Lesson 11

The purpose of this warm-up is for students to notice that figures composed of multiple arrays can be decomposed into smaller arrays, and that this is a strategy to determine the total number of dots. This will be helpful in later lessons when students decompose figures into rectangles to find the total area.

When students find ways to decompose the given arrangements of dots to find the number of dots, they practice looking for and making use of structure (MP7).

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

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

• “¿En qué se parecen las formas en las que encontraron el número de puntos de cada imagen?” // “What’s the same in how you found the number of dots in each image?” (I looked for groups of 4. I decomposed each arrangement of dots into smaller arrays.)
• Consider asking:
  • “¿Alguien puede expresar con otras palabras la forma en la que ______ vio los puntos?” // “Who can restate the way ___ saw the dots in different words?”
  • “¿Alguien vio los puntos de la misma forma, pero lo explicaría de manera diferente?” // “Did anyone see the dots the same way but would explain it differently?”
  • “¿Alguien quiere agregar algo a la forma en la que ______ vio los puntos?” // “Does anyone want to add an observation to the way ____ saw the dots?”

The purpose of this activity is for students to find missing products in the multiplication table as they consider the rectangular structure of how products are organized in the table.

• Groups of 2
• Display the four blank 5-by-5 multiplication tables, each showing a multiple of 3 in the third row, in order from 3 to 12.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Students may notice: There are numbers along the top and down the left column. In each table the number in the rectangle is going up by 3. The number in the rectangle lines up with two factors that you would multiply to get that number. Students may wonder: Why are there numbers in each rectangle? Why are there numbers along the top and on the left?)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses

• “En estas tablas, hagan sus propios rectángulos y anoten en cada una el producto que representa el rectángulo” // “Create your own rectangles in these tables and record the product in each table that represents the rectangle.”
• 3–5 minutes: independent work time
• “Compartan sus rectángulos con su compañero y expliquen su razonamiento” // “Share your rectangles with your partner and explain your reasoning.”
• 2–3 minutes: partner discussion
• Monitor for students who:
  • Create a rectangle by lining up the bottom and right edges with the corresponding factors.
  • Count the total number of squares to make sure it matches the product they record in the rectangle.
  • Record the product that represents the rectangle in the bottom right square of the rectangle.
  • Record different rectangles that both have an area of 24 square units.

• Have 3–5 previously selected students share their rectangles and how they created them.
• “Esta se llama la tabla de multiplicar. Los números de la parte de arriba y del lado izquierdo son los factores, o los números que multiplicamos” // “This is called the multiplication table. The numbers along the top and on the left side are the factors, or the numbers we are multiplying.”
• “El número que está en el cuadrado de abajo a la derecha del rectángulo es el producto. El producto corresponde al factor que está encima suyo y al factor que está a su izquierda” // “The number in the lower right square of the rectangle is the product. The product matches up with a factor above it and a factor to the left of it.”
• “Podríamos representar los factores y el producto que está en la última tabla del primer grupo de tablas con ‘3 por 4 es igual a 12’” // “We could represent the factors and the product in the last table in the first group of tables with 3 times 4 equals 12.”
• Display \(3 \times 4 = 12\)
• “¿En qué son diferentes la tabla de multiplicar y otras formas en las que hemos representado la multiplicación?” // “How is the multiplication table different than other ways we’ve represented multiplication?” (It’s harder to see the factors. There’s no equal sign. We can picture a rectangle that helps you find the product.)

The purpose of this activity is for students to find products in the multiplication table. Students are encouraged to find familiar products before working on less-familiar ones. They do not need to fill in all of the products in the table. The synthesis focuses on patterns students find in the table and how they can show the patterns with equations.

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Students may notice: Each product is in 2 places in the multiplication table. The number 4 only shows up in one place on the multiplication table. If I go down from the top 2 or across from the 2 on the left the products count by 2. The factors count by 1 as the products count by 2, like 1 and 2, 2 and 4, 3 and 6. All the numbers in the table are even. The highest product is 10. Students may wonder: Could we put other products in the table?)
• 1 minute: quiet think time
• Share responses.

• “Trabajen individualmente en el primer problema” // “Work independently on the first problem.”
• 3–5 minutes: independent work time
• Share responses.
• “Terminemos de llenar las filas y las columnas que muestran productos de dos y productos de cinco” // “Let’s finish filling in the rows and columns that show products of twos and products of fives.”
• Display table showing all the products of 2 and 5.
• “Piensen en cómo responderían los últimos dos problemas” // “Think about how you would answer the last two problems.”
• 1 minute: quiet think time
• “Ahora, con su pareja, completen los últimos dos problemas” // “Now work with your partner to complete the last two problems.”
• 5–7 minutes: partner work

• “¿Qué patrones encontraron en los productos de 5? ¿Cómo los mostraron con ecuaciones?” // “What patterns did you find in the products of 5? How did you show them with equations?”
• “¿Qué otros patrones vemos en la tabla? ¿Cómo podríamos mostrar los patrones con ecuaciones?” // “What other patterns do we see in the table? How could we show the patterns with equations?”