The purpose of this Warm-up is for students to compare ways to keep track of composing units in addition problems and in multiplication problems in which the partial products have a newly composed unit. While students may notice and wonder many things about these ways of recording, the important discussion point is to invite students to consider whatever way they currently record composing a new unit when they add, and then to use it when they multiply if there’s a need to compose a new unit when they add up the partial products.
This prompt gives students opportunities to look for and make use of structure (MP7) as they attend to ways to record newly composed units when they add partial products.
Launch
Groups of 2
Display the image.
“¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
1 minute: quiet think time
Activity
“Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
1 minute: partner discussion
Share and record responses.
¿Qué observas? ¿Qué te preguntas?
Student Response
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Advancing Student Thinking
Activity Synthesis
“En ambos ejemplos se sumaban los mismos números. ¿Cómo explicarían qué pasa al sumar las decenas y las centenas?” // “In both examples, the same numbers are added up. How would you explain what happens when adding the tens and hundreds?” (Add 5 tens and 5 tens to get 10 tens, which is 1 hundred or 100. Show that 100 with 0 in the tens place and either record a 1 at the top. To add the hundreds, 300 +700+100=1100, so record 1 in the hundreds place and then show that we also have 1000 to add.)
“¿Cómo nos ayuda el valor posicional a entender qué representa cada 1 adicional?” // “How does place value help us understand what the additional 1s represent?” (If the 1 is in the hundreds column, it represents 100. If it is in the thousands column, it represents 1000.)
“Cuando sumen los productos parciales, lleven un registro de las unidades compuestas de una forma que tenga sentido para ustedes” // “When you add up partial products, keep track of the composed units in a way that makes sense to you.”
Activity 1
Standards Alignment
Building On
Addressing
4.NBT.B.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.
In this activity, students analyze multiplication involving 2 two-digit factors. Students make sense of work where the partial products are recorded, but there is no indication of where each product came from. After making sense of work, students use the same strategy of finding and keeping track of partial products to evaluate .
This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing
Representation: Internalize Comprehension. Provide students with a partially completed version of the itemized steps as described in the Activity Synthesis. For example, offer students a page that shows four copies of . Each equation could have one of the partial products written in its corresponding color underneath. Invite students to use a green, yellow, red, and purple colored pencil to highlight the digits involved in each step and to write the corresponding multiplication expression next to each partial product. Supports accessibility for: Visual Spatial Processing, Organization, Attention
Launch
Groups of 2
“Encuentren el valor de . Compartan con un compañero cómo razonaron” // “Find the value of and share your reasoning with a partner.”
Activity Synthesis
To clarify the steps in Tyler’s notation, consider itemizing each step and using color coding, as shown here:
Also consider displaying a corresponding diagram for and inviting students to make connections to an algorithm.
“¿Por qué Tyler escribió un 1 pequeño en la posición de las unidades de mil?” // “Why did Tyler record a small 1 in the thousands place?” (It shows that we composed a new thousand when adding the values in the hundreds place because 15 hundreds is composed of 1 thousand and 5 hundreds.)
Activity 2
Standards Alignment
Building On
Addressing
4.NBT.B.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.
When using an algorithm that uses partial products, students may be inclined to pay attention only to single digits in each number and pay little attention to the value of the digits. For example, might sound like “9 times 2 is 18 and 9 times 3 is 27.” In this activity, students analyze this error (MP3) and also look at the commutativity of multiplication when finding partial products.
To help students see that the place value of the digits impacts each partial product, consider displaying a diagram that shows partial products alongside the vertical notation.
MLR8 Discussion Supports. Synthesis: Display sentence frames to support whole-class discussion: “_____ y _____ se parecen porque . . .” // “_____ and _____ are the same/alike because . . .” and “¿Por qué ustedes . . .?” // “Why did you . . .?” Advances: Speaking, Conversing
Launch
Groups of 2
Activity
“Trabajen con su compañero en el primer problema. Cada uno debe encontrar el valor de un producto” // “Work with your partner on the first problem. Each partner should find the value of one product.”
3–4 minutes: partner work time on the first problem
Pause for a discussion. Select a group whose calculations yield the same product to display their work.
“¿Los cálculos deberían mostrar el mismo resultado? ¿Por qué sí o por qué no?” // “Should the calculations show the same result? Why or why not?” (Yes, because the same two numbers are being multiplied.)
“¿En qué se diferencian las dos formas de calcular?” // “How are the two calculations different?” (The partial products are written in different orders.)
“¿Importa cuál sea el primer número que se escribe y cuál sea el segundo?” // “Does it matter which number is listed first and which is listed second?” (No)
“Completen el último problema individualmente” // “Complete the last problem independently.”
3–4 minutes: independent work time
Monitor for students who record the partial products and the newly composed hundred in different ways to correct Han’s mishap.
Decide con tu compañero quién va a encontrar cada producto. Muestra cómo pensaste. Organiza tus ideas para que los demás puedan entenderlas.
Han calculó así:
¿Qué error o errores cometió Han?
Muestra el cálculo correcto para encontrar el valor de .
Activity Synthesis
“En la síntesis de la lección, vamos a analizar distintas formas de llevar un registro de los productos parciales y vamos a compartir qué podemos aprender de los errores de cálculo de Han” // “In the Lesson Synthesis, let’s analyze different ways to keep track of partial products and share how we can learn from Han’s miscalculation”.
Lesson Synthesis
“Hoy usamos productos parciales en un algoritmo para multiplicar parejas de números de dos dígitos. Escribimos expresiones y ecuaciones que nos ayudaron a encontrar y a organizar los productos parciales cuando multiplicábamos. En algunos casos, al sumar los productos parciales, también tuvimos que llevar un registro de las nuevas centenas que se compusieron” // “Today we used partial products in an algorithm to multiply pairs of two-digit numbers. We wrote expressions and equations to help us find and keep track of the partial products as we were multiplying. Sometimes when adding partial products we also needed to keep track of a newly composed hundred.”
“Veamos estrategias para corregir el contratiempo que tuvo Han con el producto ” // “Let’s look at strategies to correct Han’s mishap with solving .”
Display one example of student work side by side with Han’s miscalculation.
“Antes de analizar este ejemplo en detalle, ¿pueden saber si la respuesta es correcta o parece más razonable que lo que hizo Han? ¿Por qué sí o por qué no?” // “Before we analyze this sample closely, can you tell whether the answer is correct or seems more reasonable than Han’s work? Why or why not?” (Han’s answer was too low because , so that means the correct answer is greater than 3,200.)
Select students to explain each step in the computation and any questions the sample raises. Consider recording the factors for each partial product and drawing a diagram to support students in making sense of the products of the decomposed factors in the vertical calculation.
Display a different calculation for side by side with the work that was discussed.
“En este trabajo se llevó un registro de los productos parciales y de su suma de otra forma. ¿Cómo se hizo?” // “How does this work keep track of partial products and their sum in a different way?”
Reiterate the importance of paying attention to the place value of each digit being multiplied as we find and record the partial products.
Standards Alignment
Building On
Addressing
Building Toward
4.NBT.B.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.
“Tómense algunos minutos para darle sentido a los cálculos de Tyler. Prepárense para explicar cómo pensaron” // “Take a few minutes to make sense of Tyler’s calculation. Be prepared to explain your thinking.”
3–4 minutes: independent work time on the first problem
MLR1 Stronger and Clearer Each Time
“Compartan con un compañero su análisis de los cálculos de Tyler. Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta ese momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share your analysis of Tyler’s calculation with a partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
2 minutes: structured partner discussion
Consider displaying these prompts to support students’ conversations:
“¿Pueden explicar cómo al multiplicar _____ y _____ se obtiene _____?” // “Can you explain how multiplying _____ and _____ gives _____?”
“¿Pueden usar las palabras ‘productos parciales’ en su explicación?” // “Can you use the phrase ‘partial products’ in your explanation?”
Repeat with 1–2 different partners.
“Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
2–3 minutes: independent work time
“Ahora intenten usar el método de Tyler para completar el último problema y usen un diagrama para comprobar su trabajo” // “Now try using Tyler’s method to complete the last problem and use a diagram to check your work.”
5 minutes: independent work time on the last problem
Tyler usa un algoritmo para encontrar el valor de .
Describe cada paso del método de Tyler. ¿Cómo crees que encontró los últimos cinco números? Registra cómo pensaste.
Usa el método de Tyler para encontrar el valor de . Luego, dibuja un diagrama para comprobar tu respuesta.
