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When dealing with areas, there are different solutions to solve for the unknown. Area mainly deals with the space covered surrounded by the perimeter. Everything has an area: the bed you sleep in, the mirror you look at, the desk you write on, and even the screen you use for your phones and gadgets.
In this post, we will refresh ourselves in how to solve for the area using a basic visual tool: tiling. Of course, there are different kinds of shapes. There are geometric shapes such as a circle, a square, a triangle, and a rectangle. There are also irregular or organic shapes such as a puddle of water, a crumpled piece of paper, and even just a blob of slime. However, with tiling, we will deal with regular shapes specifically with rectangles.
Tiling is very appropriate in finding the area of a rectangle because we will discover that rectangles consist of smaller rectangles or tiles.
Understanding a Tile
If we were to define what a tile is to the students, we can call a tile as a building block. There are various building block examples. There are the cells, which are the building blocks of life. There are atoms, which are the building blocks of matter. There are nucleotides, which are the building blocks of DNA. So, we can classify tiles as a basic unit from which something is built up. Tiles build up bigger rectangles. Your students may ask you how they can draw tiles or where can they find examples of tiling. Encourage them to look around their homes and even in the classroom you’re in. Often, we can see tiles on the floor, evenly placed like grids. They can check their notebooks and may see grid lines that form tiles. Tiles are actually not new to them since they already learned how to visually multiply using array models. Clarify to them that tiles are just arrays of equal sized squares.
A unit square is one tile. It can either be in any unit of measure as long as it is 1 by 1. A square is a four sided regular geometric shape. Remind your students that each of its sides have equal measure. A square is a type of rectangle, but a rectangle is not a square because it does not have equal sides. Unit squares can form a rectangle if you arrange them like a grid, table, or array model. Always remember the lesson on array models.
For example, 1 x 1 is equal to 1, which is the same as having 1 unit square if you draw an array. Likewise, 2 x 3 is equal to 6, which is the same as having 2 rows of 3 or 6 unit squares.
Drawing the Tiles and Getting the Area
To visualize tiling clearer, you can first use a paper with grid lines for convenience. Notice that the gridded paper is built up by unit squares. You can let your students trace each square to any quantity you like.
Here are some examples:
All the rectangles drawn above can be seen to be made up of the unit squares the grid lines form. You may ask your students to draw these on their paper and identify the area of each rectangle. Counting the unit squares inside the rectangle is identifying the area of the rectangle.
For rectangle A, ask them to draw an array that has 5 rows of 1 unit squares or a rectangle with a length of 5 units and a width of 1 unit. Once they finish drawing the rectangle, ask how many unit squares make up the rectangle.
The area of rectangle A through unit squares is 5 unit squares!
For rectangle B, ask them to draw an array that has 5 rows of 4 unit squares or a rectangle with a length of 5 units and a width of 4 units. Again, ask them to count how many unit squares make up the rectangle.
The area of rectangle B through unit squares is 20 unit squares!
For rectangle C, ask them to draw an array that has 3 rows of 3 unit squares or a rectangle with a length of 3 units and a width of 3 units. Your students may notice that the rectangle is actually a square. Unit squares can make up bigger squares. Then, ask them to count how many unit squares make up the square.
The area of rectangle C through unit squares is 9 unit squares!
For rectangle D, ask them to draw an array that has 4 rows of 7 unit squares or a rectangle with a length of 4 units and a width of 7 units. Again, ask them to count how many unit squares make up the rectangle.
The area of rectangle D through unit squares is 28 unit squares!
Lastly, for rectangle E, ask them to draw an array that has 2 rows of 5 unit squares or a rectangle with a length of 2 units and a width of 5 units. Again, ask them to count how many unit squares make up the rectangle.
The area of rectangle B through unit squares is 10 unit squares!
Just like the commutative property of multiplication, whatever orientation the rectangle may be, the area will be the same. Moreover, once we know how many unit squares a rectangle has, we can also draw a different rectangle with different lengths and widths but use the same number of tiles.
Below is rectangle D that has an area of 28 unit squares. Below it is a new rectangle F with the same area. Both rectangles are made up of the same number of unit squares.
Rectangle D is a 4 by 7 rectangle that produces an area of 28 unit squares. Rectangle is a 2 by 14 rectangle that also produces the same area as rectangle D. Therefore, rectangles can have the same number of unit squares but have different dimensions.
Let your students understand how a basic building block can produce bigger shapes than what it is. Tiles are like the puzzle pieces that make up different sizes of rectangles. It makes it easier to visually get an area of a four sided shape.
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