1PHM5eTE_02_p060-115 11/4/03 8:10 PM Page 63 2-1 Describing a Pattern 2-1 1. Plan What You’ll Learn Check Skills You’ll Need For help, go to Lesson 1-3. Lesson Preview OBJECTIVE To continue a PowerPoint 1 1–6. See margin. number pattern Order each set of decimals from least to greatest. Check Skills You’ll Need OBJECTIVE 1. 0.105, 0.0105, 10.5 2. 3.331, 3.1, 3.31 2 To write a rule for a number pattern Ordering Decimals 3. 9.06, 9.6, 9.09 4. 20.06, 26.0, 0.602 Lesson 1-3: Example 4. Extra . And Why 5. 100.1, 101.0, 100.01 6. 0.99, 0.4, 0.35 Practice p. 642. To find the number of tiles in a pattern, as in Lesson Resources Example 1 New Vocabulary • term • conjecture Optional Materials • graph paper OBJECTIVE Interactive lesson includes instant self-check, tutorials, and activities. 1 Continuing a Number Pattern Teaching Resources Practice, Reteaching, Enrichment vestigation: Recognizing and Extending Patterns Reaching All Students In Practice Workbook 2-1 The first three designs in a pattern Spanish Practice Workbook 2-1 are shown at the right. Guided Problem Solving 2-1 1. Continue the pattern. Sketch Presentation Assistant Plus! the fourth and fifth designs on Transparencies grid paper. See margin. • Check Skills You’ll Need 2-1 • Problem of the Day 2-1 2. How many squares are in the • Additional Examples 2-1 Student Edition Answers 2-1 fourth design? The fifth design? • 13 squares; 17 squares • Lesson Quiz 2-1 3. Copy and complete the table. PH Presentation Pro CD-ROM 2-1 Design Number 1 2 34567 Number of Squares 1 5 Computer Test Generator CD 913172125 4. Answers may vary. 4. Reasoning Describe how you will find the tenth design Technology ® Sample: Multiply in the pattern. Resource Pro CD-ROM (10 – 1), or 9, by 4, Computer Test Generator CD and then add 1. PH Presentation Pro CD-ROM www.PHSchool.com The numbers 1, 4, 7, 10, cform a number pattern. Each number in the Student Site Reading Math pattern is a term. For example, the third term in this pattern is 7. • Teacher Web Code: aak-5500 Read the three dots • Algebra Readiness Puzzles 45 in the pattern 1, 4, 7, 10, cas “and so on.” The three dots after the number 10 tell you that the pattern continues • Self-grading Lesson Quiz beyond what is shown. PH SuccessNet Teacher Center • Lesson Planner 2-1 Describing a Pattern 63 • Resources Plus ANT ST C H N I E C K M S E Y T S Ongoing Assessment and Intervention Before the Lesson During the Lesson After the Lesson Diagnose prerequisite skills using: Monitor progress using: Assess knowledge using: • Check Skills You’ll Need • Check Understanding • Lesson Quiz • Additional Examples • Computer Test Generator CD • Test Prep 63 1PHM5eTE_02_p060-115 5/22/03 2:49 PM Page 64 impos03 203:phmxt1:phmxt1ch02:mxt1ch02%0: 2. Teach A conjecture is a prediction about what may happen. You can use the terms you know to make a conjecture about how a pattern will continue. Math Background Real 1 EXAMPLE Finding Number Patterns World Patterns are the cornerstone of mathematics. In fact, mathematics Decorating Jacob is creating a pattern of colored tiles for a wall in his has been described as “the search bathroom. The first four designs are shown below. How many squares will for pattern.” In this lesson, be in the fifth and sixth designs? students learn to recognize, extend, and describe addition/subtraction patterns and multiplication/division patterns. In an addition/subtraction pattern, Count the squares in each design. There are 1, 4, 7, and 10 squares. Notice each term is derived from the one that each design has three more squares than the one before it. before it by consistently adding or subtracting the same amount. 3 3 Add 3 to 10 to get the fifth term. Formally, this type of pattern is 1, 4, 7, 10, 13, 16, . d called an arithmetic sequence. Add 3 to 13 to get the sixth term. In a multiplication/division So, the fifth and sixth designs will have 13 and 16 squares. pattern, each term is derived from the one before it by consistently Check Understanding 1 Write the next two terms in each number pattern. multiplying or dividing by the c c c same amount. Patterns of this type 1a. 41, 51 a. 1, 11, 21, 31, b. 56, 48, 40, 32, c. 29, 36, 43, 50, are called geometric sequences. 1b. 24, 16 d. Reasoning In Example 1, the eighth design will go all the way 1c. 57, 64 across Jacob’s bathroom wall. How many tiles will be in the Teaching Notes eighth design? 22 tiles English Learners OBJECTIVE When students hear the word pattern, they might think of a 2 Writing a Rule for a Number Pattern repeating pattern, the type of pattern they see on rugs, quilts, and other decorative items; Help One way you can describe a number pattern is to give the first term and the students understand that the rule. A rule is an explanation of how you go from one term to the next. word pattern here refers to a rule that is repeated again and again. 2 EXAMPLE Writing Number Patterns From Rules Investigation (Optional) Lead students to see the Write the first six terms in the number pattern described by this rule: following relationships: Start with 1 and multiply by 2 repeatedly. • Each design in the geometric pattern has four more squares The first term is 1. T than the preceding design. 2 2 2 2 2 • Each term in the number d Multiply each term by pattern is four greater than the 1, 2, 4, 8, 16, 32 2 to find the next term. term that precedes it. Check Understanding 2 Write the first six terms in each number pattern. 1 EXAMPLE Teaching Tip a. Start with 90 and subtract 15 repeatedly. 90, 75, 60, 45, 30, 15 After discussing Example 1, ask: b. Start with 1 and multiply by 3 repeatedly. 1, 3, 9, 27, 81, 243 How would the pattern be c. Start with 17 and add 19 repeatedly. 17, 36, 55, 74, 93, 112 different if the first term were 2? Each term would increase by 1; the pattern would be 2, 5, 8, 11, 14, . 64 Chapter 2 Patterns and Variables Reaching All Students Below Level Have students practice Advanced Learners Have students English Learners skip-counting by 2, by 3, by 4, and so on. create original geometric patterns See note on page 64. 2, 4, 6, 8, 10, ...; 3, 6, 9, 12, 15, ...; that represent number patterns, Inclusion 4, 8, 12,16, 20, ...; and so on such as the designs in the See note on page 65. Investigation and in Example 1. 64 1PHM5eTE_02_p060-115 5/21/03 5:53 PM Page 65 impos03 203:phmxt1:phmxt1ch02:mxt1ch02%0: 2 EXAMPLE Teaching Tip EXAMPLE 3 EXAMPLE Writing a Rule After discussing Example 2, ask: How would the pattern be Patterns Write the next three terms in each pattern. Then write a rule for different if the first term were 2? the pattern. It is the same pattern, but a. 53, 49, 45, 41, . without the term 1; the pattern would be 2, 4, 8, 16, 32, . 4 4 4 4 4 4 53, 49, 45, 41, 37, 33, 29 To get from one term to the next, subtract 4. 13 EXAMPLE Inclusion Stress the importance of The rule is start with 53 and subtract 4 repeatedly. identifying the two parts of the rule: the first term, and the b. 1.5, 4.5, 13.5, 40.5, . explanation of how you go from 3 3 3 3 3 3 one term to the next. Ask: When To get from one term to the next, 1.5, 4.5, 13.5, 40.5, 121.5, 364.5, 1,093.5 you write the rule for the pattern multiply by 3. in part a, why is it not enough to The rule is start with 1.5 and multiply by 3 repeatedly. say, “Subtract 4 from a term to get the next term.”? Sample: That sentence by itself could Check Understanding 3 Write the next three terms and write a rule to describe each describe a different pattern, such number pattern. See margin. as 20, 16, 12, 8, . a. 1, 7, 49, 343, c b. 10.0, 8.8, 7.6, c c. 256, 128, 64, c PowerPoint Additional Examples 1 Write the next two terms in For more practice, see Extra Practice. this number pattern. EXERCISES 5, 12, 19, 26, . 33, 40 2 Write the first six terms in the number pattern described by A Practice by Example Write the next two terms in each number pattern. this rule: Start with 47 and subtract 3 repeatedly. 47, 44, Example 1 1. 2, 6, 10, 14, c 18, 22 2. 99, 88, 77, 66, c 55, 44 41, 38, 35, 32 (page 64) 3. 1, 3, 9, 27, c 81, 243 4. 1, 1.4, 1.8, 2.2, c 2.6, 3.0 3 Write the next three terms and write a rule to describe this number pattern. Example 2 Write the first six terms in each number pattern. 2.3, 4.4, 6.5, 8.6, . 10.7, 12.8, (page 64) 14.9; rule: Start with 2.3 and 5. Start with 7 and add 4 repeatedly.