• The Write-Pair-Share activity using Number Pattern Examples may be used to assess students’ comprehension level of patterns and their ability to define and provide examples of patterns.
• The group activity, Modeling and Representing Number Patterns may be used to determine students’ level of understanding.
• Use the Lesson 1 Exit Ticket to quickly evaluate student mastery.

Part 1: Overview of Patterns

Write-Pair-Share Activity

Ask students to define a pattern. (A pattern is any regularity in a situation.) Then, ask students to define a number pattern. (A number pattern is a sequence of numbers that follow the same rule.) Students should provide examples of patterns including number patterns. Encourage students to provide a variety of representations when creating the examples. Give students 3–5 minutes to write the definitions and examples. Then, have each student share his/her ideas with a partner. After about 5 minutes, have the class reconvene. Ask one member from each group to share the definitions and examples with the whole class. Have students discuss, debate, and ask questions during this time; record student responses on the board.

(Note: The intent of this activity is to have students develop meaning for the word, pattern. Often, students are only able to provide an example of a number or shape pattern, without understanding the defining characteristics of a pattern.)

Students may provide shape patterns, as well as number patterns, in the form of sequences and input/output tables. Some students may think of real-world examples, points on a graph, or even equations.

Students may describe a pattern as something that shows repetition. Many students will say, “It is something that repeats.” One component may repeat, or a block of components may repeat. This repetition may include sounds, symbols, objects, change in numbers, etc. Patterns are found all around us, including in the designs of natural objects such as pineapples and pine cones.

A number pattern shows a particular pattern for a set of numbers. A number pattern is generated by a particular rule. The rule may include addition, subtraction, multiplication, and division. For this grade level, the focus will be on addition, subtraction, and multiplication.

After students have brainstormed the definitions and examples, provide them with the Number Pattern Examples handout (M-4-6-1_Number Pattern Examples and KEY.docx). Ask students to complete the column that explains why each sequence is indeed a number pattern. This reference sheet may also be uploaded to the class website.

Hundreds Chart Activity

As a follow-up to the Write-Pair-Share activity and as a precursor to the discussion of the generation of number patterns, provide students with a hundreds chart (M-4-6-1_Hundreds Chart.docx), and ask them to identify as many patterns as possible. Students may color/shade/highlight/circle numbers on the chart, as necessary, and write all discovered patterns at the bottom of the sheet. Students should describe what they notice about the patterns, i.e., placement of numbers in columns, diagonals, etc., as well as the rule that was used to create the pattern.

(Note: Identification of multiples is a beginning step to understanding patterns. Provide students ample time to discover the multiples and placement of the multiples on the hundreds chart. Ask them to use words to explain how the numbers relate. For example, given the multiples of 5, students may observe that 10 is 5 more than 5, 15 is 5 more than 10, 20 is 5 more than 15, and so on. This set of multiples represents a number pattern because there is a consistent rule being followed: Add 5. Students should also observe that the multiples form two shaded columns. The numbers follow a particular pattern, or repetition, on the hundreds chart.)

A sample hundreds chart is provided below, followed by the written patterns and some generalizations.

This hundreds chart shows the following patterns:

• 1, 3, 5, 7, 9, . . .
• 2, 4, 6, 8, 10, . . .
• 3, 6, 9, 12, 15, . . .

Students may notice the following:

• The set of odd numbers starts at 1, whereby each number is 2 more than the previous number. Rule: Add 2.
• The pattern of odd numbers appears in columns and comprises half of the numbers.
• Each multiple of 2 is 2 more than the previous multiple. Rule: Add 2.
• Multiples of 2 appear in columns and comprise half of the numbers.
• Each multiple of 3 is 3 more than the previous multiple. Rule: Add 3.
• Multiples of 3 form diagonal patterns in the chart.

Part 2: Generate Number Patterns

Once students have reviewed the concept of a pattern, and a number pattern, in particular, begin instruction that asks students to use given rules to generate number patterns. They will then observe characteristics in the pattern beyond simply the change in values. For example, students may recognize that every other number is even or odd. Students may recognize patterns that represent multiples of a particular number. As mentioned, such sequences provide a good starting point for discussion.

Group Activity

In this activity, students will model a number pattern using a starting value and a given rule.

Arrange students in groups of three to four. Provide each student with 10 copies of the Modeling and Representing Number Patterns resource (M-4-6-1_Modeling and Representing Number Patterns.docx). Show more examples as needed. Students will work together in each small group, but each student should perform modeling on his/her own recording sheet, while also recording his/her results.

Directions: Students will model the number pattern, using the starting number and given rule. Colored counters may be used for modeling. The value of each term will be modeled in a different frame. Students will place the correct number of colored counters into the open space of each frame on the recording sheet, with the corresponding value written in the box at the bottom of each frame. (Students may wish to draw the counters on their sheet, as well.) Students will then use these values to write a sequence. Finally, students will identify features of the pattern. All of this information will be written on each recording sheet.

Example 1

• Starting number: 4
• Rule: Add 4.

“We can create a number pattern using this information. First, we will start with the number 4 and use the rule ‘Add 4.’ Let’s write the starting number and rule at the top of our papers. How can we model this pattern using our colored counters?” (Put 4 counters in the first column.) “How many counters will we put in the second column?” (There will be 8. There will be 4, plus another 4 more in the second column. This will represent +, because 4+4=8.)

Continue similar questioning for the third, fourth, and fifth columns. (The next three numbers will be 8+4=12, 12+4=16, and 16+4=20.) Students can explain that they add 4 more counters for the next step in the columns each time. “Now let’s write those values in the boxes at the bottom of each frame.” When they finish, students should have the numbers 4, 8, 12, 16, and 20 in a sequence. “Let’s look at the sequence of numbers we have written. How can you describe this sequence?” (The sequence represents multiples of 4: 4×1=4, 4×2=8, 4×3=12, etc. Also, each term shows an even number, which also means that they are all multiples of 2.)

Example 2

• Starting number: 1
• Rule: Add 4.

“The starting number for this example is 1. The rule is ‘Add 4.’ Let’s put this information on top of our activity pages. How many counters should we start with in the first frame?” (1) “How many counters should we put in the second frame? Explain.” (5, because 1+4=5.) “Now try frames 3, 4, and 5.” Have students share and explain their answers with the person next to them. (5+4=9, 9+4=13, 13+4=17) “What pattern do you notice with these numbers?” (All the numbers are odd.)

Example 3

• Starting number: 15
• Rule: Subtract 2.

“The starting number is 15. The rule is ‘Subtract 2.’ Let’s put this information on the top of our activity pages. Go ahead and fill the rest out.” (Students should come up with
15, 13, 11, 9, and 7 showing the appropriate counters.) “Explain how you got this pattern.” (I subtracted 2 each time.) “What other patterns did you notice in this example?” (All of the numbers are odd.)

Example 4

• Starting number: 1
• Rule: Multiply by 2.

“The starting number is 1. The rule is ‘Multiply by 2.’ How does this example differ from our previous examples?” (The rule in this example uses multiplication, not addition or subtraction.) “How many counters should we put in the first frame, given that the starting number is 1?” (1) “How many counters should we put in the second frame?” (We should put 2 counters in the second frame because 1×2=2.) “Now work on the next three columns. Compare your answers with your partner. Explain how you got your answers, and describe any patterns that you see in these numbers” (1, 2, 4, 8, 16, . . . The values increase by factors of 2; the numbers are all even.) “Can you think of an instance where you will get an odd value following the fifth term? Why or why not?” (It will not happen in this example because all of the numbers are being multiplied by 2, which means they will all be even.)

Example 5

• Starting number: 2
• Rule: Add 4, and then subtract 2.

“The starting number is 2. The rule is ‘Add 4, and then subtract 2.’ How does this example differ from our previous examples?” (This rule has 2 steps. First you have to add 4 to the previous number. Then you have to subtract 2.) “As you know from previous examples, if the starting number is 2, we should put 2 counters in the first frame. How many counters should we put in the second frame? Explain.” (4 counters. The first part of the rule is add 4; 2+4=6. The second part of the rule is subtract 2; 6−2=4.) “How many counters should we put in the third frame? Explain.” (6 counters. First we add 4; 4+4=8. Then we subtract 2;
8−2=6.) “How many counters should we put in the fourth and fifth frames? Explain.” (6+4=10; 10−2=8 counters. 8+4=12; 12−2=10 counters.) “What is the overall pattern for these rules? What else do you observe?” (2, 4, 6, 8, 10, . . . All of the numbers are even.)

Example 6

• Starting number: 1
• Rule: Multiply by 3, and then subtract 1.

“Here is another example with a two-step rule. First, we must multiply the previous number by 3, and then we must subtract 1. Since the starting number is 1, we know we have to put 1 counter in the first frame. How many counters go in the second frame? Explain.” (2 because 1×3=3, and 3−1=2.) “Fill out the third, fourth, and fifth frames. What is your overall pattern, and what do you observe?” (The third frame has 5 counters because 2×3=6 and 6−1=5. The fourth has 5×3=15; 15−1=14 counters. The fifth has 14×3=42;
42−1=41 counters. The overall pattern is 1, 2, 5, 14, 41, . . . The values alternate between odd and even numbers.)

Partner Game

Once students have completed the previous small group activity, they may be placed with a partner. Students should be asked to create game cards that may be played with a partner. Each student will create a set of 10 cards. Each card will have a starting number and a rule written on the front, and the resulting sequence written on the back. Students should create three addition rules, three subtraction rules, two multiplication rules, and two multistep rules. The back of each card should also indicate at least one feature of the number pattern. The partner answering the questions must verbally list at least five numbers in the sequence, while also correctly identifying at least one feature of the pattern. (You should monitor discussions to make sure sequences and features are accurate.) You may help as needed. Correct answers for each card are worth
2 points, for a possible total of 20 points. The student with the most points wins. These game cards may be kept as resources.

Provide time for questions and discussion. Ask students about any concerns or problems with patterns and sequences. Have students complete the Lesson 1 Exit Ticket (M-4-6-1_Lesson 1 Exit Ticket and KEY.docx) at the close of the lesson to evaluate students’ level of understanding.

Extension:

• Routine: During the school year, have students identify patterns in real-world contexts, such as dates on a calendar, total amount spent on lunch after each day has passed, and total number of feet on a certain number of chickens or other animals.
• Small Groups: Students who need additional practice may by pulled into small groups to work on the Small Group Practice worksheet (M-4-6-1_Small Group Practice and KEY.docx). Students may do the worksheet together or work individually and compare answers when done.
• Expansion: Students who are prepared for a challenge beyond the requirements of the standard may be given the Expansion Work sheet (M-4-6-1_Expansion Work and KEY.docx). The sheet includes more difficult sequences, word problems, and a challenge section.