Part I: Pattern Block Designs

After the free play, refocus groups on Part I of the activity, creating a group design. As groups work, if they aren’t already considering the two questions in Part I, pose these to them.

Groups will likely discover that four squares fit together, three hexagons fit together, and six triangles fit together. Whatever cases they do find, you can point out that these blocks at least appear to fit together, but that students can’t be sure yet whether they actually fit together perfectly or just come very close. This uncertainty will help foreshadow proving the angle sum formula in Polygon Angles.

After approximately 15 minutes of exploration, get students’ attention for a brief lecture on angle. Angles can be thought of in different ways, and today’s activity looks at them from two perspectives. One perspective is dynamic, in which angle is thought of as a turn. The other is static, in which angle is thought of as a geometric figure. For most students, the dynamic concept of an angle as a turn is an easier place to start.

Begin by demonstrating a complete turn. Stand facing the class, make a complete turn, and ask, How far have I turned? You might mention the fact that you have not traveled any distance and therefore the traditional measures of length are inappropriate for measuring a turn. Some students may say that you have turned “one complete turn.” Others familiar with degree measurement may say that you have turned 360 degrees. Explain that both answers are correct and that a degree is the name for a turn that is 13601360 size 12{ { {1} over {"360"} } } {} of a complete turn. Use the symbol for degrees, writing 360° for the complete turn.

Ask students to demonstrate some other turns. For example, ask everyone to stand and do a half turn. Ask, How many degrees are in that turn? Then have students perform some other fractions of a turn.

Also go from degrees to turns. For example, ask students to turn 120°, and ask the class to describe what fraction of a whole turn that is. Most students need to develop a physical feeling for the turning concept, and this approach lets everyone get involved physically and mentally.

Emphasize that, when in doubt, students can return to the fact that a complete turn is 360°. They can always use this frame of reference to go from a fraction of a turn to degrees, and vice versa.

Ask if anyone knows a special name for a quarter turn. Introduce the term right angle and have students figure out how many degrees it must be. Also mention that an angle between 0° and 90° is called an acute angle and that an angle between 90° and 180° is called an obtuse angle.

Another important way to think about an angle is as a geometric figure (or part of one). To introduce this idea, show students a diagram such as that below, and ask, Where is the angle in this diagram?

As needed, explain that in order to think of this diagram as showing an angle in the sense of a turn, students can imagine standing at A and facing B, and then imagine turning to face C (while continuing to stand at A).

Extend the lengths of the sides of the angle and ask how this changes the angle itself. Many students confuse side lengths with the size of an angle, so it is important to bring out early and often that the angle itself remains unchanged.

Tell students that point A is called the vertex of the angle and that the rays from A through B and from A through C are called the sides of the angle. Also introduce the notation ∠BAC, read as “angle BAC.” Mention that if there is no chance for confusion, such an angle can be simply referred to as ∠A.

Make it clear that whether we start facing B or facing C, we generally assume that we turn “the short way.” Thus, if we start at A, facing B, we would turn counterclockwise to face C, rather than make almost a whole turn clockwise.

Part II: Pattern Block Angles

For Part II, students will need to be familiar with the concept of an angle in a polygon. You can introduce this concept by drawing any polygon. You may need to begin with the terms of side and vertex as applied to a polygon and introduce the plural vertices as well.

Then ask students to identify the angles in the polygon. Explain, if needed, that an angle in a polygon is an angle formed where two sides meet at a vertex. Thus, a polygon has the same number of angles as vertices (which is also the same as the number of sides). Use the special case of a square or rectangle to illustrate this fact, and ask students to find the sizes of the figure’s angles. They should be able to connect this idea with the earlier discussion and see that each angle is a quarter turn, or 90°.

Have groups now turn their attention to Part II. Explain that they are to determine the measure of the angles in degrees using only the blocks themselves. Remind them to consider what they learned about fitting blocks together to make complete turns.

As students complete Part II, encourage them to continue into Part III.

Once most groups have worked through at least a few of the pattern block angles, bring the class together for a brief discussion on methods and findings. If you have overhead pattern blocks, they will be useful here.

Most of the explanations should be straightforward, such as “I could fit six triangles together at a single point, so each angle is a sixth of a turn, which is 60°.” Students will need to do something subtler to find the large angle of the thin parallelogram, such as fit it together with the right angle from a square and an angle from the hexagon. You might again make note that the methods being used to determine these angle measurements are based on the assumption that the blocks fit together perfectly.

Part III: Pattern Block Angles with a Protractor

The protractor is a difficult tool for many students to learn to use. This activity is intended to help students develop a meaningful understanding of and experience with angles so that they can use the protractor to measure angles and get the same results. The activity assumes students will use the known angle measures of the pattern blocks to learn to read that measure on the protractor.

Students will have to learn how to align the vertex of the angle, as well as each side of the angle, with the protractor, and then how to read the measurement. Each protractor works slightly differently, so they will need to spend some time exploring with their own protractors.

Help students begin by suggesting that the protractor is a tool to measure angles. Remind them that they now know the measures of several angles—namely the angles of all the pattern blocks—and explain that they can use that knowledge to investigate how to use the protractor to obtain the same measurements.

Because pattern blocks are rather small, suggest that students trace one angle of a block onto paper and then extend its sides. In the example below (the blue rhombus), the angle is known to measure 60°. Let students learn how to use their protractors to arrive at this measure.