The purpose of this activity is for students to recall properties and types of triangles and specific language for describing them. In this activity, students draw a variety of triangles and sort them into categories of their choosing. Monitor for students choosing categories that compare angle measures and side lengths. Students may use informal language to describe their categories, such as “all of the sides are different lengths” rather than “scalene.”
Select students who use these categories to share during the whole-class discussion:
Equilateral triangles
Right triangles
Isosceles triangles
Obtuse triangles
Launch
Arrange students in groups of 4. Distribute 3 sticky notes per student. Give 1 minute of quiet work time for students to draw a triangle on each sticky note.
Tell each group to look at all of their triangles and sort them into categories. Give 2–3 minutes of group time followed by a whole-class discussion. Listen for language describing the side lengths and angles of triangles, such as “equilateral,” “right,” “obtuse,” and “acute.”
Activity
None
Draw 3 different types of triangles.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite previously selected students to share the categories they chose for the triangles. As students describe categories, organize their descriptions for all to see in a table or chart like this one:
acute (all angles acute)
right (has a right angle)
obtuse (has an obtuse angle)
scalene (side lengths all different)
isosceles (at least two side lengths are equal)
equilateral (three side lengths equal)
As students share, invite them to place their sticky note on the chart for all to see.
If any students drew triangles that fall under multiple categories, such as a right isosceles triangle, display it for all to see. If not, draw one for all to see. Ask students which categories the triangle belongs to. Students may describe an isosceles triangle as having two angles the same or two sides the same.
Ask students if they can draw any other triangles that fit into two categories. Examples include equilateral and acute, scalene and obtuse, scalene and right. Students may describe scalene as having all different side lengths or all different angles.
Ask students if it is possible to draw an equilateral triangle that has a right angle. (No, 3 right angles would not form a triangle, and all the angles are the same in an equilateral triangle.)
If time allows, continue filling out the table with an example of each type of triangle or an explanation of why that triangle is not possible.
13.2
Activity
Optional
Standards Alignment
Building On
Addressing
8.G.A.2
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
This activity is optional because it provides additional opportunity for students to practice identifying congruent figures. In this matching activity, each student receives a card with a triangle that has one angle measure marked. Students work to find partners with triangles congruent to theirs, although there are three different angles marked for each set of congruent triangles.
Students may use strategies such as applying rigid transformations to find a matching triangle, or using the structure of triangles to identify them as acute, right, or isosceles to find matches with the same features (MP7).
Launch
Provide access to geometry toolkits. Distribute one card to each student, making sure that all three cards have been distributed for each triangle. If the number of students in the class is not a multiple of three, one or two students may need to take ownership over two cards showing congruent triangles.
Explain that there are two other students who have a triangle congruent to theirs that has been re-oriented in the coordinate plane through combinations of translations, rotations, and reflections. Instruct students to look at the triangle on their card and estimate the measures of the other two angles. With these estimates and their triangle in mind, students look for the two triangles congruent to theirs with one of the missing angles labeled.
Prepare and display a table for all to see with columns angle 1, angle 2, angle 3 and one row for each group of three students. It should look something like this:
student groups
angle 1
angle 2
angle 3
Once the three partners are together, they complete one row in the posted table for their triangle’s angle measures. Whole-class discussion to follow.
Give students 1 minute quiet think time, then 3–5 minutes to find their matches and record. Follow with a whole-class discussion.
Representation: Access for Perception. Display or provide students with a physical copy of the written directions and read them aloud. Check for understanding by inviting students to rephrase directions in their own words. Consider keeping the display of directions visible throughout the activity. Supports accessibility for: Language, Memory
Activity
None
Your teacher will give you a card with a picture of a triangle.
The measurement of one of the angles is labeled. Mentally estimate the measures of the other two angles.
Find two other students with triangles congruent to yours but with a different angle labeled. Confirm that the triangles are congruent, that each card has a different angle labeled, and that the angle measures make sense.
Enter the three angle measures for your triangle on the table your teacher has posted.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is for students to describe what they notice about angles in triangles. Here are some questions for discussion:
“How did you decide that you had a match?” (I used tracing paper to see that it matched, I estimated the angle measures and found them, or I looked for a triangle with the same features.)
“What do you notice about the three angle measures for your triangle?” (They add up to 180 degrees.)
“What do all of the triangles have in common?” (They all have angles that add up to 180 degrees.)
If no students notice that the angles for each triangle have a sum of 180 degrees, add a column labeled “total” and ask students to find the sum of their group’s triangle, then display it on the table for all to see.
13.3
Activity
Standards Alignment
Building On
Addressing
8.G.A.5
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
The purpose of this activity is for students to observe that three angles that total to 180 degrees can be used to create a triangle.
Students cut out three angles that form a line, and then try to use these three angles to make a triangle. Students also create their own three angles from a line and check whether they can construct a triangle with their angles. As students observe the triangles of others in their group as well as the triangles created from their chosen angles, they use repeated reasoning to conclude that 3 angles decomposed from a straight line can form a triangle (MP8).
Monitor for students who successfully make triangles out of each set of angles and select them to share (both the finished product and how they worked to arrange the angles) during the discussion. Also monitor how students divide the blank line into angles. It is helpful if the rays all have about the same length as in the pre-made examples.
Launch
Arrange students in groups of 4. Provide access to geometry toolkits, especially scissors. Distribute 1 copy of the black line master to each group.
Instruct students to cut the four individual pictures out of the black line master. Each student will work with one of these. Instruct the student with the blank copy to use a straightedge to divide the line into three angles (different from the three angles that the other students in the group have). Demonstrate how to do this if needed.
If necessary, demonstrate “making a triangle” before beginning the activity so students understand the intent. With three cut-out 60-degree angles, for example, an equilateral triangle can be built. Here is a picture showing three 60-degree angles arranged so that they can be joined to form the three angles of an equilateral triangle. The students will need to arrange the angles carefully, and they may need to use a straightedge in order to add the dotted lines to join the angles and create a triangle.
Activity
None
Your teacher will give you a page with three sets of angles and a blank space. Cut out each set of three angles. Can you make a triangle from each set that has these same three angles?
Student Response
Activity Synthesis
If time allows, have students do a "gallery walk" at the start of the discussion. Ask students to compare the triangle they made to the other triangles made from the same angles and be prepared to share what they noticed. (For example, students might notice that all the other triangles made with their angles looked pretty much the same, but were different sizes.) If students do not bring it up, direct them to notice that all of the "create three of your own angles" students were able to make a triangle, not just students with the ready-made angles.
Ask previously selected students to share their triangles and explain how they made the triangles. To make the triangles, some trial and error is needed. A basic method is to line up the line segments from two angles (to get one side of the triangle) and then try to place the third angle so that it lines up with the rays coming from the two angles already in place. Depending on the length of the rays, they may overlap, or the angles may need to be moved further apart.
Here are some questions for discussion:
"How do you know the three angles you were given sum to 180 degrees?" (They were adjacent to each other along a line.)
"How do you know these can be the three angles of a triangle?" (We were able to make a triangle using these three angles.)
"What do you know about the three angles of the triangle you made and why?" (Their measures sum to 180 because they were the same three angles that made a line.)
Ask students if they think they can make a triangle with any set of three angles that form a line and poll the class for a positive or negative response. Tell them that they will continue to investigate this idea and emphasize that while experiments may lead us to believe this statement is true, the methods used are not very accurate and were only applied to a few sets of angles.
If time permits, perform a demonstration of the converse: Start with a triangle, tear off its three corners, and show that these three angles when placed adjacent to each other sum to a line.
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “To arrange the angles, first, I because . . .” or “I noticed so I . . . .” Supports accessibility for: Language, Social-Emotional Functioning
MLR7 Compare and Connect. After the gallery walk, lead a discussion comparing, contrasting, and connecting the different strategies for creating triangles. To amplify student language, and illustrate connections, follow along and point to the relevant parts of the displays as students speak. Advances: Representing, Conversing
Lesson Synthesis
The goal of this discussion is for students to articulate their observations that the sum of the angle measures in a triangle is 180 degrees.
Some guiding questions for the discussion include:
"What did we observe about the sum of the angles inside a triangle?" (The sum of the angles inside a triangle seem to always add up to 180 degrees.)
"Are there pairs of angles that cannot be used to make a triangle?" (Yes. If the two angles are both bigger than or equal to 90 degrees, a triangle cannot be made.)
Emphasize that we were able to see for multiple triangles that the sum of their angles is and that using several sets of three angles adding to , we were able to build triangles. In the next lesson we will investigate and explain this interesting relationship.
Student Lesson Summary
A angle is called a straight angle because when it is made with two rays, they point in opposite directions and form a straight line.
If we experiment with angles in a triangle, we find that the sum of the measures of the three angles in each triangle is — the same as a straight angle!
Through experimentation we find:
If we add the three angles of a triangle physically by cutting them off and lining up the vertices and sides, then the three angles form a straight angle.
If we have a line and two rays that form three angles added to make a straight angle, then there is a triangle with these three angles.
Standards Alignment
Building On
7.G.A.2
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
