Lesson 2

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.

• Groups of 2
• “¿Cuántas ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you see them?”
• Flash image.
• 30 seconds: quiet think time

• Display image.
• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “¿Qué patrones les ayudaron a descubrir cuántas fichas había?” // “What patterns helped you figure out how many tiles there were?” (Repeating colors, repeating symbols, repeating groups of 4 tiles)

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.

• Groups of 2

• “Recuerden el patrón hecho con figuras del primer problema. En silencio, encuentren más de una característica de ese patrón. Luego, compartan con su compañero cómo pensaron” // “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.
• “Ahora creen su propio patrón con un círculo y otra figura” // “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.)
• “Cuando hayan terminado, intercambien su patrón con el de su compañero y completen el resto del segundo problema” // “When you’re done, trade your pattern with your partner and complete the rest of the second problem.”

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

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.

• Groups of 3–4
• Assign each group member one shape in the design.
• “Anoten el nombre de la figura que les asigné en los cuatro espacios en blanco del segundo problema” // “Record your assigned shape in the four blanks in the second problem.”

• “Enumeren las figuras del patrón en orden del 1 al 12” // “Number each shape in the pattern of shapes from 1 to 12, in order.”
• “Luego, trabajen individualmente para responder las preguntas del segundo problema usando la figura que les asigné” // “Then, work independently to answer the questions in the second problem, using the shape assigned to you.”
• “Después, compartan con su grupo lo que descubrieron” // “Afterwards, share your findings with your group.”
• 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

• Invite one student who worked on each shape to share their responses and reasoning. Record the number sequences for all to see.

• “¿En qué se parecen los patrones numéricos? ¿En qué son diferentes?” // “How are the numerical patterns the same? How are they different?”

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. 

• Groups of 2
• “Clare está pensando en cierta manera de organizar unas figuras. Observemos su diseño y hagamos algunas predicciones sobre él” // “Clare has in mind a particular arrangement of shapes. Let’s see what her design looks like and make some predictions about it.”

• “Trabajen individualmente durante unos minutos y luego compartan con su compañero lo que pensaron” // “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

If students extend the pattern one shape at a time until it reaches the 31st term, consider asking: “¿Cómo nos pueden ayudar los números a encontrar la figura número 31?” // “How can the numbers help us find the 31st shape?”

• 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:
  • “¿Es posible responder las últimas dos preguntas usando el patrón numérico que representa los triángulos?” // “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.)
  • “¿Es posible usar el patrón numérico para representar los círculos?” // “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.)
• “¿Cuál sería una razón para usar el patrón numérico que representa los cuadrados?” // “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.)