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Identifying Patterns and Extending Arithmetic Sequences in Mathematics

This lesson focuses on understanding patterns and sequences in arithmetic. Students will learn to identify relationships between terms in a sequence and understand how to extend them. Through various examples, students will practice finding missing terms by recognizing patterns such as adding or subtracting constants, or multiplying and dividing values in a sequence. The course encompasses vocabulary related to sequences, such as "arithmetic sequence" and "perfect square," providing a solid foundation for further exploration in mathematics.

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Identifying Patterns and Extending Arithmetic Sequences in Mathematics

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  1. 1-7 Patterns and Sequences Course 1 Warm Up Problem of the Day Lesson Presentation

  2. Warm Up Determine what could come next. 1.3, 4, 5, 6, ___ 2. 10, 9, 8, 7, 6, ___ 3. 1, 3, 5, 7, ___ 4. 2, 4, 6, 8, ___ 5. 5, 10, 15, 20, ___ 7 5 9 10 25

  3. Learn to find patterns and to recognize, describe, and extend patterns in sequences.

  4. Vocabulary perfect square term arithmetic sequence

  5. + 3 • + 3 • + 3 Each month, Eva chooses 3 new DVDs from her DVD club. Position Value 6 9 12 The number of DVDs Eva has after each month shows a pattern: Add 3. This pattern can be written as a sequence. 3, 6, 9, 12, 15, 18, …

  6. A sequence is an ordered set of numbers. Each number in the sequence is called a term. In this sequence, the first term is 3, the second term is 6, and the third term is 9. When the terms of a sequence change by the same amount each time, the sequence is an arithmetic sequence.

  7. Helpful Hint Look for a relationship between the 1st term and the 2nd term. Check if this relationship works between the 2nd term and the 3rd term, and so on.

  8. Additional Example 1A: Extending Arithmetic Sequences Identify a pattern in each sequence and then find the missing terms. 48, 42, 36, 30, , , , . . . –6 –6 –6 –6 –6 –6 Look for a pattern. A pattern is to subtract 6 from each term to get the next term. 30 – 6 = 24 24 – 6 = 18 18 – 6 = 12 So 24, 18, and 12 will be the next three terms.

  9. +13 • +13 • +13 • +13 • +13 Additional Example 1B: Extending Arithmetic Sequences A pattern is to add 13 to each term to get the next term. 48 + 13 = 61 61 + 13 = 74 So 61 and 74 will be the next terms in the arithmetic sequence.

  10. Check It Out: Example 1A Identify a pattern in each sequence and name the next three terms. 39, 34, 29, 24, , , , . . . –5 –5 –5 –5 –5 –5 Look for a pattern. A pattern is to subtract 5 from each term to get the next term. 24 – 5 = 19 19 – 5 = 14 14 – 5 = 9 So 19, 14, and 9 will be the next three terms.

  11. Additional Example 2A: Completing Other Sequences Identify a pattern in the sequence. Name the missing terms. 24, 34, 31, 41, 38, 48, , , ,… +10 –3 +10–3+10–3 +10 –3 A pattern is to add 10 to one term and subtract 3 from the next. 48 –3 = 45 45 + 10 = 55 55 – 3 = 52 So 45, 55, and 52 are the missing terms.

  12.  4 •  4 •  4 •  4 • ÷2 • ÷2 • ÷2 Additional Example 2B: Completing Other Sequences A pattern is to multiply one term by 4 and divide the next by 2. 8 ÷ 2 = 4 4  4 = 16 16 ÷ 2 = 8 8  4 = 32 So 4 and 8 will be the missing terms in the sequence.

  13.  6 • ÷2 •  6 •  6 • ÷2 • ÷2 •  6 Check It Out: Example 2B A pattern is to multiply one term by 6 and divide the next by 2. 18 ÷ 2 = 9 9  6 = 54 54 ÷ 2 = 27 27  6 = 162 So 9 and 27 will be the missing terms in the sequence.

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