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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