Learn about tiling the plane and reasoning to find the area of regular and irregular shapes. After trying the questions, click on the buttons to view answers and explanations in text or video.
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Tiling the Plane
Let’s look at tiling patterns and think about area.
Illustrative Math Unit 6.1, Lesson 1 (printable worksheets)
1.1 - Which One Doesn’t Belong: Tilings
Which pattern doesn’t belong? Give reasons.
Tiling the plane means covering a two-dimensional region with copies of the same shape or shapes so that there are no gaps or overlaps. Which patterns are examples of tiling and which are not?
In terms of appearance, all of the patterns are different from every other pattern in some way. For example:
These are examples; any one of these patterns might not belong, for other reasons you might think of.
In terms of tiling the plane, Pattern A, Pattern B, and Pattern C are examples of tiling, while Pattern D is not because of the irregular white spaces.
1.2 - More Red, Green, or Blue?
Look at Pattern A or B.
In your pattern, which shapes cover more of the plane: blue rhombuses, red trapezoids, or green triangles? Explain how you know.
You may use the applet here to help.
Notice how the pattern is repeated. Do you need to count all the shapes in the pattern?
The shapes are different sizes. So is counting them enough? Compare the area covered by each type of shape in the pattern. How many triangles match up to one rhombus? How about trapezoids?
Can you then compare the area covered by all the triangles in the pattern with the area covered by all the rhombuses, or all the trapezoids?
“Which shapes cover more of the plane” means that we are comparing the areas covered by the different types of shapes. In these patterns, two triangles match up to one rhombus, and three triangles match up to one trapezoid.
In both Pattern A and Pattern B, the first hexagonal part is made up of 7 triangles, 4 rhombuses, and 3 trapezoids.
The first hexagonal part is repeated across the pattern, so whichever type of shape covers the most area of this first part also covers the most area in the whole pattern.
Since the triangles can be matched to the other shapes, let us say 1 triangle is 1 space, or 1 unit of area, in the pattern. So the 7 triangles cover 7 spaces.
As each rhombus is 2 triangles, rhombuses cover 4 × 2 = 8 spaces.
The trapezoids cover 3 × 3 = 9 spaces.
Hence, red trapezoids cover more of the plane in both Pattern A and Pattern B.
Lesson 1 Summary
In this lesson, we learned about tiling the plane, which means covering a two-dimensional region with copies of the same shape or shapes such that there are no gaps or overlaps.
Then, we compared tiling patterns and the shapes in them. In thinking about which patterns and shapes cover more of the plane, we have started to reason about area.
We will continue this work, and to learn how to use mathematical tools strategically.
Practice Problems
The pattern is repeated, so whichever type of square covers the most area of the first part also covers the most area in the whole pattern.
1 small square is 1 space, or 1 unit of area, in the pattern. 2 small squares are 2 spaces.
The medium squares cover 2 × 4 = 8 spaces.
The large square covers 3 × 3 = 9 spaces.
Hence, the large squares cover more of the plane compared to the other squares.
These are examples of quadrilaterals that have an area of 12 square units. Any three quadrilaterals which enclose 12 square units are valid. Notice that the sloping sides of the trapezium can form triangles which enclose half a square unit each.
The blue rectangles on the left-hand grid tile the grid. The blue rectangles on the right-hand grid do not tile the grid, since there are gaps and overlaps.
A: False. The area of the shape is the area covered or enclosed by the whole shape, not only its edges.
B: True. The area of the shape is 24 square units, the same as 24 grid squares.
C: Multiplying the side lengths is the formula for finding the area of a rectangle. The shape as presented is irregular. However, the 24 square units of this shape can be rearranged to create a rectangle which is 6 units by 4 units.
D: True. The area of a shape is the area inside a shape.
E: True. The shape can be broken down into two rectangles 4 × 3 units and 6 × 2 units, whose areas can be added together to find the area of the whole shape.
The figure can be divided into 3 rectangles.
On the left, rectangle A is 3 units tall × 5 units wide, rectangle B is (3 + 2) units tall × 3 units wide, and rectangle C is 6 units tall × 2 units wide. The total area is (3×5)+(5×3)+(6×2) = 42 units.
On the right, rectangle D is 3 units tall × 10 units wide, rectangle E is 2 units tall × (3 + 2) units wide, and rectangle F is 1 unit tall × 2 units wide. The total area is (3×10)+(2×5)+(1×2) = 42 units.
Other ways of subdividing or rearranging the figure may also give the same answer of 42 units.
To find the area of a rectangle or square, multiply the side lengths.
The area of the rectangle is 7 × ¾ = 5¼ square inches.
The area of the square is 2½ × 2½ = 6¼ square inches.
Therefore, the square has the larger area.
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