UNIT AUTHOR: Keith Beals
UNIT TOPIC: Ch 5 Quadratic functions with complex solutions
1. UNIT CONTEXT Subject/Content Area: Math
Course: Algebra II (core)
Grade Level: 10-11 (with a few seniors)
Length of Unit: 7 days (1.3 weeks) with 4 class periods, ~2 hrs long each
This is the second unit within Chaper 5 of the text. It completes the study of quadratic functions, including graphing parabolas and solving for complex roots. This unit builds upon the previous chapter graphing and solving linear equations, and provides a foundation for the next unit, graphing and solving conic sections (circles, ellipses, and hyperbolas).
2. Unit Rationale: Enduring Understandings & Essential Questions
Enduring Understandings (EU)
Explain
Essential Questions
3. STANDARDS Content & Common Core Standards
ELD Standards
4. UNIT OBJECTIVES
Cognitive, content objectives:
5. ASSESSMENT PLAN Have an assessment for every objective and standard in unit. Cross-reference the objective and standard for each assessment. Example: Assessment (Objective/Standard #)
Day 1:
Formative: Teachers and students collaborate for informal assessment during classwork.
Objective & Standard: It assesses the content standard for complex operations
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers and students provide verbal and visual feedback including additional instruction and answer checking.
Identify how assessment informs instruction: Significant problems encountered by multiple students are brought to the class for additional instruction. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers listen to pair sharing
Objective & Standard: It assesses the ELD standard for exchanging ideas
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: verbal
Feedback Strategies: Teachers provide verbal feedback, positive or encouraging
Identify how assessment informs instruction: Significant problems encountered by English learners could cause teachers to create more opportunities for group and partner discussions.
Day 2:
Formative: Teachers and students collaborate for informal assessment during classwork.
Objective & Standard: It assesses the content standard for analyzing functions
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers and students provide verbal and visual feedback including additional instruction and answer checking.
Identify how assessment informs instruction: Significant problems encountered by multiple students are brought to the class for additional instruction. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers review student homework during classwork.
Objective & Standard: It assesses the content standard for complex operations
Type: Formal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers grades are posted on school loop, significant problems or excellent work will be noted with verbal comments during grading.
Identify how assessment informs instruction: Significant problems encountered by multiple students could create a larger warm up review set for the next day. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers require written justification of vertex form usage
Objective & Standard: It assesses the ELD standard for justifying an argument
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: written
Feedback Strategies: Teachers provide verbal feedback on exit
Identify how assessment informs instruction: Significant problems encountered by English learners could cause teachers to create more opportunities writing or justification of arguments. Significant problems with the concept would cause teachers to create space for a renewed discussion of vertex form and why it’s important.
Day 3:
Formative: Teachers and students collaborate for informal assessment during classwork.
Objective & Standard: It assesses the content standard for translating between parabolic graph and equation
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers and students provide verbal and visual feedback including additional instruction and answer checking.
Identify how assessment informs instruction: Significant problems encountered by multiple students are brought to the class for additional instruction. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers review student homework during classwork.
Objective & Standard: It assesses the content standard for analyzing functions
Type: Formal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers grades are posted on school loop, significant problems or excellent work will be noted with verbal comments during grading.
Identify how assessment informs instruction: Significant problems encountered by multiple students could create a larger warm up review set for the next day. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers listen to group discussion
Objective & Standard: It assesses the ELD standard for exchanging ideas
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: verbal
Feedback Strategies: Teachers provide verbal feedback, positive or encouraging
Identify how assessment informs instruction: Significant problems encountered by English learners could cause teachers to create more opportunities for group and partner discussions.
Day 4:
Formative: Teachers and students collaborate for informal assessment during classwork.
Objective & Standard: It assesses the content standard for solving systems of inequalities graphically
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers and students provide verbal and visual feedback including additional instruction and answer checking.
Identify how assessment informs instruction: Significant problems encountered by multiple students are brought to the class for additional instruction. Large cumulative problems will affect the unit flow and could require teachers delay the summative assessment.
Formative: Teachers review student homework during classwork.
Objective & Standard: It assesses the content standard for translating between parabolic graph and equation
Type: Formal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers grades are posted on school loop, significant problems or excellent work will be noted with verbal comments during grading.
Identify how assessment informs instruction: Significant problems encountered by multiple students could create a larger warm up review set for the next day. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Students walk around room to find 6 unit review problems posted around the room (“Quadratics around the room”), and work in ad-hoc collaborative groups to solve problems. Individual student solutions are collected at the end of class for grading according to rubric.
Objective & Standard: It assesses all of the unit content standards
Type: Formal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers grades are posted on school loop, significant problems or excellent work will be noted with written comments during grading.
Identify how assessment informs instruction: Significant problems encountered by multiple students could create a larger review period before the individual, summative unit test. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers listen to group discussion
Objective & Standard: It assesses the ELD standard for exchanging ideas
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: verbal
Feedback Strategies: Teachers provide verbal feedback, positive or encouraging
Identify how assessment informs instruction: Significant problems encountered by English learners could cause teachers to create more opportunities for group and partner discussions.
6. STEPS OF INSTRUCTION
Unit Calendar – Chapter 5: Quadratic Equations (complex solutions) Block Schedule – 3 Even Days, 2 hours each
Day 1 – 5.4 Review complex numbers, addition and multiplication. Solve quadratic equations with complex numbers.
Standards
Content – Algebra II N-CN 2: Perform arithmetic operations with complex numbers.
ELD – Collaborative: Exchanging information and ideas.
Objectives:
Cognitive: Following instructor demonstration and pair interaction on complex numbers, students will be able to perform arithmetic operations on complex numbers, including addition, subtraction, and multiplication.
Language Development (Bridging):
After instructor demonstration of complex number operation, students will be able to contribute to partner discussions related to how complex numbers fit into quadratic solutions.
Student Activity :
Collaborative classwork: Students will collaborate in small groups to complete classwork problems on complex operations and solving quadratic equations with complex solutions.
Homework: Students will extend their practice at home.
Assessment
Cognitive – formative Teachers will rove the class assisting groups with problems and assessing individual comprehension.
Language development (Bridging) - formative Teachers will listen to conversations on complex numbers and assess English learners for sustained conversation. English learners that stay largely quiet will be encouraged to contribute more.
Day 2 – 5.5 Review completing the square and introduce vertex form of quadratic equations.
Standards
Content – Algebra II F-IF 8: Analyze functions using different representations. Write a function in different but equivalent forms to reveal and explain different properties of the function.
ELD – Productive: Justifying own arguments.
Objectives:
Cognitive: Following instructor demonstration, students will be able to write quadratic equations in vertex form, including equations with vertices in all four quadrants.
Language Development (Bridging):
After collaborative classwork, students will be able to write a justification for why the vertex form is useful.
Student Activity :
Collaborative classwork: Students will collaborate in small groups to complete classwork problems transforming quadratic equations into vertex form.
Homework: Students will extend their practice at home.
Assessment
Cognitive – formative Teachers will rove the class assisting groups with problems and assessing individual comprehension. Homework on complex operations will be graded during classwork.
Language development (Bridging) - formative Students will be required to write a justification for why the vertex form of a quadratic equation is useful as an exit ticket.
Day 3 – 5.8 Write quadratic functions given characteristics of their graphs (parabolas).
Standards
Content – Algebra II G-GPE 3.1: Translate between the geometric description and the equation for a conic section.
ELD – Collaborative: Exchanging information and ideas.
Objectives:
Cognitive: Following instructor demonstration, students will be able to create quadratic equations from graphs, including graphs from each quadrent.
Language Development (Bridging): After instructor demonstration, students will be able to contribute to partner discussions on how to transform parabolic graphs into quadratic equations.
Student Activity :
Collaborative classwork: Students will collaborate in small groups to complete classwork problems translating graphs into quadratic equations.
Homework: Students will extend their practice at home.
Assessment
Cognitive – formative Teachers will rove the class assisting groups with problems and assessing individual comprehension. Homework on vertex form problems will be graded during classwork.
Language development (Bridging) - formative Teachers will listen to group conversations on translating parabolic graphs into equations and assess English learners for sustained conversation. English learners that don’t participate much will be encouraged to contribute more.
Day 4 – 5.7 Solve systems of quadratic inequalities through graphing.
Standards
Content – Algebra II A-REI 11: Represent and solve equations and inequalities graphically.
ELD – Collaborative: Exchanging information and ideas.
Objectives:
Cognitive: Following instructor demonstration, students will be able to solve a system of quadratic inequalities, including systems with two inequalities written in vertex, standard, or factored form.
Language Development (Bridging): After instructor demonstration, students will be able to contribute to dynamic group discussions on how to graph and solve quadratic inequalities.
Student Activity :
Collaborative classwork: Students will collaborate in small groups to complete classwork problems graphing systems of quadratic inequalities.
Collaborative assessment: Students will move around the room to find 6 review problems, and will collaborate to solve them in ad-hoc groups.
Assessment
Cognitive – formative Teachers will rove the class assisting groups with problems and assessing individual comprehension. Homework on section 5.8 will be graded during classwork.
Cognitive – formative Teachers will collect individual responses to 6 problems students worked collaboratively as an exit criteria. This serves as the final formative assessment prior to summative unit test, and as such reassesses all content objectives.
Language development (Bridging) - formative Teachers will listen to group conversations on translating parabolic graphs into equations and assess English learners for sustained conversation. English learners that don’t participate much will be encouraged to contribute more.
ANTICIPATORY SET
Start the unit by showing the following video on Parabolas to get the students thinking about parabolas outside the classroom and to mix things up for students who might be more inclined towards visuals and music:
https://www.youtube.com/watch?v=cXOcBADMp6o
CLOSURE
“Quadratics around the room” – Students move around the room to locate 6 review problems from the unit, and collaborate to solve them in ad-hoc groups. Students turn in their individual solutions at the end of class as an exit criteria for formal grades according to the attached rubric.
Comprehensive review of this unit is excellent preparation for the next unit, which extends quadratic functions to general quadratic equations (now we can have y2) that represent conic sections.
Example LESSON PLAN
AUTHOR’S NAME _Keith Beals____________________ DATE ____19 Oct 14____________
SINGLE SUBJECT LESSON TEMPLATE
1. TITLE OF LESSON: Graphing Quadratic Inequalities (last lesson of Unit 3/ Chapter 5)
2. CURRICULUM AREA & GRADE LEVEL: Algebra II, grades 10-11
3. DATE OF LESSON/TIME NEEDED: 23 Oct 14, 4th & 6th periods, 110 min
4. RESOURCES: Power Point Presentation, White Boards, Student Notebooks, Quadratics around the room problem sheets & rubric, and Graphing parabolas flowchart
5. CA CONTENT STANDARD(S):
A-REI: Represent and solve equations and inequalities graphically
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. (3 methods to graph parabolas – see objective 1 below)
6. CA ELD STANDARD(S): Address how this lesson helps develop language
A. Collaborative
3. Offering and justifying opinions, negotiating with and persuading others in communicative exchanges. (Collaborative classwork & assessment – see objective 2 below)
7. BIG IDEA ADDRESSED/ENDURING UNDERSTANDING: Why this material is important to teach; how it fits in with the unit
The graphing of parabolas is an important tool that can be used to solve problems relevant to students’ lives. The graphing of quadratic equations is reviewed with important vocabulary terms emphasized. Then inequalities are reviewed, and systems of quadratic inequalities are solved graphically, first by the teacher then by student in groups.
This lesson is a culmination of several graphing lessons. It also builds upon the graphing linear inequalities lesson from the previous unit. Follow-on units will build upon this to solve more complicated systems of heterogeneous inequalities with 3 equations. All of the units align with common core teaching standards.
8. ESSENTIAL QUESTIONS: Open-ended, arguable questions that organize the purpose of learning
9. OBJECTIVE(S) OR LEARNING GOAL(S): Choose one: Cognitive, Affective, Psychomotor or Language Development
1. Cognitive – After reviewing graphing of quadratic inequalities, students will be able to graph two parabolas to solve the system of inequalities, on the same set of axes with a correct solution and a well labeled graph. (A-REI: Represent and solve equations and inequalities graphically 11. Find solutions approximately [by graphing])
2. Language – After reviewing graphing of quadratic ineqaulities, students collaborate with their group to complete the classwork examples, by contributing to sustained discussion asking and answering relevant questions. (A. Collaborative 3. Offering and justifying opinions)
10. ASSESSMENT(S): Choose one: Diagnostic - entry level, Formative - progress-monitoring or Summative – evaluative
1. Formative – Informal evaluation of individual students’ graphs during classwork by teacher to determine that graphs include correct shape, vertex, and intercepts. Feedback will be immediate at the class level when classwork completes, and the results will inform content and delivery of follow-on lessons for graphing quadratic equations.
2. Formative – Informal evaluation of students’ writing on an exit ticket by teacher to determine if students are developing writing skills to collaboratively describe detailed information about newly acquired vocabulary (Bridging level). Verbal feedback will be given at the end of class or during the next class for more significant issues, and results will help inform content of follow-on lessons.
3. Formative – Formal evaluation of students’ collaborative work to solve “Quadratics around the room” problems. Feedback will be written on graded problem sheets, and the results will inform content and delivery of final review prior to summative test.
11. INSTRUCTIONAL STRATEGIES: What the teacher does
1. Anticipatory Set & Objective (10 min)
Teacher models inequality systems by solving the “Big State U” problem of linear inequalities for class
· Linear programming problem with 4 constraints
· Models a college setting admissions policy
Teacher explains that the skills we are teaching you can solve problems more complicated than this!
· What are some things that fall or fly?
· Why do some of them move in arcs (parabolas)?
· How might graphing parabolas help us understand their motion better?
Teacher finishes by introducing the lesson objective (see above).
**Differentiation: the pictures help English learners and the 504 for left only hearing to have multiple ways to connect to the introductory material. An easier word arc is used to bridge the term parabola to the pictures. The discussion connects material to students backgrounds Speech throughout the lesson is slow and well enunciated, giving students time to think and connect (SIOP features 1 & 3: content objective clearly defined and content concepts appropriate for age and background).
2. Vocabulary introduction (5 min)
Teacher reviews 7 vocabulary words introduced in the uint:
· Quadratic Equation: a two variable equation with an exponent of 2
· Vertex form: y = a(x-h)2+k; vertex (h,k)
· Standard form: y = ax2+bx+c
· Intercept form: y = a(x-p)(x-q); x-intercepts (p,0), (q,0)
· The graph of a quadratic equation is called a parabola
· The vertex is the maximum or minimum point of a parabola
· The axis of symmetry is the vertical line that goes through the vertex.
· Axis of symmetry: x = -b/2a
· The solutions to the equation when y is zero are at the x intercepts of the graph. These are called roots.
** Differentiation: Special emphasis on key vocabulary (visual and verbal) should help English learners and students that are not logically inclined to reconnect with key concepts (SIOP feature 9: Key vocabulary emphasized). 2 literacy domains are used in the segment: listening and reading. Vertex form and standard form and intercepts are terms that were used in the last unit, but have expanded meanings within this unit (SIOP feature 8: links explicitly made to past learning). Alternative language from student volunteers offers English learners other words to connect to new vocabulary (SIOP feature 12: a variety of techniques).
3. Input – Modeling (system of inequalities – 10 min)
Teacher hands out worksheets and puts the notes up on the smart board. Teacher models solving the first set of inequalities:
Teachers mingle with groups to solve conceptual problems the students can’t handle collaboratively, and assesses student graphing of quadratic inequalities.
Teacher takes volunteers to lead teacher through correct solution with class watching.
** Differentiation: Not all students are logically inclined. Interpersonal activity and collaboration serve to leverage students who did connect with the lesson to broaden the variety of explanations and help to scaffold from BICS to CALP vocabulary for English learners. SIOP feature 28 & 21: review of key content concepts and an activity to apply content knowledge.
6. Unit Collaborative Review – “Quadratics around the room” (40 min)
Teacher posts 6 review problems on sheets of paper tacked to the walls around the room. Students are told to reset their desks and collaborate to solve all 6 problems. Teacher informs students that worksheets will be collected at the end of class and graded as a quiz:
[See attached problem sheets]
Teachers mingle with groups to clarify questions and assess student learning as they work through the problems.
Teacher collects worksheets as students exit class.
** Differentiation: Not all students are logically inclined. Interpersonal activity and collaboration serve to leverage students who did connect with the lesson to broaden the variety of explanations and help to scaffold from BICS to CALP vocabulary for English learners. SIOP features 29 & 30: regular feedback to students and assessment of student comprehension and learning
12. STUDENT ACTIVITIES: What the students do
1. Anticipatory Set & Objective
Students listen to teacher and consider what the skills they have been learning can be used to accomplish.
Objective criteria: students should start to connect that we will be revisiting inequalities in class today
2. Vocabulary introduction
Students listen to instructions and concentrate on the reviewing vocabulary.
3. Input – Modeling
Students listen to and think about new material. Students follow example problem and imagine doing it themselves.
Objective criteria: students should start to connect their work on linear inequalities with their more recent work graphing quadratic equations.
4. Check for Understanding
Students listen to and think about new material. Students follow example problem and imagine doing it themselves.
Objective criteria: students should start to connect their work on linear inequalities with their more recent work graphing quadratic equations.
5. Collaborative practice
Students rearrange desks for collaboration, and collaborate with their group to make their own graph.
Students follow along, contribute to class discussion of solution, and correct their work as applicable.
Objective criteria: students will meet the objective if they obtain the capability of plotting both parabolas on the same graph, shading the solution area, and labeling the graph correctly.
6. Collaborative assessment
Students rearrange desks and walk about the room finding the 6 problems and collaborating with other students to solve them.
Students turn in worksheets before leaving class.
Objective criteria: students will meet the objective if they obtain the capability of graphing parabolas through one of the three methods.
__________________________________________________ [End Example Lesson]
7. MATERIALS/RESOURCES
Day 1: Lecture notes, Unit homework assignment
Day 2: Lecture notes
Day 3: Lecture notes
Day 4: Lecture notes, Graphing flowchart, Quadratics around the room problem sheets & rubric
8. REFLECTION
Most students need repetition, vocabulary emphasis, a variety of perspectives, and clear direction. Some students benefit from the opportunity to talk or move around. My lessons incorporate all of these activities in an effort to be as engaging as possible to everyone.
The strengths lie in the variety and group work opportunities. The weakness, every math teachers Achilles' heel, is creating lessons that students want to engage in that also meet the standards objectives, for students that are not interested nor inclined towards math.
We plan to collect the collaborative assessment, which should be a decent gauge as to how effective the unit has been for student learning.
A lot of work goes into planning and differentiating lessons for students, but I know I will have to keep at it in terms of finding engaging content.
9. RUBRIC WITH SELF-ASSESSMENT
[See Cougar Courses Forum]
UNIT TOPIC: Ch 5 Quadratic functions with complex solutions
1. UNIT CONTEXT Subject/Content Area: Math
Course: Algebra II (core)
Grade Level: 10-11 (with a few seniors)
Length of Unit: 7 days (1.3 weeks) with 4 class periods, ~2 hrs long each
This is the second unit within Chaper 5 of the text. It completes the study of quadratic functions, including graphing parabolas and solving for complex roots. This unit builds upon the previous chapter graphing and solving linear equations, and provides a foundation for the next unit, graphing and solving conic sections (circles, ellipses, and hyperbolas).
2. Unit Rationale: Enduring Understandings & Essential Questions
Enduring Understandings (EU)
Explain
- Students will explain what a parabola is and describe the three equation forms, vertex, standard, and factored.
- Students will be able to interpret complex and real solutions to quadratic equations
- Students will be able to apply graphing and algebraic skills to solve quadratic functions for real and complex solutions
- Students will be able to consider quadratic equations from alternative forms (vertex and factored) and transform them to standard form
- Students will be able create quadratic functions based upon a description of a system or problem
- Students will be able to help other students visualize and create quadratic equations
- Students will be able to assess equations and confidently use the procedure they are most comfortable with to factor, graph, and solve them
Essential Questions
- How can I interpret this parabola, equation, or system of equations?
- How can I apply my graphing and algebraic skills to develop my interpretation?
- How can I apply my interpretation to help create practical solutions?
3. STANDARDS Content & Common Core Standards
- Algebra II N-CN 2: Perform arithmetic operations with complex numbers.
- Algebra II F-IF 8: Analyze functions using different representations. Write a function in different but equivalent forms to reveal and explain different properties of the function.
- Algebra II G-GPE 3.1: Translate between the geometric description and the equation for a conic section.
- Algebra II A-REI 11: Represent and solve equations and inequalities graphically.
ELD Standards
- Collaborative: Exchanging information and ideas.
- Productive: Justifying own arguments
- Collaborative: Offering and justifying opinions, negotiating with and persuading others in communicative exchanges
4. UNIT OBJECTIVES
Cognitive, content objectives:
- Review complex numbers, addition and multiplication. Solve quadratic equations with complex numbers.
- Following instructor demonstration, students will be able to write quadratic equations in vertex form, including equations with vertices in all four quadrants.
- Following instructor demonstration, students will be able to create quadratic equations from graphs, including graphs from each quadrant.
- Following instructor demonstration, students will be able to solve a system of quadratic inequalities, including systems with two inequalities written in vertex, standard, or factored form.
- After instructor demonstration of complex number operation, students will be able to contribute to partner discussions related to how complex numbers fit into quadratic solutions.
- After collaborative classwork, students will be able to write a justification for why the vertex form is useful.
- After instructor demonstration, students will be able to contribute to partner discussions on how to transform parabolic graphs into quadratic equations.
- After instructor demonstration, students will be able to contribute to dynamic group discussions on how to graph and solve quadratic inequalities.
5. ASSESSMENT PLAN Have an assessment for every objective and standard in unit. Cross-reference the objective and standard for each assessment. Example: Assessment (Objective/Standard #)
Day 1:
Formative: Teachers and students collaborate for informal assessment during classwork.
Objective & Standard: It assesses the content standard for complex operations
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers and students provide verbal and visual feedback including additional instruction and answer checking.
Identify how assessment informs instruction: Significant problems encountered by multiple students are brought to the class for additional instruction. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers listen to pair sharing
Objective & Standard: It assesses the ELD standard for exchanging ideas
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: verbal
Feedback Strategies: Teachers provide verbal feedback, positive or encouraging
Identify how assessment informs instruction: Significant problems encountered by English learners could cause teachers to create more opportunities for group and partner discussions.
Day 2:
Formative: Teachers and students collaborate for informal assessment during classwork.
Objective & Standard: It assesses the content standard for analyzing functions
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers and students provide verbal and visual feedback including additional instruction and answer checking.
Identify how assessment informs instruction: Significant problems encountered by multiple students are brought to the class for additional instruction. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers review student homework during classwork.
Objective & Standard: It assesses the content standard for complex operations
Type: Formal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers grades are posted on school loop, significant problems or excellent work will be noted with verbal comments during grading.
Identify how assessment informs instruction: Significant problems encountered by multiple students could create a larger warm up review set for the next day. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers require written justification of vertex form usage
Objective & Standard: It assesses the ELD standard for justifying an argument
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: written
Feedback Strategies: Teachers provide verbal feedback on exit
Identify how assessment informs instruction: Significant problems encountered by English learners could cause teachers to create more opportunities writing or justification of arguments. Significant problems with the concept would cause teachers to create space for a renewed discussion of vertex form and why it’s important.
Day 3:
Formative: Teachers and students collaborate for informal assessment during classwork.
Objective & Standard: It assesses the content standard for translating between parabolic graph and equation
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers and students provide verbal and visual feedback including additional instruction and answer checking.
Identify how assessment informs instruction: Significant problems encountered by multiple students are brought to the class for additional instruction. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers review student homework during classwork.
Objective & Standard: It assesses the content standard for analyzing functions
Type: Formal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers grades are posted on school loop, significant problems or excellent work will be noted with verbal comments during grading.
Identify how assessment informs instruction: Significant problems encountered by multiple students could create a larger warm up review set for the next day. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers listen to group discussion
Objective & Standard: It assesses the ELD standard for exchanging ideas
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: verbal
Feedback Strategies: Teachers provide verbal feedback, positive or encouraging
Identify how assessment informs instruction: Significant problems encountered by English learners could cause teachers to create more opportunities for group and partner discussions.
Day 4:
Formative: Teachers and students collaborate for informal assessment during classwork.
Objective & Standard: It assesses the content standard for solving systems of inequalities graphically
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers and students provide verbal and visual feedback including additional instruction and answer checking.
Identify how assessment informs instruction: Significant problems encountered by multiple students are brought to the class for additional instruction. Large cumulative problems will affect the unit flow and could require teachers delay the summative assessment.
Formative: Teachers review student homework during classwork.
Objective & Standard: It assesses the content standard for translating between parabolic graph and equation
Type: Formal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers grades are posted on school loop, significant problems or excellent work will be noted with verbal comments during grading.
Identify how assessment informs instruction: Significant problems encountered by multiple students could create a larger warm up review set for the next day. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Students walk around room to find 6 unit review problems posted around the room (“Quadratics around the room”), and work in ad-hoc collaborative groups to solve problems. Individual student solutions are collected at the end of class for grading according to rubric.
Objective & Standard: It assesses all of the unit content standards
Type: Formal, formative
Purpose: Assess skills and knowledge
Implementation: paper and pencil, verbal
Feedback Strategies: Teachers grades are posted on school loop, significant problems or excellent work will be noted with written comments during grading.
Identify how assessment informs instruction: Significant problems encountered by multiple students could create a larger review period before the individual, summative unit test. Large cumulative problems will affect the unit flow and could require teachers pull in slack days to extend the unit.
Formative: Teachers listen to group discussion
Objective & Standard: It assesses the ELD standard for exchanging ideas
Type: Informal, formative
Purpose: Assess skills and knowledge
Implementation: verbal
Feedback Strategies: Teachers provide verbal feedback, positive or encouraging
Identify how assessment informs instruction: Significant problems encountered by English learners could cause teachers to create more opportunities for group and partner discussions.
6. STEPS OF INSTRUCTION
Unit Calendar – Chapter 5: Quadratic Equations (complex solutions) Block Schedule – 3 Even Days, 2 hours each
Day 1 – 5.4 Review complex numbers, addition and multiplication. Solve quadratic equations with complex numbers.
Standards
Content – Algebra II N-CN 2: Perform arithmetic operations with complex numbers.
ELD – Collaborative: Exchanging information and ideas.
Objectives:
Cognitive: Following instructor demonstration and pair interaction on complex numbers, students will be able to perform arithmetic operations on complex numbers, including addition, subtraction, and multiplication.
Language Development (Bridging):
After instructor demonstration of complex number operation, students will be able to contribute to partner discussions related to how complex numbers fit into quadratic solutions.
Student Activity :
Collaborative classwork: Students will collaborate in small groups to complete classwork problems on complex operations and solving quadratic equations with complex solutions.
Homework: Students will extend their practice at home.
Assessment
Cognitive – formative Teachers will rove the class assisting groups with problems and assessing individual comprehension.
Language development (Bridging) - formative Teachers will listen to conversations on complex numbers and assess English learners for sustained conversation. English learners that stay largely quiet will be encouraged to contribute more.
Day 2 – 5.5 Review completing the square and introduce vertex form of quadratic equations.
Standards
Content – Algebra II F-IF 8: Analyze functions using different representations. Write a function in different but equivalent forms to reveal and explain different properties of the function.
ELD – Productive: Justifying own arguments.
Objectives:
Cognitive: Following instructor demonstration, students will be able to write quadratic equations in vertex form, including equations with vertices in all four quadrants.
Language Development (Bridging):
After collaborative classwork, students will be able to write a justification for why the vertex form is useful.
Student Activity :
Collaborative classwork: Students will collaborate in small groups to complete classwork problems transforming quadratic equations into vertex form.
Homework: Students will extend their practice at home.
Assessment
Cognitive – formative Teachers will rove the class assisting groups with problems and assessing individual comprehension. Homework on complex operations will be graded during classwork.
Language development (Bridging) - formative Students will be required to write a justification for why the vertex form of a quadratic equation is useful as an exit ticket.
Day 3 – 5.8 Write quadratic functions given characteristics of their graphs (parabolas).
Standards
Content – Algebra II G-GPE 3.1: Translate between the geometric description and the equation for a conic section.
ELD – Collaborative: Exchanging information and ideas.
Objectives:
Cognitive: Following instructor demonstration, students will be able to create quadratic equations from graphs, including graphs from each quadrent.
Language Development (Bridging): After instructor demonstration, students will be able to contribute to partner discussions on how to transform parabolic graphs into quadratic equations.
Student Activity :
Collaborative classwork: Students will collaborate in small groups to complete classwork problems translating graphs into quadratic equations.
Homework: Students will extend their practice at home.
Assessment
Cognitive – formative Teachers will rove the class assisting groups with problems and assessing individual comprehension. Homework on vertex form problems will be graded during classwork.
Language development (Bridging) - formative Teachers will listen to group conversations on translating parabolic graphs into equations and assess English learners for sustained conversation. English learners that don’t participate much will be encouraged to contribute more.
Day 4 – 5.7 Solve systems of quadratic inequalities through graphing.
Standards
Content – Algebra II A-REI 11: Represent and solve equations and inequalities graphically.
ELD – Collaborative: Exchanging information and ideas.
Objectives:
Cognitive: Following instructor demonstration, students will be able to solve a system of quadratic inequalities, including systems with two inequalities written in vertex, standard, or factored form.
Language Development (Bridging): After instructor demonstration, students will be able to contribute to dynamic group discussions on how to graph and solve quadratic inequalities.
Student Activity :
Collaborative classwork: Students will collaborate in small groups to complete classwork problems graphing systems of quadratic inequalities.
Collaborative assessment: Students will move around the room to find 6 review problems, and will collaborate to solve them in ad-hoc groups.
Assessment
Cognitive – formative Teachers will rove the class assisting groups with problems and assessing individual comprehension. Homework on section 5.8 will be graded during classwork.
Cognitive – formative Teachers will collect individual responses to 6 problems students worked collaboratively as an exit criteria. This serves as the final formative assessment prior to summative unit test, and as such reassesses all content objectives.
Language development (Bridging) - formative Teachers will listen to group conversations on translating parabolic graphs into equations and assess English learners for sustained conversation. English learners that don’t participate much will be encouraged to contribute more.
ANTICIPATORY SET
Start the unit by showing the following video on Parabolas to get the students thinking about parabolas outside the classroom and to mix things up for students who might be more inclined towards visuals and music:
https://www.youtube.com/watch?v=cXOcBADMp6o
CLOSURE
“Quadratics around the room” – Students move around the room to locate 6 review problems from the unit, and collaborate to solve them in ad-hoc groups. Students turn in their individual solutions at the end of class as an exit criteria for formal grades according to the attached rubric.
Comprehensive review of this unit is excellent preparation for the next unit, which extends quadratic functions to general quadratic equations (now we can have y2) that represent conic sections.
Example LESSON PLAN
AUTHOR’S NAME _Keith Beals____________________ DATE ____19 Oct 14____________
SINGLE SUBJECT LESSON TEMPLATE
1. TITLE OF LESSON: Graphing Quadratic Inequalities (last lesson of Unit 3/ Chapter 5)
2. CURRICULUM AREA & GRADE LEVEL: Algebra II, grades 10-11
3. DATE OF LESSON/TIME NEEDED: 23 Oct 14, 4th & 6th periods, 110 min
4. RESOURCES: Power Point Presentation, White Boards, Student Notebooks, Quadratics around the room problem sheets & rubric, and Graphing parabolas flowchart
5. CA CONTENT STANDARD(S):
A-REI: Represent and solve equations and inequalities graphically
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. (3 methods to graph parabolas – see objective 1 below)
6. CA ELD STANDARD(S): Address how this lesson helps develop language
A. Collaborative
3. Offering and justifying opinions, negotiating with and persuading others in communicative exchanges. (Collaborative classwork & assessment – see objective 2 below)
7. BIG IDEA ADDRESSED/ENDURING UNDERSTANDING: Why this material is important to teach; how it fits in with the unit
The graphing of parabolas is an important tool that can be used to solve problems relevant to students’ lives. The graphing of quadratic equations is reviewed with important vocabulary terms emphasized. Then inequalities are reviewed, and systems of quadratic inequalities are solved graphically, first by the teacher then by student in groups.
This lesson is a culmination of several graphing lessons. It also builds upon the graphing linear inequalities lesson from the previous unit. Follow-on units will build upon this to solve more complicated systems of heterogeneous inequalities with 3 equations. All of the units align with common core teaching standards.
8. ESSENTIAL QUESTIONS: Open-ended, arguable questions that organize the purpose of learning
- Where do quadratic equations show up in real life?
- How could graphing parabolas help us solve problems?
- How are quadratic inequalities different from linear inequalities?
9. OBJECTIVE(S) OR LEARNING GOAL(S): Choose one: Cognitive, Affective, Psychomotor or Language Development
1. Cognitive – After reviewing graphing of quadratic inequalities, students will be able to graph two parabolas to solve the system of inequalities, on the same set of axes with a correct solution and a well labeled graph. (A-REI: Represent and solve equations and inequalities graphically 11. Find solutions approximately [by graphing])
2. Language – After reviewing graphing of quadratic ineqaulities, students collaborate with their group to complete the classwork examples, by contributing to sustained discussion asking and answering relevant questions. (A. Collaborative 3. Offering and justifying opinions)
10. ASSESSMENT(S): Choose one: Diagnostic - entry level, Formative - progress-monitoring or Summative – evaluative
1. Formative – Informal evaluation of individual students’ graphs during classwork by teacher to determine that graphs include correct shape, vertex, and intercepts. Feedback will be immediate at the class level when classwork completes, and the results will inform content and delivery of follow-on lessons for graphing quadratic equations.
2. Formative – Informal evaluation of students’ writing on an exit ticket by teacher to determine if students are developing writing skills to collaboratively describe detailed information about newly acquired vocabulary (Bridging level). Verbal feedback will be given at the end of class or during the next class for more significant issues, and results will help inform content of follow-on lessons.
3. Formative – Formal evaluation of students’ collaborative work to solve “Quadratics around the room” problems. Feedback will be written on graded problem sheets, and the results will inform content and delivery of final review prior to summative test.
11. INSTRUCTIONAL STRATEGIES: What the teacher does
1. Anticipatory Set & Objective (10 min)
Teacher models inequality systems by solving the “Big State U” problem of linear inequalities for class
· Linear programming problem with 4 constraints
· Models a college setting admissions policy
Teacher explains that the skills we are teaching you can solve problems more complicated than this!
· What are some things that fall or fly?
· Why do some of them move in arcs (parabolas)?
· How might graphing parabolas help us understand their motion better?
Teacher finishes by introducing the lesson objective (see above).
**Differentiation: the pictures help English learners and the 504 for left only hearing to have multiple ways to connect to the introductory material. An easier word arc is used to bridge the term parabola to the pictures. The discussion connects material to students backgrounds Speech throughout the lesson is slow and well enunciated, giving students time to think and connect (SIOP features 1 & 3: content objective clearly defined and content concepts appropriate for age and background).
2. Vocabulary introduction (5 min)
Teacher reviews 7 vocabulary words introduced in the uint:
· Quadratic Equation: a two variable equation with an exponent of 2
· Vertex form: y = a(x-h)2+k; vertex (h,k)
· Standard form: y = ax2+bx+c
· Intercept form: y = a(x-p)(x-q); x-intercepts (p,0), (q,0)
· The graph of a quadratic equation is called a parabola
· The vertex is the maximum or minimum point of a parabola
· The axis of symmetry is the vertical line that goes through the vertex.
· Axis of symmetry: x = -b/2a
· The solutions to the equation when y is zero are at the x intercepts of the graph. These are called roots.
** Differentiation: Special emphasis on key vocabulary (visual and verbal) should help English learners and students that are not logically inclined to reconnect with key concepts (SIOP feature 9: Key vocabulary emphasized). 2 literacy domains are used in the segment: listening and reading. Vertex form and standard form and intercepts are terms that were used in the last unit, but have expanded meanings within this unit (SIOP feature 8: links explicitly made to past learning). Alternative language from student volunteers offers English learners other words to connect to new vocabulary (SIOP feature 12: a variety of techniques).
3. Input – Modeling (system of inequalities – 10 min)
Teacher hands out worksheets and puts the notes up on the smart board. Teacher models solving the first set of inequalities:
Teachers mingle with groups to solve conceptual problems the students can’t handle collaboratively, and assesses student graphing of quadratic inequalities.
Teacher takes volunteers to lead teacher through correct solution with class watching.
** Differentiation: Not all students are logically inclined. Interpersonal activity and collaboration serve to leverage students who did connect with the lesson to broaden the variety of explanations and help to scaffold from BICS to CALP vocabulary for English learners. SIOP feature 28 & 21: review of key content concepts and an activity to apply content knowledge.
6. Unit Collaborative Review – “Quadratics around the room” (40 min)
Teacher posts 6 review problems on sheets of paper tacked to the walls around the room. Students are told to reset their desks and collaborate to solve all 6 problems. Teacher informs students that worksheets will be collected at the end of class and graded as a quiz:
[See attached problem sheets]
Teachers mingle with groups to clarify questions and assess student learning as they work through the problems.
Teacher collects worksheets as students exit class.
** Differentiation: Not all students are logically inclined. Interpersonal activity and collaboration serve to leverage students who did connect with the lesson to broaden the variety of explanations and help to scaffold from BICS to CALP vocabulary for English learners. SIOP features 29 & 30: regular feedback to students and assessment of student comprehension and learning
12. STUDENT ACTIVITIES: What the students do
1. Anticipatory Set & Objective
Students listen to teacher and consider what the skills they have been learning can be used to accomplish.
Objective criteria: students should start to connect that we will be revisiting inequalities in class today
2. Vocabulary introduction
Students listen to instructions and concentrate on the reviewing vocabulary.
3. Input – Modeling
Students listen to and think about new material. Students follow example problem and imagine doing it themselves.
Objective criteria: students should start to connect their work on linear inequalities with their more recent work graphing quadratic equations.
4. Check for Understanding
Students listen to and think about new material. Students follow example problem and imagine doing it themselves.
Objective criteria: students should start to connect their work on linear inequalities with their more recent work graphing quadratic equations.
5. Collaborative practice
Students rearrange desks for collaboration, and collaborate with their group to make their own graph.
Students follow along, contribute to class discussion of solution, and correct their work as applicable.
Objective criteria: students will meet the objective if they obtain the capability of plotting both parabolas on the same graph, shading the solution area, and labeling the graph correctly.
6. Collaborative assessment
Students rearrange desks and walk about the room finding the 6 problems and collaborating with other students to solve them.
Students turn in worksheets before leaving class.
Objective criteria: students will meet the objective if they obtain the capability of graphing parabolas through one of the three methods.
__________________________________________________ [End Example Lesson]
7. MATERIALS/RESOURCES
Day 1: Lecture notes, Unit homework assignment
Day 2: Lecture notes
Day 3: Lecture notes
Day 4: Lecture notes, Graphing flowchart, Quadratics around the room problem sheets & rubric
8. REFLECTION
Most students need repetition, vocabulary emphasis, a variety of perspectives, and clear direction. Some students benefit from the opportunity to talk or move around. My lessons incorporate all of these activities in an effort to be as engaging as possible to everyone.
The strengths lie in the variety and group work opportunities. The weakness, every math teachers Achilles' heel, is creating lessons that students want to engage in that also meet the standards objectives, for students that are not interested nor inclined towards math.
We plan to collect the collaborative assessment, which should be a decent gauge as to how effective the unit has been for student learning.
A lot of work goes into planning and differentiating lessons for students, but I know I will have to keep at it in terms of finding engaging content.
9. RUBRIC WITH SELF-ASSESSMENT
[See Cougar Courses Forum]