Lesson 14

The purpose of this warm-up is to elicit the idea that while there are multiple ways to represent 2 groups of 12, some ways are more useful than others. While students may notice and wonder many things about the images, how 2 images show the groups of 12 have been organized using place value and how this type of decomposition can be helpful in finding the total are the important discussion points.

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “La imagen de la izquierda es un dibujo de grupos iguales. Las otras imágenes son diagramas en base diez. ¿En qué se parecen y en qué son diferentes estas representaciones?” // “The image on the left is a drawing of equal groups. The other images are base-ten diagrams. What is the same and different about these representations?” (They all show 12. They all show 2 groups of the same size. In the base-ten diagrams you can see the tens easier. It’s harder to see the tens in the first drawing.)

The purpose of this activity is for students to see how, when multiplying a number larger than ten, the distributive property can be used to decompose the factor into tens and ones, creating two smaller products. Base-ten blocks are used to help students visualize what is happening when a factor is decomposed to make two more easily known products. Factors slightly larger than ten can be naturally decomposed into a ten and some ones using place value. This will be useful in subsequent lessons as students progress towards fluent multiplication and division within 100.

When students see that you can decompose a teen number into tens and ones and use this to multiply teen numbers, they look for and make use of structure (MP7).

• Groups of 2
• Give students base-ten blocks.

• “Tómense unos minutos para examinar la estrategia de Tyler y decidir si están de acuerdo o en desacuerdo con ella” // “Take a few minutes to look at Tyler’s strategy and decide if you agree or disagree with it.”
• 2–3 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for students who connect the expressions \(7 \times 10\) and \(7 \times 3\) to the tens and ones portion of the place value diagram and \(7 \times 13\) to the entire diagram.
• Have students share why they agreed or disagreed with Tyler’s strategy with a focus on using the place value diagram as a justification.
• Consider asking:
  • “¿En qué parte del diagrama de valor posicional vemos \(7 \times 13\)?” // “Where do we see the \(7 \times 13\) in the place value diagram?”
  • “¿De dónde vienen el 10 y el 3?” // “Where do the 10 and the 3 come from?”
  • “¿Cómo podrían usar el diagrama de valor posicional para averiguar cómo encontrar el valor de \(7 \times 13\)?” // “How could you use the place value diagram to figure out how to find the value of \(7 \times 13\)?”
• “Ahora, solos, usen el método de Tyler para encontrar el valor de \(3 \times 14\)” // “Now use Tyler’s method on your own to find the value of \(3 \times 14\).”
• 2–3 minutes: independent work time
• “Compartan su solución y su razonamiento con su compañero” // “Share your solution and your reasoning with your partner.”
• 2–3 minutes: partner discussion

If students say they don’t see \(7\times10\) and \(7\times3\) in Tyler's diagram, consider asking:

• “¿En qué parte del diagrama ves \(7\times13\)?” // “Where do you see \(7\times13\) in the diagram?”
• “Si separamos las decenas y las unidades, ¿qué expresión podemos usar para describir las decenas?, ¿las unidades?” // “If we separate the tens and ones, what expression could we use to describe the tens? The ones?”

• Display base-ten blocks or place value diagrams that students used to solve. As students explain their work, write multiplication expressions to represent them.
• “¿Cómo muestra este diagrama (o los bloques en base diez) que el método de Tyler se puede usar para multiplicar \(3 \times 14\)?” // “How does this diagram (or the base-ten blocks) show how Tyler’s method could be used to multiply \(3 \times 14\)?” (We can see there are 3 tens which is 30. We can see that there are 3 groups of 4 ones which is 12. \(30 + 12\) is 42. The whole diagram represents \(3 \times 14\).)
• If there is time, ask students to find the value of \(4 \times 12\) and \(5 \times 16\) using the base-ten blocks and Tyler’s strategy.

The purpose of this activity is for students to make sense of different ways of representing multiplication of a teen number. Students analyze a gridded area diagram, base-ten blocks, and an area diagram labeled with side lengths. When they discuss how the different diagrams represent the same product, students reason abstractly and quantitatively (MP2).

• Groups of 2
• “Vamos a examinar tres formas diferentes en las que algunos estudiantes mostraron la misma expresión. ¿Qué observan? ¿Qué se preguntan?” // “We’re going to look at three different ways students showed the same expression. What do you notice? What do you wonder?” (Students may notice: You can see all the squares in the first 2 diagrams, but not in the last one. The middle diagram looks like base-ten blocks. Students may wonder: Why would you choose to use one of these diagrams? What numbers were they multiplying?)
• 1 minute: quiet think time
• Share responses.

• “Con su compañero, digan en qué parte ven los factores en cada diagrama y en qué parte ven el producto en cada diagrama” // “Work with your partner to tell how you see the factors in each diagram and how you see the product in each diagram.”
• 5–7 minutes: partner work time

• “¿En qué se parecen estas formas de representar \(3 \times 15\)?” // “How are these ways of representing \(3 \times 15\) the same?” (They all represent 3 times 15. They all show the 15 being decomposed into 10 and 5. They are all shaped like a rectangle.)
• “¿En qué son diferentes estas formas de representar \(3 \times 15\)?” // “How are these ways of representing \(3 \times 15\) different?” (Clare used base-ten blocks, but Andre and Diego used rectangles. Diego didn’t show the squares in his rectangle, but Clare and Andre did.)
• “¿Cómo podríamos representar con expresiones las estrategias que se muestran en todos los diagramas?” // “How could we represent the strategy shown in all the diagrams with expressions?” (\(3 \times 10\) and \(3 \times 5\) or \(10 \times 3\) and \(5 \times 3\).)