Problem Description:
The task is to tessellate each of the given polygons six times in an isometric lattice. The goal is to arrange the shapes such that they fit together without any gaps or overlaps, adhering to the principles of tessellation.
Given Information:
1.
Polygons to Tessellate:
- A parallelogram
- A hexagon
- An irregular quadrilateral (trapezoid-like shape)
- An irregular pentagon
2.
Isometric Lattice: The grid provided is an isometric grid, which consists of equilateral triangles arranged in a honeycomb pattern. This type of grid is ideal for tessellations involving regular and irregular polygons.
3.
Tessellation Requirement: Each polygon must be repeated six times in the lattice.
4.
Objective: Understand how to tessellate shapes by fitting them together seamlessly.
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Solution Approach:
To solve this problem, we need to:
1.
Understand the Isometric Grid: The isometric grid is composed of equilateral triangles, which means that any shape placed on this grid must align with its structure to avoid gaps or overlaps.
2.
Identify Key Properties of Each Polygon:
-
Parallelogram: Can tessellate easily because it can be translated and rotated to fit adjacent spaces.
-
Hexagon: Naturally tessellates due to its symmetry and the fact that six hexagons can meet at a single point.
-
Irregular Quadrilateral: May require careful placement to ensure no gaps or overlaps.
-
Irregular Pentagon: Requires strategic placement to tessellate, as not all pentagons tessellate naturally.
3.
Repeat Each Shape Six Times: Ensure that each polygon is repeated exactly six times in the grid.
4.
Verify Tessellation: Check that there are no gaps or overlaps between the shapes.
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Step-by-Step Solution:
#### 1.
Parallelogram
- Place the parallelogram on the isometric grid.
- Translate (slide) the parallelogram horizontally and vertically to fill the grid.
- Rotate the parallelogram if necessary to fit adjacent spaces.
- Repeat this process until six parallelograms are placed without gaps or overlaps.
#### 2.
Hexagon
- Place the hexagon on the grid.
- Use the natural tessellation property of hexagons: six hexagons can meet at a single point.
- Arrange the hexagons in a honeycomb pattern.
- Repeat this pattern until six hexagons are placed.
#### 3.
Irregular Quadrilateral
- Carefully place the irregular quadrilateral on the grid.
- Adjust its orientation to fit adjacent spaces.
- Use translation and rotation to fill the grid.
- Ensure that the edges align perfectly with the grid lines.
- Repeat until six quadrilaterals are placed.
#### 4.
Irregular Pentagon
- Place the irregular pentagon on the grid.
- Experiment with translations and rotations to find a pattern that avoids gaps or overlaps.
- Use the grid's structure to guide the placement.
- Repeat until six pentagons are placed.
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Final Answer:
The solution involves carefully placing each polygon six times on the isometric grid, ensuring that they fit together without gaps or overlaps. The exact arrangement will depend on the specific properties of each polygon and their alignment with the grid.
\boxed{\text{Follow the steps above to tessellate each polygon six times on the isometric grid.}}
Parent Tip: Review the logic above to help your child master the concept of tessellation worksheet 7th grade answers.
