Number and Shape Patterns

Part 1: Introduction Number and Shape Patterns Develop Skills and Strategies CCSS 4.OA.C.5 You have used rules to describe patterns in numbers. In this lesson, you will explore patterns further. Take a look at this problem. What are the next two numbers in the pattern below? 5, 10, 15, 20, 25,, Explore It Use the math you already know to solve the problem. What are two ways to get from 5 to 10? How do you get from 10 to 15? What rule works for all of the numbers in the pattern? How can you find the number that comes after 25? What are the next two numbers in the pattern? 66

Part 1: Introduction Find Out More Is add 5 the only relationship between the numbers in the pattern? Look at the numbers, along with the model of the numbers, and try to find a different pattern. 5, 10, 15, 20, 25 If you look only at the ones digits, you see that they alternate between 5 and 0. From the model, you can see that the number of dots alternates between odd and even. The pattern below also follows the rule add 5. 22, 27, 32, 37, 42 Do the other relationships you found in the first pattern apply to this pattern, too? The numbers in this pattern also alternate between odd and even, but the ones digits alternate between 2 and 7 instead of between 5 and 0. So, sets of numbers can share some patterns or rules, but have others that are different. Reflect 1 Describe a pattern that you have noticed in the real world. 67

Part 2: Modeled Instruction Read the problem below. Then explore different ways to understand it. Orlando does push-ups every day. Each day, he wants to do 4 more push-ups than the day before. Find out how many push-ups Orlando will do each day this week if he does 20 on Monday. Picture It You can use a table to help understand the problem. Day Monday Tuesday Wednesday Thursday Friday Number of Push-ups 20 +4 +4 +4 +4 Model It You can also use a number line to help understand the problem. 14 14 14 14 20 24 28 32 36 Monday Tuesday Wednesday Thursday Friday Start at 20, the number of push-ups done on Monday, and then count 4 more for each day. 68

Part 2: Guided Instruction Connect It Now you will explore the problem from the previous page further. 2 How many push-ups did Orlando do each day? Monday: Tuesday: Wednesday: Thursday: Friday: 3 What is the rule for the pattern? 4 What does the pattern tell you about what happens when you start with an even number and add an even number? 5 What other pattern(s) do you see in this set of numbers? 6 Explain how you found the additional pattern(s). Try It Use what you just learned to solve these problems. 7 Lori scored 100 points in a game, then doubled her score each of the next 3 times she played. What were Lori s scores the first 4 times she played the game? 8 What is one additional pattern in Lori s scores? 69

Part 3: Modeled Instruction Read the problem below. Then explore different ways to understand it. Camille made a shape pattern with pattern blocks that goes back and forth between a triangle and a square. Draw the pattern that Camille made. Picture It You can use models to help understand the problem. Start by describing the pattern with words. Repeat the pattern at least 3 times. triangle square triangle square triangle square Now draw the shapes in the order you named them. Model It You can also use pattern blocks to help understand the problem. Use pattern blocks in the shapes Camille used to create her pattern. 70

Part 3: Guided Instruction Connect It Now you will explore the shape pattern from the previous page further. 9 How many sides does a triangle have? 10 How many sides does a square have? 11 How could you describe the pattern using the number of sides the shapes have? 12 What would the 10th shape in the pattern be? 13 Explain how you can figure out what the 85th number in the pattern would be without drawing all 85 shapes. Try It Use what you just learned to solve this problem. 14 Describe any rules you see in the shape pattern below. 71

Part 4: Guided Practice Study the model below. Then solve problems 15 17. The student used the rule add 2 because each sandwich is $2 more than the one before. Student Model Hungry Heath s sells four different sizes of sandwiches: small, medium, large, and jumbo. The small sandwich costs $3. Each size after that costs $2 more than the size before it. How much does each sandwich cost? Look at how you could show your work using a picture. small medium large jumbo 1 2 1 2 1 2 Pair/Share Are there any other patterns in this set of numbers? Solution: small: $3, medium: $5, large: $7, jumbo: $9 There is more than one pattern in these shapes! 15 Draw the next two shapes in the shape pattern shown below. Pair/Share Are your shapes the same as your partner s? Solution: 72

Part 4: Guided Practice 16 Eva drew a shape pattern that goes back and forth between rectangles and ovals. There are several ways to describe a pattern! What are two other patterns shown in this set of shapes? Solution: Pair/Share Did you partner describe the pattern in the same way you did? 17 Lana wrote the pattern below. 7, 14, 21, 28, 35 If the pattern continues, what would be the next number in the pattern? Circle the letter of the correct answer. You can check your answer by working backward! A 40 B 42 C 49 D 70 Diego chose D as the correct answer. How did he get that answer? Solution: Pair/Share What would the next 3 numbers in the pattern be? 73

Part 5: Common Core Practice Solve the problems. 1 What would be the 99th number in the pattern shown below? 10, 20, 30, 40, 50 A 99 B 900 C 909 D 990 2 Nia used pattern blocks to make the shape pattern shown below. Which does NOT describe Nia s shape pattern? A B C Each shape has one more side than the shape before it. The shapes in the odd numbered spots have an odd number of sides. The sides in a shape are all the same length. D The hexagon only appears in spots that are multiples of 4. 3 Choose Yes or No to tell whether the pattern follows the rule: 17. a. 7, 17, 27, 37 Yes No b. 1, 7, 49, 343 Yes No c. 3, 10, 17, 24 Yes No d. 7, 77, 777, 7777 Yes No e. 7, 14, 21, 28 Yes No 74

Part 5: Common Core Practice 4 Tell whether each sentence is True or False. a. A number pattern that follows the rule add 3 has both odd and even numbers. True False b. A number pattern starts with 5 cannot include the number 3. True False c. The number pattern that follows the rule start at 20 and subtract 4 has only even numbers. True False d. A number pattern that follows the rule multiply by 2 must have even numbers only. True False 5 Draw a shape pattern that follows the rule that the shapes go back and forth between four sides and five sides. Show your work. Answer 6 Write a number pattern that follows the rule subtract 6 and also has all odd numbers. Show your work. Answer Self Check Go back and see what you can check off on the Self Check on page 37. 75

Develop Skills and Strategies (Student Book pages 66 75) Number and Shape Patterns Lesson Objectives Use rules to generate or extend a number pattern. Use manipulatives or drawings to show a shape pattern. Analyze and describe patterns in numbers and shapes. Prerequisite SkilLs Use addition, subtraction, multiplication, and division. Recognize simple patterns. Extend simple patterns. Understand even and odd. The Learning Progression This lesson builds on students previous work with number and shape patterns, including identifying patterns in addition and multiplication tables. In this lesson, students reason about number patterns and see that there may be more than one pattern in a sequence of numbers. For example, a list of numbers with the rule Add 5 also alternates between odd and even numbers. They connect a rule to a sequence of numbers or shapes and either extend or generate numbers or shapes that follow the pattern. The work in this lesson helps build a foundation for examining relationships between ordered pairs and coordinate graphs, and, later on, studying proportional relationships and functions in middle school. Vocabulary There is no new vocabulary. Review the following key terms. rule: a procedure that is followed to go from one number or shape to the next in a pattern shape pattern: a series of shapes that follow a rule to repeat or change number pattern: a series of numbers that follow a rule to repeat or change Teacher Toolbox Prerequisite Skills Ready Lessons Tools for Instruction Interactive Tutorials Teacher-Toolbox.com 4.OA.C.5 4.OA.5 CCSS Focus 4.OA.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule Add 3 and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. ADDitionaL standards: 4.OA.B.4 (See page A42 for full text.) standards FOR MatheMaticaL Practice: SMP 2, 3, 4, 5, 7 (See page A9 for full text.) 72

Part 1: Introduction At a glance Students explore one way to describe a pattern in a sequence of numbers. Part 1: introduction number and shape Patterns Develop skills and strategies ccss 4.oa.c.5 Step By Step Tell students that this page reviews the idea of using a rule to extend a number pattern. Have students read the problem at the top of the page. Work through Explore It as a class. Ask students to first look at the first two numbers and ask themselves what they could do to the number 5 to get the next number (10) in the pattern. [either add 5 or multiply by 2] Have them keep this in mind as they look at the next couple of numbers in the pattern to see if 5 is added or if the number is multiplied by 2. you have used rules to describe patterns in numbers. in this lesson, you will explore patterns further. take a look at this problem. What are the next two numbers in the pattern below? 5, 10, 15, 20, 25,, explore it use the math you already know to solve the problem. What are two ways to get from 5 to 10? add 5 or multiply by 2 How do you get from 10 to 15? add 5 What rule works for all of the numbers in the pattern? add 5 How can you find the number that comes after 25? add 5 to 25 What are the next two numbers in the pattern? 30, 35 Ask student pairs or groups to explain their answers for what the rule is for this pattern. 66 Concept Extension If students need more review with Add 5 patterns, write the sequence 2, 7, 12, 17, 22 on the board. Ask students to work in pairs and apply the thinking steps they just used on the previous problem to describing this pattern. Remind them to check to see if the rule works for all the numbers. Encourage them to look at the numbers to see if there are any other patterns that they notice (for example, the ones digits alternate between 2 and 7). Mathematical Discourse How can you prove that the rule for this pattern is Add 5? Test each number to make sure it is 5 more than the previous number. How else could you describe this pattern? Does it repeat? Is it growing? Decreasing? Observations may include, for example, that every other number ends in 5, or the tens digit repeats once and then increases by one. 73

At a glance Part 1: Introduction Students explore finding more than one pattern in a sequence of numbers. Part 1: introduction Find out More Step By Step Read Find Out More as a class. Ask students to look at both the number sequence and the array model. Help students make the connection between the number sequence and the model by asking questions such as, How do the arrays model the numbers 5, 10, 15, and so on? How could we write 1 set of 5 and 2 sets of 5? [5 3 1 and 5 3 2]. How are these numbers connected to multiplication? [They are multiples of 5.] Ask students to work in pairs to find at least one more pattern in the sequence 0, 5, 10, 15 and have them share. Expect students to describe how the arrays grow (one more column, dots in rows increase by one), the pattern of alternating 0 and 5 in the ones digit, and the pattern of alternating even and odd. Ask students to share what they notice first about the numbers 21, 18, 15, etc. Have students work in pairs to find at least two patterns in this sequence. [Possible patterns: even/odd and subtract 3.] Is add 5 the only relationship between the numbers in the pattern? Look at the numbers, along with the model of the numbers, and try to find a different pattern. 5, 10, 15, 20, 25 If you look only at the ones digits, you see that they alternate between 5 and 0. From the model, you can see that the number of dots alternates between odd and even. The pattern below also follows the rule add 5. 22, 27, 32, 37, 42 Do the other relationships you found in the first pattern apply to this pattern, too? The numbers in this pattern also alternate between odd and even, but the ones digits alternate between 2 and 7 instead of between 5 and 0. So, sets of numbers can share some patterns or rules, but have others that are different. reflect 1 Describe a pattern that you have noticed in the real world. Possible answer: house numbers on a street 67 Real-World Connection Have students continue to work in pairs to think of at least one more pattern for their Reflect response. You may wish to ask if they are wearing anything that shows a pattern. If possible, when students share patterns, help them see that it could also be a numerical pattern. 74

Part 2: Modeled Instruction At a glance Students use tables and number lines to help them explore a pattern and use it to solve a problem. Part 2: Modeled instruction read the problem below. then explore different ways to understand it. Step By Step Read the problem at the top of the page as a class. For the Picture It discussion, quickly draw the table on the board. Ask questions such as, Why does the table begin with the number 20? Do you expect the numbers to increase or decrease as you fill in the table? Why? Ask students to read Model It and to study the number line shown on the page. Fill in the table together and also count by 4 on the number line to model both ways of solving the problem. Ask questions such as, When using a table for this problem, what information has to be in the table? Why? When using a number line to solve this problem, what do you put on the number line? Why? Orlando does push-ups every day. Each day, he wants to do 4 more push-ups than the day before. Find out how many push-ups Orlando will do each day this week if he does 20 on Monday. Picture it you can use a table to help understand the problem. Day Monday tuesday Wednesday thursday Friday number of Push-ups 20 +4 +4 +4 +4 Model it you can also use a number line to help understand the problem. 14 14 14 14 20 24 28 32 36 Monday Tuesday Wednesday Thursday Friday Start at 20, the number of push-ups done on Monday, and then count 4 more for each day. When students find additional patterns, ask them to make convincing arguments that it is a pattern. For this pattern, students may say the numbers are all even. Point out that this describes the numbers in the pattern, but this is not the same as finding the rule. 68 Concept Extension Multiplying and Dividing Patterns Provide exposure to and practice with multiplying and dividing patterns. Write this pattern on the board: 1, 2, 4, 8,. Give the rule Multiply by 2 and have students test it out on all the numbers in the pattern. Ask for the next number in the pattern. [16] Write this pattern on the board: 24, 12, 6,. Ask students to look at the whole pattern is it increasing? decreasing? repeating? Have them look at how the first two numbers relate [the second is half of the first number, or it s divided by 2] and test that rule out on the rest of the numbers. Ask them for the next number in the pattern. [3] Point out to students that they will come across patterns that could be based on adding, subtracting, multiplying, or dividing. Mathematical Discourse How can tables be used to help figure out patterns? Tables help organize the numbers to see how they relate to each other. How can number lines be helpful in figuring out patterns? They can help to see the size of the skip between each number and how the numbers are related. 75

At a glance Part 2: Guided Instruction Students review the solution from the problem on page 68 and look for additional patterns in the numbers. They use what they ve learned to solve a similar problem. Step By Step Discuss problems 2 and 3 together to state the solution and to describe the rule. It s important to point out that this is a growing (or increasing) pattern. Make sure that students understand the difference between a repeating and growing pattern. To assess understanding, draw two patterns on the board: 24, 36, 24, 36 and 22, 44, 66, 88. Point to each set of numbers and have students raise 1 finger if the pattern repeats, and 2 fingers if it is growing. SMP Tip: Facilitate discussion in which you encourage students to listen and respectfully respond to, build on, or critique each other s ideas as they share answers to problems 5 and 6. (SMP 3) Ask pairs to find additional patterns in the numbers on the page and then share how they found them. Ask questions such as, What did you look at first in the numbers when looking for a different pattern? How can you convince us that it really is a pattern? Part 2: guided instruction connect it TRY it solutions now you will explore the problem from the previous page further. 2 How many push-ups did Orlando do each day? Monday: 20 Tuesday: 24 Wednesday: 28 Thursday: 32 Friday: 36 3 What is the rule for the pattern? add 4 4 What does the pattern tell you about what happens when you start with an even number and add an even number? all of the numbers in the pattern will be even. 5 What other pattern(s) do you see in this set of numbers? Possible answer: 6 Explain how you found the additional pattern(s). Possible explanation: i looked at what the numbers in the pattern have in common, and they are all multiples of 4. try it because the first number in the pattern was a multiple of 4, and you add 4 to get from one number to the next, all of the other numbers in the pattern are multiples of 4. use what you just learned to solve these problems. 7 Lori scored 100 points in a game, then doubled her score each of the next 3 times she played. What were Lori s scores the first 4 times she played the game? 100, 200, 400, 800 8 What is one additional pattern in Lori s scores? Possible answer: each number in the pattern is a multiple of 100. 7 Solution: 100, 200, 400, 800; Students may show a number line and show the doubling or create a chart to show the four scores. 69 Visual Model ERROR ALERT: Students who wrote 100, 200, 400, You may wish to explore with students how a doubling pattern looks using arrays. Write the numbers 2, 4, 8 on the board. Ask students how you could show these numbers with arrays of dots. Show 1 row of 2 dots, then 2 rows of 2 dots, then 4 rows of 2 dots. Ask students to describe what they see. Ask questions such as, What would the next array look like in the pattern? Can you show what is doubling in the model? Is there another way to model this pattern using arrays? 600 may have been confused by doubling and, instead, added 200 to make the last two scores. The student also did not check to see if adding 200 worked for all the numbers. 8 Solution: Possible descriptions include: double the number or multiply by 2. 76

Part 3: Modeled Instruction At a glance Students use words, drawings, and models to explore different ways to represent repeating shape patterns. Step By Step Read the problem at the top of the page as a class. Work on Picture It together. Explain to students that although this pattern is a simple one, they can develop strategies for analyzing patterns that will help them with more complicated patterns. Instruct students to name the shapes they see out loud using words and then draw it. Explain that saying what you see out loud (hearing the pattern) can be a strategy for figuring out a pattern. Be sure students understand that the shapes show a repeating pattern. Part 3: Modeled instruction read the problem below. then explore different ways to understand it. Camille made a shape pattern with pattern blocks that goes back and forth between a triangle and a square. Draw the pattern that Camille made. Picture it you can use models to help understand the problem. Start by describing the pattern with words. Repeat the pattern at least 3 times. triangle square triangle square triangle square Now draw the shapes in the order you named them. Model it you can also use pattern blocks to help understand the problem. Use pattern blocks in the shapes Camille used to create her pattern. 70 Hands-On Activity You may wish to have students use pattern blocks or other models to create and describe other repeating shape patterns. Instruct students to work in pairs and use no more than 3 different blocks to create a pattern. Choose several pairs to share a variety of repeating patterns with the class. The class says the pattern out loud using words, and then students create the pattern on their own using the blocks. Ask the pairs to describe the pattern. Mathematical Discourse If you make your own pattern, how can you prove that your shapes make a repeating pattern? Students may use a variety of arguments. They may talk about overlapping one set of shapes or using a set of labels that apply over and over. 77

At a glance Part 3: Guided Instruction Students revisit the problem on page 70 to look for additional ways to describe the repeating shape pattern and predict what the 85th term will be. They use what they have learned to describe a new shape pattern. Step By Step Discuss problems 9 through 11 as a class. Point out that the faces of the pattern blocks are triangles and squares but the actual threedimensional blocks are prisms: triangular prisms and square prisms. Have students work in pairs for problems 12 and 13; then discuss as a class. Encourage students to share their reasoning, listen respectfully, and question each other when they disagree or do not understand. Guide students to see that the number of a position can tell you what shape will be in that position. Here, even numbered positions are squares and odd numbered positions are triangles. SMP Tip: Students develop abstract and quantitative reasoning (SMP 2) by connecting the numerical pattern of odd and even to a pattern of two alternating shapes. To challenge students further, ask them to predict the 30th and 31st shape in a pattern of three repeating shapes (for example: triangle, square, circle). [The 30th shape will be a circle and the 31st a triangle.] Ask students to work in pairs on the Try It problem. Encourage them to describe the shapes out loud as a strategy to distinguish what characteristics are repeating. Part 3: guided instruction connect it now you will explore the shape pattern from the previous page further. 9 How many sides does a triangle have? 3 10 How many sides does a square have? TRY it solutions 11 How could you describe the pattern using the number of sides the shapes have? 12 What would the 10th shape in the pattern be? 13 Explain how you can figure out what the 85th number in the pattern would be without drawing all 85 shapes. Possible explanation: there is a triangle in all the odd-numbered spots and a square in all the even-numbered spots. 85 is an odd number, so it would be a triangle. try it the shapes alternate between 3 and 4 sides. use what you just learned to solve this problem. 14 Describe any rules you see in the shape pattern below. Possible answer: the pattern goes back and forth between a shape with no curves and a shape with all curves. 14 Solution: alternating shape pattern of figures with no curves (star and arrow) and figures with all curves (circle and S). Four figures: star, circle, arrow, S; Students may use words or draw the four figures that repeat. ERROR ALERT: Students who can t find a pattern may be looking for polygons instead of looking at the attributes of the shapes. 4 square 71 78

Part 4: Guided Practice Part 4: guided Practice Part 4: guided Practice study the model below. then solve problems 15 17. Student Model 16 Eva drew a shape pattern that goes back and forth between rectangles and ovals. There are several ways to describe a pattern! The student used the rule add 2 because each sandwich is $2 more than the one before. Hungry Heath s sells four different sizes of sandwiches: small, medium, large, and jumbo. The small sandwich costs $3. Each size after that costs $2 more than the size before it. How much does each sandwich cost? Look at how you could show your work using a picture. small medium large jumbo 1 2 1 2 1 2 What are two other patterns shown in this set of shapes? Solution: Possible answer: the shapes go back and forth between straight sides and curves, and the oddnumbered spots have rectangles and the even-numbered spots have ovals. Pair/share Did you partner describe the pattern in the same way you did? Pair/share Are there any other patterns in this set of numbers? Solution: small: $3, medium: $5, large: $7, jumbo: $9 17 Lana wrote the pattern below. 7, 14, 21, 28, 35 If the pattern continues, what would be the next number in the pattern? Circle the letter of the correct answer. a 40 You can check your answer by working backward! There is more than one pattern in these shapes! 15 Draw the next two shapes in the shape pattern shown below. b 42 c 49 Pair/share Are your shapes the same as your partner s? Solution: D 70 Diego chose D as the correct answer. How did he get that answer? Solution: Diego looked at 7 and 14 and thought the pattern was to multiply by 2 instead of add 7. Pair/share What would the next 3 numbers in the pattern be? 72 73 At a glance Students look at relationships between numbers and shapes to describe or extend a repeating or growing pattern. Step By Step Ask students to solve the problems individually. Direct their attention to the hints given to help them think about and solve the problems. When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group. solutions Ex The price of each sandwich is represented pictorially, with 2 more dollar bills shown in each set of bills. Students could also use a table or list to show the pattern 3, 5, 7, 9. 15 Solution: heptagon and pentagon; Students should continue the pattern of sides (5, 7, 7, 5, 7,... ) and draw a 7-sided polygon and then a 5-sided polygon. (DOK 1) 16 Solution: Possible answer: alternating curves and straight edges; Students may also note that the even-numbered positions have ovals and odds have rectangles. (DOK 2) 17 Solution: B; Diego looked at 7 and 14 and thought the pattern was to multiply by 2 instead of add 7. Explain to students why the other two answer choices are not correct: A is not correct because only 5 is added instead of 7. C is not correct because it s the number that would come after the correct answer of 42. (DOK 3) 79

Part 5: Common Core Practice Part 5: common core Practice Part 5: common core Practice Solve the problems. 1 What would be the 99th number in the pattern shown below? 10, 20, 30, 40, 50 A 99 B 900 C 909 D 990 2 Nia used pattern blocks to make the shape pattern shown below. 4 Tell whether each sentence is True or False. a. A number pattern that follows the rule add 3 has both odd and even numbers. 3 True False b. A number pattern starts with 5 cannot include the number 3. True 3 False c. The number pattern that follows the rule start at 20 and subtract 4 has only even numbers. 3 True False d. A number pattern that follows the rule multiply by 2 must have even numbers only. True False 5 Draw a shape pattern that follows the rule that the shapes go back and forth between four sides and five sides. 3 Show your work. Which does NOT describe Nia s shape pattern? A Each shape has one more side than the shape before it. B The shapes in the odd numbered spots have an odd number of sides. C The sides in a shape are all the same length. D The hexagon only appears in spots that are multiples of 4. 3 Choose Yes or No to tell whether the pattern follows the rule: 17. a. 7, 17, 27, 37 Yes No b. 1, 7, 49, 343 Yes No 3 c. 3, 10, 17, 24 Yes No d. 7, 77, 777, 7777 Yes No 3 3 3 3 e. 7, 14, 21, 28 Yes No Answer Possible pattern: square, pentagon, square, etc. 6 Write a number pattern that follows the rule subtract 6 and also has all odd numbers. Show your work. Answer Possible pattern: 71, 65, 59, 53, 47 self check Go back and see what you can check off on the Self Check on page 37. 74 75 At a glance Students analyze, describe, and extend patterns in numbers and shapes that might appear on a mathematics test. solutions 1 Solution: D; The useful pattern is that each number is its position times 10, so the 99th term is 99 3 10 or 990. (DOK 2) 2 Solution: A; Each shape has one more side than the shape before it until it goes from the hexagon to the triangle. (DOK 3) 3 Solution: a. No; b. No; c. Yes; d. No; e. Yes (DOK 2) 4 Solution: a. True; b. False; c. True; d. False (DOK 2) 5 Solution: Answers will vary. Sample: square, pentagon, square, pentagon. (DOK 1) 6 Solution: Answers will vary. Sample: 71, 65, 59, 53, 47. Any sequence that begins with an odd number and subtracts 6 will fit the rule. (DOK 2) 80

Differentiated Instruction Assessment and Remediation Ask students to write the number that comes next in the following pattern: 9, 13, 17, 21,. [25] For students who are struggling, use the chart below to guide remediation. After providing remediation, check students understanding. Ask students to write the number that comes next in the following pattern: 21, 18, 15, 12,. [9] If a student is still having difficulty, use Ready Instruction, Level 3, Lesson 7. If the error is... Students may... To remediate... 24 think the next number is a multiple of 4 since the pattern is to add 4. They may not think about how the pattern starts with a number that is not a multiple of 4. Hands-On Activity Represent number patterns. Model how to approach finding and following the rule for a pattern. Write the steps in the process on the board. For example: 1. Say the pattern out loud. 2. Ask: Do I hear anything repeating? 3. Ask: Are the numbers getting bigger or smaller? 4. Look at first two numbers. By how much are they getting bigger or smaller? 5. Ask: Is the pattern to add, subtract, multiply, or divide? 6. Ask: Does the rule work for all of the numbers in the pattern? 7. Apply the rule to the last number in the pattern to find the next number. Challenge Activity Explore doubling patterns. Materials: square tiles or grid paper Have students explore a variety of number patterns using tiles. For patterns involving larger numbers, use grid paper instead. For example, have students represent the pattern 2, 5, 8, 11 with rows of 2, 5, 8, and 11 square tiles. These rows should form a step pattern. Ask students to describe the pattern the tiles show. Have them build the next row and explain how they knew how many tiles to put in that row. They may base their reasoning on the geometric arrangement, to make each step the same size, or they may realize that the pattern is to add 3. Repeat with other number patterns such as 24, 20, 16, 12, or challenge them with an alternating add/ subtract pattern such as 5, 8, 7, 10, 9, 12, 11. In each case, after students build or draw rows for each pattern, ask them to describe and extend the pattern, and then compare and discuss with a partner. Give students an opportunity to explore doubling patterns. First, have them find and write the first ten terms of a pattern beginning with 2 and doubling to get the next term. [2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024] Help students visualize how quickly the numbers grow by stacking sheets of paper to equal to each number. (Helpful hint: one pack of printer paper is 500 sheets.) For fun, have them estimate how big the 15th term would be, and then continue the pattern for another 5 terms to check their estimate. 81