Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Angle (∠)
- Complementary (∠)
- Supplementary (∠)
Purpose and Instructional Strategies
In the elementary grades, students were introduced to acute, right, obtuse, straight reflex angles and solved real-world and mathematical problems involving angle measures. They also used angles to classify triangles and quadrilaterals. In grade 8, students solve problems involving supplementary, complementary, vertical and adjacent angles. In Geometry, students will extend the learning from this benchmark to prove relationships and theorems involving lines and angles.
- This benchmark is foundational to help develop the understanding of angles and connections related to parallel lines cut by a transversal.
- In order for students to learn relationships between angles, it is important to provide an opportunity to connect complementary and supplementary angles to work with triangles. Students should draw or be given a right triangle to explore rearranging the angles to show both the 90 and 180 degrees that can be created for a right angle and a straight line, respectively.
- To support the concept of adjacent angles, students should have examples and non-examples to write their own definition and revise it based on critiques from others (MTR.4.1). Students should trace each angle with different colors to ensure that there isn’t overlap, but has a common side.
- When discussing vertical angles, use a model of two strips of paper with a small brad at the center where they cross. Then, moving the paper to create different sized angles, measure each angle to show the vertical angle measures to lead to understanding that the vertical angles will have the same measure.
- Vertical angles can be explored using the same activity as adjacent angles with examples and non-examples. The criteria could include the following:
- Formed from exactly 2 straight intersecting lines
- Pair of angles
- Common vertex
- It is important to have students’ reasoning supported. This can be done by making statements with reasoning such as “always true, sometimes true, never true.”
- For example, a linear pair of angles (a type of adjacent angles) are always supplementary because they form a straight line.
- Once conceptual understanding and definitions are built, introduce algebraic concepts for students to write and solve equations using facts about the angle relationships. Students should be able to generate equations written in different forms.
- For example, if students are provided the figure below, they can generate multiple equivalent equations to represent their thinking. For this figure, three possible equations are:
180 = 147 + 2 + 3
180 − 147 = 2 +3
2 + 150 = 180
Common Misconceptions or Errors
- Students may invert the definition of complementary and supplementary.
Strategies to Support Tiered Instruction
- Instruction includes co-constructing a graphic organizer with students to measure, label and record the angle measurement of two intersecting lines. The teacher labels the angles, and measures and record the angle measurements. The teacher then leads a discussion and documents the relationships between different angle pairs.
- Instruction includes erasing or covering part of a line for students to visually see the supplementary angles within two intersecting lines.
- Instruction includes co-creating a graphic organizer identifying the relationships between supplementary, complementary, vertical, and adjacent angles. Include a strategy for solving problems involving each type of angle pair, such as setting vertical angle measures equal.
Instructional Task 1 (MTR.1.1, MTR.2.1)
Complete the table below that includes the following types of angles:Instructional Task 2 (MTR.4.1)
Determine if each of the following statements is always true, sometimes true or never true. For each statement that you chose as “sometimes true”, provide an example and non-example.
a. The sum of the measures of two supplementary angles is 180°.
b. Vertical angles are also adjacent angles.
c. Two adjacent angles are complementary.
d. If two lines intersect, each pair of vertical angles are complementary.
Instructional Item 1
The measure of angle 1 is 12 more than the measure of angle 2. What is the degree measure of angle 3?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.