# Lesson 2

Patterns that Repeat

## Warm-up: How Many Do You See: Colorful Tiles (10 minutes)

### Narrative

This warm-up encourages students to look for structure in the ways the symbols or colors repeat and to use grouping strategies or the structure they see (MP7) to quantify something that would be tedious to count individually. The work here prepares students to analyze and describe patterns formed by repetition later in the lesson.

### Launch

• Groups of 2
• “How many do you see? How do you see them?”
• Flash image.
• 30 seconds: quiet think time

### Activity

• Display image.
• 1 minute: partner discussion
• Record responses.

### Student Facing

How many tiles do you see? How do you see them?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “What patterns helped you figure out how many tiles there were?” (Repeating colors, repeating symbols, repeating groups of 4 tiles)

## Activity 1: Patterns that Repeat (20 minutes)

### Narrative

In this activity, students analyze a pattern with repeating shapes and look for as many features of the pattern as they can find. They then extend the pattern based on their observations. Students also generate an original pattern of shapes that repeat by following a rule. The work here prepares students to reason about such patterns represented numerically in the next activity.

MLR8 Discussion Supports. During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . .” Original speakers can agree or clarify for their partner.

• Groups of 2

### Activity

• “Take a few quiet minutes to find more than one feature of the pattern made with shapes in the first problem. Then, share your thinking with your partner.”
• 2 minutes: quiet think time on the first part of the first problem
• 3–4 minutes: partner work time on the rest of the first problem
• Pause for a discussion before students proceed to the second problem. Invite students to share their responses.
• “Now, create a pattern of your own by repeating a circle and another shape.” (Alternatively, students can create a pattern that uses only two colors.)
• “When you’re done, trade your pattern with your partner and complete the rest of the second problem.”

### Student Facing

1. Here is a pattern made by arranging shapes.

1. Look for as many features of patterns as you can and describe them to your partner.
2. What rule might this pattern follow?

3. Use the rule to extend the pattern so that it repeats one more time.
2. Create a new pattern that uses only a circle and one other shape, and that follows a new rule.

1. Trade your pattern with your partner. Look for as many features of patterns as you can and describe them.
2. What rule might your partner have followed to create their pattern?

3. Use the rule to extend their pattern so that it repeats one more time.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• If time permits, invite 1–2 groups of students to share their patterns.

## Activity 2: Numbered Patterns (15 minutes)

### Narrative

In this activity, students number the shapes in the pattern from the first activity, examine and use features of the numerical patterns to predict the shapes in particular spots.

### Launch

• Groups of 3–4
• Assign each group member one shape in the design.
• “Record your assigned shape in the four blanks in the second problem.”

### Activity

• “Number each shape in the pattern of shapes from 1 to 12, in order.”
• “Then, work independently to answer the questions in the second problem, using the shape assigned to you.”
• 5 minutes: independent work time
• 5 minutes: small-group discussion
• Monitor for the different ways students answer the questions. They may, for instance:
• use skip-counting (4, 8, 12, . . .)
• reason additively (add 2 or 4 each time) or use “_____ more” language
• reason multiplicatively or use the term “multiples” (multiples of 2 or 4, or groups of 2 or 4)
• write addition or multiplication expressions or equations

### Student Facing

Here is the pattern of shapes you saw earlier.

1. Number the shapes 1 to 12.
2. Your teacher will assign you a shape. Write it in every blank space and answer the questions.

1. What numbers were written for the ____________________s?
2. If you extend the pattern, what numbers will be written for the next two

_____________________s?

3. What number will the tenth ____________________ have? Explain or show your reasoning.

4. Will the 30th shape be a ____________________? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite one student who worked on each shape to share their responses and reasoning. Record the number sequences for all to see.
• “How are the numerical patterns the same? How are they different?”

## Activity 3: Clare’s Pattern (15 minutes)

### Narrative

This optional activity prompts students to generate a shape pattern given a rule and to describe the numerical patterns that are created when they number the shapes. They make predictions about whether a certain value or shape would appear in a particular position of the pattern (MP2).

While students may make predictions in a number of ways, during the synthesis, highlight reasoning that is based on the idea of multiples or adding a certain multiple to a number.

Action and Expression: Develop Expression and Communication. Provide students with alternatives to writing on paper. Students can share their learning verbally.
Supports accessibility for: Language, Conceptual Processing

### Launch

• Groups of 2
• “Clare has in mind a particular arrangement of shapes. Let’s see what her design looks like and make some predictions about it.”

### Activity

• “Work independently for a few minutes, and then share your thinking with your partner.”
• 5–7 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for the ways students:
• describe the rule
• reason about the 31st shape in Clare’s pattern
• reason about whether a square could be the last one of 40 shapes

### Student Facing

Clare created a pattern using 3 shapes—a triangle, a circle, and a square—that repeat in that order.

1. Draw the first 10 shapes in Clare’s pattern.
2. Clare numbered her shapes. What numbers are the first 5 squares?
3. What rule is the numerical pattern following?
4. What is the 31st shape in Clare’s pattern? Explain or show your reasoning.
5. Clare wants to use 40 shapes in her pattern and the last shape to be a square. Is this possible? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

If students extend the pattern one shape at a time until it reaches the 31st term, consider asking: “How can the numbers help us find the 31st shape?”

### Activity Synthesis

• Select students to share their responses and reasoning to the last two problems.
• If all students used the numerical pattern that represents squares to help them find the 31st shape or to reason about the last shape in a list of 40, ask:
• “Is it possible to answer the last two questions using the numerical pattern that represents triangles?” (Yes. The numbers would be 1, 4, 7, . . . We could keep adding 3 or multiples of 3 to one of these numbers to see if we get 31 at some point.)
• “Is it possible to use the numerical pattern to represent the circles?” (Yes. The numbers would be 2, 5, 8, . . . We could keep adding 3 or multiples of 3 to one of these numbers.)
• “Why might you want to use the numerical pattern that represents the squares?” (The numbers are all multiples of 3, which are familiar numbers. We can reason using only multiplication, instead of adding repeatedly.)

## Lesson Synthesis

### Lesson Synthesis

“Today we explored patterns created by shapes that repeat according to a rule. When we numbered the shapes, we created a numerical pattern.”

Display numerical patterns that represent each shape in the last activity:

Triangles: 1, 3, 5, 7, 9

Circles: 2, 6, 10, 14, 18

Squares: 4, 8, 12, 16, 20

“Here are some numerical patterns we saw in this lesson. How would you find the 50th number in each pattern, without listing all 50 numbers?” (Sample responses:

• For triangles: Add $$45 \times 2$$ to 9, or find $$9 + (45 \times 2)$$
• For circles: Add $$45 \times 4$$ to 18, or find $$18 + (45 \times 4)$$
• For squares: Find the 50th multiple of 4, or $$50 \times 4$$.)

## Cool-down: Happy Faces (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.