Lesson 12

The purpose of this warm-up is to elicit the idea that 3 groups of 40 can also be seen as 12 groups of 10, which will be useful when students multiply one-digit whole numbers by multiples of 10 in a later activity. While students may notice and wonder many things, seeing that the total can be decomposed into rows of 30 and further decomposed into units of 10 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 cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “¿Qué valor está representado por el diagrama?” // “What is the value the diagram represents?” (120)
• “¿Cómo nos puede ayudar ver los grupos de diez a encontrar el número total de cuadrados?” // “How could noticing groups of ten help us find the total number of squares?” (There are 3 groups of 4 tens, which is 12 tens. There are 4 groups of 30, which is 12 tens. We could count by tens to find the total. We know 12 tens would be 120.)
• Record equations that reflect student thinking such as \(3 \times 4 \times 10 = 12 \times 10\) and \(4 \times 30 = 12 \times 10\).

The purpose of this activity is for students to work with products of whole numbers and multiples of 10 in a concrete and familiar context before reasoning more abstractly about them. Given some numbers of dollar bills (for instance, four \$20 bills), students write expressions to represent the amount (\(4 \times 20\)) and then find its value using strategies that make sense to them. For example, they may count by 20 four times, think of \$20 in terms of two \$10 bills and find \(4 \times 2 \times 10\) (or \(8 \times 10\)). Consider giving students access to play money, if available, to help them visualize the quantities and support their reasoning.

The reasoning here prompts students to use strategies based on place value and properties of operations (especially the associative property). It prepares students to work more flexibly with products involving factors and multiples of 10 in which the product is greater than 100.

• Groups of 2
• “Vamos a resolver un problema sobre un juego en el que se usa dinero de juguete. ¿Qué saben sobre los juegos en los que se usa dinero de juguete?” // “We’re going to solve a problem about a game that involves play money. What do you know about games that involve play money?”
• 1 minute: quiet think time
• Share responses.
• Give students access to base-ten blocks, grid paper, and play money, if available.

• “Completen los problemas con su compañero” // “Work with your partner to complete the problems.”
• 7-10 minutes: partner work time
• In the last problem monitor for students who use the following strategies to highlight in the synthesis:
  • Count by multiples of 10 to find a total, such as 50, 100, 150, 200, 250, 300.
  • Use place value to find a total, such as knowing that 14 tens is 10 tens or 100, and 4 more tens or 40, which makes 140.

If students don’t find the product of one-digit whole numbers and multiples of 10 in the last problem, consider asking:

• “¿Qué has intentado hasta ahora para encontrar el producto?” // “What have you tried so far to find the product?”
• “¿Cómo podrías representar el producto con bloques en base diez?” // “How could you represent the product with base-ten blocks?”

• Invite students to share different combinations of the same bill that could be used to make \$150. Record and display expressions for each combination.
• Select previously identified students to share their strategies for how they found one of the totals in the last problem.

The purpose of this activity is for students to continue to reason about products of a whole number and a multiple of 10, this time using base-ten blocks to support their thinking. They analyze two strategies for multiplying. Both strategies are based on place value, but the second strategy also uses the associative property to think about \(8 \times 30\) as \(8 \times 3 \times 10\) or \(24 \times 10\).

• Groups of 2
• “Durante un momento, examinen las estrategias de Jada y de Kiran para multiplicar \(8 \times 30\)” // “Take some time to look at Jada and Kiran’s strategies for multiplying \(8 \times 30\).”
• 30 seconds: quiet think time
• “Hablen con su compañero sobre cómo podemos ver las estrategias de Jada y de Kiran en el diagrama” // “Talk to your partner about how we can see Jada and Kiran’s strategies in the diagram.” (We can see Jada’s skip counting by 30 in the rows. The 8 in Kiran’s strategy is the 8 rows and the 3 is the 3 tens in each row, so there are 24 tens.)
• 2–3 minutes: partner discussion
• Share responses.
• Give students access to grid paper and base-ten blocks.

• “Trabajen en el primer problema con su compañero” // “Work with your partner on the first problem.”
• 2–3 minutes: partner discussion
• Invite students to share how the strategies are alike and how they’re different.
• “¿Cómo pudo Kiran convertir \(8 \times 30\) en \(24 \times 10\)?” // “How was Kiran able to turn \(8 \times 30\) into \(24 \times 10\)?” (Eight times 30 is like 8 groups of 3 tens, so that’s like 24 tens. You can see \(8 \times 30\) and \(24 \times 10\) in the same diagram, so they are the same amount.)
• “Ahora encuentren el valor de los demás productos con su compañero” // “Now, work with your partner to find the value of other products.”
• 5–7 minutes: partner work time
• Monitor for students who use the associative property as a strategy to highlight during the synthesis.

• Select 2–3 students who used a strategy based on the associative property (for example, thinking of \(7 \times 40\) as 28 tens) to share their responses.
• Consider asking:
  • “¿En qué parte del trabajo de _____ vemos la expresión original?” // “Where do we see the original expression in _____’s work?”
  • “¿Cómo cambió _____ la expresión original para que fuera más fácil encontrar el total?” // “How did _____ change the original expression to make it easier to find the total?”
  • “¿Cómo funciona la estrategia para multiplicar que usó _____?” // “How does _____’s strategy for multiplying work?”