Tiling the Plane
Students start the first lesson of the school year by recalling what they know about area (note that students studied the areas of rectangles with whole-number side lengths in grade 3 and with fractional side lengths in grade 5). The mathematics they explore is not complicated, so it offers a low threshold for entry. The lesson does, however, uncover two important ideas:
- If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
- The area of a region does not change when the region is decomposed and rearranged.
At the end of this lesson, students are asked to write their best definition of area. It is important to let them formulate their definition in their own words. For English learners, it is especially important that they be encouraged to use their own words and also to use words of their peers. In the next lesson, students will revisit the definition of area as the number of square units that cover a region without gaps or overlaps.
As the first set of problems in a problem-based curriculum, students will also begin their year-long work on making sense of problems and persevering in solving them (MP1). This opening lesson leaves space for teachers to begin setting classroom routines and their expectations for mathematical discourse (MP3).
In all of the lessons in this unit, students should have access to their geometry toolkits, which should contain tracing paper, graph paper, colored pencils, scissors, and an index card. Students may not need all (or even any) of these tools to solve a particular problem. However, to make strategic choices about when to use which tools (MP5), students need to have opportunities to make those choices. Apps and simulations should supplement rather than replace physical tools.
Notes on terminology. In these materials, when we talk about a figure such as a rectangle, triangle, or circle, we usually mean the boundary of the figure (e.g., the sides of the rectangle), not including the region inside. However, we also use shorthand language such as “the area of a rectangle” to mean the “the area of the region inside the rectangle.” The term shape could refer to a figure with or without its interior. Although the terms figure, region, and shape are used without being defined precisely for students, help students understand that sometimes our focus is on the boundary (which in this unit will always be composed of black line segments), and sometimes it is on the region inside (which in this unit will always be shown in color and referred to as “the shaded region”).
- Compare (orally) areas of the shapes that make up a geometric pattern.
- Comprehend that the word “area” (orally and in writing) refers to how much of the plane a shape covers.
Let’s look at tiling patterns and think about area.
Assemble geometry toolkits. It would be best if students have access to these toolkits at all times throughout the unit. Toolkits include tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles.
- I can explain the meaning of area.
Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.
For example, the area of region A is 8 square units. The area of the shaded region of B is \(\frac12\) square unit.
A region is the space inside of a shape. Some examples of two-dimensional regions are inside a circle or inside a polygon. Some examples of three-dimensional regions are the inside of a cube or the inside of a sphere.
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