Lesson 8

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

When students use equal groups and a known quantity to find an unknown quantity, they are looking for and making use of structure (MP7).

• Groups of 2
• “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you 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.
• Repeat for each image.

• “¿Qué números fueron fáciles de ver en las imágenes?” // “What numbers were easy to see in the images?” (4, 5, 6)
• “¿Cómo les ayudó la primera imagen a encontrar el número de puntos de las siguientes dos imágenes?” // “How did the first image help you find the number of dots in the next 2 images?” (I know each group in the second image has 1 more dot than each group in the first image, so I figured out 4 groups of 5, then added 4 more dots. For the last image, I subtracted 4 from 20, since one dot was missing in each group.)

The purpose of this activity is for students to solve an area problem with a partially tiled rectangle. This encourages students to multiply to solve problems involving area, but still provides some visual support to see the arrangement of the rows and columns. This problem includes a product of ten, with which students should be increasingly comfortable. The number of square inches is large in order to discourage one-by-one counting.

• Groups of 2
• Display images of the painted tiles.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Students may notice: There are many square tiles. That is a lot of tiles. The tiles are painted. Students may wonder: How many tiles were used? What else could be tiled? How long did it take to tile that building?)
• 1 minute: quiet think time
• Share and record responses.
• “Estos son ejemplos de baldosas pintadas que se llaman azulejos de Portugal. En Portugal, se han usado por mucho tiempo para decorar paredes, pisos y hasta techos. También muestran eventos de la historia portuguesa” // “These are examples of painted tiles called azulejos from Portugal. In Portugal, they have been used for a very long time to decorate walls, floors, and even ceilings. They also show events in Portuguese history.”
• “En este problema, queremos encontrar el área de un proyecto de arte que está parcialmente recubierto con baldosas cuadradas. Piensen cuántas baldosas se necesitan para recubrir todo el rectángulo” // “This problem involves finding the area of an art project that is partially tiled with square tiles. Think about how many tiles are needed to tile the whole rectangle.”
• 1 minute: quiet think time

• 3–5 minutes: partner work time
• Monitor for students who use multiplication to find the number of tiles.
• Consider asking:
  • “¿Cómo encontraron cuántas baldosas necesitarían para cubrir el rectángulo?” // “How did you find how many tiles it would take to cover the rectangle?”
  • “¿Hay alguna manera que sea más rápida que contar todos los cuadrados, uno por uno?” // “Is there a way that’s faster than counting every square one by one?”

If students count one by one to find the total number of square tiles, consider asking:

• “¿Cuántos cuadrados habría en cada fila (o columna)?” // “How many squares would be in each row (or column)?”
• “¿Cómo podríamos usar esto para encontrar el área?” // “How could we use this to find the area?”

• “¿Cómo supieron cuántos cuadrados habría en cada fila o columna?” // “How did you know how many tiles would be in each row or column?” (The first row had 10 tiles, so I know every other row has 10 tiles because I could put more tiles to fill in the rows. It’s like an array. Each column has to have the same number of tiles, so there is 9 in each column.)
• “¿Cómo encontraron el número total de cuadrados que se necesitaban?” // “How did you find the total number of tiles needed?” (I counted by ten 9 times. I multiplied 9 times 10.)
• “La actividad menciona que los cuadrados son de 1 pulgada en cada lado. ¿El lado de cada cuadrado es realmente de 1 pulgada?” // “The activity mentions that the tiles are 1 inch on each side. Is the side of each tile actually 1 inch long?” (No). “¿Cómo lo saben?” // “How can you tell?” (Sample response: We know how long 1 inch is and can see that the sides of the tiles are less than 1 inch. There are 10 tiles across. If they really are 1 inch wide, the image won’t fit on the paper.)
• “Algunas veces veremos imágenes marcadas con unidades que no son exactamente del tamaño que dice la marca. De todas maneras, podemos usar esas imágenes para representar la situación de la que hablamos” // “Sometimes we will see images labeled with units that are not exactly the size the label says. We can still use these images to represent the situation we are talking about.”

In this activity, students find the area of rectangles that are not tiled but whose sides are marked with equally spaced tick marks. The tick marks give students the side lengths of the rectangle, help students visualize a tiled region, and enable them to confirm that multiplying the side lengths give the number of square units in the rectangle. The work here serves to transition students to using only side lengths to find area.

• Groups of 2
• Display the first problem and the image.
• Read the first sentence in the first problem.
• “El enunciado dice que las marcas están a 1 metro de distancia. ¿Realmente están a un 1 metro de distancia?” // “The statement says the tick marks are 1 meter apart. Are they really 1 meter apart?” (No, the spaces between them represent 1 meter each.)
• “En una hoja de papel de tamaño estándar, no podríamos dibujar un rectángulo que realmente estuviera en metros porque sería mucho más grande que el papel. Podemos dibujar este rectángulo para representar ese rectángulo más grande” // “We could not draw a rectangle that is actually in meters on a standard piece of paper because it would be much larger than the paper. We can draw this rectangle to represent that larger rectangle.”
• “¿En qué son diferentes este rectángulo y los otros rectángulos de los que hemos encontrado el área?” // “How is this different from other rectangles whose area we’ve found?” (Before we had squares or tiles to count or we used actual tiles filled into a shape to count to find the area.)
• 30 seconds: quiet think time
• 1 minutes: partner discussion
• Share responses.
• Give students access to rulers or straightedges.

• “Con su compañero, encuentren el área de este rectángulo” // “Work with your partner to find the area of this rectangle.”
• 3–5 minutes: partner work time
• Circulate and consider asking:
  • “¿Cómo describirían las filas y las columnas si se imaginaran los cuadrados en el rectángulo?” // “How would you describe the rows and columns if you pictured the squares in the rectangle?”
  • “¿Cómo encontrarían el número total de metros cuadrados?” // “How could you find the total number of square meters?”
• Monitor for students who draw the missing grid lines before they multiply to find the area of the rectangle.
• Invite 2–3 students to share how they found the area of this rectangle.
• Display the second problem.
• “Este rectángulo tiene marcas que están a 1 metro en la parte de arriba y está marcado con metros en el lado. Piensen en cómo podrían encontrar el área de este rectángulo” // ”This rectangle is marked off in meters along the top and is labeled with meters on the side. Think about how you might find the area of this rectangle.”
• 1 minute: quiet think time
• 3 minutes: partner work time
• Monitor for students who create the missing grid lines before they multiply to find the area of the rectangle.

If students count one by one to find the total number of square units, consider asking:

• “¿Cómo encontraste el área del rectángulo?” // “How did you find the area of the rectangle?”
• “¿Cuántos cuadrados habría en cada fila (o columna)? ¿Cómo podríamos usar esto para encontrar el área?” // “How many squares would be in each row (or column)? How could we use this to find the area?”

• Display samples of student work where students created the missing grid lines.
• “Cuando queremos ver los grupos que hacen falta, ¿cómo puede ayudarnos crear una cuadrícula a partir de las marcas?” // “How can creating the grid from the tick marks help us see the missing groups?” (We can see all the squares in a row or in a column. We can see how many rows and columns there are.)
• “¿Tuvieron que trazar las líneas de la cuadrícula para encontrar el área de estos rectángulos?” // “Did you have to fill in the grid lines to find the area of these rectangles?” (No, since counting the squares gives the same area as multiplying the side lengths, we can just find the length of each side.)