Instructor: Elizabeth Proll 9th grade Algebra 1 Common Core
Lesson Title:Linear vs. Exponential Function Word Problems Date: Wednesday, November 19th
- LESSON CONTEXT AND RATIONALE
Culture and Context / The target class population consists of nine students, with 44% Hispanic, 33% Black, and 22% Caucasian and 66.6% of which receive free or reduced lunch.
I have designed this lesson to reinforce problem solving when approached with a word problem. We have currently covered a unit on both linear and exponential functions, which both included solving word problems. Our current “mini” unit is comparing linear functions to exponential functions, and thus is the focus of the word problems presented to the class. Throughout the school year, I have encouraged and helped my students begin forming positive attitudes towards mathematics in order to enhance lesson meaning through dialogue based on prior knowledge and experiences of the students’ (Wlodkowski & Ginsberg, 1995).During this lesson along with my assistance, students will review their knowledge of how to set up both linear and exponential equations, as well as determine which equation is the most appropriate to use from the verbal description given. Word problems are very relevant to the real world, as they are examples that the students themselves may someday encounter, in variation to be given an equation, table, or graph. This lesson relates such real world examples to the lives of students and asks them to use mathematics in order to arrive at a solution.
Identity and Intellectual Development / This lesson offers students an opportunity to compare and contrast linear and exponential function word problems. Key words to represent each kind of function will be presented, as a problem solving technique, as well as assistance setting up the equation based off the given information in the word problem. Furthermore, this lesson promotes conceptual understanding which pertains to the everyday life of all students.
Lesson Rationale / This lesson is the concluding lesson to our current “mini” unit of comparing linear and exponential functions to one another. We have previously focused two units on linear and exponential functions separately, and this unit is a combination of the two former. We have focused the past week on comparing equations, graphs, and tables of both linear and exponential functions and thus will conclude with word problem comparisons. The following day of class will consist of a test review day leading up to this unit test the day after.
My intent for teaching this topic in this manner is that I have recently experimented with small group, whole class, and individual classroom assignments. While students work in groups, I have observed that they are often off task unless I am assisting their group directly and get confused often and easily. Additionally, when assigned individual work, the students do not demonstrate appropriate autonomy, and instead feel as though they are able to chat with their friends and complete any assigned class work for homework instead. Therefore, these notes and classroom discussion will be delivered whole group, but still not teacher-centered. I plan to ask many questions of my students and encourage discussion of the differences between linear and exponential functions.
Management / During this lesson, the students will transition from an individual warm up activity, to whole group notes/discussion. In order to transition students between each lesson component, I will make sure to warn students of the upcoming transitions with time limit warnings and ensure that directions are clearly explained. Specifically for the warm up, a timer will be presented on the board with a countdown from two minutes to ensure students are aware of their remaining time to complete the problem.
- STANDARDS & REQUIREMENTS
CCGSP or GPS Standard(s) / MCC9-12.F.IF.1Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).
MCC9-12.F.BF.1Write a function that describes a relationship between two quantities.
MCC9-12.F.LE.1Distinguish between situations that can be modeled with linear functions and with exponential functions.
Individual Education Plan Goal(s) and Benchmarks / As this is a mathematics resource class, each student in this class has individual education plan goals. The majority of the target class population have not met a standard score on middle grades CRCT testing and have also scored below average on the Wechsler Individual Achievement Test in mathematics. Additionally, a majority of the students within this class have curriculum mastery goals outlined in their individual education plans, such as evaluating and performing operations with real numbers and algebraic expressions/equations as measured by 80% accuracy.
Modification(s) /Accommodation(s) / All students are offered access to the use of a calculator, as defined in their IEP accommodations. Additionally, the majority of students within this class are provided with breaking material into manageable parts, additional time to respond, extra time if effort is shown, and checking work frequently to ensure understanding. Several students also require giving short and concise directions, providing a copy of all notes/study guides, graphic organizers, reading aloud, as well as presenting information visually.
Differentiation / Each student will participate in the whole group discussion/note taking. I will observe student work for understanding and learning, give multiple means of representation, and promote autonomy of each student in order for all to be successful.“One approach to meeting each student’s needs is to provide tasks within each student’s zone of proximal development and to ensure that each student in the class has the opportunity to make a meaningful contribution to the class community of learners… instruction within the zone of proximal development allows students, whether with guidance from the teacher or by working with other students, to access new ideas that are beyond what the students know but within their reach” (Small, 2010, p. 2). I often call on students to answer questions aloud, beginning with a question that I know they are capable of answering. Then, I continue to ask them questions to arrive at a more difficult answer, that requires deeper conceptual thinking, thus enhancing their zone of proximal development.
- RESOURCES
Academic Language / The key language demand of this lesson is writing and speaking. Each student will discuss with the teacher and their peers to solve the given example word problems.
Language Functions / The primary language functions of this lesson are to predict, evaluate, and explain. Each student pair will predict whether they feel the given problem represents a linear or exponential function, write the appropriate equation in response, and finally check their work to ensure they have the correct equation.
Language Vocabulary / Key Vocabulary to be used throughout this lesson:
Exponent / Variable / Coefficient
Y-Intercept / Interest / Rate
Slope / Equation / Growth
Decay / Power
Materials /
- Class Notes and Homework assignment
- Calculator
- Projector
Technology / Promethean Board- Unfortunately, the use of the classroom Promethean Board is unavailable for this lesson as the smart board pen is unresponsive when attempted to use.
Calculator- All students will have access to the use of a calculator if needed. A majority of the students have an accommodation of a calculator, which are kept at the front of the classroom for easy access.
- LESSON PROCEDURE
Enduring Understanding / Understand how to interpret linear and exponential functions that arise in applications in terms of the context.
Understand how to construct and compare linear and exponential models and solve problems.
Understand how to interpret expressions for functions in terms of the situation they model.
Essential Question(s) / How do I interpret functions that arise in applications in terms of context?
How do I build a linear or exponential function that models a relationship between two quantities?
How can we use real-world situations to construct and compare linear and exponential models and solve problems?
Lesson Objective(s) / Students will construct and compare linear and exponential models and preserve in solving them using real-world contexts.
Assessment Link / Students will demonstrate their learning by completing their homework to compare linear and exponential models as they write functions to describe linear or exponential relationships between two quantities.
Introduction to Lesson / A warm up will be presented at the opening of the class period which asks students to examine a table of values and determine if the relationship is linear or exponential as well as write an equation to express the relationship. This warm activity is relevant due to the need to interpret and compare linear vs. exponential functions using multiple representations: graphs, equations, tables, and verbal descriptions.
Following the warm up activity which reinforces prior student knowledge, I will distribute the class notes for the day and begin the discussion to compare linear and exponential verbal descriptions.
The introduction/warm up activity should take no more than 5 minutes to complete.
Body of Lesson / Students will participate in the whole group discussion of linear and exponential word problems. I will use Socratic questioning in order for the work and problem solving to be student-centered, as I guide them towards the end goal. I will also circulate the room and oversee dialogue and student work individually. The group discussion and student participation will also be used to assess student comprehension.
The body of the lesson will take approximately30 minutes to complete, depending on student needs and level of comprehension.
Closure / Students will summarize what they have learned through filling in a comparison chart of things to look for in word problems regarding linear and exponential functions. The closing discussion should take approximately 5 to 10 minutes to complete.
If any additional class time remains at the conclusion of the days notes, I will distribute the homework assignment to each student and use it as a ticket out the door instead.
Students will also be assigned an additional homework assignment of a test review guide in order to begin preparing for their test on Friday. The review should be started this evening so that questions may be asked the day before the test for clarification.
- ASSESSMENT
Evaluation (Assessment Plan for IEP Goals and/or 504 Plans) / I will assess student learning through evaluating student participation, conceptual understanding, and appropriate language use as they discuss and answer questions with myself and the whole class. Feedback will be provided when questions are answered, as well as when I circulate the classroom and observe student work. The student homework assigned for the evening is an additional form of assessment, which will include additional feedback during its return, while a summative assessment will be given in two days at the conclusion of the current unit.
- REFLECTION
Analyzing Teaching Effectiveness
References
Small, M. & Lin, A. (2010). More Good questions: Great Ways to differentiate secondary
mathematics instruction, Ch.1: Why and how to differentiate math instruction, 1-16. Teacher College Press: USA.
Wlodkowski, R. & Ginsberg, M.B.J. (1995). A framework for culturally responsive teaching. Strengthening Student Engagement, 53(1), pp. 17-21.