Lesson 3

The purpose of this warm-up is to draw students' attention to different ways of covering a plane figure with squares and reinforce the idea that tiling involves covering a region without gaps and overlaps. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. During the synthesis, ask students to explain the meaning of any terminology they use, such as rows, columns, area, gaps, overlap, and tiling.

• Groups of 2
• Display the image.
• “Escojan uno que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “¿Cómo podrían usar los cuadrados que tiene cada uno de estos rectángulos para encontrar el área de cada rectángulo?” // “How could you use the squares in each of these rectangles to find the area of each rectangle?” (In C, I can just count the tiles. In B, I could finish tiling the rectangle and count the tiles. In D I would need to straighten out the tiles so they cover all of the rectangle. In A, I could count the blue tiles and double the number since in each row there are the same number of white tiles as there are blue tiles.
• Consider saying:
  • “Encontremos al menos una razón por la que cada uno es diferente” // “Let’s find at least one reason why each one doesn’t belong.”

The purpose of this activity is for students to use square tiles to find the area of rectangles. They use their knowledge of tiling to complete the tiling that is started in each rectangle. Students may use physical tiles on copies of the blackline master or reason directly on the images in the student book, which may not be the right size for physical tiles. The synthesis focuses on solidifying the idea that area is the number of square units that cover a flat figure with no gaps or overlaps.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing

• Each group of 2 needs 24 square tiles.

• Groups of 2
• Give each student 1 copy of the blackline master.
• Give students access to inch tiles.
• “Piensen durante un minuto en cómo terminarían de medir el área de estos rectángulos que están parcialmente recubiertos” //  “Take a minute to think about how you would finish measuring the area of these rectangles that are partially tiled.”
• 1 minute: quiet think time

• “Describan con su compañero cómo usar fichas cuadradas para encontrar el área de cada rectángulo. Si lo necesitan, pueden usar las fichas cuadradas y reorganizar lo que se muestra en cada rectángulo. Después, completen el último problema individualmente” // “Work with your partner to describe how to use square tiles to find the area of each rectangle. You can use the square tiles and rearrange what’s shown in each rectangle, if needed. Then complete the last problem independently.”
• 5–7 minutes: partner work time

• “¿Por qué las fichas cuadradas del primer, tercer y cuarto rectángulo se debían acomodar antes de que pudiéramos terminar de encontrar el área del rectángulo?” //“Why did the square tiles in the first rectangle, the third rectangle, and the fourth rectangle need to be adjusted before we could finish finding the area of the rectangle?” (In the first rectangle, the square tiles had to be moved over because they weren’t going to fill the whole rectangle if we left them in the center. In the third rectangle, the squares in the second row needed to be lined up with the first row so there would be the same number of squares in each row. In the fourth rectangle, the squares need to be adjusted so they are not crooked or overlapping and one square needs to be removed from the first row.)
• “Si alguien les dijera que en la parte superior del rectángulo caben cuatro cuadrados, pero que en la parte inferior del rectángulo solo caben tres cuadrados, ¿cómo sabrían que esto no tiene sentido?” // “If someone told you four squares would fit across the top of the rectangle, but only three squares would fit across the bottom of the rectangle, how would you know this didn’t make sense?” (The top and bottom have the same length so they should fit the same number of squares.)

MLR1 Stronger and Clearer Each Time

• “Compartan con su compañero su respuesta al último problema. 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 response to the last problem with your 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.
• 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
• “Acabamos de encontrar el área de algunos rectángulos. Aprendimos que cuando cubrimos el rectángulo con fichas cuadradas, las fichas no pueden tener espacios ni superposiciones. Es importante que tengamos en cuenta estas mismas ideas para cualquier figura plana. El área es el número de unidades cuadradas que se necesitan para cubrir una figura plana sin espacios ni superposiciones” // “We just found the area of rectangles, and learned that when we cover the rectangle with square tiles, the tiles can’t have gaps or overlaps. The same ideas are important with any flat figure. Area is the number of square units that it takes to cover a flat figure without gaps or overlaps.”

The purpose of this activity is for students to recognize that different shapes can have the same area. Students first sort the cards in any way that makes sense to them and then by area. After the cards are sorted by area, students create another rectangle that would fit into one of the categories (by having a particular area). A sorting task prompts students to look for structure and make connections across the representations and statements being analyzed (MP7).

Students may start to notice that the organization of the squares in rectangles makes it efficient to count: The squares can be grouped by row, column, or in other ways. As students sort and create rectangles with certain areas, monitor for students who leverage the structure of a rectangle to find area. Invite them to share in the synthesis.

In this activity, the squares on the gridded rectangles are not the same size as the square tiles, but students could still use tiles as a support. Provide students access to square tiles if they would like to use them, but encourage them to draw what they create on the grid provided.

• Create a set of cards from the blackline master for each group of 2. 

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Students may notice: There are 3 rectangles. One of the rectangles is made up of square tiles. One of the rectangles is shaded and the other rectangle isn’t. They all have 12 squares. They all have an area of 12 square units. Students may wonder: Why are there 3 rectangles? Why is one rectangle shaded and the other one isn’t? Do the blue squares show tiles?)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses.
• “Estas son maneras en las que podemos representar un rectángulo que tiene 12 unidades cuadradas. Cuando los cuadrados están sombreados en la imagen parecen fichas cuadradas. Pero también podemos hacer un rectángulo en una cuadrícula y decir que tiene un área de 12 unidades cuadradas porque contiene 12 cuadrados” // “These are ways that we can represent a rectangle with 12 square units. When the squares are shaded in the image they look like square tiles, but we can also make a rectangle on a grid and say that it has an area of 12 square units, because it contains 12 squares.”
• “Dibujen en la cuadrícula un rectángulo que tenga un área de 8 unidades cuadradas” // “Draw a rectangle with an area of 8 square units on the grid.”
• 30 seconds: independent work time
• Share responses.
• Distribute one set of pre-cut cards to each group of students.
• Give students access to inch tiles.

• “Clasifiquen las tarjetas en categorías con su compañero. Prepárense para explicar cómo clasificaron sus tarjetas” // “Work with your partner to sort the cards into categories. Be prepared to explain how you sorted your cards.”
• 5 minutes: partner work time
• Select groups to share their categories and how they sorted their cards.
• If no groups sort their rectangles by area, give students 2-3 minutes to do so and then ask them to share their new categories.
• “Piensen durante un minuto en qué otros rectángulos podrían ser incluidos en estas categorías” // “Take a minute to think about what other rectangles might fit into these categories.”
• 1 minute: quiet think time
• “Ahora, junto con su compañero, hagan por lo menos un rectángulo distinto que tenga la misma área que los rectángulos de cada grupo. Prepárense para compartir cómo saben que sus rectángulos pertenecen a cada grupo” // “Now, work with your partner to create at least one different rectangle that has the same area as the rectangles in each group. Be prepared to share how you know your rectangles belong in each group.”
• 5 minutes: partner work time
• Monitor for the strategies students use to find the area of rectangles.

• Invite students to share the rectangles they created for each category.
• Consider asking:
  • “¿Cómo supieron que su rectángulo pertenecía a esta categoría?” // “How did you know that your rectangle belongs here?”
  • “¿Cómo supieron que el rectángulo que hicieron tenía la misma área que los otros rectángulos de esa categoría?” // “How did you know that the rectangle you created had the same area as the other rectangles in that category?”
• Discuss some of the efficient ways that students counted to find the area of rectangles.