Private: Learning Math: Geometry

Professional Development > Private: Learning Math: Geometry > 2. Triangles and Quadrilaterals > 2.1 Part A: Different Triangles (20 minutes)

Mathematics

K-2, 3-5, 6-8

Triangles and Quadrilaterals Part A: Different Triangles (20 minutes)

Session 2, Part A

In This Part

Drawing Triangles
Classifying Triangles

In the following problems, you will draw a variety of triangles and take note of their differences. Note 2

Drawing Triangles

Problem A1

a. Draw any triangle on your paper.

b. Draw a second triangle that is different in some way from your first one. Describe in just a word or two how it is different.

Problem A2

Draw a third triangle that is different from both of your other two. Describe how it is different.

Problem A3

Draw two more triangles, different from all the ones that came before.

Problem A4

To make “different” triangles, you have to change some feature of the triangle. Make a list of the features of triangles that you changed.

Classifying Triangles

The following shows how triangles can be classified according to some of their features:

Angles:

Right Triangles	Acute Triangles	Obtuse Triangles

One right (90°) angle	All angles less than 90°	One angle more than 90°

Sides:

Equilateral Triangles	Isosceles Triangles	Scalene Triangles

All three sides are the same length.	Two sides are the same length.	All three sides are different lengths.

Size:

Similar Triangles	Congruent Triangles

Same shape, possibly different size	Same size and shape

Problem A5

More than one feature can be combined into a triangle. Decide which of the following combinations are possible. If the combination is possible, draw a sketch on a piece of paper. If not, explain why not.

scalene right triangle
an isosceles right triangle
an equilateral right triangle
a right triangle that is also obtuse

Notes

Note 2

If you’re working in a group, first work on Problems A1-A5 individually. Afterwards, break into small groups to compare the lists of the features of triangles that everyone created.

Solutions

Problem A1

a. Answers will vary. The following is one example:

b. Answers will vary. One example is a triangle whose side lengths are longer.

Problem A2

Answers will vary. One example is a triangle that, unlike the previous two, has one right angle.

Problem A3

Answers will vary. Some examples are the following:

Problem A4

Answers will vary, but generally, either the measures of angles or lengths of sides need to be changed to make “different” triangles.

Problem A5

a. This is possible.

b. This is possible.

c. This is impossible. Inside a triangle, equal angles correspond to equal sides, so for a right triangle to be equilateral, it would have to have three right angles. Two right angles next to each other, however, form parallel lines, which would mean it would not be possible to complete such a triangle.

d. This is impossible. The picture below shows the right angle and obtuse angle next to each other, with the side in between laid out horizontally. The side that extends from the right angle is vertical, while the side that extends from the obtuse angle is pointed away from the side that extends from the right angle. Because these sides must be connected to form a triangle, this kind of triangle is impossible to make.

Series Directory

Private: Learning Math: Geometry

Credits

Produced by WGBH Educational Foundation. 2003.

ISBN: 1-57680-597-2

Sections

2.1 Part A: Different Triangles (20 minutes)

Session 2, Part A

2.2 Part B: Linkage-Strip Constructions (40 minutes)

Session 2, Part B

2.3 Part C: Building Towers (45 minutes)

Session 2, Part C

2.4 Homework

Session 2, Homework

Sessions

Session 1 What Is Geometry?

Explore the basics of geometric thinking using rich visualization problems and mathematical language. Use your intuition and visual tools for geometric construction. Reflect on the basic objects of geometry and their representation.

Session 2 Triangles and Quadrilaterals

Learn about the classifications of triangles, their different properties, and relationships between them. Examine concepts such as triangle inequality, triangle rigidity, and side–side–side congruence, and look at the conditions that cause them. Compare how these concepts apply to quadrilaterals. Explore properties of triangles and quadrilaterals through practical applications such as building structures.

Session 3 Polygons

Explore the properties of polygons through puzzles and games, then proceed into a more formal classification of polygons. Look at mathematical definitions more formally, and explore how terms can have different but equivalent definitions.

Session 4 Parallel Lines and Circles

Use dynamic geometry software to construct figures with given characteristics, such as segments that are perpendicular, parallel, or of equal length, and to examine the properties of parallel lines and circles. Look past formal definitions and discover the properties and relationships among geometric figures for yourself.

Session 5 Dissections and Proof

Review and explore transformations such as translation, reflection, and rotation. Apply these ideas to solve more complex geometric problems. Use your knowledge of properties of figures to reason through, solve, and justify your solutions to problems. Analyze and prove the midline theorem.

Session 6 The Pythagorean Theorem

Continue to examine the idea of mathematical proof. Look at several geometric or algebraic proofs of one of the most famous theorems in mathematics: the Pythagorean theorem. Explore different applications of the Pythagorean theorem, such as the distance formula.

Session 7 Symmetry

Investigate symmetry, one of the most important ideas in mathematics. Explore geometric notions of symmetry by creating designs and examining their properties. Investigate line symmetry and rotation symmetry; then learn about frieze patterns.

Session 8 Similarity

Examine your intuitive notions of what makes a "good copy" and then progress toward a more formal definition of similarity. Explore similar triangles and look into some applications of similar triangles, including trigonometry.

Session 9 Solids

Explore various aspects of solid geometry. Examine platonic solids and why there are a finite number of them. Investigate nets and cross-sections for solids as a way of establishing the relationships between two–dimensional and three–dimensional geometry.

Session 10 Classroom Case Studies, K-2

This is the final session of the Geometry course! In this session, we will examine how geometry as a problem-solving process might look when applied to situations in your own classroom. This session is customized for three grade levels.

Session 11 Classroom Case Studies, 3-5

Session 12 Classroom Case Studies, 6-8

Watch Videos 11 and 12 in the 10th session for grade 6–8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6–8 teachers (former course participants) who have adapted their new knowledge to their classrooms.

Private: Learning Math: Geometry

Triangles and Quadrilaterals Part A: Different Triangles (20 minutes)

Session 2, Part A

In This Part

Drawing Triangles

Problem A1

Problem A2

Problem A3

Problem A4

Classifying Triangles

Problem A5

Notes

Solutions

Problem A1

Problem A2

Problem A3

Problem A4

Problem A5

Series Directory

Credits

Sections

2.1 Part A: Different Triangles (20 minutes)

2.2 Part B: Linkage-Strip Constructions (40 minutes)

2.3 Part C: Building Towers (45 minutes)

2.4 Homework

Sessions

Session 1 What Is Geometry?

Session 2 Triangles and Quadrilaterals

Session 3 Polygons

Session 4 Parallel Lines and Circles

Session 5 Dissections and Proof

Session 6 The Pythagorean Theorem

Session 7 Symmetry

Session 8 Similarity

Session 9 Solids

Session 10 Classroom Case Studies, K-2

Session 11 Classroom Case Studies, 3-5

Session 12 Classroom Case Studies, 6-8

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