The purpose of this activity is to use the fact that the sum of the angles all the way around a point is 360 degrees to reason about the measure of other angles. In this activity, students use the structure of pattern blocks to explore configurations that make 360 degrees and to solve for angles of the individual blocks (MP7). For this activity, there are multiple configurations of blocks that will accomplish the task.

This activity works best when each student has access to the pattern blocks. If pattern blocks are not available, consider using the digital version of the activity. In the digital version, students use an applet to dynamically manipulate shapes to determine angle measures.  

Direct students’ attention to the reference created using Collect and Display. Ask students to share their strategies for figuring out each angle measure for the pattern blocks. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “The angles in the triangles are all the same” already on it and be updated with the more precise phrase “The triangles have 1 unique angle.”)

As students describe the angle measures for each shape, write an equation to represent how their angles add up to 360 degrees. Listen carefully for how students describe their reasoning, and make your equation match the vocabulary that they use. For example, students might have reasoned about 6 green triangles by thinking , or , or .

When an angle from one block is known, it can be used to help figure out angles for other blocks. For example, students may say that they knew the angles on the yellow hexagon measured because they could fit two of the green triangles onto one corner of the hexagon, and , or . There are many different ways students could have reasoned about the angles on each block, and it is okay if they didn’t think back to for every angle.

Before moving on to the next activity, ensure that students know the measure of each interior angle of each shape in the set of pattern blocks. Display these measures for all to see throughout the remainder of the lesson.

In this activity, students use the pattern block angles to determine the measure of other given angles. Students also recognize that a straight angle can be considered an angle and not just a line. Students are asked to find different combinations of pattern blocks that form a straight angle to connect the algebraic representation of summing angles and the geometric representation of joining angles with the same vertex.

If students find only one combination of pattern blocks that form each angle, encourage them to look for more combinations.

As students work on the task, monitor for students who use different combinations of blocks to form a straight angle.

This activity works best when each student has access to the pattern blocks. If pattern blocks are not available, consider using the digital version of the activity. In the digital version, students use an applet to dynamically manipulate shapes to determine angle measures.

The goal of this discussion is for students to be exposed to many different examples of angle measures that add up to 180 degrees.

First, instruct students to compare, with a partner, their answers to the first question and to share their reasoning until they reach an agreement. To help students see Angle as a 180-degree angle and not just as a straight line, consider using only the smaller angle on the tan rhombus blocks to measure all three figures: Composing four tan rhombuses gives an angle measuring a degrees, seven rhombuses give an angle measuring degrees, and six rhombuses give an angle measuring  degrees.

Next, select previously identified students to share their solutions to the second question. For each combination of blocks that is shared, invite other students in the class to write an equation, displayed for all to see, that reflects the reasoning.

The purpose of this activity is to address the common error of reading a protractor from the wrong end. The problem gives students the opportunity to critique someone else’s thinking and make an argument if they agree with either students’ claim (MP3).

Ask students to indicate whether they agree with Priya or Tyler. Invite students to explain their reasoning until the class comes to an agreement that the measurement of angle  is 40 degrees.

Ask students how Tyler could know that his answer of 140 degrees is unreasonable for the measure of angle . Possible discussion points include:

• “Is angle acute, right, or obtuse?” (acute)
• “Where is there an angle that measures 140 degrees in this figure?” (adjacent to angle , from side to the other side of the protractor)

Make sure that students understand that a protractor is often labeled with two sets of angle measures, and they need to consider which side of the protractor they are measuring from.