Step 1 Lesson Plan



Author (s): Ben Pope and Jared ShaferTeam Members: Ben Pope and Jared ShaferTitle of Lesson: Sequences and SeriesLesson # 1Date lesson will be taught: 2-25 & 2-27Grade level: 11-12Lesson Source (kit, lesson): Power point from Elizabeth DennisConcepts/Main Idea – in paragraph form give a broad, global statement about the concepts and vocabulary you want students to understand as a result of doing this activity:We want students to recognize patterns by writing recursive and explicit formulas. We want the students to identify and describe geometric and arithmetic sequences. We want the students to calculate arithmetic and geometric means. Objective/s- Be specific; prioritize; include higher-order objectives; be sure they are measurable. Write objectives in SWBAT form…The Students Will Be Able To:Evaluation In the space below, explain the type(s) of evaluation that will provide evidence that students have learned the objectives of the lesson (formative and summative). You will provide student copies at the end of the lesson.-Recognize patterns using recursive and explicit formulas.-Use a formula for finding the nth term of a sequence.-Identify and describe geometric and arithmetic sequences.-Calculate arithmetic and geometric means.We will be asking them to do example problems in pairs 2, and we will be checking their answers and reasoning. We will also give a post-test after each day to check their understanding.NGSS and Common Core StandardsMath Lessons: Must include a minimum of one Common Core Math Practice Standard (number and name of standard)CCSS.Math.Practice.MP8?Look for and express regularity in repeated reasoning.Must include a minimum of one Common Core Math Content Standard (domain, cluster, standard)CCSS.Math.Content.HSF-BF.A.2?Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.A minimum of one NGSS Science and Engineering Practice (number and name of practice)Practice 5: Using Mathematics and Computational Thinking. Students are expected to use mathematics to represent physical variables and their relationships, and to make quantitative predictions.Materials list (BE SPECIFIC about quantities) for Whole Class:used projector and doc camper Group:per Student:printed off handoutAdvance preparation:Print off handout at the end of this document and draw in a few progressing polygons with equal dashed lines as sides. Go over power points to be sure what is being covered and when.Include handouts at the end of this lesson plan document (blank page provided)Accommodations: Include a general statement and any specific student needsWe are having information given orally and visually for visually or hearing impaired students. Safety: Include a general statement and any specific safety concerns Make sure each group is working safely together, both physically and non physically. Everyone is working fairly with each other. Follow proper fire drill procedure (a fire drill happened when we taught this, and we had to follow those rules).Engagement: Estimated Time: 10 minWhat the teacher does AND how will the teacher direct students: (Directions)Probing Questions: Critical questions that will connect prior knowledge and create a “Need to know”Expected Student Responses AND Misconceptions - think like a student to consider student responses INCLUDING misconceptions:Bell RingersDay 1: Start by writing the numbers 1-10 on the board, then 1-20 (abbreviated with …), and then 1-100 (abbreviated with …). Tell the students they have 5 minutes to find the sum of each of the 3 lists of numbers. At the end, ask for the answers and have the students explain how they came up with those answers. If one calculated 1-100 then make sure they explain how. Afterwards go over how you can recognize patterns to make solving the problem easier. Suggest using the number of terms to help solve the problem. Use questions to have the students bring out the answer for a formula. (n+1) x n/2.Day 2: Pass out the bell ringer sheet attached at the end of the lesson. Allow 5 minutes for the students to complete it. Once done, collect them. After all have been collected, lead a discussion about what answers they put and why. Use this as a review of the material given from day 1.What methods did you use to solve it? Did you just add all of the terms up? Is there an easier way? How did you get your answer?How many terms are there? What relationship does the number of terms have to the sum of the sequence?What did you get for this problem? Why did you get that answer? How was finding the explicit formula? What did you do to find it? What is the recursive formula (even though it isn’t asked for)? What is the difference between recursive and explicit formulas in sequences?Students should answer with: “I added all of the terms up and got the answer.” “The number of terms is used to answer the problems.” Students’ answers will vary on the answers. Some might give the recursive formula instead of the explicit. Make sure they remember the difference between the two. Teacher Decision Point Assessment:The majority of the class can provide the correct reasoning to the answer.Exploration: Estimated Time: 30 minWhat the teacher does AND what the teacher will direct students to do: (Directions)Probing Questions: Critical questions that will guide students to a “Common set of Experiences”Expected Student Responses AND Misconceptions - think like a student to consider student responses INCLUDING misconceptions:Day 1:Use the power point attached to this document as a guide for the lesson. It will start with “the cell phone” example to discover what a formula is. Then move on to the next slide describing what a sequence is. Continue to go over the example of a model sequence. Have the students answer the following questions on their own. Discuss the importance of “n”, and continue into the recursive formula. Go over the examples and practice problems. Continue into the explicit formula. Go over the examples and practice problems.Day 2: We will use the power point labeled day 2 attached to this document to guide the lesson. It will start with arithmetic sequences what they are and are not. Then we cover arithmetic mean. We will attempt to have the students discover a method for this. Then we will move onto geometric series and geometric mean. We will attempt to have the students discover a method for the geometric mean. This will be mixed with the explanation because we will have students explain how they got their answers during each slide.How did you generate that pattern? Is there a set way to create all formulas to patterns? Do all patterns look the same? What does the word sequence mean? Can you give an example? How did you know what to draw next for the triangle? What does the word recursive mean? What does “n” represent? What if it is a subscript to a letter? What about a number as a subscript instead of “n”? Where have you seen the recursive formula? How do you find the next terms with this formula? What does the word explicit mean? Where have you seen the explicit formula? How do you find terms with the explicit formula? What is the difference between explicit and recursive formulas? How does the triangle sequence picture trick you into thinking it is arithmetic? What is the recursive or explicit formula for the arithmetic \ geometric sequence? How can you verify your method for finding arithmetic \ geometric mean is correct? Is there an alternative way of looking at this sequence? Can the sequences be wrote differently?Answers will vary for all questions. Some students might mix up the definition of explicit and recursive. Students might get the wrong definition of explicit and recursive. Some students might not be able to see where the next terms are generated. Some students might not be able to understand the difference between explicit and recursive formulas.Day 2: Students will likely pick up on arithmetic sequences quickly. Hopefully some of them will be able to figure out a method to find the arithmetic mean and share it with the class. Students will likely pick up on the simple geometric sequences, but may struggle on the more complex sequences. We need to ask leading questions that help them look at the sequences in different perspectives if they are struggling.Teacher Decision Point Assessment:Day 1: The majority of the class can explain the reasoning behind their answers.Explanation: Estimated Time:What the teacher does AND what the teacher will direct students to do: (Directions)Clarifying Questions: Critical questions that will help students “Clarify their Understanding” and introduce information related to the lesson concepts & vocabulary – check for understanding (formative assessment)Expected Student Responses AND Misconceptions - think like a student to consider student responses INCLUDING misconceptions:Day 1: This is very intertwined with the exploration. It is all written up there.Day 2: Same as day one.Day 1: Questions are written in the exploration.Day 1: Responses are written in the exploration.Teacher Decision Point Assessment:Day 1: Listed above under exploration.Elaboration: Estimated Time: 5 minWhat the teacher does AND what the teacher will direct students to do: (Directions)Probing Questions: Critical questions that will help students “Extend or Apply” their newly acquired concepts/skills in new situationsExpected Student Responses AND Misconceptions - think like a student to consider student responses INCLUDING misconceptions:Day 2: We will use the slide on the power point that asks. Can you right a series or sequence that defines multiplication?We will show farther elaboration of this concept by showing them examples of series and sequences being used in higher levels of mathematics. Such as matrix multiplication.Is your series geometric or arithmetic?Why are some of your answers different?Can you defend your answer?How do you know your equation works?Is this explicit or recursive?Is this geometric or arithmetic?Is this a different kind of series or sequence than we have been talking about?What are the strengths and weaknesses of your equation?What do you think this says about the methods of writing equations we learned previously?Why do you think this is an important mathematical concept?Students may not be able to write one that is completely correct because we never covered exactly how to do this. Although they may not get it exactly right they have the tools to get close. Students may accidently use some form of multiplication in their equation. This should not be allowed.Teacher Decision Point Assessment:Evaluation: Estimated Time: 5 minCritical questions that ask students to demonstrate their understanding of the lesson’s performance objectives.Formative Assessment(s): In addition to the final assessment (bell ringer or exit slips), how will you determine students’ learning within this lesson: (observations, student responses/elaborations, white boards, student questions, etc.)? We have a bell ringer for each day. We also are interacting with the class by having them explain their answers. If the students are not able to explain well then we know that we need to continue practicing these types of problems.Walking around and checking student work. Also have students give reasoning to problems to the class. Summative Assessment: Provide a student copy of the final assessment/exit slips or other summative assessments you use in the lesson Summative Assessment is included in the power point as the last slide. Have students answer it on a separate piece of paper.\sBell-RingerName: ______________Patterns:Draw the next step:Draw a quick sketch of step 27 (you can write the number of each line used):Complete the table:Step n123451027# of linesWrite the explicit formula: ................
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