<p>Students will create number patterns. Students will:&nbsp;<br>- define and present examples of patterns.&nbsp;<br>- identify number patterns.&nbsp;<br>- create sequences using a rule and a beginning number.&nbsp;<br>- determine the obvious features of sequences.&nbsp;</p>

<p>- How are relationships represented mathematically?<br>- How may data be arranged and represented to reveal the link between quantities?<br>- How can mathematics help us communicate more effectively?&nbsp;<br>- How may patterns be used to describe mathematical relationships?&nbsp;<br>- How can probability and data analysis be used to make predictions?&nbsp;<br>- How may detecting repetition or regularity assist in solving problems more efficiently?<br>- How can mathematics help to quantify, compare, depict, and model numbers?</p>

<p>- Factor: The number or variable multiplied in a multiplication expression.&nbsp;<br>- Multiple: A number that is the product of a given integer and another integer (e.g., 6 and 9 are multiples of 3).&nbsp;<br>- Patterns: Regularities in situations such as those in nature, events, shapes, designs and sets of numbers (e.g., spirals on pineapples, geometric designs in quilts, number sequence 3, 6, 9, 12, . . .).</p>

<p>- one copy of the Number Pattern Examples (M-4-6-1_Number Pattern Examples and KEY) per student<br>- one copy of the Modeling and Representing Number Patterns worksheet (M-4-6-1_Modeling and Representing Number Patterns) per student<br>- one copy of the Lesson 1 Exit Ticket (M-4-6-1_Lesson 1 Exit Ticket and KEY) per student<br>- copies of the Small Group Practice worksheet (M-4-6-1_Small Group Practice and KEY) as needed<br>- copies of Expansion Work (M-4-6-2_Expansion Work and KEY) as needed<br>- copies of the Hundreds Chart (M-4-6-1_Hundreds Chart) as needed</p>

<p>- The Write-Pair-Share exercise with a Number Pattern Examples can be used to examine students' understanding of patterns as well as their ability to define and present instances of patterns.&nbsp;<br>- The group exercise Modeling and Representing Number Patterns can be used to assess students' understanding.&nbsp;<br>- Use the Lesson 1 Exit Ticket to swiftly assess students' mastery.</p>

<p>Scaffolding, Active Engagement, Metacognition, Modeling, Explicit Instruction, and Formative Assessment&nbsp;<br>W: Students will recognize number patterns and use rules and starting numbers to create sequences. Students will also identify sequence features.&nbsp;<br>H: Students will engage with this lesson by brainstorming definitions and examples of patterns. The Write-Pair-Share activity allows students to generate a conceptual explanation for a pattern. The Hundreds Chart activity allows students to visually detect number patterns on a chart. Students will be required to identify the rules and attributes related with the number patterns.&nbsp;<br>E: The lesson focuses on creating sequences using rules and starting numbers, and identifying their qualities. Students will begin by defining and offering examples of patterns. Students will then identify numerical patterns on a hundreds chart. Students will then work through examples that provide rules and starting numbers, as well as ask for sequence generation. Finally, students will have the option to participate in a pair game in which they will construct rules, starting numbers, and sequences while identifying pattern aspects.&nbsp;<br>R: The Write-Pair-Share activity initiates discussion opportunities in the class. Throughout the class, students will discuss ideas with their group members, prompting them to reconsider and revise their understanding. The partner game will also allow students to construct their own rules, starting numbers, and sequences, as well as identify at least one feature in each sequence. A discussion session precedes the Exit Ticket. If you need more practice or a challenge, use the Expansion Worksheet.&nbsp;<br>E: Observing students throughout the Write-Pair-Share activity might help assess their comprehension level. Before the lesson ends, administer the Lesson 1 Exit Ticket to assess students' knowledge.&nbsp;<br>T: Use the Extension section to personalize the lesson to the needs of each student. The Routine section includes ideas for reviewing course concepts throughout the year. The Small Groups section offers options for students who could benefit from additional training or practice. The Expansion section may be presented to students who are willing to take on a challenge that goes beyond the standards.&nbsp;<br>O: The lesson is scaffolded to help students establish conceptual meaning for patterns. Then, students will detect number patterns on a hundreds chart. Finally, they will design sequences based on predetermined rules and starting numbers. Students will also recognize aspects of the sequences that are not expressly stated in the rules. This lesson focuses on numerical patterns. The following lesson in the unit will cover shape patterns, provided rules, and starting diagrams. The final session will cover input/output tables, missing values, and the rules used to generate each table.&nbsp;</p>

<h3><strong>Part 1: Overview of Patterns</strong></h3><p><strong>Write-Pair-Share Activity</strong><br><br>Ask students to identify a <i>pattern</i>. (A pattern refers to any regularity in a situation.) Then, ask students to define a <i>number pattern</i>. (A number pattern is a sequence of numbers that follow the same rule.) Students should present examples of patterns, including number patterns. Encourage students to use a variety of representations while building their examples. Allow students 3-5 minutes to create definitions and examples. Then, have each student discuss his/her thoughts with a companion. After about 5 minutes, the class should reconvene. Ask one member of each group to share the concepts and examples with the entire class. During this time, encourage students to talk, argue, and ask questions; record their comments on the board.<br><br>(Note: The goal of this assignment is for students to create meaning for the word, <i>pattern</i>. Frequently, students can just present an example of a number or shape pattern, without grasping the distinguishing characteristics of a pattern.&nbsp;<br><br>Students can offer shape and number patterns in the form of sequences and input/output tables. Some students may consider real-world examples, points on a graph, or even equations.<br><br>Students may characterize a <i>pattern</i> as something that repeats. Many students will remark, "It is something that repeats." A single component or an entire block of components may repeat. This repetition can involve noises, symbols, objects, changes in numbers, and so on. Patterns may be found all around us, including the motifs on natural objects like pineapples and pine cones.&nbsp;<br><br>A number pattern represents a specific pattern for a set of numbers. A particular rule determines the number pattern. The rule may cover addition, subtraction, multiplication, and division. This grade level will focus on addition, subtraction, and multiplication.<br><br>After students have discussed the definitions and examples, give them the Number Pattern Examples handout (M-4-6-1_Number Pattern Examples and KEY). Ask students to fill out the column explaining why each sequence is a number pattern. This reference sheet can also be shared to the class website.</p><p><strong>Hundreds Chart Activity</strong><br><br>As a follow-up to the Write-Pair-Share activity and a prelude to a discussion of number pattern generation, hand out a hundreds chart (M-4-6-1_Hundreds Chart) and ask students to identify as many patterns as possible. Students may color/shade/highlight/circle numbers on the chart as needed, and record any patterns identified at the bottom of the sheet. Students should describe what they see about the patterns, such as the placement of numbers in columns and diagonals, as well as the rule employed to construct the pattern.<br><br>(Note: Identifying multiples is a first step toward comprehending patterns. Allow students adequate opportunity to discover the multiples and place them on the hundreds chart. Ask them to use words to describe how the numbers are related. For example, given the multiples of 5, students would notice that 10 is 5 more than 5, 15 is 5 more than 10, 20 is 5 more than 15, and so on. This sequence of multiples illustrates a number pattern since it follows a consistent rule: add 5. Students should also notice how the multiples form two shaded columns. On the hundreds chart, the numbers follow a specific pattern of repetition.<br><br>A sample hundreds chart is presented below, followed by the textual patterns and some generalizations.<br><br><img src="https://storage.googleapis.com/worksheetzone/images/Screenshot_51.png" width="609" height="324"></p><p>This hundred chart shows the following patterns:<br>1, 3, 5, 7, 9, . . .<br>2, 4, 6, 8, 10, . . .<br>3, 6, 9, 12, 15, . . .</p><p>Students may notice the following:<br>The set of odd numbers begins at 1, with each number being 2 more than the previous number. Rule: Add 2.<br>The pattern of odd numbers appears in columns and makes up half of the numbers.&nbsp;<br>Each multiple of 2 is 2 more than the previous one. Rule: Add 2&nbsp;<br>Multiples of 2 appear in columns and makes up half of the numbers.&nbsp;<br>Each multiple of 3 is three times the previous multiple. Rule: Add 3.<br>Multiples of 3 form diagonal patterns in the chart.</p><h3><strong>Part 2: Generate Number Patterns</strong></h3><p>After students have studied the concept of a pattern, specifically a number pattern, begin instruction by asking them to generate number patterns using given guidelines. They will then notice qualities in the pattern that go beyond just the shift in values. For example, students may notice that every other number is even or odd. Students may recognize patterns representing multiples of a specific number. As previously indicated, such sequences make a good beginning point for discussion.<br><br><strong>Group Activity</strong><br><br>In this activity, students will create a number pattern based on a starting value and a provided rule.&nbsp;<br><br>Divide students into groups of three to four. Give each student ten copies of the Modeling and Representing Number Patterns resource (M-4-6-1_Modeling and Representing Number Patterns). Show further examples if needed. Students will collaborate in small groups, but each student should model on his or her own recording sheet while also documenting his or her findings.<br><br>Directions: Students will model the number pattern using the initial number and the provided rule. Colored counters can be used for modeling. The value of each word will be represented in a distinct frame. Students will place the appropriate number of colored counters in the open space of each frame on the recording sheet, with the matching value typed in the box at the bottom. (Students may also choose to draw the counters on their sheets.) Students will then utilize these values to construct a sequence. Finally, students will identify the pattern's features. All of this information will be documented on each recording sheet.<br><br><strong>Example 1</strong><br><br>Starting number: 4.&nbsp;<br>Rule: Add 4.&nbsp;<br><strong>"With this information, we can generate a number pattern. First, we will begin with the number 4 and apply the rule 'Add 4.' Let's write the starting number and rule on the top of our papers. "How can we model this pattern with our colored counters?"</strong> (<i>Place 4 counters in the first column.</i>) <strong>"How many counters will we put in the second column?"</strong> (<i>There will be 8. There will be 4 plus 4 more in the second column. This will indicate +, as 4+4=8.</i>)<br><br>Continue with similar questions for the third, fourth, and fifth columns. (The following three numbers will be 8+4 = 12, 12+4=16, and 16+4=20.) Students might explain that for the next stage, they add 4 more counters to the columns each time. <strong>"Now let's write those values in the boxes at the bottom of each frame."</strong> When they're finished, students should have the numbers 4, 8, 12, 16, and 20 in order. <strong>"Look at the sequence of numbers we've written. "How would you describe this sequence?"</strong> (<i>The sequence symbolizes multiples of 4: 4×1= 4, 4×2=8, 4×3=12, and so on. Also, each term displays an even number, implying that they are all multiples of 2.</i>)<br><br><strong>Example 2</strong><br><br>Starting number: 1.&nbsp;<br>Rule: Add 4.&nbsp;<br><strong>"The beginning number for this example is 1. The rule is: 'Add 4.' Let's place this information at the top of our activity pages. "How many counters should we start with in the first frame?"</strong> (<i>1</i>) <strong>"How many counters should we use in the second frame? Explain."</strong> (<i>5, because 1+4=5</i>). <strong>"Now try frames 3, 4, and 5."</strong> Encourage students to share and explain their answers with the person next to them. (<i>5+4=9, 9+4=13, 13+4=17</i>) <strong>"What pattern do you notice with these numbers?"</strong> (<i>All the numbers are odd.</i>)<br><br><strong>Example 3</strong><br><br>Starting number: 15.&nbsp;<br>Rule: Subtract 2.&nbsp;<br><strong>"The starting number is 15." The rule is 'Subtract 2.' Let's highlight this information at the top of our activity pages. Go ahead and fill in the rest."</strong> (Students should identify the numbers 15, 13, 11, 9, and 7 as appropriate counters.) <strong>"Explain how you got this pattern."</strong> (<i>I subtract 2 each time.</i>) <strong>"What other patterns did you notice in this example?"</strong> (<i>All of the numbers are odd.</i>)<br><br><strong>Example 4</strong><br><br>Starting number: 1.<br>Rule: Multiply by 2.&nbsp;<br><strong>"The starting number is 1. The rule is 'Multiply by 2.' How does this example differ from our previous ones?"</strong> (<i>In this case, the rule uses multiplication not addition or subtraction.</i>) <strong>"How many counters should we put in the first frame, given that the starting number is 1?"</strong> (<i>1</i>) <strong>"How many counters should we put in the second frame?"</strong> (<i>Use 2 counters in the second frame since 1×2=2</i>). <strong>"Now, focus on the next three columns. Compare your responses with your partner. Explain how you obtained your results, and describe any patterns you find in these numbers."</strong> (<i>1, 2, 4, 8, 16, . . . The values grow by factors of 2, and the numbers are all even.</i>) <strong>"Can you think of an example in which you will receive an odd value after the fifth term? Why, or why not?"</strong> (<i>It will not happen in this example since all of the numbers are multiplied by two, therefore they will all be even.</i>)<br><br><strong>Example 5&nbsp;</strong><br><br>Starting number: 2<br>Rule: Add 4, and then subtract 2.<br><strong>"The starting number is 2. The rule is 'Add 4, then subtract 2. "How does this example differ from our previous ones?"</strong> (<i>This rule has two steps. First, add 4 to the preceding number. Then you need to subtract 2.)</i> <strong>"As you know from previous instances, if the starting number is 2, we need include two counters in the first frame. How many counters should we use in the second frame? Explain."</strong> (<i>4 counters. The first element of the rule is to add 4; 2+4=6. The second step of the rule is to subtract 2; 6−2=4</i>). <strong>"How many counters should we have in the third frame? Explain."</strong> (<i>6 counters. First, we add 4; 4+4=8. Then we subtract 2: 8-2=6.</i>) <strong>“How many counters should we put in the fourth and fifth frames? Explain.”</strong> (<i>6+4=10; 10-2=8 counters. 8+4=12; 12-2=10 counters.</i>) <strong>“What is the overall pattern for these rules? What else do you observe?”</strong> <i>(2, 4, 6, 8, 10, … All of the numbers are even.</i>)<br><br><strong>Example 6</strong><br><br>Starting number: 1.&nbsp;<br>Rule: Multiply by 3 then subtract 1.&nbsp;<br><strong>"Here's another example using a two-step rule. First, we must multiply the preceding amount by 3 and then subtract 1. Because the initial number is 1, we know we need to include 1 counter in the first frame. How many counters appear in the second frame? Explain."</strong> (<i>2 because 1×3=3, and 3−1=2</i>). <strong>"Complete the third, fourth, and fifth frames. What is your overall pattern, and what are your observations?"</strong> (<i>The third frame contains 5 counters since 2×3=6 and 6−1=5. The fourth includes 5×3=15 and 15−1=14 counters. The fifth has 14x3=42; 42-1=41 counters. The overall pattern is 1, 2, 5, 14, 41. The values change between odd and even numbers.</i>)<br><br><strong>Partner Game:</strong><br><br>Once students have completed the previous small group exercise, they may be assigned a partner. Students should be asked to design game cards that can be played with a partner. Each student will make a set of ten cards. Each card will have a starting number and a rule written on the front, followed by the resulting sequence on the back. Students should develop three addition rules, three subtraction rules, two multiplication rules, and two multistep rules. Each card's back should also include at least one component of the number pattern. The partner who answers the questions must verbally list at least five numbers from the sequence and accurately identify at least one pattern feature. The partner who answers the questions must verbally list at least five numbers from the sequence and accurately identify at least one pattern feature. (You should watch discussions to ensure sequences and features are correct.) You can aid as needed. Correct answers on each card are awarded 2 points, for a total of 20 points. The student who scores the most points wins. These game cards could be preserved as resources.&nbsp;<br><br>Allow time for questions and discussions. Inquire if students have any issues or challenges with patterns and sequences. Students should complete the Lesson 1 Exit Ticket (M-4-6-1_Lesson 1 Exit Ticket and KEY) at the end of the lesson to assess their level of understanding.<br><br><strong>Extension:</strong><br><br><strong>Routine:</strong> Throughout the school year, have students notice patterns in real-world contexts, such as calendar dates, total amount spent on lunch after each day, and total number of feet on a specific number of chickens or other animals.&nbsp;</p><p><strong>Small Groups:</strong> Students who need more practice may be divided into small groups to work on the Small Group Practice worksheet (M-4-6-1_Small Group Practice and KEY). Students may complete the task together or independently, and then compare their solutions.&nbsp;</p><p><strong>Expansion:</strong> Students who are prepared for a challenge above the requirements of the standard may be assigned the Expansion Work sheet (M-4-6-1_Expansion Work and KEY). The worksheet contains more difficult sequences, word problems, and a challenge area.</p>