In this lesson, students will identify numerous types of triangles and other plane figures. Students will: 
- categorize quadrilaterals according to the number of parallel and perpendicular lines they contain. 
- categorize quadrilaterals based on their internal angles. 
- categorize triangles depending on their angles. 

- How do spatial relationships, such as shape and dimension, help to create, construct, model, and portray real-world scenarios or solve problems? 
- How may geometric properties and theorems be applied to describe, model, and analyze problems? 
- How can patterns be used to describe relationships in mathematical situations? 
- How might recognizing repetition or regularity help to solve problems more efficiently? 
- How might using geometric shape features help with mathematical reasoning and problem solving?

- Acute Angle: An angle measuring less than 90˚. 
- Acute Triangle: A triangle made up of 3 acute angles. 
- Angle: A geometric figure formed by two rays that share a common endpoint. 
- Line: A straight path that extends infinitely in both directions. 
- Line Segment: A straight path with a finite length. 
- Obtuse Angle: An angle measuring more than 90˚. 
- Obtuse Triangle: A triangle made up of 1 obtuse angle and 2 acute angles. 
- Point: A specific location in a geometric plane with no shape, size, or dimension. 
- Ray: A straight path that begins at an endpoint and extends infinitely in 1 direction. 
- Right Angle: An angle measuring exactly 90˚. 
- Right Triangle: A triangle with 1 right angle and 2 acute angles.

- one copy of the Lesson 2 Entrance Ticket (M-4-5-2_Lesson 2 Entrance Ticket and KEY) for each student
- one copy of the Triangle Classifications sheet (M-4-5-2_Triangle Classifications and KEY) for each student
- one copy of the Categorizing Shapes worksheet (M-4-5-2_Categorizing Shapes and KEY) for each student

- Use the Lesson 2 Entrance Ticket to assess students' knowledge of triangles. 
- Students may be graded based on their completion of the Categorizing Shapes worksheet (M-4-5-2_Categorizing Shapes and KEY). 
- Students may also be graded on their performance on the Triangle Classifications worksheet (M-4-5-2_Triangle Classifications and KEY).

Scaffolding, Active Engagement, Modeling, and Formative Assessment 
W: Students will expand their knowledge of geometric shapes and definitions. They will apply what they know about simple shapes to classify newer, more diverse, and complicated shapes. Students will learn about several sorts of quadrilaterals and triangles. 
H: Begin with a common shape, such as a square, to engage students. Students work as a class to discover what makes the shape a square, and then develop the definition from there. 
E: Students will apply their previous knowledge of angles, parallel and perpendicular lines, and elementary shapes to learn new shapes. 
R: Students can review the properties that define each quadrilateral, as many of the shapes share attributes. This allows individuals to reconsider and revise their knowledge of each idea by presenting multiple, diverse examples. 
E: Students' comprehension will be assessed using the Categorizing Shapes worksheet and observation while working with classmates. 
T: Use the Extension section to personalize the lesson to the students' specific requirements. The Routine section includes strategies for reviewing course concepts throughout the year. The Small Group section is designed for students who might benefit from more learning or practice opportunities. The Expansion section contains suggestions for students who are ready for a challenge that exceeds the criteria of the standard. 
O: The lesson begins with an exploration of a shape that students are familiar with. This should engage students and help them understand that, even if they are familiar with squares, there is still more to learn about them. Each task starts with a known shape and progresses to explore less familiar shapes. 

Activity 1

Draw a square on the board. "What shape is this?" Most students should recognize it as a square. "How do you know it's a square?" Students may recognize that all four sides are the same. Write that information on the board. "What can you tell me about the angles in the square?" Students should be able to recognize them as right angles; if not, revisit the three types of angles discussed in Lesson 1. Note on the board that a square has 4 right angles.



Then, highlight two opposing sides of the square. "What can you tell me about the line segments? If they were lines that went on forever, would they ever cross?" (no) Next, highlight the other two sides of the square and pose the same question. Note that a square has two sets of parallel sides.



"All of these things—that the sides are all the same length, that there are 4 right angles, and that there are 2 sets of parallel sides—are all characteristics of a square. When you draw a square, it must have all of these characteristics."

Now draw a rectangle on the board. Make sure the shape is clearly a rectangle, not a square, with a length that exceeds the width.



"Is this a square?" (no) "Why not?" Students should note that all 4 sides are not the same length. Ask about the various properties assigned to a square: "Are there four right angles?" (yes) "Are there two sets of parallel sides?" (yes) "What do we call this shape?" (rectangle) Make a note of its rectangle shape and its two attributes.

"So it's almost the same as a square. A rectangle just has to have two sets of parallel sides and four right angles. That's what defines it as a rectangle; as long as any shape has these two properties." Discuss whether a square has the two properties required to be a rectangle. Discuss how a square is a special type of rectangle, but a rectangle is not a square since it lacks all of the attributes associated with a square.

Next, draw a parallelogram. "Is this a square?" (No.) "Is this a rectangle?" (No.) "Why not?" Students will see that it does not have four right angles. "Does it share any properties with a rectangle?" Students should take note that it does have two sets of parallel sides. Label the figure parallelogram and note its only feature: two sets of parallel sides. Point out that the word parallel is right in the name parallelogram, and it is the only property to consider when deciding whether anything is a parallelogram.


"Is a rectangle a parallelogram?" If students are unsure, reword the question as "Does a rectangle have two sets of parallel sides?" (yes) "Then it is a parallelogram."

“Is a square a parallelogram?” (Yes.)

Point out that a square is also a rectangle and a parallelogram.

Finally, create a trapezoid on the board. "Is this a parallelogram?" (No.) "Why not?" (Because it does not have two parallel sides.) Point out one pair of parallel sides (the bases), but show that the other two sides are not parallel by extending them until they clearly intersect one another. "So, if it's not a parallelogram, can it be a rectangle?" (No.) "And can it be a square?" (No.) 

"So this shape is sort of by itself. It does not fit in with the other shapes we have studied about so far. It's called a trapezoid and just has one property." Write the term trapezoid and describe its properties: There is exactly one set of parallel sides.



Activity 2

"The previous activity involved categorizing shapes with four sides. They go by many names, including square, rectangle, parallelogram, trapezoid, and some that we didn't learn about. Polygons with three sides are a little easier. First, how do we refer to a shape with three sides?" (Triangle.) 

Draw a right triangle on the board. "Describe the angles in this triangle." Students should notice that there are two acute angles and one right angle. "This triangle is known as a right triangle because it has only one right angle. It makes no difference what the other angles look like. A right triangle is any triangle with a right angle." Under the picture, write the term right angle and has 1 right angle.



Have a volunteer come to the board. "Draw a triangle that is not a right triangle. Your triangle should not contain a right angle." Depending on the type of triangle drawn, use the discussion about acute or obtuse triangles below. 

Following that conversation, request that a volunteer come up and draw a triangle that does not suit any of the two descriptions provided so far.

For acute triangles:

"Describe the three angles in this triangle." Students should recognize them all as acute angles. "Because this has three acute angles, we call it an acute triangle." Write the term and its definition next to the picture. "Notice that our right triangle has two acute angles. That isn't enough to make an acute triangle. Acute triangles must have three acute angles."



For obtuse triangles:

"Describe the three angles in this triangle." Students should recognize two acute and one obtuse angles. "Because this has one obtuse angle, we call it an obtuse triangle." Write the term and definition next to the picture. “Like our right triangle, we just care about having one obtuse angle. We don't particularly care about the other two angles.”

Ask that a volunteer come up and draw a triangle that doesn't fit any of the three descriptions provided so far. (This is an impossible task.) If students think they have one, let them to draw it and then lead the class through the categorization process (or explain why the figure is not a triangle at all, such as having curved sides, not being closed, or having more than three sides). 

Point out that every triangle fits into one of these three categories.

Students should work in pairs to complete the Categorizing Shapes worksheet (M-4-5-2_Categorizing Shapes and KEY).

Students may be assigned the Triangle Classifications sheet (M-4-5-2_Triangle Classifications and KEY) to complete if time permits, or as homework to assist them further explore this skill.

This course reinforced previous knowledge of geometric shapes and their properties. Students learnt how to classify quadrilaterals and triangles using their attributes and meanings. Students should understand that some figures can be classified into multiple categories, such as a square, quadrilateral, rectangle, or parallelogram.

Extension:

Routine: To assist students remember what they learned about triangles, have them redo 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 center (to form quadrants). Then say, "Now, number them one through four in the upper left corner of each quadrant. In the first quadrant please draw 3 triangles that are right triangles." Give students less than a minute to complete this. Then you can say, "In the second quadrant draw 3 acute triangles." Give them less than a minute to do this. Then you might say, "In the third quadrant draw 3 obtuse triangles." Give them less than a minute to do this. Following this, you should say, "In the last quadrant, draw 1 triangle of each type and label each of them correctly." 
As students progress through their geometry studies, they will encounter several opportunities to refresh their definitions of geometric figures and categorize shapes by qualities.

Small Group: Some students may benefit from more practice or discussion of lesson concepts in small groups. 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. Students can carry out a similar assignment for each of the four types of quadrilaterals mentioned. 

Expansion: The lesson can be expanded as follows: 
Students can investigate triangle classification based on side length rather than angle measure (scalene, isosceles, equilateral) and then classify triangles using both types of classification (for example, isosceles right triangles).
Students can investigate more quadrilaterals (kites and rhombuses) and draw a Venn diagram illustrating the relationships between different types of quadrilaterals. 
Students can investigate both convex and concave quadrilaterals. 
Students can also practice recognizing geometric forms using the game on the website:
http://www.bbc.co.uk/bitesize/ks2/maths/shape_space/2d_shapes/play/.