Understanding By Design Unit Template



Understanding By Design Unit Template

|Title of Unit |     Series, Sequences, and Mathematical Induction |Grade Level |     8-12 |

|Curriculum Area |     Honors Pre-Calculus |Time Frame |3/26/13-4/23/13 |

|Developed By |     Kristina Messina |

|Identify Desired Results (Stage 1) |

|Content Standards |

|P8.1- Know, explain, and use sigma and factorial notation. |

|P8.2- Given an arithmetic, geometric, or recursively defined sequence, write an expression for the nth term when |

|possible. Write a particular term of a sequence when given the nth term. |

|P8.3- Understand, explain, and use the formulas for the sums of finite arithmetic and geometric sequences. |

|P8.4- Compute the sums of infinite geometric series. Understand and apply the convergence criterion for geometric series. |

|P8.5- Understand and explain the principle of mathematical induction and prove statements using mathematical induction. |

|Understandings |Essential Questions |

|Overarching Understanding |Overarching |Topical |

| Arithmetic sequences can reduce the amount of time it takes to find the sum of a sequence of numbers with a |If a sequence is arithmetic what does it mean? | What are arithmetic sequences? |

|common difference. |If a sequence is geometric what does it mean? |What are geometric series? |

|Geometric sequences can reduce the amount of time it takes to find the sum of a sequence of numbers with a |How can we use arithmetic sequences and |What does it mean to find the sum of an |

|common ratio. |geometric sequences in the real world? |arithmetic series? How can you find the sum? |

|Mathematical Induction can be used to prove statements involving positive integers. |Why is learning mathematical induction |What does it mean to find the sum of a |

| |important? |geometric series? How can you find the sum? |

| |What does mathematical induction allow us to do|What is a recursive formula? When can a |

| |with series and sequences general formulas? |recursive formula be useful? |

|Related Misconceptions | | |

|Mathematical Induction- students will freak about using a style of proof such as mathematical induction and why | | |

|it is important. Students may think that it is the same as a general formula which is not the case, because | | |

|mathematical induction validates that it works for all case. A general formula can be proven wrong, but | | |

|mathematical induction proves that it works for all cases possible for the limits and given information of the | | |

|situation. | | |

|Students will get confused on the difference between geometric and arithmetic series. Students might get | | |

|confused on what it means for arithmetic series to have a common difference and a geometric series to have a | | |

|common ratio. | | |

|Infinite geometric series- Students will first be a little confused on how to generate their own infinite | | |

|geometric series equation and what it actually means. Students will get confused how to use the infinite | | |

|geometric series. | | |

|Objectives |

|Knowledge |Skills |

|Students will know… |Students will be able to… |

|Students will know the difference between an arithmetic sequence and a geometric sequence. Students will realize|Students will be able to write arithmetic and geometric formulas. |

|that an arithmetic sequence has a common difference and a geometric sequence has a common ratio. |Students will be able to find the sum and partial sum of arithmetic and geometric sequences. |

|Students will know that arithmetic and geometric sequences as well as their sums can be used to solve real-life |Students will be able to use mathematical induction to prove that a sequence works for all |

|situations. |cases. |

|Students will know that mathematical induction validates that a conjecture or in this particular unit a series | |

|is valid and works for all cases. | |

|Assessment Evidence (Stage 2) |

|Performance Task Description |

|Goal |Students will know the difference between an arithmetic sequence and a geometric sequence. Students will realize that an arithmetic sequence has a common |

| |difference and a geometric sequence has a common ratio. |

| |Students will know that arithmetic and geometric sequences as well as their sums can be used to solve real-life situations. |

| |Students will know that mathematical induction validates that a conjecture or in this particular unit a series is valid and works for all cases. |

| |Students will be able to write arithmetic and geometric formulas. |

| |Students will be able to find the sum and partial sum of arithmetic and geometric sequences. |

| |Students will be able to use mathematical induction to prove that a sequence works for all cases. |

|Role |It is a test that covers series, sequences, and mathematical induction (9.2-9.4) |

|Audience | I will be evaluating the test, looking on the completion, and the work students provide for each question. I will grade the tests and really pick through |

| |students’ work to gauge students’ understanding after taking this summative assessment to see where students ended after the test. |

|Situation |The summative assessment will be completed in class March 19, 2013. |

|Product/Performance |It will be a completed test that will cover arithmetic sequences, geometric sequences, and mathematical induction. |

|Standards |P8.1- Know, explain, and use sigma and factorial notation. |

| |P8.2- Given an arithmetic, geometric, or recursively defined sequence, write an expression for the nth term when |

| |possible. Write a particular term of a sequence when given the nth term. |

| |P8.3- Understand, explain, and use the formulas for the sums of finite arithmetic and geometric sequences. |

| |P8.4- Compute the sums of infinite geometric series. Understand and apply the convergence criterion for geometric series. |

| |P8.5- Understand and explain the principle of mathematical induction and prove statements using mathematical induction. |

|Other Evidence |

|Every lesson I will be asking assessing questions to see where students are in their understanding then use that information to determine if I need to spend more time of certain concepts before moving on to|

|the next concept. |

|Entry slips- I am going to use entry slips sometimes to see where my students are at conceptually wise with understanding the previous lesson content. I will probably have a discussion about the entry slips|

|so that I can get immediate feedback on what they know and build from there. |

|Exit slips- sometimes I will use exit slips to see what the students learned in the lesson, and use that information to address misconceptions the next day. |

|Learning Plan (Stage 3) |

|Day in Unit |Lesson Topic |Lesson Learning Objective |Description of how lesson contributes to unit-level |Assessment activities |

| | | |objectives | |

|(1) Friday, March 8, 2013 |Arithmetic Sequences|To recognize, write, and find the nth terms of |This lesson contributes to the following objectives |Assessing questions |

| | |arithmetic sequences |and overarching understanding: |Essential questions: |

| | |To find nth partial sums of arithmetic sequences |Arithmetic sequences can reduce the amount of time |If a sequence is arithmetic what does it mean? |

| | |To use arithmetic sequences to model and solve |it takes to find the sum of a sequence of numbers |How can we use arithmetic sequences and geometric |

| | |real-life problem |with a common difference. |sequences in the real world? |

| | | |Students will know the difference between an |What are arithmetic sequences? |

| | | |arithmetic sequence and a geometric sequence. |What does it mean to find the sum of an arithmetic |

| | | |Students will realize that an arithmetic sequence |series? How can you find the sum? |

| | | |has a common difference and a geometric sequence has|Activity: Find the sum of 1 to 100. Write a formula |

| | | |a common ratio. |that works from 1 to n. Why does this work? What |

| | | |Students will know that arithmetic and geometric |type of sequence is this? Why is it true? How can we|

| | | |sequences as well as their sums can be used to solve|use this n formula to find sums of any arithmetic |

| | | |real-life situations. |sequence? |

| | | |Students will be able to write arithmetic and | |

| | | |geometric formulas. | |

| | | |Students will be able to find the sum and partial | |

| | | |sum of arithmetic and geometric sequences. | |

|(2) Monday, March 11, 2013 |Arithmetic Sequences|To recognize, write, and find the nth terms of |This lesson contributes to the same objectives as |Assessing questions |

| | |arithmetic sequences |above. |Essential questions: |

| | |To find nth partial sums of arithmetic sequences | |If a sequence is arithmetic what does it mean? |

| | |To use arithmetic sequences to model and solve | |How can we use arithmetic sequences and geometric |

| | |real-life problem | |sequences in the real world? |

| | | | |What are arithmetic sequences? |

| | | | |What does it mean to find the sum of an arithmetic |

| | | | |series? How can you find the sum? |

| | | | |Exit Ticket: In the last two lessons we talked about|

| | | | |arithmetic sequences and how to find the sum of |

| | | | |arithmetic sequences. With the knowledge that we |

| | | | |covered in the last two days what do you think we |

| | | | |will be learning tomorrow? Also: Name one sequence |

| | | | |that would not be arithmetic and explain why that is|

| | | | |a true statement. |

|(3) Tuesday, March 12, 2013|Geometric Sequences |To recognize, write, and find the nth terms of |This lesson contributes to the following objectives | Assessing questions |

| | |geometric sequences |and overarching understandings: |Essential questions: |

| | |To find sums of infinite geometric series |Geometric sequences can reduce the amount of time it|If a sequence is geometric what does it mean? |

| | | |takes to find the sum of a sequence of numbers with |How can we use arithmetic sequences and geometric |

| | | |a common ratio. |sequences in the real world? |

| | | |Students will know the difference between an |What are geometric series? |

| | | |arithmetic sequence and a geometric sequence. |What does it mean to find the sum of a geometric |

| | | |Students will realize that an arithmetic sequence |series? How can you find the sum? |

| | | |has a common difference and a geometric sequence has|Activity: Double the penny activity. Can the n |

| | | |a common ratio. |formula represent an arithmetic sequence? Why or why|

| | | |Students will know that arithmetic and geometric |not? What do you notice about the difference between|

| | | |sequences as well as their sums can be used to solve|each number in the sequence? Why do you think it is |

| | | |real-life situations. |a common difference? (Introduction to geometric |

| | | |Students will be able to write arithmetic and |sequences) |

| | | |geometric formulas. | |

| | | |Students will be able to find the sum and partial | |

| | | |sum of arithmetic and geometric sequences. | |

| | | | | |

|(4) Wednesday, March 13, |Geometric Sequences |To recognize, write, and find the nth terms of |This lesson contributes to the same objectives as | Assessing questions |

|2013 | |geometric sequences |above. |Essential questions: |

| | |To find sums of infinite geometric series | |If a sequence is geometric what does it mean? |

| | | | |How can we use arithmetic sequences and geometric |

| | | | |sequences in the real world? |

| | | | |What are geometric series? |

| | | | |What does it mean to find the sum of a geometric |

| | | | |series? How can you find the sum? |

| | | | |Self-Assessment Exit Ticket: 3-2-1(3 things that you|

| | | | |know, 2 questions, and 1 concepts you want some time|

| | | | |to review) |

| | | | |Entry Ticket: Create your own arithmetic sequence |

| | | | |and geometric sequence. What is the difference |

| | | | |between geometric and arithmetic sequences? |

|(5) Thursday, March 14, 2013|Mathematical |To use mathematical induction to prove statements |This lesson contributes to the following objectives |Assessing questions |

| |Induction |involving a positive integer n |and overarching understandings: |Essential questions: |

| | |To find sums of powers of integers |Mathematical Induction can be used to prove |Why is learning mathematical induction important? |

| | |To find finite difference of sequences |statements involving positive integers. |What does mathematical induction allow us to do with|

| | | |Students will know that mathematical induction |series and sequences general formulas? |

| | | |validates that a conjecture or in this particular |What is mathematical induction? Why do we use |

| | | |unit a series is valid and works for all cases. |mathematical induction? |

| | | |Students will be able to use mathematical induction | |

| | | |to prove that a sequence works for all cases. | |

|(6) Friday, March 15, 2013 |Mathematical |To use mathematical induction to prove statements |This lesson contributes to the same objectives as |Assessing questions |

| |Induction |involving a positive integer n |above. |Essential questions: |

| | |To find sums of powers of integers | |Why is learning mathematical induction important? |

| | |To find finite difference of sequences | |What does mathematical induction allow us to do with|

| | | | |series and sequences general formulas? |

| | | | |What is mathematical induction? Why do we use |

| | | | |mathematical induction? |

| | | | |Peer Assessment: Have students work on using |

| | | | |mathematical induction to prove the warm-up |

| | | | |conjecture is true then have students swap them and |

| | | | |critique each other’s work on mathematical |

| | | | |conjecture by stating what was good about their use |

| | | | |of mathematical induction and what needs improvement|

| | | | |on using mathematical induction. Then have them |

| | | | |sign it before giving the papers back to the |

| | | | |rightful owner. |

|(7) Monday, March 18, 2013 |Review |To self-assess the material that you understand and|This is allows students to assess their |Review Assignment |

| | |what you still don’t understand |understanding of the unit objectives and overarching| |

| | | |understanding to see what they understand and what | |

| | | |they still don’t get. | |

|(8) Tuesday, March 19, 2013|Quest 9.1-9.4 |To be tested on the series, sequences, and |This quest tests on all the knowledge and skills |Quest 9.2-9.4 |

| | |mathematical induction unit. |objectives of the unit. | |

| | | | | |

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