Lesson 4
Numerical Patterns
Warmup: Which One Doesn’t Belong: Stacked Squares (10 minutes)
Narrative
Launch
 Groups of 2
 Display image.
 “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
 1 minute: quiet think time
Activity
 “Discuss your thinking with your partner.”
 2–3 minutes: partner discussion
 Share and record responses.
Student Facing
Which one doesn’t belong?
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 “What features of the diagrams did you look at when you tried to find which one doesn’t belong?” (Sample response: the number of small squares in each shape)
 If time permits, ask students to think about which one doesn’t belong if they pay attention to the number of rows or the perimeter rather than the number of squares.
Activity 1: Count by 10 and by 9 (20 minutes)
Narrative
This activity prompts students to examine patterns in multiples of 10 and 9, and to notice that the digits in the multiples of 9 can be reasoned in relation to the morefamiliar multiples of 10. Students use what they know about the place value and operations to explain the patterns in these multiples (MP7). For instance, students may reason that, because 9 is 1 less than 10, to find \(12 \times 9\) is to find the \(12 \times 10\) and then subtract 1 twelve times (or subtract \(12 \times 1\)) from the product.
The reasoning in this activity prepares them to notice patterns in the multiples of 100 and 99 in the next lesson.
Advances: Listening, Representing
Supports accessibility for: Conceptual Processing, Visual Spatial Processing
Launch
 Groups of 2
 Read the opening paragraph as a class.
 “Which do you prefer, counting by 10 or counting by 9? Why?” (Counting by 10 because I've been doing that since kindergarten.)
 30 seconds: partner discussion
 “Let’s look at numbers we get by counting by 9 and by 10 and see what patterns we can find.”
Activity
 “Take a few quiet minutes to work on the first few problems. Then, share your thinking and complete the rest of the activity with your partner.”
 5–6 minutes: independent work time
 5–6 minutes: partner discussion
 Monitor for students who:
 can clearly explain the pattern of the digits in multiples of 10 in terms of place value
 notice connections between the values in the two columns and use them to explain the patterns in the digits in multiples of 9
Student Facing
Andre’s class is choral counting by 10 and then by 9. The left column shows the numbers they say when counting by 10.

Complete the right column with the first ten numbers the class will say when counting by 9.
What patterns do you notice about the features of the numerical patterns? Make at least two observations about each list of numbers.
counting by 10 counting by 9 10 20 30 40 50 60 70 80 90 100 
For the numbers in the “counting by 10” column, why do you think:
 the digits in the tens place change the way they do?
 the digits in the ones place are the way they are?
 For the numbers in the “counting by 9” column, why do you think the digits in the ones place change the way they do? Explain your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Advancing Student Thinking
Students may not see that the digit in the ones place decreases by 1 each time the count goes up by 9. Consider asking:
 “What patterns do you see in the pair of numbers in each row?”
 “How do you think the numbers in the ‘counting by 9’ column are related to those in the ‘counting by 10’ column?”
Activity Synthesis
 Display the completed table.
 Invite students to share the patterns they noticed in the numbers in each column. Record their observations by annotating the numbers in the table.
 If no students mentioned that the two sets of numbers are multiples of 10 and multiples of 9, ask them about it.
 Select previously identified students to share their responses to the last two problems.
 If no students reason that counting by 9 can be thought of as counting by 10 and subtracting 1 each time, bring this to their attention.
 In other words, counting by 9 once means \(10 1\) , which is 9. Counting by 9 again means adding another \(10 1\) to 9, or \(9 + 10  1\), which is 18. Counting by 9 a third time means \(18 + 10  1\), which is 27. And so on.
 “Counting by 9 eight times is the same as counting by 10 eight times and subtracting 1 eight times, or \((8 \times 10)  (8 \times 1)\), which is \(80  8\) or 72.”
Activity 2: Count by 99 (15 minutes)
Narrative
Launch
 Groups of 2
 “Earlier we counted by 9 and found some patterns in the numbers. Now let’s see what patterns we can find when we count by 99.”
Activity
 “Work with your partner to complete the activity.”
 8–10 minutes: partner work time
 Monitor for students who:
 Identify different patterns in the numbers
 reason about the numbers in the “counting by 99” column (multiples of 99), by reasoning about multiples of 100
Student Facing
Andre’s class did a choral count by 99. Here are the first six numbers they said.
 Study the list of numbers. Make at least 3 observations about features of the pattern.
counting by 99 99 198 297 396 495 594  Extend the list with the next four multiples of 99. Be prepared to discuss how you know what numbers to write.

Why do you think the digits in the numbers change the way they do?
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 Select students to share the features of the patterns they noticed, making sure to highlight the digits in each the hundreds, tens, and ones place. Annotate the numbers as needed.
 Select other students to share how they extended the patterns and record their responses. If no students mentioned using multiples of 100 as a strategy, discuss this with students.
 “Counting by 99 five times is the same as counting by 100 five times and subtracting 1 five times, or \((5 \times 100)  (5 \times 1)\).”
 “How can we use the pattern to find the 20th multiple of 99?” (Find \(20 \times 100\) and subtract \(20 \times 1\) from it.)
Activity 3: Count by 15 [OPTIONAL] (20 minutes)
Narrative
In this optional activity, students investigate patterns in multiples of 15 and analyze and describe features of the digits in the tens and ones place. The activity also prompts them to consider why those features exist and to predict whether a given number could be a multiple of 15. The goal here is not to elicit clear justifications, but rather to encourage students to use their understanding of place value and numbers in baseten to reason more generally about patterns in numerical patterns.
This activity uses MLR5 Cocraft Questions. Advances: writing, reading, representing
Launch
 Groups of 2
MLR5 CoCraft Questions
 Display only the opening sentence and list of numbers, without revealing the question(s).
 “Write a list of mathematical questions that could be asked about this situation.”
 2 minutes: independent work time
 2–3 minutes: partner discussion
 Invite several students to share one question with the class. Record responses.
 “What do these questions have in common? How are they different?”
 Reveal the task (students open books), and invite additional connections.
 “Let’s see what patterns we can find when we count by 15.”
Activity
 “Take a few quiet minutes to work on the activity. Afterwards, discuss your responses with your partner.”
 5 minutes: independent work time
 5 minutes: partner discussion
 Monitor for:
 the different patterns students notice
 the different ways they explain the patterns
 the ways students reason about whether 250 could be a number being called out
Student Facing
Elena counted by 15 and recorded the numbers she counted:
 15
 30
 45
 60
 75
 90

Write the next four numbers she’d record if she kept going.
 What patterns do you see? Describe as many as you can.
 Choose one pattern that you noticed and explain why you think it happens.
 Could 250 be a number that Elena calls out if she continued to count by 15? Explain or show your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Advancing Student Thinking
Activity Synthesis
 Invite students to share the patterns they noticed and their explanations for the patterns. Record them for all to see.
 Select other students to share their explanation on whether 250 could be a number that Elena calls said. Highlight explanations that make use of the structure in the numbers.
Lesson Synthesis
Lesson Synthesis
“Today we saw different features of patterns in the numbers that we get when counting by 9, 10, 99, and 100.” (Include 15, if students completed the optional activity).
“What new ideas did you have about patterns in this section?”
“What are you still wondering about patterns?”
Cooldown: Count by 8 (5 minutes)
CoolDown
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Student Section Summary
Student Facing
In this section, we looked at different patterns of shapes and patterns of numbers. We saw shapes that grew or repeated by certain rules, and we used numbers to help us see how the shapes changed. Here are some examples of the patterns:
 Shapes that grow by a rule: add 1 row of equalsize squares
Area of the rectangle: 4, 6, 8, 10, . . .

Shapes that repeat by a rule: triangle, circle, triangle, square, repeat
▲ : 1, 3, 5, 7, . .
◯ : 2, 6, 10, . . .
▨ : 4, 8, 12, . . .
 Rectangles that change by a rule: increase the length of the rectangle by 5 inches
Side length:
5, 10, 15, 20, . . .Area:
15, 30, 45, 60, . . .Perimeter:
16, 26, 36, 46, . . . 
Numbers that change by a rule
 Add 9: 9, 18, 27, 36, 45
 Add 10: 10, 20, 30, 40, 50
 Add 99: 99, 198, 297, 396, 495
 Add 100: 100, 200, 300, 400, 500
We learned to extend the patterns by first finding their rule. Sometimes we can use addition and multiplication to represent a rule and then extend the pattern. Other times we can see how the digits in the numbers change to make predictions.