Lesson Title: Graphing Linear and Exponential Functions Date: _____________ Teacher(s): ____________________ Course: Common Core Algebra 1, Unit 2 Start/end times: _________________________ Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which Mathematical Practices do you expect students to engage in during the lesson? F.IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear functions and show intercepts. e. Graph exponential functions, showing intercepts and end behavior. MP1: MP2: MP3: MP4: MP5: MP6: MP7: MP8: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Common Core Algebra I, Unit 2 Lesson Launch Notes: Exactly how will you use the first five minutes of the lesson? Day 1: Attached Warm-Up Day 2: Have students exchange their matching activities with a partner. The attached warm-up is an alternate activity. (Look for evidence of MP2 and MP4.) Lesson Closure Notes: Exactly what summary activity, questions, and discussion will close the lesson and connect big ideas? List the questions. Provide a foreshadowing of tomorrow. Day 1: Attached Exit Slip. (Look for evidence of MP4 and MP7.) Day 2: 1. Look back at the objectives and evidence of success. What part of today’s lesson helped you learn to graph linear and exponential functions using just the equation? 2. What common mistakes do you think students make when graphing from equations? 3. Graphs are one of the representations we can use to study functions. Do you remember all the others? Which is your favorite and why? (This question leads to standard F.IF.C.9.) Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations, problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic connections to appropriate mathematical practices. Day 1: 1. Group students and distribute one or more of these activities to each student to explore: a. “F.IF.7 Resource Option 1:” Transformation/Interactive Graphing Using a TI-84 Calculator. In this activity, students will observe the effect changing the parameters of an equation has on the graph. From there, students will describe how to graph linear and exponential functions given just the equation. See attached resource sheet, available at http://education.ti.com. (Look for evidence of MP4.) b. “F.IF.7 Resource Option 2:” Discovering the Relationship Between a Function and its Graph. Students will explore a set of matching functions and graphs and a set of non-matching. Students will describe the relationship between graphs and functions. (See attached resource sheet.) (Look for evidence of MP7.) HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement. Portsmouth, NH: Heinemann. Lesson Title: Graphing Linear and Exponential Functions Course: Common Core Algebra 1, Unit 2 Date: _____________ Teacher(s): ____________________ Start/end times: _________________________ c. “F.IF.7 Resource Option 3:” Using TI-Nspire Software To Observe Relationships Between Graphs and Functions. (For linear functions only.) (Look for evidence of MP7.) 2. Discuss the conclusions. 3. Have students complete “F.IF.7 Graphic Organizer,” which asks students to record their conclusions. Have students keep this in their notebooks or circulate and monitor students’ progress to gage student learning. After some private think/work time, review the organizer as a class. Students can then create instructional materials (posters, foldables, etc.) to reinforce the relationship between graphs and equations. 4. Students should now work independently, or in pairs, on “F.IF.7 Equation and Graph Matching.” (Look for evidence of MP7.) Day 2: 1. Present the following two problems and have students choose one to complete independently. Have them create two graphs: a. Half-life – Iodine 131 is a byproduct of weapons testing and nuclear facility problems, like the recent Fukushima meltdown in Japan, and can have negative effects on health. The half-life of iodine 131 is approximately 8 days. Create a graph that illustrates the amount of Iodine 131 remaining vs. the number of days. Iodine 131 occurs in very small amounts; for simplicity, 1 gram can be used as a starting amount. (Look for evidence of MP1, MP4 and MP6.) b. Planning – An old building is being renovated, which means it needs to be brought up to code. It needs a new wheelchair accessible ramp. Code dictates that for every 1 foot of vertical distance, the ramp must extend horizontally 12 feet. Many ramps of varying heights need to be installed throughout the building. Illustrate this algebraically and graphically. (Look for evidence of MP1, MP4 and MP6.) 2. Pair students and ask them to share their graphs (or consider a gallery walk). Have them reflect on the graphs’ accuracy and how important that is. Brainstorm ways of verifying and improving accuracy. 3. As a class, develop a variety of self-checking methods. Use this time to work on problem solving strategies, such as a problem setting (What key features need to be checked?), visualizing the solution (What methods and technologies do we already have that can show us this information)? and reflection (Is this a method you can and will use in the future? Is it the best method to check your work?) Methods may include: creating a table of values, graphing using a calculator or CAS technology, and justification within the context (e.g., if the graph is correct, there should be a point that corresponds with a fact that can be extrapolated from the context). It may be helpful to create a permanent record of the methods, posted on chart paper, made into a foldable, or recorded in a notebook by each student. (Look for evidence of MP1.) 4. Have students correct their graphs using one or more of the methods developed as a class. Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I measure student success? That is, deliberate consideration of what performances will convince you (and any outside observer) that your students have developed a deepened and conceptual understanding. Day 1: Students will relate key features of linear and exponential graphs with their equations. Day 2: Students will graph linear and exponential functions given their equations. Students will develop strategies to check the accuracy of their graphs using various technologies. Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc. Previous vocabulary: Words to describe key features of graphs, such as y-intercept, slope and end behavior Words to describe key features of functions/equations, such as rate of change, initial value, increase vs. decrease New vocabulary: HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement. Portsmouth, NH: Heinemann. Lesson Title: Graphing Linear and Exponential Functions Course: Common Core Algebra 1, Unit 2 Date: _____________ Teacher(s): ____________________ Start/end times: _________________________ Slope-intercept form: equations written in the form y mx b, where m is the slope and b is the y-intercept Connections: In the warm up, students practice previous skills of connecting contexts to graphs. Students should also have experience working between sets or tables of values and graphs. Common Mistakes/ Typical misconceptions: Reversing m and b. When using the TI-84 software, students will use the general form y ax b, which may be confusing. This is necessary because the software needs the variables to be in alphabetical order. Students may struggle initially with how to develop points on a graph without using a table of values. Determining the y-intercept will be a natural starting point, but developing further points (by using the rate of change) may be difficult at first. Notes: In Unit 1, students learned that “the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve,” but this will be students’ first experience graphing directly from an equation. Present this to students as a challenge. Consider planning out groupings ahead of time, to account for ability, personality and learning style. Resources: What materials or resources are essential for students to successfully complete the lesson tasks or activities? Homework: Exactly what follow-up homework tasks, problems, and/or exercises will be assigned upon the completion of the lesson? Graphing Calculators Computer with TI-nSpire software Resource Sheets http://education.ti.com/calculators/downloads/US/Activities /Detail?id=5558 for downloading instructions for using the Transformation Graphing App Day 1: Create a matching activity with two linear and two exponential functions and their graphs. You will exchange your activity with a partner tomorrow. Day 2: Complete the Day 2 Extension Activity. Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson? Can students describe how changes to an equation will affect the graph? Can students use key features of a graph to pick the correct equation? Do students know how to verify their graphs to an acceptable degree of accuracy? Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Sources: http://www.physics.isu.edu/radinf/natural.htm HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student achievement. Portsmouth, NH: Heinemann.