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How can we use polynomials to solve quadratic functions?

Common Core Standards 2017-2018 Print Unit 7
High School - Math - Algebra I - Unit 7
Quadratic Functions
*How to evaluate and simplify square roots *Solve quadratic equations * Add, subtract, and multiply polynomials *Factor and solve polynomials
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Students will use square root concepts to solve quadratic functions, graph quadratic functions, and use the quadratic formula to model x-intercepts and real-world situations. Students will also identify, compute and factor polynomial expressions.

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-Operations of Polynomials (linear and quadratic) (A-APR.1, A.SSE.1, A.SSE.2)
-Factor polynomials – GCF, trinomials (A.SSE.2, A.SSE.3a, b)
-Graph quadratic functions by using intercepts and maximum/minimum values. (F.IF.7a, F.IF.8a)
-Write quadratic functions modeling relationships between two quantities. (F.BF.1)
-Solve quadratic equations by factoring (A.SSE.3a,b, A.REI.4 F.IF.8a)
-Solve quadratic square root method (A.REI.4, A.SSE.3b)
–Solve quadratic equations by completing the square (A.SSE.3b, A.REI.4, F.IF.8a)
-Solve quadratic equations by quadratic formula (A.REI.4, A.SSE.3b)
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I can identify that the sum, difference, or product of two polynomials will always be a polynomial, which means that polynomials are always closed under the operations of addition, subtraction, and multiplication.
I can define “closure.”
I can add, subtract, and multiply polynomials.
A.SSE.1 a
I can define and recognize parts of an expression, such as terms, factors, and coefficients.
I can interpret parts of an expression, such as terms, factors and coefficients.
A.SSE1 b
I can define and recognize parts of an expression, such as terms, factors, and coefficients.
I can interpret complicated expressions, by viewing one or more of their parts as a single item.such as terms, factors, and coefficients.
I can identify ways to rewrite expressions such as difference of squares, factoring a GCF, and regrouping.
I can identify various structures of expressions.
I can use the structure of an expression to identify ways to rewrite it.
I can classify expressions by structure and develop strategies to assist in classification.
I can apply models for factoring and multiplying polynomials to rewrite expressions.
I can factor a quadratic expression.
I can use the zero product property to solve a quadratic equation.
I can interpret values in a quadratic expression.
I can predict whether a quadratic function will have a minimum or a maximum based on the value of a.
I can interpret a quadratic expression.
I can graph quadratic functions and show/label intercepts, maxima, and minima of the graph.
I can identify different forms of a quadratic expression.
I can write functions in equivalent forms using the process of factoring.
I can identify zeros, extreme values, and symmetry of the graph of a quadratic function.
I can identify the relationship between quantities in a real-world problem.
I can apply algebraic transformations of parent functions.
I can write a function that describes a real-world problem.
I can combine two functions using the operations of addition, subtraction, multiplication and division.
I can evaluate the domain of the combined function.
I can build standard functions to represent relevant relationships/quantities in real-world or mathematical situations.
I can determine which operations should be performed to build the appropriate combined function.
I can relate the combined function to the context of a real-world or mathematical problem.
I can factor a quadratic expression.
I can use the zero product property to solve a quadratic equation.
I can interpret values in a quadratic expression.
I can solve quadratic equations using different methods.
I can complete the square on a quadratic expression to produce an equivalent form of an expression.
I can interpret a quadratic expression.
I can solve quadratic equations using different methods.
I can complete the square and transform any quadratic equation in x to its factored form.
I can solve quadratic equations in one variable.
I can derive the quadratic formula by completing the square.
I can interpret different but equivalent forms of a function defined by an expression in terms of a context.
I can 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.
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Square root, perfect square, radical expression, quadratic equation in standard form, leading coefficient, simplest form of a radical expression, parabola, vertex, axis of symmetry, roots, quadratic formula, discriminant, monomial, binomial, trinomial, foil pattern, factor of quadratic expression, prime factor, complete factor, degree

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Educator Uploaded Plans (These are educators specific templates with included information and specifics)
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Common core standards, textbook, geometry sketchpad, zoom math, TI-84 graphing calculators,
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Thinking Strategies for Readers
Researchers who have studied the thinking processes of proficient readers conclude that if teachers taught the following strategies instead of much of the traditional skills-based reading curriculum, students who use the strategies would be better equipped to deal with a variety of texts independently (Keene and Zimmerman, 1997). These strategies are use­ful for composing meaning at both a text and word level.
Monitoring for Meaning
at a text level, readers . . .
pause to reflect on their growing understandings
recognize when they understand the text, and when they don’t
identify when and why the meaning of the text is unclear
identify the ways in which a text becomes gradually more understandable by reading past an unclear portion and by rereading text
decide if clarifying a particular confusion is critical to overall understanding
explore a variety of means to remedy confusion
consider, and sometimes adjust, their purpose for reading
check, evaluate and make revisions to their evolving interpretation(s) of text
at a word level, readers . . .
identify confusing words
employ a range of options for reestablishing meaningful reading (e.g., rereading, reading on, using words around the unknown word, using letters and sounds, using a meaningful substitution)
Activating, Utilizing and Building Background Knowledge (Schema)
at a text level, readers . . .
activate relevant, prior knowledge before, during and after reading
build knowledge by deliberately assimilating new learning with their related prior knowledge
clarify new learning by deleting inaccurate schema
relate texts to their world knowledge, to other texts and to their personal experiences
activate their knowledge of authors, genre, and text structure to enhance understanding
recognize when prior knowledge is inadequate and take steps to build knowledge necessary to understand
at a word level, readers . . .
apply what they know about sounds-letter relationships and word parts to make sense of unknown words
Asking Questions
at a text-level, readers . . .
generate questions before, during and after reading about the text’s content, structure and language
ask questions for different purposes, including clarifying their own developing understandings, making predictions, and wondering about the choices the author made when composing
realize that one question may lead to others
pursue answers to questions
consider rhetorical questions inspired by the text
distinguish between questions that lead to essential/deeper understandings and “just curious” types of questions
allow self-generated questions to propel them through text
contemplate questions posed by others as inspiration for new questions
at a word level, readers . . .
pose self-monitoring questions to help them understand unknown/unfamiliar words (e.g., “What would make good sense?”, “What would sound like language?”, “What would sound right and match the letters?”, “Is this a word I want to use as a writer? If so, how am I going to remember it?”)
Drawing Inferences
at a text level, readers . . .
draw conclusions about their reading by connecting the text with their schema
make, confirm, and/or revise reasonable predictions
know when and how to infer answers to unanswered questions
form unique interpretations to deepen and personalize reading experiences
extend their comprehension beyond literal understandings of the printed page
make judgments and create generalizations about what they read
create a sense of expectation as they read
at the word level, readers . . .
use context clues and their knowledge of language to predict the pronunciation and meaning of unknown/unfamiliar words
Determining Importance
at a text level, readers . . .
identify key ideas, themes and elements as they read
distinguish between important and unimportant information using their own purpose(s), as well as the text structures and word cues the author provides
use text structures and text features to help decide what is essential and what is extraneous
use their knowledge of important and relevant parts of text to prioritize what they commit to long-term memory and what they retell and/or summarize for others
consider the author’s bias/point of view
use the filter of essential/other to clarify usefulness when applying other cognitive strategies to their reading
at a word level, readers . . .
determine which words are essential to the meaning of the text
know when choosing to skip words/phrases of text will or will not impact their overall un­derstanding
make decisions about when unknown/unclear words need clarification immediately and accurately, and when substitutions can be used to maintain meaning and fluency
Creating Sensory Images
at a text level, readers . . .
immerse themselves in rich detail as they read
create images connected to the senses of sight, hearing, taste, touch and smell to enhance and personalize understandings
attend to “heart” images – feelings evoked while reading
revise their images to incorporate new information and new ideas revealed in the text
adapt their images in response to the images shared by other readers
at a word level, readers . . .
use visual, auditory and kinesthetic modes when learning how words work
use what they know about a word’s appearance (e.g., length, spacing above and below the line) to understand unknown words
ask themselves “Does that look right?” and “Does that sound right?” whencross-checking unknown words
Synthesizing Information
at a text level, readers . . .
continually monitor overall meaning, important concepts and themes while reading
recognize ways in which text elements fit together to create larger meaning
create new and personal meaning
develop holistic and/or thematic statements which encapsulate the overall meaning of the text
capitalize on opportunities to share, recommend and criticize books
attend to the evolution of their thoughts across time while reading a text, and while reading many texts
at a word level, readers . . .
select specific vocabulary from the text(s) to include in their synthesis because they know that specific language is highly meaning-laden
know when certain vocabulary is critical to the text’s overall meaning, and therefore, must be understood if comprehension is to be achieved
Problem Solving
at a text level, readers . . .
know that once meaning has broken down, that any of the other cognitive behaviors can be employed to repair understanding
use information from the three deep surface structure systems to repair text meaning
at a word level, readers . . .
use information from the three surface structure systems to solve word issues
select from a wide range of word strategies (e.g., skip and read on, reread, use context clues, use the letters and sounds, speak to a peer reader) to help make sense of unknown words
develop reading fluency
Print Version
Thoughtful Strategies by Learning Style
Self – Expressive
Utility (Can be used in multiple styles)
Fact or Fiction
Spider/Fist List
Word Association
Word Wall
Reading for Meaning
Interactive Lecture
Group & Labeling
Memory Box
Write to Learn
Building Writing
Reciprocal Learning
Give one, Get one
Collaborative Summarizing
Anticipation Guides
Concept Attainment
Yes, No, Why
Graduated Difficulty
Comprehension Menu
Task Rotation
Voc Notebook
Carousel Brainstorming
Reader’s Theatre
Vocabulary Code
4-2-1 Free Write
The biggies…. The following strategies require a little more planning to use. They are all very effective. Your school has folders and materials that specifically explain these strategies.
Word Works-Cracking Vocabulary’s CODE – 4 phases of vocabulary learning
  • Connect – new words to prior knowledge
  • Organize – new words to find relationships
  • Deep-processing – internalizing the words, deepen the understanding
  • Exercise – or rehearse their knowledge of new words
Reading for Meaning – strategy that helps students become proficient at making claims, finding main ideas, and using reasoning and details to support their ideas.
· Students are presented with a series of statements before they read the text, they need to either agree/disagree with the statement.
· After reviewing the statements, the students read the text and collect evidence either for or against the statements.
Task Rotation – creating activities that fit students’ learning styles:
· Mastery
· Interpersonal
· Understanding
· Self-expressive
Interactive Lecture – strategy that increases the students’ ability to remember, comprehend, and think actively about a lectures’ content. It engages the students by moving them through the four phases:
  • Connect – hook students’ attention
  • Organize – use graphic organizer/note taking procedure to help organize info
  • Deep-processing – pause every 5-7 min during lecture to allow students time to process information with questions in the different learning styles
  • Exercise – use review questions and have students use notes in a synthesis or application task.
These activities or tools are easy to slip in anywhere within your unit plan. They can be used for the CODE strategies, class openers, to brainstorm, to review, or as energizers for those “glazed over” moments. They are categorized somewhat, but several of these activities can be used in more than one category. Many of the strategies are referenced (in parenthesis) if you want more information on each of these activities.
Class Openers
Fact or Fiction/ Three’s a Crowd – Students decide which word/fact of three doesn’t belong and explain why. (Tool book p.10)
Anticipation Guides – Teacher prepares 3-5 statements based on the content that the students will be reading. Students are asked to decide which statements they believe the text will support. Teacher develops a class tally for each statement and discusses opinions. Students then read text. (Tool book p. 40)
Give one, Get one- students generate ideas from a question posed by the teacher, then have to collect a predetermined number of ideas from their classmates. (Tool book, p.11)
KWL- Tool to assess students’ prior knowledge, help generate questions about what they want to learn, and encourage reflection about what they have learned. (Tool book, p.28)
Spider List/Fist list/Fishbone- teacher provides a category in palm of hand/belly of spider and the students brainstorm ideas to fill in the fingers/legs/bones of hand print, spider, or fish sketch or vice versa.
Word Association/3-way tie- students select 3 words from a unit vocabulary and arrange them in a triangle. They then connect the words with lines then write a sentence that describes the relationship between the words that are connected. (binder)
Content Teaching
Word Wall – Collection of words on the wall for students to use during their reading and writing (Binder)
Reciprocal Learning/Peer Practice Strategy- students work in pairs (player and coach) to review or read & summarize concepts.
Think/pair/share-teacher poses a question, the students think and construct a response, then share their ideas with a neighbor, teacher records/collects their ideas. (Tool book, p.10)
Vocabulary Notebook- Where students collect critical vocabulary In the notebook students can write their initial “educated” definitions, then they can write the dictionary definition, and maybe a visual image as well. There is a lot of variations to this one. (Tool book, p. 92; binder)
Group & Labeling – students examine a list of vocabulary words and place them into groups based on common characteristics. For each group that students create, they devise a label that describes
Etch-a-sketch – students draw pictures, symbols, or icons to represent ideas presented in a lecture, reading, or other form of presentation (Tool book, p. 60)
Collaborative Summarizing – After lecture or reading, the students are asked to identify the 3-6 most important ideas. Students then pair up and compare their lists and come up with a consensus of the most important ideas with their partner. They (the group of two) pair up with another group of two and compare lists and once again come with an agreed upon list of 3-6 important ideas. These four use their list to create a collaborative summary. (Tool book, p.78)
Jigsaw- students work cooperatively with each student having an assigned task within the group to accomplish/perform.
Carousel Brainstorming – teacher generates different styles of questions & posts them around the room. The students work in groups of 3-5, rotates around the room to reading the question, the other responses, and either expands on existing ideas or develops a new idea. (Tool book, p.19)
4-2-1 Free Write – students identify 4 important ideas previously presented in the lesson. Each student meets with another student to compare ideas and decide on the two most important from their lists of four. This pair meets with another pair. They compare their ideas, then come to a consensus on the most important idea. The students then take this and do a free write, explaining all they know about the big idea. (Tool book, p. 82)
Concept Attainment – teacher presents examples and non-examples of a concept in a class discussion, the students use these to brainstorm the key attributes/characteristics of the concept.
Compare/Contrast – comparing likenesses and differences. The Georgia website I sent you has several different variations on this that are interesting.
Kindling – F - Find a question that can be explored
I - Internalize the question
R - Record your thoughts (sketch, write…)
E - Exchange ideas with a neighbor
S - Select and record your ideas in public (Tool Book p.74)
Comprehension Menu – an abbreviated version of Task Rotation. Teacher creates at least four questions in the four learning styles about the content. (Tool book, 162)
Review Activities/Tools
1,2,3,4 – teacher stops 5 minutes before end of class period to allow the students to reflect upon what was presented by writing in this format:
1 – What was the big idea
2 – Important details discussed
3 – Personal connections discovered
4 – Questions students have about the content
Boggle – students review notes for 2 minutes, then list as many ideas or details they can remember for 2-5 minutes, then students share their ideas with 1 or 2 other students and can add to their lists. Lastly, students pair up and compete with another student using the Boggle technique (They earn a point for every idea that their Boggle partner doesn’t have). Then the students go back to their study teams and compute the team score. (Tool book p.134)
Memory Box – a box usually put on a test where students can take the first five minutes to list as many things they can remember(formulas, definitions, etc…)
TGT – (Teams Game Tournament) teacher creates vocabulary/fact cards. The teacher divides the students heterogeneously by academic ability. This is the study group. After they have studied for a while, they then move to homogenous groups established by the teacher and compete against each other. They follow the points system to see how many points they take back to their study teams. This is a great review activity, it takes some time to prepare it, but it is well worth it.
Jeopardy - This follows the same format as the game show. It is great to use on the active board. There are many already developed on the Ashland Schools website.
Categories – technique for forming groups and reviewing content. (Tool book,
p. 138)
Tool Book refers to: Silver & Strong, 2001 Tools for Promoting Active, In-Depth Learning, Thoughtful Education Press.
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