# Lesson 7

Mejoremos nuestra fluidez al multiplicar

## Warm-up: Observa y pregúntate: La misma solución (10 minutes)

### Narrative

The purpose of this warm-up is to elicit the idea that there are different ways to calculate a product, using the standard algorithm, which will be useful when students find products of a 3-digit number and a 2-digit number in a way that makes sense to them.

### 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.

### Student Facing

¿Qué observas? ¿Qué te preguntas?

### Activity Synthesis

• “¿Por qué piensan que los resultados de los dos cálculos son los mismos?” // “Why do you think the results of the two calculations are the same?” (The factors are the same, just in a different order. The order of the factors does not change the result of multiplication.)
• “¿De qué forma prefieren encontrar el valor de $$417 \times 28$$?” // "Which way would you prefer to find the value of $$417 \times 28$$?" (I like the one with two partial products as there is less adding up to do.)
• “Hoy van a escoger cómo encontrar productos de números de 3 dígitos por números de 2 dígitos” //  "Today you will get to choose how to find products of a 3-digit number and a 2-digit number.

## Activity 1: El mayor producto (15 minutes)

### Narrative

Students use their understanding of place value to generate expressions that have the greatest product. They take turns selecting number cards 0 through 9 and place them strategically to make the largest  product of a 2-digit number and a 3-digit number. Students will need to think about how the different place values influence the value of the product and choose where to put their digits accordingly. There is also a large element of chance since they do not know in advance which numbers they will select.

Action and Expression: Develop Expression and Communication. Give students access to grid paper for finding the product.
Supports accessibility for: Visual-Spatial Processing, Organization

### Required Materials

Materials to Copy

• Number Cards (0-10)
• Greatest Product, Spanish

### Launch

• Groups of 2
• Give each group a set of number cards and 2 copies of the blackline master.
• “Saquen las tarjetas que muestran 10 y déjenlas a un lado” // “Remove the cards that show 10 and set them aside.”
• “Vamos a jugar un juego que se llama ‘El mayor producto’. Leamos las instrucciones y juguemos una ronda juntos” // “We’re going to play a game called Greatest Product. Let’s read through the directions and play one round together.”
• Read through the directions with the class and play a round against the class using the diagram in the student workbook:
• Display each number card.
• Think through your choices aloud.
• Record your move and score for all to see.

### Activity

• “Ahora jueguen con su compañero” // “Now, play the game with your partner.”
• 8–10 minutes: partner work time

### Student Facing

Instrucciones:

• En la primera ronda, el compañero A escoge una tarjeta de números y escribe el número en alguno de sus espacios en blanco.
• El compañero B hace lo mismo.
• Repitan lo anterior hasta que cada compañero tenga un problema de multiplicación de un número de dos dígitos por un número de tres dígitos.
• Encuentren el producto.
• El compañero que tenga el mayor producto gana un punto.
• Gana el que tenga más puntos después de 5 rondas.

### Activity Synthesis

• Display a blank image from game board.
• “Si este es su tablero de juego y escogen un 8, ¿en dónde escribirían el 8? ¿Por qué?” // “If this is your game board and you pick an 8, where would you write the 8? Why?” (I would either put it in the hundreds place of the top number or the tens place of the bottom number because it’s a big number and those are the biggest place values.)
• Write the 8 in the hundreds place of the top number.
• “¿Qué pasa si después seleccionan un 1? ¿En dónde escribirían el 1? ¿Por qué?“ // “What if you next select a 1? Where would you write the 1? Why?” (I would put it in the ones place of one of the numbers because I want to put a bigger number in the tens place.)
• “¿Qué les pareció retador en el juego?” // “What did you find challenging about the game?” (Since I didn’t know what numbers I was going to get on later picks, I sometimes wasn’t sure where to put a number because I didn’t know if I would get a bigger number on a later pick.)

## Activity 2: Buscando desesperadamente 9 unidades nuevas en base diez [OPTIONAL] (20 minutes)

### Narrative

The purpose of this activity is for students to  explore the number of new units that can be composed when using the standard algorithm. As students experiment with the given numbers, they will find examples of 1, 2, 3, 4, 5, 6, 7, or 8 new units composed. Here they address the question of whether or not it is possible to compose 9 or more new units when using the standard algorithm. In order to make sense of and persevere in solving this problem (MP1), there are several types of arguments students may make, all of which highlight how the standard algorithm works:

• they may calculate $$999 \times 9$$ and see that while they get very close to composing 9 new units, they fall 1 short
• the biggest product that can be made from multiplying a pair of 1-digit numbers is 81, which would mean that 8 units are composed
• this 81 has to be combined with whatever new units were composed before, but since that’s 8, that means the largest number you can form at each step in the standard algorithm is 89, which is 1 short of the 90 you would need to compose 9 new units
MLR8 Discussion Supports. Synthesis: At the appropriate time, give groups 2–3 minutes to plan what they will say when they present to the class. “Practiquen lo que van a decir cuando compartan sus soluciones con toda la clase. Hablen sobre qué es importante decir y decidan quién va a compartir cada parte” // “Practice what you will say when you share your solutions with the class. Talk about what is important to say, and decide who will share each part.”

• Groups of 2

### Activity

• 1–2 minutes: quiet think time
• 3–5 minutes: partner work time

### Student Facing

Tyler observa que cuando usa el algoritmo estándar y compone una nueva unidad en base diez, a veces hay 1 nueva unidad, a veces 2, y así hasta llegar a 8. Él no ha visto un ejemplo en el que se compongan 9 de la nueva unidad.

1. En cada uno de estos productos, ¿cuántas de cada nueva unidad en base diez se componen?

1. $$256 \times 5$$
2. $$587 \times 8$$
3. $$809 \times 9$$
2. ¿Crees que es posible componer 9 de una nueva unidad en base diez usando el algoritmo estándar de multiplicación?

### Student Response

If students are not fluent with their multiplication facts, offer them a multiplication table. As they use the standard algorithm to find the products in problem 1, ask them to circle each new unit. If they are not sure if it is possible to compose 9 new units, ask them to use the algorithm to find the product of $$99 \times 9$$ and  $$999 \times 99$$.

### Activity Synthesis

• Invite students to share the number of units they composed in the calculations of problem 1.
• “¿Faltó algún número del 1 al 8?” // “Was anything missing from 1 to 8?” (No, we composed everything from 1 to 8 new units.)
• “¿Alguien pudo componer 9 decenas nuevas en algún producto? ¿Por qué?” // “Was anyone able to compose 9 new tens in a product? Why?” (No. The new tens come when I take the product of the ones. The biggest that can be is $$9 \times 9$$ which gives me 8 tens.)
• “¿Alguien pudo componer 9 centenas nuevas? ¿Por qué?” // “Was anyone able to compose 9 new hundreds? Why?” (No. I could get 8 new hundreds if I have $$9 \times 90$$, giving me 810. I might also have to combine that with the new tens from the product of the ones, but that would give me at most 890, not 900.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy usamos el algoritmo estándar para encontrar productos de números sin poner un límite al número de nuevas unidades en base diez que se componen. También analizamos cuántas nuevas unidades se pueden componer” // “Today we used the standard algorithm to find products of numbers with no restriction on the number of newly composed units and we examined how many new units can be composed.”

“¿Creen que el algoritmo para multiplicar números enteros va a funcionar para multiplicar cualquier pareja de números enteros? ¿Por qué sí o por qué no? Discutan con un compañero” // “Do you think the algorithm for multiplying whole numbers will work for any and all whole numbers? Why or why not? Discuss with a partner.” (I think so but if the numbers are big it will take up a lot of space and there could be a lot more new units to compose. I think it might work but it would take a long time if the numbers are big and I think it would be easy to make a mistake.)

Ask students to share their thinking.

“¿Qué se preguntan todavía sobre el algoritmo estándar para multiplicar números enteros?” // “What do you still wonder about the standard algorithm for multiplying whole numbers?” (When is it a good strategy to use? Are there other ways that work well or better?)