Lesson 1

This warm-up prompts students to analyze a visual pattern and the mathematics involved in how each step in the pattern changes. They also familiarize themselves with a kind of pattern they will investigate closely later in the lesson.

• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “What could go in the third step? Why is that?” (Three rows of 3 circles. The step number seems to correspond to how many rows of 3 there are.)
• “Is there a rule this pattern follows?” (Add a row of 3 circles each time.)
• “If we follow the rule, what will the fifth step look like?” (Five rows of 3 circles)

This activity invites students to look for structure in visual diagrams and describe possible patterns in them (MP7). Because only the first two steps of the pattern are given, students could draw different conclusions about the rule the pattern follows and the subsequent steps. They may say, for instance, that the number of caps are increasing by 5 each time, doubling each time, or increasing by 5 the first time, by 6 the second time, and so on. In the last question, students represent the visual patterns numerically and begin to notice patterns in the numbers as well.

Students may choose to describe the pattern they see using expressions or in terms of operations but are not expected to do so. They may describe their observations using words, numbers, or diagrams.

• Groups of 2–4
• “What are some patterns you see around your neighborhood, at home or on the way to school?”
• 30 seconds: quiet think time
• 1 minute: partner discussion

• “Take a few quiet minutes to look at Han’s pattern and answer the first two problems.”
• “Share your thinking with your group before continuing to the last problem.”
• 5 minutes: independent work time
• Monitor for the different ideas students have about the rule that Han might have in mind.
• Identify students with different ideas and ways of representing or describing their ideas, to share during the activity synthesis.

If students find only one rule for the pattern, consider asking:

• “What do you know about the relationship between 5 and 10? Think of as many ways as you can about how they are related.”
• “How can those relationships help you find another rule?”

• Select previously identified students to share their responses, including the numbers they wrote to represent their visual patterns.
• Display all of the numerical patterns students generated.
• “How can we tell from each numerical pattern what the rule is?” (We look at how the numbers change. They might:
  • increase or decrease by the same amount each time
  • be multiplied by the same factor each time
  • change by 1 more than the previous change.)
• “Besides the rule, what other interesting features do you notice about each number pattern?” (Alternating odd and even numbers, multiples of 5 and 10)

In this activity, students analyze a new visual pattern, describe its features, and make predictions about what they would see if the pattern continues (MP7). As in the first activity, students may show their reasoning using words, numbers, expressions, or equations. Unlike in the first activity, some elements in the steps remain constant, and students are given the rule the pattern follows.

Monitor for the ways students reason about the number of square blocks (partner A) or the number of all blocks (partner B) in the tenth step of the pattern. For the square blocks, students may:

• Start with only the number of square blocks in the legs of the giraffe (4) and reason:
  • We can skip-count by 2 ten times, starting from 4.
  • At each step, 2 square blocks are added to the 4 for the neck, so at the tenth step, there are 10 times 2, or 20, more square blocks.
  • \(4 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2\)
  • \(4 + (10 \times 2)\)
• Start with the number of square blocks in the first step (6) and reason:
  • We can add 2 to 6 nine times.
  • \(6 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2\)
  • \(6 + (9 \times 2)\)
• Start with the number in the fifth step (14) and reason:
  • We can add 2 to 14 five times.
  • \(14 + 2 + 2 + 2 + 2 + 2\)
  • \(14 + (5 \times 2)\)

In the activity synthesis, highlight the connections across the different representations (words, addition expressions, multiplication expressions) used to describe and extend the pattern.

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

• Consider preparing a set of pattern blocks for building the first two or three steps of the giraffe pattern. The set should include 6 hexagons, 6 triangles, 3 trapezoids, and 24 squares.

• Groups of 2
• “Work with your partner on this activity. One person should be partner A and the other person partner B.”
• Give access to pattern blocks.

• “Take a few quiet minutes to work on your part. Afterwards, share your responses with your partner.”
• 6–7 minutes: independent work time
• “When it’s your turn to share, explain your thinking so that it is clear to your partner.”
• “When it’s your turn to listen, pay close attention to your partner’s explanation. If you disagree or are unclear about a statement they make, ask questions or discuss the disagreement.”
• 4–5 minutes: partner discussion
• Identify students who reason differently about the number of blocks in the tenth step (as noted in the Activity Narrative) and about whether 25 could be a number  in each pattern.
• Consider asking them to create a visual display that shows their reasoning and include details to help others understand their thinking.

• Select previously identified students to share their visual displays or otherwise share their responses and reasoning to the second question (about the number of blocks in the tenth step).
• Sequence students’ explanations to go from the more concrete (using words or lists of numbers) to the more abstract (using expressions and operations).

MLR7 Compare and Connect

• Keep the visual displays visible or record students’ reasoning for all to see.
• “What is the same and what is different between the different strategies shared?”
• 2 minutes: partner discussion
• Highlight the similarities and differences across the various solution paths and representations used to reason about the same question. For example:
  • Alike: The pattern can be described by using skip-counting, words, addition, or multiplication. The pattern was expressed in terms of an increase by a fixed amount each time.
  • Different: We could use different numbers in the sequence as a starting point for figuring out the 10th term or to see if 25 is a value in the sequence.