Question

Use dot paper to create a tessellation using the given polygons. trapezoids

Short answer

Create and replicate trapezoids on dot paper to fill the space seamlessly.

Step-by-step solution

Understand Tessellation

Tessellation involves covering a plane with repeated polygons without any gaps or overlaps. We will use trapezoids for this exercise.

Create the Initial Trapezoid

On a sheet of dot paper, draw an initial trapezoid. Make sure it is not leaning and has at least one pair of parallel sides.

Replicate the Trapezoid

Copy the initial trapezoid by drawing identical trapezoids on the dot paper. Align the new trapezoid so that it shares one complete side with the original trapezoid.

Fill the Plane

Continue replicating the trapezoid across the dot paper, ensuring each new shape shares a side with an already-drawn trapezoid. Keep adding until the paper is covered, creating a seamless pattern.

Check for Gaps and Overlaps

Inspect the tessellation. Make sure all the trapezoids fit perfectly without any gaps or overlapping areas.

Key concepts

Trapezoid

A trapezoid, in geometry, is a four-sided polygon with at least one pair of parallel sides. These sides are typically referred to as the 'bases' of the trapezoid, while the non-parallel sides are termed 'legs'. The unique feature of a trapezoid is its ability to tile a plane in tessellation exercises. This is possible because trapezoids can be arranged in such a way that they fit perfectly side by side without leaving any gaps or requiring overlaps.

When creating your trapezoid, start by ensuring it has a solid base structure - a straightforward parallel side setup. This will simplify aligning and replicating the shape on the dot paper. You can experiment with different trapezoid configurations, such as right trapezoids with one or two right angles, or isosceles trapezoids, which have the legs of equal length.

Understanding these properties and shapes helps in predicting how they will fit together in tessellation.

Plane Covering

Plane covering refers to the method of filling an entire surface, or plane, with a repeated pattern of shapes, without any empty spaces or overlaps. In this exercise, the goal is to completely cover the surface using trapezoids.

To achieve a successful tessellation, the crucial step is ensuring that each tile shares full sides with the previously drawn trapezoids. This creates a continuous and uninterrupted pattern throughout the plane. It is important to be precise in the placement of each shape so that the edges align perfectly.

Consider starting from one corner of your dot paper, and work your way across diagonally or in rows. This strategy helps maintain a consistent progression and makes it easier to spot any misalignments before they accumulate. With thoughtful placement, you can cover the plane seamlessly, ensuring a perfect tessellation.

Remember, there are various ways to approach plane covering, but the key is consistency and precision in shape placement.

Dot Paper

Dot paper is an extremely helpful tool when learning and practicing tessellation. It features a grid of dots that provide reference points for drawing and aligning shapes perfectly.

By using dot paper, you can easily measure precise distances and angles required for creating trapezoids. Each vertex of the trapezoid can be placed on an intersection of dots, which helps in maintaining symmetry and uniformity across the tessellation.

When you start drawing on dot paper, aim for consistency in the pattern by replicating the same shape repeatedly across the sheet. The dots facilitate aligning the bases and legs of the trapezoids accurately.

Another advantage of dot paper is that it helps in easily spotting any gaps or overlaps in your tessellation. If your shapes drift off the intended alignment, the dot grid makes it easier to adjust and correct the pattern as needed.