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 the columns. This problem includes a product of 10, with which students should be increasingly comfortable. The total number of square inches is large in order to discourage one-by-one counting although many students may begin with this approach.

Monitor for and select students with the following approaches to share in the Activity Synthesis:

• Draw to show the tiles in each row (or column) for the first few rows before switching to a different approach.
• Count the tiles in either the first column or the first row, and count on by whichever number they find first.
• Count the tiles in the first row and in the first column, and choose to count on by 10 for each row because it's an easier count.
• Count the tiles in the first column and in the first row, and multiply .

The approaches are sequenced from more concrete to more abstract to help students make connections between their understanding of area measurement, counting methods, and multiplication. It is likely that students will switch their approach as they work on the problem. Students may not yet use multiplication to find the area or represent their thinking. If it does not come up in this activity, there is an opportunity to bring it up in the next. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently.

• Invite previously selected students to share in the given order. Record or display their work for all to see.
• Connect students’ approaches by asking:
  • “¿En qué se parecen estas maneras de encontrar el área?” // “How are these ways of finding the area the same?” (All found the same area. All show they found the same number in each row or column.)
  • “¿En qué son diferentes?” // “How are they different?” (Some just counted what was in 1 row or 1 column and then counted on. Some found how many were in a row and in a column before they picked a way to find the area. Some counted by 10 nine times. Some counted by 9 ten times. One way multiplied 9 times 10.)
• Connect students’ approaches to the learning goal by asking:
  • “¿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 are 9 tiles in each column.)
• “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). “How can you tell?” (We know the length of 1 inch 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.)
• “A veces vamos a ver 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 areas of rectangles that are not tiled but the sides of which 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 gives the number of square units in the rectangle. The work here serves to transition students to using only side lengths to find an area.

• “¿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 students’ work in which students created the missing grid lines.
• “Para ver los grupos que faltan, ¿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.)