Lesson 1

Tiling the Plane

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Problem 1

Which square—large, medium, or small—covers more of the plane? Explain your reasoning.

A plane composed of a series of squares. There are 5 large squares, 10 medium squares, and 10 small squares.

Solution

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Problem 2

Draw three different quadrilaterals, each with an area of 12 square units.

Image of a grid.

Solution

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Problem 3

Use copies of the rectangle to show how a rectangle could:

a. tile the plane.

shaded rectangle on a grid. length = 3 units, height = 2 units.

b. not tile the plane.

shaded rectangle on a grid. length = 3 units, height = 2 units.

Solution

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Problem 4

The area of this shape is 24 square units. Which of these statements is true about the area? Select all that apply.

A figure on a grid.
A:

The area can be found by counting the number of squares that touch the edge of the shape.

B:

It takes 24 grid squares to cover the shape without gaps and overlaps.

C:

The area can be found by multiplying the sides lengths that are 6 units and 4 units.

D:

The area can be found by counting the grid squares inside the shape.

E:

The area can be found by adding \(4 \times 3\) and \(6 \times 2\).

Solution

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Problem 5

Here are two copies of the same figure. Show two different ways for finding the area of the shaded region. All angles are right angles.

A geometric figure with straight sides and right angles.
A geometric figure with straight sides and right angles.

 

Solution

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Problem 6

Which shape has a larger area: a rectangle that is 7 inches by \(\frac 34\) inch, or a square with side length of \(2 \frac12\) inches? Show your reasoning.

Solution

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