For this lesson, the students need a set of pattern blocks. (Only the yellow hexagons, red trapezoids, blue rhombuses, and green triangles are needed. The students do not use the orange triangle or the tan rhombus for this lesson.) If the students are seated at tables, one complete set of pattern blocks should serve an entire group. The most common regions studied at the elementary grade levels are the rectangle and circle. The "region" represents the "whole," and parts of the region are all congruent. The students should be exposed to a variety of shapes and not limited to the rectangle and circle. It is important that the students work with a variety of regions so that they do not think of the region as only "pieces of a pie." For this reason, pattern blocks are an appropriate tool for work with the region model.
Have students work in pairs to explore relationships among the four shapes. The Questions for Students at the end of this lesson facilitate the exploration and help students focus on the mathematical concepts of these lessons.
The students should use pattern blocks to answer the questions. If overhead pattern blocks (for use on overhead projectors) are available, the two pattern blocks being compared can be displayed on the overhead projector.
If you prefer, give printed copies of the Region Relationships 1 activity sheet to all students. An overhead transparency of this worksheet can be made for use with the entire class. You may want to color a transparent overhead of the pattern block shapes with a permanent marker to create overhead pattern blocks to use for demonstration purposes.
Guide the students through each question on the activity sheet. For the first example, you may want to show students how two green triangles can be placed to exactly cover one blue rhombus, thereby showing that they are equivalent. After that, you may wish to have students perform the demonstrations for one another using the pattern blocks on the overhead projector.
How many green triangles  are in one blue rhombus  ? [Two.] |
How many green triangles are in one red trapezoid  ? [Three.] |
How many green triangles are in one yellow hexagon  ? [Six.] |
| How many blue rhombuses are in one yellow hexagon ? [Three.] |
| How many red trapezoids are in one yellow hexagon ? [Two.] |
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The students might notice that there is one blue rhombus and one green triangle in one red trapezoid. This discovery could lead to a rich discussion of equivalency. If the students do not discover this relationship on their own, guide them in seeing this relationship. For example, you could ask, "Is there a way to represent the red trapezoid using blue and green pattern blocks?" The students should state that they could construct the trapezoid with one green triangle and one blue rhombus. You could then ask, "Could we cover the red trapezoid using only one color?" The students should indicate that the red trapezoid could be covered with three green triangles. And you could also ask, "What does this tell us about the relationship between the blue rhombus and the green triangle?" The students should state that there are two green triangles in one blue rhombus. The students may continue discovering other such relationships using two or more pattern blocks and exchanging them for one pattern block.
Have the students record as many fraction relationships as possible. You may choose to have them record the relationships in a math journal to which they may refer later. Each pair should record relationships on chart paper to share with the whole class. As each pair shares, have the students add to their journal any relationships that they may have missed.
As the students work to understand fraction relationships using the region model, it is appropriate to work with concepts on a continuum from concrete to abstract. This lesson first exposes the students to a concrete representation of the region model through work with pattern blocks. As the students move toward more abstract work, it is appropriate to introduce semi‑concrete representations. Having the students record fraction relationships pictorially gives them the opportunity to be exposed to such a model.
