills to establish and maintain positive relationships with other classmates. Work

with students to create a system of group discussion that values each group member and provides opportunities

to maintain an objective, nonjudgmental tone during disagreements.

For example, on day 4 students may have differing views on a statement such as, “The function g(x) = 2x + 3x2 is

not quadratic,” If this happens, have students decide if they agree, disagree, or are unsure about the statement.

Form small groups that will go through three rounds of discussing the statement.

https://www.mathsisfun.com/algebra/quadratic-equation.html

Unit 8 Outline


Round 1- Each group member shares if they agree, disagree, or are unsure about the statement and

explains why. Students do not comment on the responses of others, and everyone must give a why.

Round 2- Each group member shares whether they now agree, disagree, or are unsure about their own

statement or someone else’s statement and says why. Again, there is no commenting on the responses

given in this round.

Round 3- In this final round, students state whether they agree, disagree, or are unsure about the original

statement. Reasons are not given in this round. Students may change their minds from round 1

This type of open environment for discussion can give students the safety they need to be open and to learn to

trust their peers. For more information on this type of discussion model, visit:

http://cheesemonkeysf.blogspot.com/2014/07/tmc14-gwwg-talking-points-activity.html

http://cheesemonkeysf.blogspot.com/2014/07/tmc14-gwwg-talking-points-activity.html

Unit 8 Outline


Week 2 – Vertex Form & Completing the Square Unit 8: Quadratic Functions

Learning Goals This week, students will work with the vertex form of a quadratic function, as well as explore translating between

standard and vertex form.

Graph a quadratic function given in vertex form, identifying the key features of the graph.

(F-IF.4, F-IF.7.a, F-BF.3)

Write quadratic functions given in standard form in vertex form by completing the square.

(A-SSE.3.b, F-IF.8.a)

Identify and graph transformations of the parent quadratic function.

(F-IF.4, F-IF.7.a)

Edgenuity Digital Lessons Quadratic Functions: Vertex Form

Completing the Square

Completing the Square (Continued)

Week at a Glance

Day 1 Open the class period with a discussion question.

Which key features of quadratic functions are easily identified when the function is written in factored

form? Which key features are easier to identify in standard form?

Encourage students to draw pictures—using examples and non-examples—and evaluate the claims of others

during the discussion. If time and resources are available, create classroom charts based on the information

discussed on this day. Save these charts for Day 3. Then have students work on “Quadratic Functions: Vertex

Form” for the remainder of the class period. Monitor students who are struggling and provide individual

attention as needed.

Unit 8 Outline


Day 2 Begin by allowing students to complete the quiz from “Quadratic Functions: Vertex Form.” In the second half of

class, use data to identify students who did not pass the quiz from the lesson “Quadratic Functions: Vertex Form.”

These students will be Group A. Students who passed the quiz will be group B. During the first part of the class

period, pull Group A together for re-teaching (see suggestions below) while Group B students work on the next

digital lesson (“Completing the Square”).


The vertex form for quadratic functions is often perplexing to students because of the difference in

operations associated with h and k. Students may struggle to identify the signs of the coordinates of the

vertex from this form when they do not take the time to rewrite a given function in the proper form.

Discuss how to rewrite each of the functions below and identify the vertex for each. While it certainly

isn’t necessary to show 𝑓(𝑥) = −(𝑥 + 1)2 as 𝑓(𝑥) = −(𝑥 − (−1))2 + 0, the extra step clearly

illustrates why h = –1 and k = 0.

𝑓(𝑥) = 2(𝑥 − 5)2 − 3

𝑓(𝑥) = −(𝑥 + 1)2 + 4

𝑓(𝑥) = −5(𝑥 + 6)2 − 8

𝑓(𝑥) = 7𝑥2 + 4

𝑓(𝑥) = 2(𝑥 − 1)2

Day 3 Developing Higher Order Thinking

Open the class period with a discussion question around the three different equation forms of a

quadratic function.

Which key features are easily identified when the function is written in vertex form?

What makes each form-standard, factored, and vertex- different? When might you want to use

one form over the other?

Unit 8 Outline


Encourage students to draw pictures—using examples and non-examples—and evaluate the claims of

others during the discussion. Add this information to the charts created on Day 1 and display the charts

for reference in the classroom.

After the discussion have students work on “Completing the Square” for the remainder of the class period.

Monitor students who are struggling and provide individual attention as needed.

Day 4 Students will work independently on the digital lesson “Completing the Square.” If students have completed this

lesson already, have them begin the next lesson: “Completing the Square (Continued).” Work with individual

students independently as needed.


Students often come away from lessons on completing the square with an algorithm to perform but

without a lot of thought about what they are doing and why. Encourage students to begin tasks in these

lessons with the end in mind. Students are translating quadratic functions from standard form to vertex

form to make them more useful in identifying key attributes of the function.

The purpose of completing the square is to figure out what goes in those boxes.

As students work to think about how to complete the square when 𝑎 = 1, remind them of the work

they did with tiles in the lesson. The side length of their square was x plus half of 𝑏, because the x pieces

were evenly divided between the two sides. This is how the number in the blue box is found. It is half

of 𝑏 . Because 8 positive unit tiles had to be added to the model to complete the square, the number –

8 must be used in the orange box to keep the equation balanced.

Unit 8 Outline


Day 5 Some students will need this day to finish the week’s required digital lessons. Other students will be finished.

Refer to the work for early finishers (see suggestions below) for those that have completed the required lessons.



Students will complete a Venn Diagram of quadratic

functions in vertex form and quadratic functions in

standard form. Discuss findings as a class.

Retell the process used to identify the three different

equation forms of a quadratic function using Think

Aloud and Teacher Modeling. See links for think aloud

and teacher modeling. Students will use key, technical

vocabulary.

http://ell/ThinkAlouds.pdf

http://ell/ThinkAlouds.pdf

http://ell/TeacherModeling.pdf

Unit 8 Outline


Work for Early Finishers Have students consider writing a quadratic function given in standard form in vertex form using −𝑏

2𝑎 to find the

coordinates of the vertex.

What additional work must be done when approaching writing functions in vertex form this way?

Are there certain advantages to this method over completing the square? Are there any disadvantages?

Students may wonder why they are learning to complete the square when this method works just as well.

Interested students may try using the method of completing the square to write 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 in vertex form.

Ask students how this work can be used to justify using −𝑏

2𝑎 as the x-coordinate of a quadratic function.

Allow early finishers to begin the first digital lesson from Week 3, in which they will explore modeling with

quadratic functions.


Math Skills

Scaffold the work with graphing using translations by using color and building the number of steps students are

expected to complete. Encourage students to rewrite given functions using different colors to show which parts

of a function indicate vertical translations, horizontal translations, and reflections of the parent function

𝑓(𝑥) = 𝑥2.

Stage 1: Practice with only vertical translations by graphing functions such as 𝑓(𝑥) = 𝑥2 + 3. Use blue to indicate

these translations.

Stage 2: Move to graphing fuctions such as 𝑓(𝑥) = (𝑥 + 1)2 + 3 using vertical and horizontal translations. Use

red to indicate horizontal translations.

Stage 3: Increase the complexity of the quadratic functions graphed by including reflections. Use orange to

indicate reflections as shown below:

Unit 8 Outline


Social Emotional


The process of completing the square is a useful yet sometimes tedious series of steps. It isn’t unreasonable to

think that students may experience frustration as they work towards successfully completing this process.

Managing that frustration when it arises is a part of developing self-management skills to achieve success. Work

with students to identify ways to overcome frustration without invalidating it. Teach students that it is

understandable to be frustrated at times with a task, but that dwelling in the frustration can detract from the

students’ ability to solve problems. Work with a small group or the class to develop a troubleshooting guide for

the unit that students can turn to when they are overwhelmed. Have students identify stress management skills

that work best for them and to share self-calming techniques.

Unit 8 Outline


Week 3 – Modeling with Quadratic Functions Unit 8: Quadratic Functions

Learning Goals Main Task: Students will finish the final lesson in the unit and complete the Unit Review and Unit Test.

Use quadratic functions to model and solve real-world and mathematical problems.

(A-REI.4.b, F-IF.4, F-IF.7.a, F-IF.8.a)

Edgenuity Digital Lessons Modeling with Quadratic Functions

Unit Review

Unit Test

Week at a Glance

Day 1 Students will work independently on the digital lesson “Modeling with Quadratic Functions.” Monitor students

who are struggling and provide individual attention as needed.

Day 2 Use data to identify students who did not pass the quiz from the lesson “Modeling with Quadratic Functions.”

These students will be Group A. Students who passed the quiz will be group B. During the first part of the class

period, pull Group A together for re-teaching (see suggestions below) while Group B students work on the Unit

Review.


Although assessment items are structured in way that doesn’t require students to determine what form

of a quadratic function to use to model a situation, the process of making that choice can help students

organize information as they solve problems. Use the charts created in week 2 to help students review

the key features that are readily identified in the vertex and factored forms of quadratics. Then discuss

Unit 8 Outline


how the given information in an exercise in this week’s lesson can be used to predict which form of a

quadratic will be used in the correct answer.

When the vertex is given, the likely model to use is vertex form.

When both zeroes (or x-intercepts) are given, the likely model to use is factored form.

Often students will have difficulties identifying known information because the way the information is

presented is unfamiliar. Review different ways to talk about key attributes of a quadratic function.

“zeros” means x-intercepts

“maximum value”/”minimum value” indicates the vertex and the direction of the graph, which can

be used as a check of the sign of a once a is calculated.

“initial value” indicates the value of f(0) or the y-intercept.

“after…” or “… years later” are clues to finding additional points on the graph

Day 3 Have students complete the Unit Review activity. Then ask students to work in small groups to make a poster

that teachers others how to graph a quadratic function given in each of three equation forms. Each poster should

list the key features of the quadratic function.

Day 4 Have students share their work from the previous day with one other group. Invite each group to share positive

feedback with the whole class about the other’s group’s work.

Day 5 Have all students take the Unit Test.



Students will review for the exam by completing a sort

of quadratic equations in vertex and standard form.

As a review, students will complete a Word Cloud

activity.

Unit 8 Outline


This activity covers much of the vocabulary associated

with quadratic functions.

Work for Early Finishers If students complete the Unit Test before the entire class is done, encourage them to journal or discuss the

questions below with other students:

What professions would likely need to use quadratic functions to truly understand their field and be

excellent at their work? What is a sample situation that these professions would deal with that would

highlight their need for understanding quadratic functions?

One approach to this work would be to reconsider the application that was presented on the first day of the unit.

The real world example suggested was very basic to the point of being almost unrealistic. Students can spend time

looking for more applicable examples. http://math.slu.edu/~may/ExcelCalculus/sec-2-2-

ModelingRevenueCostProfit.html


Math Skills

One key to success in this week’s work lies in comfort in substituting values for multiple variables and solving for

any remaining unknown in an equation. Give students an opportunity to work with this skill by presenting students

with explicit information on the values of variable in one general form of a quadratic function. The focus of this

work will only be substitution and solving within the context of quadratic functions. Scaffold the work by starting

with whole numbers only and then moving to integers, fractions, and decimals. Then move to a different form of

a quadratic.

𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘

vertex: (ℎ, 𝑘)

Find the value of 𝑎 when…

𝑥 = 5, 𝑓(𝑥) = 13, ℎ = 2, 𝑘 = 4

𝑥 = −2, 𝑓(𝑥) = −20, ℎ = −7, 𝑘 = −30

𝑥 = 1, 𝑓(𝑥) = −1, ℎ =1

2, 𝑘 =

1

4

𝑓(𝑥) = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞)

zeros of the function: 𝑧1, 𝑧2

Find the value of 𝑎 when…

𝑥 = 4, 𝑓(𝑥) = 12, 𝑧1 = 2, 𝑧2 = 3

𝑥 = 0, 𝑓(𝑥) = 14, 𝑧1 = −2, 𝑧2 = 1

𝑥 = 2.5, 𝑓(𝑥) = −4.9, 𝑧1 = 1.8, 𝑧2 = −0.5

http://math.slu.edu/~may/ExcelCalculus/sec-2-2-ModelingRevenueCostProfit.html

http://math.slu.edu/~may/ExcelCalculus/sec-2-2-ModelingRevenueCostProfit.html

Unit 8 Outline


Social Emotional


Encourage students to take time before and after the unit test to reflect on how attitude and preparation can

impact success.

Before the Unit Test- Students can spend time considering how any feelings of anxiety about the unit test could

influence their performance. Give students time to discuss in small groups how they deal with test anxiety.

Encourage students to use the unit review to identify areas of strength and weakness in order to alleviate stress.

Work with students to identify resources they have for improving work on specific objectives. Have students

complete a brief writing task before the unit test in which they list any stress-relieving techniques or content

resources they used to get ready for the test.

After the Unit Test After students have had a chance to review their results on the unit test, return the writing

task to them. Ask students to reflect on what was and was not helpful and to identify ways they can continue

success and improve on the next assessment.

Alternatively, pair students in twos or small groups to discuss how math influences the decisions in their life. Do

they think about math when solving day to day problems or does it just come naturally? Did students realize that

math was a part of decision making? What are drastic changes that could occur in one’s life if problems could not

be solved?

quadratic functions - edgenuity inc.€¦ · this unit provides a firm foundation for students to...

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