Patterns, Relationships, and Algebraic Thinking

Transcription

1 Patterns, Relationships, and Algebraic Thinking Activity: TEKS: Sequences, Sequences Everywhere! (8.5) Patterns, relationships, and algebraic thinking. The student uses graphs, tables, and algebraic representations to make predictions and solve problems. The student is expected to: (B) find and evaluate an algebraic expression to determine any term in an arithmetic sequence (with a constant rate of change). (8.15) Underlying processes and mathematical tools. The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models. The student is expected to: (A) communicate mathematical ideas using language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models; and Overview: Materials: Students will describe and extend arithmetic sequences using both symbols and words; create artifacts in the form of tables, graphs, and notes that may be used to document participation and understanding; and use both oral and written communication skills to justify their solutions. Sequences Transparency Master Sequence Exploration Questions Student Handout Arithmetic Sequences: From Tables to Graphs Graph paper Graphing calculators Grouping: Whole group and small groups of 2-3 Time: 90 minutes Lesson: 1. Review the following vocabulary terms with the class: sequence constant rate of change element term arithmetic sequence iteration recursion constant Have students take notes during the vocabulary discussion component of the lesson. The following terms may be brand new to the students, in which case you will need to provide an example and definition. iteration- Repeating a set of rules Sequences, Sequences Everywhere! Page 1

2 Ask students to provide a definition and examples for each vocabulary term. or steps over and over. One step is called an iterate recursion- Given some starting information and a rule for how to use it to get new information, the rule is then repeated using the new information. 2. Present a few elements of a sequence to students and have them determine what should come next. Ask the class, "If I listed the following numbers, what would come next: 5, 10, 15, 20...?" If a student answers "25," then have the student suggest why he/she knew that was the next number. Ask the students what is being added or multiplied to get each new number. Assist the students in understanding that each number is obtained by adding 5 to the previous number. Summarize what has taken place thus far: This list of numbers that we have been discussing is a sequence. A sequence is a list of numbers in which each number depends on the one before it. We added a number to get from one element to the next in each sequence so we call it an arithmetic sequence. 3. Place the Sequences Transparency Master on the overhead. 2, 6, 10, 14, 18, , -3, -1, 1, 3 1, 4, 7, 10, 13, 16-1, 10, 21, 32, 43, Note: It is important to verify that all of the students made progress toward understanding the concepts presented in this component of the lesson. You may do this in one of several ways: Sequences, Sequences Everywhere! Page 2

3 3, 0, -3, -6, -9, ways: Grouping: partner or in groups of 3 Task 1: Explain in words how to determine the next 3 terms for each of the sequences. Task 2: Are these are examples of arithmetic sequences.? Support your reasoning with mathematical proof. Bring the class together and share some of the answers that the students obtained for each item on the transparency. Sample response for Task 1: First sequence: To get the first term, we start with 2 and add on 4's. To get to the second term we start with 2 and add one 4. To get to the third term, we start with 2 and add two 4's. To get the fourth term we start with 2 and add three 4's. Second sequence: To get the first term, we start with -5 and add on 2's. To get to the second term we start with -5 and add one 2. To get to the third term, we start with -5 and add two 2's. To get the fourth term we start with -5 and add three 2's. To get to the n th term, we start with -5 and add n - 1 twos. Let the students write a brief definition of a sequence on paper and provide an example to ensure that they have understood the lesson. Task 2: Ask the following questions to debrief this task. Is each arithmetic sequences? yes How do you describe difference between consecutive terms? It is always constant. How else may you describe the difference between consecutive Sequences, Sequences Everywhere! Page 3

4 terms? This difference may also be described as the constant rate of change. Provide the constant rate of change for each sequence. 4. Grouping: partner or groups of 3 Students have provided verbal responses, now they are to match algebraic representations to their verbal responses. Write these expressions on the board oneby-one (in this order) and ask students to match them to their verbal responses and check to see if they work to produce the next three terms in the sequence (n - 1) Question: How can we take this reasoning we explained orally and think of it in an algebraic representation or how to find the n th term? Key: 2 + 4(n - 1) Sequence n th term 3-3(n - 1) 1 + 3(n - 1) (n - 1) 2,6,10,14,18, (n - 1) -5,-3,-1, 1, (n - 1) 1, 4, 7,10,13, (n - 1) -1, 10, 21,32,43, (n - 1) 3, 0,-3,-6,-9, (n - 1) Discussion/Teacher notes: Students may want to express this as adding 4 each time, adding 2 each time, etc. Point out to students that the idea behind arithmetic sequences is to determine a common difference as well as and add-on values used to produce the sequence. Sequences, Sequences Everywhere! Page 4

5 A recursive definition, since each term is found by adding the common difference to the previous term is a k+1 = a k + d. For any term in the sequence, we've added the difference one less time than the number of the term. For example, for the first term, we haven't added the difference at all (0 times). For the second term, we've added the difference once. For the third term, we've added the difference two times. This recursive idea can be shown with technology. Type in the starting term of a sequence and enter. Now add the constant difference. For this example, the common difference is four. Sequences, Sequences Everywhere! Page 5

6 Notice however, it is easy to loose count of how many iterations have been made. A more descriptive form of sequence on the home screen is possible. Using curly braces, input {1,2}. The first number is going to serve as our counter and the second number is the starting value of our sequence. Now instruct the calculator to add one to the first value (the term number) and 4 to the second value (the value of that term in the sequence). In the syntax Ans(1) references the first value in the previous line and the Ans(2) references the second value in the previous line. Sequences, Sequences Everywhere! Page 6

7 Each time you enter, another term of the sequence is calculated along with the position in the sequence (term number). 5. Grouping: Independent Pass out Arithmetic Sequences: From Tables to Graphs handout. Have students complete the table using information from the previous lesson components. Start with the sequence of 2, 6, 10, 14, 18, Have students fill in the table and emphasize what should go in the process column. Remind them again that the idea behind arithmetic sequences is to determine a common difference as well as add-on values used to produce the sequence. In this case, we are adding 4 each time. Sequences, Sequences Everywhere! Page 7

8 6. Grouping: Independent Graphing activity. Have students create graphs using the data in the tables. Because these are terms of a sequence, students should not connect the dots with a line. This is discrete data. There is not point between terms 2 and 3 or terms 5 and Student Reflection: Give an example of an arithmetic sequence. You must provide mathematical proof that it is an arithmetic sequence by identifying that all consecutive terms have a common difference. Provide a non-example of an arithmetic sequence. Again, you must use mathematics to prove that it is not an arithmetic sequence. In your explanation, you must use the following vocabulary terms: Discussion/Teacher note: An arithmetic sequence is a linear function. Point this out to students once they have completed their graphs. Ask them to describe the Shape of the graph and also compare graphs to others. While it is not necessary to express this to students, it is important for the teacher to know at this point instead of y = mx + b, we write a n = dn + c where d is the common difference and c is a constant (not the first term of the sequence, however). Students will deal with this concept and representation much later, but the needed foundation is now being developed. Monitor students as they respond to the reflection activity. Remind them to use work from today s lesson to support their reflections. sequence constant rate of change element term arithmetic sequence iteration recursion constant Sequences, Sequences Everywhere! Page 8

9 Homework: Consider the following sequences from today s lesson: -5, -3, -1, 1, 3 1, 4, 7, 10, 13, 16-1, 10, 21, 32, 43, , 0, -3, -6, -9, Have students select 2 sequences and create a tabular representation which includes the process column and then draw the companion graphical representation. Term Process Number Assessment: Extensions: Resources: The reflection activity acts as an assessment for this lesson. Use graphing technology to demonstrate to students how this graph may be created. A nice variety of graph paper can be found at the following sights. or or Modifications: 1. Pass out Sequence Exploration Questions student handout. After you have explored the sequences, have students work in groups to answer the questions from the Sequence Exploration Questions handout. Some students may have difficulty with sequences that involve integers. Allow groups to work together to discuss and answer the questions. Conduct a brief class discussion after about 5 minutes of group discussion. 2. Use models with the table building component of the lesson for those students who may need the additional visual stimuli to make sense of the pattern. Counters may be used to model the sequences which do not include negative integers. Sequences, Sequences Everywhere! Page 9

10 Sequences Transparency Master Explain how to determine the next 3 terms for each of the following sequences: 2, 6, 10, 14, 18, , -3, -1, 1, 3 1, 4, 7, 10, 13, 16-1, 10, 21, 32, 43, , 0, -3, -6, -9, Sequences, Sequences Everywhere! Page 10

11 Sequence Exploration Questions Student handout 1. What effect does a negative starting number have on the sequence? 2. What effect does a large negative starting number have on the sequence? 3. What effect does a positive starting number have on the sequence? 4. What effect does a large positive number have on the sequence? 5. What effect does a negative add-on have on the sequence? 6. What effect does a positive add-on have on the sequence? Sequences, Sequences Everywhere! Page 11

12 Arithmetic Sequences: From Tables to Graphs Term Process Number n Sequences, Sequences Everywhere! Page 12

13 12 y-axis x-axis n term Sequences, Sequences Everywhere! Page 13

14 Arithmetic Sequences: From Tables to Graphs Key Term Process Number (2+4)+4 or (2+4+4)+4 or ( )+4 or ( )+4 or n n = 2 + 4(n - 1) Sequences, Sequences Everywhere! Page 14