Unit 6, Lesson 11
Productos parciales y el algoritmo estándar
Grado 4
Warm-up
10 mins
Conversación numérica: El valor de los dígitos
Standards Alignment
Building On
Addressing
4.NBT.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Building Toward
Instructional Routines
Number TalkMaterials
NoneActivity Narrative
This Number Talk routine encourages students to think about decomposing factors by place value to multiply multi-digit numbers by one-digit numbers. As students look for and make use of structure, they may notice that multiplying the ones place has a result in the pattern of 5, 10, and 15. Students may use the partial products equations to multiply.
This routine helps students pay attention to the value of the digits when multiplying. This is important for setting up the conversation about the standard algorithm, in which students will find partial products mentally and use what they know about the value of the digits to condense the number of steps to multiply by multi-digit numbers.
Launch
- 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
Activity
- Record answers and strategy.
- Keep expression and work displayed.
- Repeat with each expression.
Student Task Statement
Encuentra mentalmente el valor de cada expresión.
Activity Synthesis
- “¿Qué conexiones o relaciones observan entre las expresiones?” // “What connections or relationships do you see between each expression?” (Each problem involves one more group of 5 or sets or 100 more.)
Activity 1
20 mins
Dos algoritmos para multiplicar
Instructional Routines
NoneMaterials
NoneActivity Narrative
This activity introduces students to the standard algorithm for multiplication. Students make sense of it by comparing and contrasting it to an algorithm that uses partial products for multiplying three- and four-digit numbers by one-digit numbers where no regrouping is necessary. When they interpret the given student work showing the standard algorithm students construct a viable argument for what Kiran did in his calculation (MP3). They also have an opportunity to make use of the structure they notice to compute the value of other products.
MLR8 Discussion Supports. Synthesis: Create a visual display of Diego’s algorithm. As students share their strategies, annotate the display to illustrate connections. For example, next to each step, write the multiplication expression used to find each partial product.
Advances: Speaking, Representing
Advances: Speaking, Representing
Launch
- Groups of 2
Activity
- “Traten de entender los algoritmos de Kiran y de Diego. Cuéntenle a su compañero en qué se parecen y en qué son diferentes los algoritmos y cómo creen que Kiran encontró su respuesta” // “Make sense of Kiran and Diego’s algorithms. Talk to your partner about how they are alike and different, and how Kiran might have found his answer.”
- 3 minutes: partner discussion
- Pause for a discussion.
- Invite students to share their conjectures on how Kiran might have reasoned about the product.
- Alternatively, consider displaying a few scenarios and polling students on which one might be closest to their conjecture, if any. For example:
- A: Kiran found
- B: Kiran used the same method as Diego but added the 9, 30, and 2,100 mentally, without writing them down.
- C: Kiran drew a diagram and did the computation on another sheet of paper and wrote the result here.
- D: Kiran multiplied the single-digit 3 with each digit in 713 and wrote each partial product in a single line.
- A: Kiran found
- Explain that Kiran had multiplied 3 by each digit in 713, but instead of reasoning about
- Demonstrate Kiran’s process:
- Because the 3 in 713 means 3 ones, he wrote the result of
- Because the 1 means 1 ten, he wrote the result of
- Because the 7 means 7 hundreds, he wrote the result of
- Because the 3 in 713 means 3 ones, he wrote the result of
- “Traten de usar el algoritmo de Kiran para encontrar el valor de los últimos dos productos” // “Try using Kiran’s algorithm to find the value of the last two products.”
- 3 minutes: independent work time
- 2 minutes: partner discussion
Activity 2
15 mins
Comparemos algoritmos
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this activity is for students to compare the standard algorithm for multiplication and an algorithm that uses partial products. The focus of the Lesson Synthesis is on the convention used for composing a new unit and how it connects to their work with the standard algorithm for addition.
Engagement: Internalize Self-Regulation. Synthesis: Provide students an opportunity to self-assess and reflect on their own progress. Remind students what kinds of multiplication problems they were working on before this unit, and ask them to notice how the multiplication they are working on now is different. Invite them to list the new strategies they have learned in this section.
Supports accessibility for: Conceptual Processing, Social-Emotional Functioning
Supports accessibility for: Conceptual Processing, Social-Emotional Functioning
Launch
- Groups of 2
Activity
- “Trabajen con su compañero en el primer problema” // “Work with your partner on the first problem.”
- 3–4 minutes: partner work time
- Pause for brief discussion. Invite students to share how the two methods are alike and how they are different.
- “¿Por qué en el algoritmo de Kiran hay un 1 escrito encima del 2 que está en la columna de las decenas?” // “In Kiran's algorithm, why is a 1 written above the 2 in the tens column?” (It represents 1 ten from the number 12. It helps us remember to add it to the tens place because we are doing a lot of calculations in our heads.)
- “¿Dónde hemos visto antes esta notación?” // “Where have we seen this notation before?” (When we add using the standard algorithm, we use this notation to show that we have more than 9 ones, tens or hundreds, and so on, in a given place, we add each group of 10 units to the place value to the left.)
-
“Kiran usó el algoritmo estándar de multiplicación. Traten de usarlo para encontrar el valor de
- 3 minutes: independent work time
- 1–2 minutes: partner discussion
Lesson Synthesis
“Hoy comparamos el algoritmo estándar de multiplicación con un algoritmo en el que se usan productos parciales. Veamos cómo encontraríamos el valor de
Display:
“¿Cómo encontrarían el valor del producto?” // “How would you find the value of the product?” (I know that 500 x 3 is 1500 and 12 x 3 is 36 and 1500 + 36 = 1,536. Three times 2 is 6, so that goes in the ones place. 3 times 1 is 3, so that goes in the tens place. 3 times 5 is 15, so that goes in the thousands and hundreds place.)
Assure students that they are not expected to use a particular method for multi-digit multiplication in IM Grade 4. Explain that they will study this algorithm more in IM Grade 5. Invite them to try to use it to multiply as we continue to work through lessons.
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