Lesson 5

This Number Talk encourages students to decompose factors and to rely on the distributive property to mentally solve. The strategies elicited here will be helpful later in the lesson when students multiply up to four-digit numbers by one-digit numbers, and later in the section when they multiply 2 two-digit numbers by decomposing factors.

• 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 les ayudaron las tres primeras expresiones a encontrar el valor de \(15 \times 30\)?” // “How did the first three expressions help you find the value of \(15 \times 30\)?” (I know that \((5 \times 30) + (10 \times 30)\) is \(15 \times 30\). I doubled the value of \(8 \times 30\) to get the value of \(16 \times 30\) and then I subtracted 30 from it to get the value of \(15 \times 30\).)

In this activity, students build on grade 3 work with arrays to consider how to find the total number in an array without counting by 1. Students are not asked to find the answer, but instead share their strategies for doing so. This allows teachers to observe how students make sense of multiplying larger numbers.

Students may decompose the larger array of stickers into two smaller arrays using the distributive property to determine the product (MP7). They may also use the idea of doubling and tripling to find the product. (For instance, they may start with \(13 \times 2\) and triple the result to get \(13 \times 6\).) 

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

• Groups of 2
• Give each group tools for creating a visual display.

• “Tómense unos minutos en silencio para responder la pregunta. Luego, comparen su estrategia con la de su compañero” // “Take a few quiet minutes to answer the question. Then, compare your strategy with your partner’s.”
• 3 minutes: independent work time
• 2 minutes: partner discussion

• “Creen una presentación que muestre sus ideas. Escríbanlas de manera que los demás puedan entenderlas” // “Create a display that shows both of your ideas. Record your thinking so that it can be followed by others.”
• 5 minutes: partner work time
• Monitor for students who:
  • decompose the two-digit factor by place value and use a drawing or an expression to show the decomposition (for example: partition the 13 columns in the array into 10 and 3 columns)
  • write expressions that involve the distributive or associative properties (as noted in Student Responses)
• 3 minutes: gallery walk

If students choose to count the number in the first column or the first row and then skip-count by that number, consider asking them how they might record that strategy (other than writing “skip-count by 6” or “skip-count by 13”).

• “¿Qué ideas fueron las que más vieron durante su recorrido de hoy?” // “Which ideas did you see the most as you walked around today?”
• 30 seconds: quiet think time
• Invite students to share their strategies and reasoning.
• To highlight similarities and differences, consider comparing and contrasting the strategies using the term “descomponer” // “decompose.”

In this activity, students use strategies and representations that make sense to them to find products beyond 100. As before, the context of stickers lends itself to be represented with an array. The factors are large enough, however, that doing so would be inconvenient, motivating other representations or strategies (MP2). Look for the ways that students extend or generalize previously learned ideas or representations to find multiples of larger two-digit numbers. While many of the student responses are written with expressions, students are not expected to represent their reasoning using equations and expressions as this time. Teachers may choose to represent student reasoning using equations and expression so students can start connecting representations.

After students work on the first problem, pause to discuss some possible representations for finding the number of stickers. Each of the representations show different ways to represent the decomposition of a factor and students may decompose the factors in a variety of ways.

A. I created an array and decomposed it into smaller arrays.

B. I drew a diagram and decomposed it into smaller sections.

C. I decomposed the 21 and wrote one or more expressions or equations.

\((9 \times 20) + (9 \times 1)\)
\((9 \times 10) + (9 \times 10) + (9 \times 1)\)

D. I decomposed the 9 and wrote one or more expressions or equations. 

\((3 \times 21) + (3 \times 21) + (3 \times 21)\)
\((4 \times 21) + (4 \times 21) + (1 \times 21)\)

Consider using the “four corners” structure to allow for movement and for interactions among students who might not typically interact. Post each of the four strategies in a different corner of the classroom. For the representations that use arrays or rectangular diagrams, it may help to give examples of decomposing based on 1–2 samples of student work that you observe during the activity. Then, ask students to move to a corner based on their reasoning strategy and representation. 

• Create 4 posters showing the 4 representations shown in the activity narrative.

• As a class, read the first problem about Elena’s stickers.
• “Hagan una estimación: ¿Creen que Elena tiene menos de 100 calcomanías, entre 100 y 200, o más de 200?” // “Make an estimate: Do you think Elena has fewer than 100 stickers, between 100 and 200, or more than 200?”
• 30 seconds: quiet think time
• Poll the class on their estimates (fewer than 100, between 100 and 200, or more than 200).
• “Hablen con su compañero y explíquenle cómo hicieron su estimación” //  “Turn to your partner and explain how you made your estimate.”
• 1 minute: partner discussion

• “Tómense unos minutos en silencio para encontrar el número exacto de calcomanías que tiene Elena. Expliquen o muestren cómo razonaron” // “Take a few quiet minutes to find the exact number of stickers Elena has and explain or show your reasoning.”
• 2–3 minutes: independent work time
• Display the four representations shown in the activity narrative.
• “¿Cuál es la representación que mejor describe su estrategia? Si ninguna lo hace, creen una presentación que muestre cómo pensaron” // “Which representation best describes your approach? If none of them does, create a display that shows your thinking.”
• Poll the class on their representations. Select a student who uses each strategy to explain more fully how they solved the problem.
• “Ahora respondan la última pregunta usando alguna de estas representaciones u otra que tenga sentido para ustedes” // “Now answer the last question using any of these representations or another one that makes sense to you.”
• 5–7 minutes: group work time
• Monitor for the strategies students use to find \(3 \times 48\) and \(7 \times 23\).

If students start to draw arrays to represent \(3 \times 48\) and \(7 \times 23\), ask them if they could represent the factors in another way that doesn’t require drawing individual dots for the stickers.

• See lesson synthesis.