Lesson 4

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).

• Groups of 2
• Display image.
• “Escojan uno que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “¿Qué características de los diagramas examinaron cuando trataron de encontrar cuál era diferente?” // “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.

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.

• Groups of 2
• Read the opening paragraph as a class.
• “¿Qué prefieren: contar de 10 en 10 o contar de 9 en 9? ¿Por qué?” // “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
• “Exploremos los números que obtenemos al contar de 9 en 9 y de 10 en 10, y veamos qué patrones podemos encontrar” // “Let’s look at numbers we get by counting by 9 and by 10 and see what patterns we can find.”

• “En silencio, trabajen unos minutos en los primeros problemas. Luego, compartan cómo pensaron y completen el resto de la actividad con su compañero” // “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

Students may not see that the digit in the ones place decreases by 1 each time the count goes up by 9. Consider asking:

• “¿Qué patrones ves en la pareja de números de cada fila?” // “What patterns do you see in the pair of numbers in each row?”
• “¿Cómo crees que los números de la columna ‘contando de 9 en 9’ se relacionan con los números de la columna ‘contando de 10 en 10’?” // “How do you think the numbers in the ‘counting by 9’ column are related to those in the ‘counting by 10’ column?”

• 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.
• “Contar ocho veces de 9 en 9 es lo mismo que contar ocho veces de 10 en 10 y restar 1 ocho veces, es decir, \((8 \times 10) - (8 \times 1)\). Esto es \(80 - 8\), es decir, 72” // “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.”

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 \(12 \times (10 - 1) = (12 \times 10) - (12 \times 1)\).

• Groups of 2
• “Antes, contamos de 9 en 9 y encontramos algunos patrones en los números. Ahora veamos qué patrones podemos encontrar cuando contamos de 99 en 99” // “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.”

• “Completen la actividad con su compañero” // “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

• 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.
• “Contar cinco veces de 99 en 99 es lo mismo que contar cinco veces de 100 en 100 y restar 1 cinco veces; es decir, \((5 \times 100) - (5 \times 1)\)” // “Counting by 99 five times is the same as counting by 100 five times and subtracting 1 five times, or \((5 \times 100) - (5 \times 1)\).”
• “¿Cómo podemos usar el patrón para encontrar el 20º. múltiplo de 99?” // “How can we use the pattern to find the 20th multiple of 99?” (Find \(20 \times 100\) and subtract \(20 \times 1\) from it.)

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.

This activity uses MLR5 Co-craft Questions. Advances: writing, reading, representing

• Groups of 2

MLR5 Co-Craft Questions

• Display only the opening sentence and list of numbers, without revealing the question(s).
• “Escriban una lista de preguntas matemáticas que se pueden hacer sobre esta situación” // “Write a list of mathematical questions that could be asked about this situation.”
• 2 minutes: independent work time
• 2–3 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “¿En qué se parecen estas preguntas? ¿En qué son diferentes?” // “What do these questions have in common? How are they different?”
• Reveal the task (students open books), and invite additional connections.
• “Veamos qué características podemos encontrar en el patrón que se forma cuando contamos de 15 en 15” // “Let’s see what patterns we can find when we count by 15.”

• “En silencio, trabajen unos minutos en la actividad. Después, discutan sus respuestas con su compañero” // “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

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.

• 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.