This activity invites students to identify features of a growing shape pattern that are not explicit in a given rule. They are first asked to continue a given shape pattern and informally describe features of the pattern that they notice. Then students are asked to generate a pattern given a new rule. Both patterns use the structure of an array and the familiar factors of 5 and 10 to encourage all students to notice, describe, and explain features of the patterns that may include:

• Whether the total amount of bottle caps in each step is even or odd.
• Patterns in the digits in the numbers that represent the total amount of bottle caps in each step.
• Differences in how much the patterns grow from one step to the next. 

Although students are asked to describe or draw the patterns, they may also create the patterns with counters. 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. Students will revisit rules for patterns that involve doubling in a later lesson with numerical patterns. 

This activity uses MLR2 Collect and Display. Advances: Conversing, Reading, writing. Consider updating the visual display throughout the following lessons as students notice, describe, and explain new features of the patterns they generate.

• Invite students to share how they completed the steps for Han and Lin’s patterns. Consider recording student thinking on separate charts for reference in later lessons.
• Invite previously selected students to share what they noticed, including any numbers they wrote to represent their visual patterns.
• Display the collection of students’ words and phrases.
• “¿Qué otras palabras o frases importantes deberíamos incluir en nuestra presentación?” // “Are there any other words or phrases that are important to add to our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• “¿Qué otras características interesantes observan acerca de los patrones de Han y de Lin?” // “What other interesting features do you notice about Han and Lin’s patterns?”
• “¿En qué se parecían los patrones? ¿En qué eran diferentes?” // “How were the patterns the same? How were they different?” (The first two steps are the same. They both show arrays with 5 bottle caps in each row. Lin’s array grows faster because she’s not just adding 1 row each time, she’s doubling what’s there.)

In this activity, students generate a new visual pattern and describe what they notice. They create a list of numbers that represents a count of either the total number of squares or the total number of blocks. This encourages students to look for and make use of the structure of the pattern as they use what they notice to anticipate what they would see as 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 the shapes in the visual pattern are not arranged in an array.

Monitor for and select students with the following approaches to share in the Activity Synthesis for determining whether a step will have 25 square blocks or 25 total blocks, including:

• Attempt to continue the pattern with blocks first, before adjusting their approach to count on by 2 with a list of numbers.
• Reason about even and odd numbers of blocks using words or lists of numbers. 
• Add by 2 or add on multiples of 2 using expressions and operations (such as informally explaining ).

The approaches are sequenced from more concrete to more abstract to help students identify and explain features of the visual pattern that weren’t explicit in the given rule. Not having enough blocks to continue the pattern beyond step 3 prompts students to adjust their strategy based on their observations. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently and include at least one example from Partner A and one example from Partner B. For examples of approaches with lists of numbers, words, or expressions and operations, look at the Student Responses.

• Invite previously selected students to share in the given order. Include examples from both Partner A and Partner B for how they determined whether there would be 25 square blocks or 25 total blocks in the pattern. Record or display their work for all to see.
• Connect students’ approaches by asking:
  • “¿Cómo se relacionan las distintas maneras de explicar si en un paso habrá 25 fichas cuadradas o 25 fichas en total?” // “How are the different ways of explaining whether a step will have 25 square blocks or 25 total blocks related?” (They all show a list of numbers and were based on observations. Some explained using words and some used expressions.)
• Connect students’ approaches to the learning goal by asking:
  • “¿En qué se parecen la lista de números que escribió el compañero A y la lista de números que escribió el compañero B?” // “How are the list of numbers that Partner A wrote and the list of numbers that Partner B wrote the same? (They’re both showing counts of something from the same pattern. They both show numbers that increase by 2.)
  • “¿En qué son diferentes?” // “How are they different?” (They start at different numbers. They are showing counts of different things from the pattern. One has even numbers and the other has odd numbers.)
  • “Puede que hayan observado números pares o impares, o hayan usado lo que saben sobre contar de 2 en 2 hacia adelante. ¿Cómo les ayudaron estas cosas a predecir si habría 25 fichas cuadradas o 25 fichas en total?” // “How did noticing even or odd numbers or what you know about counting on by 2 help you predict whether there would be 25 square blocks or 25 total blocks?” (You didn’t have to build it with blocks or list all the numbers to know if 25 was in the pattern.)