University:
Jackson College
Course:
MAT-039 | Beginning Algebra
Academic year:

2024
Views:

256

Pages:

44
Author:

Aspen Harper

Arithmetic Sequences Lesson #1: Investigating Patterns and Sequences Overview In this unit we investigate patterns used to define types of sequences. We then focus on arithmetic sequences and apply formal language to increasing and decreasing linear patterns. We derive a rule for determining the general term of an arithmetic sequence and explore problems relating to arithmetic growth and decay. As an extension we explore arithmetic series. Investigation 1 Jesse is making a tower using playing cards. The top three rows of the tower are shown. The top row (Row 1) requires three playing cards. Row I Row 2 Row 3 The second row (Row 2) requires six additional playing cards. Continue the pattern for two more rows, and complete Tables A and B below. Row Number 12 3 45 Table A Number of Additional Cards in the Row 3 Table B Investigation 2 Row Number Number of Triangles in the Row 6 9 12 15 1 2 3 4 5 13579 Each row in the triangle, named after French mathematician Blaise Pascal, begins and ends with the number 1. Apart from the ones, every other number in a particular row can be determined by adding the two numbers diagonally above it. Continue the pattern for rows 5, 6, and 7, and complete Table C. Table C Row Number Sum of the Numbers in the Row 1 In grades 8 and 9 you learned how to determine a formula From a specifie pattern. In this lesson, He foous on investigatin patterns to Wentify different types of Sequences, a skill which is useful in patterns of numbers which are more difficult to identify In the next lesson we develope a fermula and use this formula for patterns of numbers which are arithmetic. 14 6 4 15 10 10 51 1 6 15 20 15 61 1 2 3 4 5 6 7 1248 16 32 64 Row 1 2 2 4 5 67 Copyright by Absolute Value Publications. This book is NOT covered by the Cancopy agreement 506 Arithmetic Sequences Lesson #1: Investigating Patterns and Sequences Investigation 3 Triangle 1, shown in Diagram 1, is an equilateral triangle with sides of length 32 cm. A smaller triangle, Triangle 2, is placed inside Triangle 1 by joining the midpoints of Triangle 1 (as illustrated in Diagram 2). The pattern is continued as illustrated in Diagram 3. 32 16 Diagram I Diagram 2 Diagram 3 a) Diagrams 4 and 5 in the sequence have not been completed. Complete each diagram. Diagram 4 (incomplete) Diagram 5 (incomplete) 1 b) Complete Tables D and E below. Table D Triangle Number 2 3 4 5 Length of Side of Triangle (cm) 32 16 842 3 Table E Diagram Number 1 1 2 4 5 Number of Triangles of any Size in the Diagram 59 13 17 Investigation 4 The mean daily temperature in Prince George on Dec 1 was 8°C. For the next six days, the mean daily temperature decreased by 4°C each day. Complete Table F below. Table F Day Number Temperature (°C) 1 2 84 345 0-4-8 Copyright by Absolute Value Publications. This book is NOT covered by the Cancopy agreement Arithmetic Sequences Lesson #1: Investigating Patterns and Sequences Group Investigation The following investigation can be used as a lead-in to the next lesson. The solution can be obtained by drawing a grid and using counters or coins; however, greater understanding may be obtained by having students act out the situation. A B C 1 S S 2 S S 3 S S Eight students are arranged in a 3x3 grid. Students can move either left/right or up/down into an empty cell. A "move" consists of any one student moving into an adjacent cell. a) Determine the smallest number of "moves" required for the student in cell A1 to end up in cell C3. 13 moves b) Extend your thinking by developing a strategy for determining the smallest number of "moves" required to move a student from cell A1 to cell Z26 in a 26x26 grid. Grid # of Moves 2x2 5 3x3 13 4x4 21 Looks like an Arithmetic Sequence (add 8 to the previous term) → Extend the pattern for a 26x26 grid. You will get 197 moves There is also a rule to get from grid size to # of moves. [(grid size)(8)] - 11 = # of moves [(26 x 8) - 11] = 197 moves 516 Arithmetic Sequences Lesson #1: Investigating Patterns and Sequences Answer Key 1. a) 2. a) b) 8, 12, 16 e) i) adding four to the previous term 11)14.1520,1624 Hi) arithmetic, linear b) 1,4,16,64 c) i) multiplying the previous term by 4 11) 1464,15256, 161024 iii) geometric, non-linear 3. a) arithmetic b) geometric e) geometric d) geometric e) neither f) arithmetic infinite 1) infinite 4. a) finite b) infinite c) infinite c) 1500,7500, 37 500 d) finite e) 5,8,13 5. b) 1,-1,1 6. a) Arithmetic. 20, 25. Add 5 to the previous term. b) Geometric. 40, 80. Multiply the previous term by 2. e) Arithmetic. 1,4. Add 3 to the previous term. d) Geometric. 10.5. e) Arithmetic. 4, -3. Add-7 to the previous term. f) Arithmetic. -17, -20. Add-3 to the previous term. Multiply the previous term by 1) 100, 50,0 7.) 8. a) b) 3,9,27 c) Geometric d) 2187 b) 2, 3, 5, 8, 13 c) Neither d) 8 e) 21 9. If the graph is linear, the sequence is arithmetic. Sequence 2 is arithmetic. 10. For sequence 1, multiply the previous term by 2. For sequence 2, add -5 to the previous term. 11. B 12. 1 2 8 Group Investigation ) 13 b) 2 x 2 grid 5 moves, 3 x 3 grid 13 moves. 4 x 4 grid 21 moves. Extend the pattern: 26 x 26 grid 197 moves. Copyright by Absolute Value Publications. This book is NOT covered by the Cancopy agreement.

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