Lesson 2

The purpose of this How Many Do You See is for students to use grouping strategies to describe the images they see.

In the synthesis, students describe how two images can be used to describe a multiplicative comparison and connect the images to a multiplication equation.

• Groups of 2
• “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see and how you do see them?”
• Flash the image.
• 30 seconds: quiet think time

• Display the image.
• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses. Use multiplication equations when appropriate.
• Repeat for each image.

• “¿Cómo muestra esto que el segundo rectángulo tiene 2 veces la cantidad que tiene el primer rectángulo?” // “How does this show that the second rectangle has 2 times as many as the first rectangle?”
• “¿Cómo podemos escribir una ecuación que muestre esta comparación?” // “How could we write an equation that shows this comparison?” (\(6 = 2 \times 3\) or \(2 \times 3 = 6\) or \(3 \times 2 = 6\))

The purpose of this activity is for students to analyze and describe how images and diagrams can show “\(n\) times as many”. Students generate ideas for how to use a multiplication equation to represent the comparison.

Students begin by interpreting an image in which the multiplier (3) and the numbers are given. They explain how some number of times of the smaller amount can be seen in the larger amount. Next, they create their own diagram and see different ways of representing the iterations (groups of) the smaller amount to create the larger amount.

During the activity, make connecting cubes accessible for students who may choose to use them for problem solving—either to reason about the quantities or to explain their reasoning.

• Groups of 2
• Give students access to connecting cubes.
• Display the image of Mai's cubes and Kiran's cubes.
• “¿De qué manera estos cubos representan 3 veces una cantidad?” // “How do these cubes represent 3 times as many?” (Mai has 6 cubes and Kiran has 2. Mai has 3 groups of 2 cubes. Mai has 6 cubes and Kiran has 2. Three times as many as 2 is 6, or 3 times 2 is 6.) 
• Give students access to cubes.
• Read the statement about Jada and Kiran’s cubes as a class. 
• 1 minute: quiet think time

• “Creen una presentación visual que muestre cómo pensaron en los cubos de cada problema e incluyan detalles para ayudar a otros a entender sus ideas” // “Create a visual display that shows your thinking about the cubes in each problem and include details to help others understand your thinking.”
• 6–8 minutes: independent or group work
• 3 minutes: gallery walk 
• “¿Cómo muestra cada representación ‘varias veces’ una cantidad?” // “How does each representation show ‘times as many’?”
• 30 seconds quiet think time
• 1 minute: partner discussion
• Monitor for students who create diagrams that are similar to connecting cube images and discrete tape diagrams to share in the synthesis. 

Students may compare additively instead of multiplicatively when using cubes or drawing diagrams. Consider asking: “¿Cómo usarías los cubos para mostrar 3 más?” // “How would you use the cubes to show 3 more?” and “¿Cómo usarías los cubos para mostrar 3 veces?” //  “How would you use the cubes to show 3 times as many?”

• Have selected students share diagrams and explain how they show “times as many”.
• If needed, use cubes to represent statements.
• “¿Qué ecuación escribirían para comparar los cubos de Kiran con los de Jada?” // “How could you write an equation to compare Kiran’s and Jada’s cubes?”
• “¿Qué representan en la situación los números de la ecuación?” // “What do the numbers in the equation represent in the situation?” (Four is the “4 times as many”. Two is how many Kiran had. Eight is how many Jada had.) 
• Write equations for each situation and ask about what students notice about the relationships.

The purpose of this activity is for students to deepen their understanding of how diagrams and multiplication equations can represent “\(n\) times as many”. Students explain how the diagrams and equations represent the situation. In order to match situations, diagrams, and equations, students reason abstractly and quantitatively (MP2).

• Groups of 2
• “Por turnos, lean una descripción y encuentren un diagrama y una ecuación que también representen la situación. Explíquenle su razonamiento a su compañero” // “Take turns reading a description and finding a diagram and an equation that also represent the situation. Explain your reasoning to your partner.”

• 5–7 minutes: partner work time
• Monitor for students who connect the factors and product of the equation to the situation and diagram
• If students finish early, give them blank index cards. Ask them to make several sets of matching representations, shuffle the cards, and trade them with another group that is also creating their own representations.

Students may consider the total amount represented on a card rather than making comparisons. For example, on card K, they might see 4 groups of 3 in the two quantities combined. Consider asking:

• “¿Cómo compararías las dos cantidades que se muestran en la tarjeta?” // “How would you compare the two quantities shown on the card?”
• “¿En qué parte de la ecuación de la tarjeta E ves que se están comparando dos cantidades?” // “Where might you see two quantities being compared in the equation on card E?”

• Select students to share their matches. 
• Record student explanations to show how they connected the diagrams and equations (display or draw the diagrams and equations and annotate).