Lesson 1

The purpose of this Number Talk is to elicit strategies and understandings students have for multiplication of two- and three-digit numbers that are multiples of 10 or 100. These understandings help students develop fluency and will be helpful later in this lesson when students use multiplication to estimate products. This work also prepares them for the work of the standard algorithm for multiplication in which each product is a product of single-digit multiples of powers of ten.

In this activity, students have an opportunity to look for and make use of structure (MP7) because the basic fact they are using is \(5 \times 6 = 30\), and each successive product is ten times larger.

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

• “¿Cómo podemos usar \(5 \times 6\) para encontrar el valor de cada producto?” // “How can we use \(5 \times 6\) to find the value of each product?” (\(50 \times 6 = 5 \times 6 \times 10\), \(50 \times 60 = 5 \times 6 \times 10 \times 10\), \(50 \times 600 = 5 \times 6 \times 10 \times 100\), \(600 \times 500 = 6 \times 5 \times 100 \times 100\))
• “¿Por qué cada producto de la conversación numérica tiene un cero más que el producto anterior?” // “Why does each product in the number talk have one more 0 in it than the previous product?” (Because one of the factors has an additional 0.)

The purpose of this activity is for students to make a reasonable estimate for a given product. In addition to estimating the product, students also decide whether the estimate is too large or too small. In the activity synthesis, students consider about how far their estimate is from the actual product. In the next activity, students will evaluate the expressions using a strategy of their choice.

Students choose between several different possible estimates and justify their choice before they calculate the product (MP3).

• Groups of 2

• 5–7 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who:
  • relate the given expression to each proposed answer by rounding or changing one or both factors.
  • estimate by rounding the factors.
  • use benchmark numbers.
  • use place value reasoning or the properties of operations to explain why their estimate is reasonable.

If students do not choose a reasonable estimate for \(18 \times 149\), ask, “¿Puedes explicar cómo escogiste tu estimación?” // “Can you explain how you chose your estimate?”

• Invite students to share their estimates for the product and their reasoning.
• “¿Por qué piensan que \(3,\!000\) es una buena estimación?” // “Why do you think \(3,\!000\) is a good estimate?” (18 is close to 20 and 149 is very close to 150 and I know \(20 \times 150 = 3,\!000\).)
• “¿El valor de \(18 \times 149\) es mayor que o menor que \(3,\!000\)? ¿Cómo lo saben?” // “Is the value of \(18 \times 149\) greater than or less than \(3,\!000\)? How do you know?” (Less, because 18 is less than 20 and 149 is less than 150.)
• “¿Pueden estimar cuánto menos?” // “Can you estimate how much less?” (About 300, because I added 2 to 18 to get 20 and \(2 \times 149\) is about 300.)
• Display: \(18 \times 149\) is about \(2,\!700\).
• “En la siguiente actividad, vamos a comprobar qué tan buena es nuestra estimación” // “In the next activity, we’ll check to see how good our estimate is.”

The purpose of this activity is for students to multiply a three-digit number by a two-digit number using a strategy that makes sense to them. The expressions are scaffolded so that students can use one calculation to help with the next, particularly when they look for the final product, which is \(18 \times 149\).

Students may:

• draw diagrams to help visualize the calculations.
• use a form of partial products and the distributive property.
• round 49 to 50 or 149 to 150 and then compensate.

This activity encourages students to compare the strategy they used with the strategies that their classmates used, and to discuss the similarities and differences. The intent of this activity is not to create a list of strategies for students to choose from. Instead, students have an opportunity to think about how the properties of operations and place value understanding were used in each strategy.

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

• Groups of 2

• 1–2 minutes: quiet think time
• 5–8 minutes: partner work time

• “Creen una presentación visual que muestre cómo pensaron en cada uno de los problemas. Incluyan detalles, como notas, diagramas, dibujos, etc., para ayudar a los demás a entender sus ideas” // “Create a visual display that shows your thinking about each of the problems. You may want to include details such as notes, diagrams, drawings, and so on, to help others understand your thinking.”
• 2–5 minutes: independent or group work
• Monitor for students who:
  • use diagrams to show and keep track of their calculations
  • use their solution from one problem to solve a different problem
  • find a related product like \(18 \times 10\) or \(18 \times 50\) and use that to find the value of \(18 \times 9\) or \(18 \times 49\)

If students do not use the value of the product \(18 \times 9\) to help them find the value of \(18 \times 49\), ask, “¿Cómo te puede ayudar el valor del producto \(18 \times 9\) a encontrar el valor de \(18 \times 49\)?” // “How could you use the value of the product \(18 \times 9\) to help you find the value of \(18 \times 49\)?”

• 5–7 minutes: gallery walk
• “¿En qué se parecen y en qué se diferencian las estrategias para resolver los problemas?” // “What is the same and what is different between the approaches to solve the problems?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Invite students to share their diagrams.
• Highlight at least one of the diagrams or use the diagram from the student solutions. “¿Cómo ayuda el diagrama a llevar la cuenta de los cálculos?” // “How does the diagram help keep track of the calculations?” (For \(18 \times 9\) I know I need to find \(10 \times 9\) and then \(8 \times 9\) and add them.)