Lesson 4

Numerical Patterns

Warm-up: Which One Doesn’t Belong: Stacked Squares (10 minutes)

Narrative

This warm-up prompts students to carefully analyze and compare representations of patterns. Listen for the language students use to describe and compare the elements of each pattern and give them opportunities to clarify what they mean when they use numbers to describe the patterns (MP6).

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

• 2–3 minutes: partner discussion
• Share and record responses.

Student Facing

Which one doesn’t belong?

Student Response

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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 more-familiar 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.

MLR8 Discussion Supports. Use multimodal examples to show the patterns of both columns. Use verbal descriptions along with gestures, drawings, or concrete objects to show the connection between the multiples of 9 and 10.
Representation: Access for Perception. Synthesis: Use pictures of the long rectangle base-ten blocks to help students visualize the patterns. For example, display a picture of eight long rectangle base-ten blocks. Count by 10 while pointing to each block. Then, cross out one unit in each block and discuss how this shows that counting by 9 is like multiplying by 10 and subtracting.
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.

1. 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
2. For the numbers in the “counting by 10” column, why do you think:

1. the digits in the tens place change the way they do?
2. the digits in the ones place are the way they are?
3. 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

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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

In this activity, students continue to analyze patterns in numbers. This time, they look at the relationship between multiples of 100 and multiples of 99. As in the previous activity, they rely on their understanding of place value and operations to explain the patterns in the digits of the numbers (MP7). Although the use of the distributive property is not expected or made explicit, the work in both activities in this lesson develops students’ intuition for seeing, for instance, that $$12 \times (10 - 1) = (12 \times 10) - (12 \times 1)$$.

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.

1. Study the list of numbers. Make at least 3 observations about features of the pattern.
counting by 99
99
198
297
396
495
594

2. Extend the list with the next four multiples of 99. Be prepared to discuss how you know what numbers to write.
3. Why do you think the digits in the numbers change the way they do?

Student Response

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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 base-ten to reason more generally about patterns in numerical patterns.

Launch

• Groups of 2

MLR5 Co-Craft Questions

• Display only the opening sentence and list of numbers, without revealing the question(s).
• 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?”
• “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
1. Write the next four numbers she’d record if she kept going.

2. What patterns do you see? Describe as many as you can.
3. Choose one pattern that you noticed and explain why you think it happens.
4. Could 250 be a number that Elena calls out if she continued to count by 15? Explain or show your reasoning.

Student Response

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To answer the last question, students may try to count up by 15 and see if they would reach 250. Encourage students to see if any of the patterns they noticed could help them answer the question.

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?”

Cool-down: Count by 8 (5 minutes)

Cool-Down

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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 equal-size 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.