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A Story of Functions

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A Story of Functions

A Close Look at Grade 9 Module 4

- Answer the following and discuss your responses with a neighbor:
- Why should students spend so much time studying quadratics?
- Why are quadratics (polynomials of degree 2) called quadratics anyway?
- Can any u-shaped graph be represented by a quadratic function?

A Story of Functions

A Close Look at Grade 9 Module 4

- Classroom teacher
- Math trainer
- Principal or school leader
- District representative / leader
- Other

- Experience and model the instructional approaches to teaching the content of Grade 9 Module 4 lessons.
- Articulate how the lessons promote mastery of the focus standards and how the module addresses the major work of the grade.
- Make connections from the content of previous modules and grade levels to the content of this module.

- Orientation to Materials (if needed)
- A Foundation for the Study of Quadratics
- Examination and exploration of:
- Topic A
- Fluency Exercises – The Rapid White Board Exchange
- Mid-Module Assessment
- Topic B
- Topic C
- End of Module Assessment

- Teacher Materials
- Module Overview
- Topic Overviews
- Daily Lessons
- Assessments
- Student Materials
- Daily Lessons with Problem Sets
- Copy Ready Materials
- Exit Tickets
- Fluency Worksheets / Sprints
- Assessments

- Problem Set
Students and teachers work through examples and complete exercises to develop or reinforce a concept.

- Socratic
Teacher leads students in a conversation to develop a specific concept or proof.

- Exploration
Independent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge.

- Modeling
Students practice all or part of the modeling cycle with real-world or mathematical problems that are ill-defined.

- Teacher Materials Lessons
- Student Outcomes and Lesson Notes (in select lessons)
- Classwork
- General directions and guidance, including timing guidance
- Bulleted discussion points with expected student responses
- Student classwork with solutions (boxed)

- Exit Ticket with Solutions
- Problem Set with Solutions
- Student Materials
- Classwork
- Problem Set

- Orientation to Materials (if needed)
- A Foundation for the Study of Quadratics
- Examination and exploration of:
- Topic A
- Fluency Exercises – The Rapid White Board Exchange
- Mid-Module Assessment
- Topic B
- Topic C
- End of Module Assessment

- What is the next number in the sequence?
- 4, 7, 10, 13, 16, …
- 4, 5, 8, 13, 20, 29, …
- 2, 4, 9, 22, 48, 102, …
- What does the leading diagonal look like for each of the following:
- 1: 1, 1, 1, 1, 1, ….
- n: 1, 2, 3, 4, 5, 6, …
- n2: 1, 4, 9, 16, 25, 36, …
- n3: 1, 8, 27, 64, 125, 216, …
- n2 +n:

- Do heavier objects fall through the air faster than lighter objects of the same shape and size? Consider a real elephant and a life-sized paper Mache model of an elephant.
- What Galileo hoped to do was prove they fall at the same rates and with a constant acceleration.
- Imagine what experiment / data he would need?
- How do we measure speed? Acceleration?
- Based on our work in Part 1, if acceleration is constant, the formula defining the height at time , must be a quadratic.

Tape the ends of the chain so that the lowest part of the chain falls right at the origin.

Identify several other points that the chain goes through.

Create a quadratic equation that goes through the points you identified.

- Orientation to Materials (if needed)
- A Foundation for the Study of Quadratics
- Examination and exploration of:
- Topic A
- Fluency Exercises – The Rapid White Board Exchange
- Mid-Module Assessment
- Topic B
- Topic C
- End of Module Assessment

- Topic A: Quadratic Expressions, Equations, Functions, and their Connection to Rectangles
- Reversing multiplication yields factored expressions (recall geometric models); practice factoring of quadratics.
- When combined with the Zero Product Property we have a new power to solve quadratic equations.
- What does the graph of a quadratic equation look like? It’s symmetric (and u-shaped).
- Factored form + symmetry makes graphing simple.
- Relating quadratic equations and their graphs to real-world context, giving contextual interpretations of key features.

- Topic B: Using Different Forms for Quadratic Functions
- Other ways to see structure in quadratics –solving by completing the square; the quadratic formula.
- Why does completing the square yield something we call “vertex form”? The relationship between vertex form and transformations; the helpfulness of vertex form in graphing.
- Further examination of quadratic functions and their graphs in context.

- Topic C: Function Transformations and Modeling
- The square root function and its relationship to the basic quadratic function; the cube root and cubic functions.
- Transformations of all of these types of functions.
- Analyzing and comparing functions represented in different forms, all done in context.

- Experience multiplying with polynomials using the distributive property (G9-M1)
- Experience relating the distributive property to an area model or an modified area model (the tabular method) (G9-M1)
- Experience writing a sum as a product of two factors (G7-M3) and factoring out a greatest common factor (G6-M2)
- Experience transforming graphs, transforming functions and relating the transformed function to the transformed graph (G9-M3)

- Opening Exercise
- Example 1
- Extension: Is there another option? How many possible answers are there?

- Prime numbers can be related to ‘counting by’ instead of factors
- (Before presenting the given description) How can we describe what we mean by a factor being prime? How could I describe what it means when you can’t factor it any more than you already have?
- Write a simple binomial. Now write that binomial as a product of two other polynomials. (Remember, even a simple integer is a polynomial)

- Why are they called quadratics anyway?
- Note the scaffold box at the top of page 31
- Exercises 7-8

- Lesson 3 Opening Exercise
- Continue to use the tabular model as needed.
- Encourage students to verbalize their process of finding factors that work.
- Lesson 4 Problem Set #3, an example of MP.1

- Lesson 5 Opening Exercise, Exercises 1-4 lead students to know and apply the zero product property.
- Example 1 provides context for its application.
- Reasoning through a problem is still a valid approach. Factoring is only one means to the end.
- Lesson 7 calls upon students to build their own equations from context. (Work through exercises 5-7.)

- Scaffold: If needed, begin this lesson with an opportunity to graph a selection of relatively simple quadratic functions, allowing students to work in pairs on a problem of appropriate complexity.
- What do you notice about these graphs? Allow students to notice the symmetry, and begin with an informal description of the vertex.
- This lesson brings up the question, are all u-shaped curves represented by quadratics.
- Work the Extension question after Exploratory Challenge 2.

- Lesson 9 Opening Exercise
- Lesson 9 Example 2. Pose the questions to students, how on earth did they come up with this formula.
- Scaffold in formal terms by using contextual everyday language and then repeating with more formal words.
- Lesson 10 Example 1. Ask the students to think critically about the reasonableness of this graph for this situation.
- Lesson 10 Example 2. Spend ample time challenging the students with the ‘How do you know’ question.

- Consider having students come up with their own summaries for how they approach factoring /solving/ graphing a quadratic.
- It’s better to study deeply a given application problem and the analysis of its graph’s features than to do multiple problems.
- Introduce concepts like domain, range, increasing, decreasing, average rate of change, etc. by using words that feel natural in the context, and then repeat the statement or question using the more formal words.
- Scaffolds are a critical tool for successful implementation. In addition to those given in the module, consider the ones we explored in this session. (Take time now to reflect and take note of them.)

- Orientation to Materials (if needed)
- A Foundation for the Study of Quadratics
- Examination and exploration of:
- Topic A
- Fluency Exercises – The Rapid White Board Exchange
- Mid-Module Assessment
- Topic B
- Topic C
- End of Module Assessment

Factoring trinomials

- Orientation to Materials (if needed)
- A Foundation for the Study of Quadratics
- Examination and exploration of:
- Topic A
- Fluency Exercises – The Rapid White Board Exchange
- Mid-Module Assessment
- Topic B
- Topic C
- End of Module Assessment

Work with a partner on this assessment

- Orientation to Materials (if needed)
- A Foundation for the Study of Quadratics
- Examination and exploration of:
- Topic A
- Fluency Exercises – The Rapid White Board Exchange
- Mid-Module Assessment
- Topic B
- Topic C
- End of Module Assessment

- A valuable scaffold even with the opening exercise is to use a geometric model of a square. Knowing that that you are attempting to factor it such that you are creating a perfect square develops students capacity to do so.
- Example 1.
- Alternative for opening Lesson 11:
- Solve by inspection: , , ,
- Discourage use of the sign. Instead model less abstract ‘or’ that emulates our thinking. “If something squared is 9, then either that something equals 3 or that something equals -3. “ This scaffold helps students follow their own thinking all the way through to a final answer.

- What strategies can we offer up for completing the square of
- Try Lesson 12 Examples 1 and 2.
- An optional scaffold again relies on the context of solving quadratic equations.
- Using this scaffold means students won’t have the benefit of being able to use their completing the square skill to get into vertex form when working with a quadratic function. (However, there are ways to get into vertex form.)
- Try Lesson 13 Exercises 1-4

- Deriving the quadratic formula
- Algebraic approach.
- Using geometric square model scaffold.

- Lesson 15: Exercises 1-5 & Discussion: Students are asked to reflect on the quadratic formula to generalize about how many real solutions a quadratic equation will have; they then relate their findings to features of graphs.
- Lesson 16: Starting with simple horizontal and vertical translations students explore the graph of the function and transformations thereof.
- Students discover this ‘vertex form’ makes identifying the vertex a simple task.
- Ask students to summarize, challenging their capacity to articulate the somewhat counter-intuitive nature of horizontal translations.
- Note the scaffold at the bottom of page 172.

- Lesson 17:
- Work the Opening Exercise, then challenge students to develop their own ‘general strategy’ for graphing a quadratic function before reviewing what is provided before Example 1.
- Example 1 provides another opportunity to ask, ‘How do you suppose the math class was able to determine this formula?’
- It is not explicitly asked or stated, but is suggested, ‘How can I put a function into vertex form?’
- Have students to come up with their general approach to graphing on their own before considering the approach provided.

- Completing the square has a geometric meaning.
- A scaffold for completing the square when the leading coefficient is not 1 involves multiplying the equation through first by the leading coefficient (if not already a perfect square) and then by the factor , if the coefficient of the term is not easily halved.
- This same scaffold used with the geometric model provides an alternative to the purely algebraic derivation of the quadratic formula.
- The final lesson should include a reflection on the student’s general strategy for graphing quadratic functions.
- Lessons 16-21 in Topics B and C provide a second opportunity for students to master transformations of functions.

- Orientation to Materials (if needed)
- A Foundation for the Study of Quadratics
- Examination and exploration of:
- Topic A
- Fluency Exercises – The Rapid White Board Exchange
- Mid-Module Assessment
- Topic B
- Topic C
- End of Module Assessment

- Exercise 1
- Exercise 2
- Exercise 3
- Suggestion: Don’t give away the relationship between the graphs of these inverse functions. Ask the question, then spend ample time letting students contemplate and articulate to the best of their ability what they notice.

- Make use of technology to demonstrate and apply previous understanding of transformations of functions.
- Completing the square when working with a function or an equation in two variables.

- Lesson 22, Exercises 1-3
- Lesson 23, the mathematics of objects in motion.
- All free-falling objects on Earth accelerate toward the center of the earth (downward) at a constant rate (rate of acceleration, not rate of speed).
or

- For this reason, the leading coefficient for a quadratic function that models the position of a falling, launched, or projected object must be or .
- Reflection: note the phrase, “without a power source”. Were the dolphins in Lesson X without a power source?

- All free-falling objects on Earth accelerate toward the center of the earth (downward) at a constant rate (rate of acceleration, not rate of speed).
- Lesson 23 Example 1
- Lesson 23 Example 2
- Lesson 24 Opening Exercise

- Comparing features of functions provided in different forms deepens and consolidates student understanding of the relationship between the structure of expressions and equations, the graphs of equations and functions, and the contexts they model.
- Students should walk away from quadratics understanding that a primary use of these functions is in modeling height over time of projectile objects, that they are naturally related to rectangular area problems, and that there are also used in an early study of business applications.

- Answer the following and discuss your responses with a neighbor:
- Why should students spend so much time studying quadratics?
- Why are quadratics (polynomials of degree 2) called quadratics anyway?
- Can any u-shaped graph be represented by a quadratic function?

- Students are called upon to Look for and make use of structure (MP.7) as they choose equivalent forms of quadratics to gain insight into the function’s behavior and its graph.
- Students are called upon to reason abstractly and quantitatively (MP.2) as they decontextualize and work with quadratic equations representing real-world contexts and then re-contextualize as they analyze and interpret the key features of the function and its graph in the context of the problem.
- Note that the physics contexts have the same coefficients due to the mathematics of objects in motion.

- Orientation to Materials (if needed)
- A Foundation for the Study of Quadratics
- Examination and exploration of:
- Topic A
- Fluency Exercises – The Rapid White Board Exchange
- Mid-Module Assessment
- Topic B
- Topic C
- End of Module Assessment

Work with a partner on this assessment

- End of Module assessment are designed to assess all standards of the module (at least at the cluster level) with an emphasis on assessing thoroughly those presented in the second half of the module.
- Recall, as much as possible, assessment items are designed to asses the standards while emulating PARCC Type 2 and Type 3 tasks.
- Recall, rubrics are designed to inform each district / school / teacher as they make decisions about the use of assessments in the assignment of grades.

What are your biggest takeaways from the study of Module 4?

How can you support successful implementation of these materials at your schools given your role as a teacher, trainer, school or district leader, administrator or other representative?