Quadratic and exponential functions are important ways of modeling real-world situations and processes. Students need to become comfortable working with these new types of functions.
How is the quadratic formula useful?
How is the exponential function useful?
What is inverse variation?
√ How do we use the quadratic formula?
√How do you write and graph equations for exponential growth and decay functions?
√How do you identify the domain and range of an exponential function?
√What is the meaning of the exponent in an exponential function?
√How do you distinguish between linear, quadratic, and exponential equations?
TEKS Specificity – Intended Outcomes
A.04.A The student is expected to find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations.
A.10.A The student is expected to solve quadratic equations using concrete models, tables, graphs, and algebraic methods.
A.10.B The student is expected to make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function.
A.11.A The student is expected to use patterns to generate the laws of exponents and apply them in problem-solving situations.
A.11.C The student is expected to analyze data and represent situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods.
A.11.B The student is expected to analyze data and represent situations involving inverse variation using concrete models, tables, graphs, or algebraic methods.
A.08.AThe student is expected to analyze situations and formulate systems of linear equations in two unknowns to solve problems.
A.08.BThe student is expected to solve systems of linear equations using concrete models, graphs, tables, and algebraic methods.
A.08.CThe student is expected to interpret and determine the reasonableness of solutions to systems of linear equations.
“I CAN” statements highlighted in yellow and italicize should be displayed for students
solve quadratic equations using concrete models, tables, graphs, and algebraic methods
make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function
use patterns to generate the laws of exponents and apply them in problem-solving situations
analyze data and represent situations involving inverse variation using concrete models, tables, graphs, or algebraic methods
analyze situations and formulate systems of linear equations in two unknowns to solve problems
solve systems of linear equations using concrete models, graphs, tables, and algebraic methods
1E – Internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment.
2D – Monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
3D – speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
3E – Share information in cooperative learning interactions
4F – Use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language.
College and Career Readiness Standards
I.B.1 Perform computations with real and complex numbers
II.A.1 Explain and differentiate between expressions and equations using words such as solve, evaluate and simplify
II.B.1 Recognize and use algebraic properties, concepts, procedures, and algorithms to combine, transform and evaluate expressions
Evidence of Learning (Summative Assessment)
Given an end of unit quiz, students apply the quadratic formula correctly 8 out of 10 times.
Given an end of unit quiz, students are able to model exponential functions 8 out of 10 times.
Given an end of unit quiz , studentsare able to model inverse variation.
Given a variety of problems (different content and context), students will apply a problem solving process 100% of the time.
The Teaching Plan
Instructional Model & Teacher Directions
The teacher will…
So students can…
Follow the prescribed countdown to prepare for TAKS. There is a 17 day countdown.
The TAKS test uses 5 days.
Make sure and read the Prepare Instruction for the given Topic before the lesson. This area contains Goals and Objectives, Topic at a Glance, Prerequisite Skills, Resources and Language support for the given Topic. Also make sure and read Delivering Instruction for the given Topic and block. This area contains Agile Mind materials, opening the lesson, framing questions, lesson activities, and further questions for the given block. Both Prepare Instruction and Delivering Instruction can be found in the Advice for Instruction area. You can find the Advice for Instruction area by clicking Show Professional Support under the View tab. It is also a good idea to make copies of Student Activity sheets and use them in class as a notetaking template. Also remember Agile Assessment has more TAKS formatted problems if needed.
The Quadratic Formula
Blocks 1- 6
Block 1 introduces the concept of irrational numbers, developing the ideas through a geometric approach, and is supported by the Overview and the Exploring "The geometry of square roots." Grounding irrational numbers in the concept of length provides a historical perspective and a concrete representation that students can then connect to the more traditional algebraic manipulations of square roots that will come in Block 2.
Block 2 focuses on the skills required to simplify expressions involving square roots. It is supported by the Exploring "The algebra of square roots." The activities in this block tie the concrete representations of square roots to their simplified algebraic representations by connecting to students' prior work in middle school with similar figures and scaling. The knowledge and skills presented in this block are important in the routine manipulation of many algebraic expressions. However, do explain that in many applied problems square roots can be approximated with technology or by other means. Allow time to add numerical examples and extend them to symbolic expressions.
Block 3 builds on students' prior work in solving quadratic equations by developing the quadratic formula as a second method (with completing the square) for solving equations with irrational solutions. The quadratic formula is derived and then applied to solve quadratic equations.
Block 4 uses the Summary and the Guided assessment as a formative assessment of student knowledge.
Block 5 provides time for a topic-level assessment.
Block 6 provides time for reteaching.
TEKS covered A.4.A, A.10.A, A.10.B
Modeling with the Exponential Function
Block 1 introduces students to exponential functions and compares them to linear functions using a problem situation. This block is supported by Overview and Student Activity Sheet 1.
In Block 2, students compare linear and exponential growth as they continue to explore the fruit fly and fire ant situations. This block is supported by the Exploring "Exponential growth" and Student Activity Sheet 2.
Block 3 builds on the paper-folding activity to give students more experience with exponential functions. Students compare different exponential models. This block is supported by the Exploring "Paper folding," the Constructed response, and Student Activity Sheet 3.
In Block 4, students share solutions to the Constructed response and work on the Guided assessment to reinforce their understanding of exponential functions. Students will benefit from access to computers, if possible, for their work on the Guided assessment so that they can receive targeted feedback.
Block 5 employs the Summary to review key concepts and terminology and allows time for assessment to make sure that students understand the connections among all the representations of linear and exponential functions. The Multiple choice questions can be used for such an assessment.
Block 6 provides time for reteaching
TEKS covered A.11.A, A.11.C
Modeling with Inverse variation
Block 1 introduces the concept of inverse variation through contrast with direct variation and then provides a scenario in which students can fully explore and connect representations (graphical, tabular, and algebraic) of inverse variation. The Overview, the first Exploring, "Traveling by train," and Student Activity Sheets 1 and 2 support this block.
Block 2 is supported by the materials in the second Exploring, "Bus problem," and provides students an opportunity to compare and contrast a situation that exhibits inverse variation with a related situation that does not. Student Activity Sheet 3 also supports this block.
Block 3 is supported by the materials in the Guided assessment. This block also provides the opportunity for students to revisit linear and exponential functions as they compare the growth patterns of those two functions to that of inverse variation.
Block 4 employs the Summary to review key concepts and terminology. The Constructed response allows time for assessment to make sure that students understand the connections among all the representations of linear, exponential, and inverse functions.
Block 5 allows time for topic level assessment to make sure that students understand the connections among all the representations of inverse variation.
Block 6 provides time for reteaching
TEKS covered A.11.B
Other Methods of Solving Systems
Block1 motivates the need for analytic solution techniques to complement graphical and tabular methods for solving systems. This block also introduces the substitution method and builds connections between the solutions found by this method to solutions found using graphing and tables. The Overview, pages 1-4 of the Exploring "Substitution method," and Student Activity Sheets 1 and 2 support this block.
Block2 continues the discussion of the substitution method as students begin to analyze its efficiency for certain kinds of systems. Pages 5-9 of the Exploring "Substitution method" and Student Activity Sheet 2 support this block.
Blocks 3 and 4 address the mechanics of the linear combination method for solving systems. The Exploring "Linear combination method" and Student Activity Sheet 3 support this block.
Block 5 addresses the three solution cases for systems of linear equations and connects algebraic and graphical results. The Guided assessment supports this block.
TEKS covered A.08.A, A.08.B, A.08.C
Semester Review and Final
Vocabulary: (Pertinent to the learning – specific)
Anticipated Skills for SAT/ACT/College Board/Career/Life
FMA May 2009
2006 TAKS 9th Grade problem 16 (A.4A)
TAKS 2004 Grade 9 (A.4A)
James purchased a hybrid golf cart for $5000 and its value decreases by 20% each year. The value, in dollars, of the golf cart years from the date of purchase is given by the function, where. How many years from the date of purchase will the value of the golf cart be $3200?
Power Standards represent the essential knowledge and skills students need for success in high school and beyond. Power Standards must be mastered to successfully pass the required assessments at each grade level. All TAKS eligible knowledge and skills are identified as Power Standards.