In this lesson, students will categorize many types of triangles. Students will use a variety of strategies to measure the side lengths and angles of various triangles in order to determine what type(s) of triangle(s) they are. Students will: 
- discover similarities and differences between several types of triangles.
- understand the differences and similarities between the many types of triangles. 
- practice measuring with a ruler and a protractor. 
- find the number of degrees in each triangle.

- How can we use the relationship between area and volume to draw, construct, model, and represent real-world situations, as well as solve problems of area and volume?

- Acute Triangle: A triangle whose interior angles all measure less than 90°. 
- Equilateral Triangle: A triangle whose three sides have equal lengths. 
- Isosceles Triangle: A triangle that has at least two sides of equal lengths. 
- Obtuse Triangle: A triangle that has one interior angle that measures more than 90°. 
- Right Triangle: A triangle that has one interior angle that measures 90° exactly. 
- Scalene Triangle: A triangle with no two sides of equal length.

- student copies of the Lesson 2 Entrance Ticket (M-6-2-2_Lesson 2 Entrance Ticket and KEY) 
- ruler for each student (group), with centimeters. 
- protractor for each student (group) 
- pencils for each student 
- student copies of Triangle Classifications sheet (M-6-2-2_Triangle Classifications and KEY)

- Use the Triangle Classification sheet and question/answer time to assess student understanding. 
- Formally evaluate students based on the brief summary of the lesson written by each student following the Types of Triangles activity. 

Scaffolding, Active Engagement, Modeling and Explicit Instruction 
W: Students will become familiar with the differences and similarities of several triangle types via discovery and modeling. They will also learn how to total the angle measurements for each given triangle. 
H: Use the entrance ticket to evaluate previous knowledge. Once the collective knowledge level has been assessed, training is given on different types of triangles and their definitions. 
E: Students categorize numerous types of triangles and practice measuring angles and side lengths. They should also conclude that the total of the interior angles of any triangle equals 180 degrees. 
R: Students work in pairs to explore discrepancies on the Types of Triangles worksheet. This will serve to reinforce what students already know and allow them to learn from one another. The writing task at the end of the lesson will help students bring together new information and fit it into what they already knew. 
E: During the Types of Triangles activity, the teacher evaluates students' understanding informally by observation. After that, collect the written paragraphs and scan them to find any learning gaps or to determine which students may benefit from additional instruction and/or practice. 
T: Tailor the lesson to match the needs of diverse students using extension suggestions, such as small groups for those who need more learning opportunities or expansion activities for students in need of a challenge. 
O: In this lesson, students learn about different types of triangles and practice evaluating different triangles to find their suitable classification. They also learn that the sum of the measures of the angles is always 180 degrees. 

As students enter the room, hand them all a copy of the Lesson 2 Entrance Ticket (M-6-2-2_Lesson 2 Entrance Ticket and KEY). Allow students to work alone at their seats. After a few minutes, ask them to discuss their answer to question 1 with a partner. Then, ask each group to share one fact about triangles (keep a list on the board). Allow them to share until everyone has written everything they know about triangles on the board (and then have them add new facts to their list). Review the answers to question 2 (demonstrating that they can "measure" a 90 degree angle with the corner of a piece of paper). After finishing the angle classifications, review the answers to questions 3 and 4. Emphasize that in obtuse and right triangles, only one angle can be obtuse or right, whereas in an acute triangle, all three angles have to be acute. If there is time, pose the question: "Can you have a triangle with both an obtuse angle and a right angle?"

Finally, explain, "We were looking at the triangles and grouping them according to their angle measurements. How else could we group them?" Wait for students to respond. "Correct, side lengths. How many different types of triangles do we have based on their side lengths?" As students come up with the three categories, write them on the board, leaving space below to draw an example. 
Optional: Have a student come up to the board and draw an example from one of the three categories. After drawing an example of each, challenge someone to draw a different triangle that is still in the same category.

Continue with another activity focusing on different types of triangles. Students can work independently or together in pairs. Each group or student will require pencils, a ruler, and a protractor. Give each student a copy of the Triangle Classifications activity sheet (M-6-2-2_Triangle Classifications and KEY). First, ask a few (7) volunteers to read the different types of triangles and their definitions. Next, have someone read the directions. Following that, you'll explain, "In summary, for each of these triangles in the chart, you (and your partner) must use the ruler and measure all three of the side lengths of each triangle in centimeters, and record your results in the appropriate place." Draw a basic triangle on the board and label the three corners A, B, and C. Then label the side lengths freely (for example, 3cm, 4cm, and 5cm). Then, to the right, write the notation . Ask, "Which number corresponds to which side?" What is the length of the side ? How do you know?" Make sure students understand that the sides are labeled with the endpoints. "After you've measured the side lengths, you'll need to determine how many degrees each angle of the triangle measures. Make sure to sum up all three angles to determine their total value. You can divide the work up so that each of you do two triangles, or you can do them all at once. But both partners must measure the side lengths and angles of at least two triangles. You have 20 minutes to do this task. Make sure to write the triangle type(s) in the last column." If necessary, remind students that extending one of the triangle's sides with a straight edge will make measuring the angles easier. 

While students are working, take the time to walk around and assist individuals who are having difficulty using a ruler and protractor properly. 

When the time is over, ask students to share their answers with the rest of the class. Some questions to ask are:

"How many degrees were the angles equal in each triangle? 
Can a triangle have just one classification? 
Can the longest side equal or be longer than the sum of the other two sides? Shorter than their differences?" 
Once finished, have each student write one paragraph (which can include drawings) about what he or she did in class that day and what they learned. This should be a brief overview of the lesson. There is a 3 minute time limit, and this must be collected immediately. 

Extension:

Routine: To assist students remember what they learned about triangles, have them retake the lesson activity by drawing additional triangles. Begin by handing out blank sheets and saying, "Please divide the sheet of paper into four quadrants like so." Draw a rectangle on the board to represent the paper, followed by horizontal and vertical lines through the middle (to form quadrants). Then say, "Now number them from one to four in the upper left corner of each quadrant. In the first quadrant, please create a triangle that is both a RIGHT and SCALENE triangle." Give students less than a minute to complete this. Next, say, "In the second quadrant draw a triangle that is EQUILATERAL, but NOT obtuse." Give them less than a minute to do this. Following that, say, "In the third quadrant draw a triangle that is OBTUSE and EQUIANGULAR." Give them less than a minute to do this. Following that, say, "In the last quadrant, 4, draw a triangle that is ACUTE and ISOSCELES." Once students have completed their drawings, go through them and have two to three students come forward to draw their triangles on the board. Make sure to remind out that there are numerous possible correct answers. When you reach the second quadrant, ask about how the triangles that meet these requirements can change (just in size). Also, ask why the stipulation "NOT obtuse" was essential, and why not (because it is equilateral all of the angles must equal 60 degrees and therefore acute, which is not obtuse; point out that right is also not obtuse) . For the third one, ask students to create triangles that fit the provided parameters. After that, show them that because the triangle is equiangular, all of the angles equal 60 degrees and so cannot be obtuse. It's impossible. Then go through the last one, as the first. 

Small Group: Students who do not instantly understand the concepts may benefit from additional practice in a small group. Have groups create classroom posters illustrating each type of triangle. Each group will be responsible for designing a "Type of Triangle" poster. Each poster must be titled with the type of triangle, feature a textual definition (in their own words), and include a "large" drawing of the triangle to represent the definition. On the drawing, students must indicate the length of each side (to nearest tenth of a centimeter) and angle. Then, at the bottom of the poster, they should put a "Disclaimer" that lists the alternative categories into which the triangle could be classified. After they've finished, ask each group to present their triangle poster verbally to describe each type of triangle.

Expansion: Finding Area Activity. Students searching for a challenge can try this activity. Each student should trace or draw three different triangles (with a meter stick) on chart paper. They will need to measure the lengths of all three sides as well as the angles, and then classify the triangle based on the side lengths and angle measurements. They must also note the sum of the angles and the perimeter of the triangle. Then they must calculate the height of each triangle, remembering that it "drops down" from the top to the base at a 90-degree angle. They can choose which side they want to be the base and draw the altitude accordingly. The height will next be measured and used to calculate the triangle's area (A = \(1 \over 2\) bh). The same process should be followed for all three triangles.

Technology: If computers are available, have students complete the small-group exercise on the computer by creating a PowerPoint of their type of triangle, or use another program (e.g., Notebook, Microsoft Publisher, Adobe) to create posters to display to the class. Students may find the following Web site useful in designing a triangle with dimensions (to capture a screen shot, SnagIt, or Ctrl+F4 on a Mac) to include in their poster or presentation:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=142