Private: Learning Math: Geometry

Professional Development > Private: Learning Math: Geometry > 2. Triangles and Quadrilaterals

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.

In This Session

Part A: Different Triangles
Part B: Linkage-Strip Constructions
Part C: Building Towers
Homework

In this session, you will build triangles and quadrilaterals to explore their properties. Classification is an important part of geometry and other areas of mathematics. By creating and manipulating triangles and quadrilaterals, you will develop a sense of logical classifications and how different figures are related.

For information on required and/or optional materials for this session see Note 1 below.

Learning Objectives

In this session, you will learn the following:

How to classify triangles according to some of their features
A rule that describes when three given lengths will make a triangle and when they will not
A rule that describes when four given lengths will make a quadrilateral and when they will not
How to use what you’ve learned about geometry and the properties of figures to help with a construction task

Key Terms

Previously Introduced

Altitude: An altitude of a triangle is a line segment connecting a vertex to the line containing the opposite side and perpendicular to that side.

Perpendicular Bisector: The perpendicular bisector of a line segment is perpendicular to that segment and bisects it; that is, it goes through the midpoint of the segment, creating two equal segments.

New in This Session

Acute Triangle: An acute triangle is a triangle with all three angles less than 90°.

Congruent: Two figures are congruent if all corresponding lengths are the same, and if all corresponding angles have the same measure. Colloquially, we say they “are the same size and shape,” though they may have different orientation. (One might be rotated or flipped compared to the other.)

Equilateral Triangle: An equilateral triangle is a triangle with three equal sides.

Isosceles Triangle: An isosceles triangle is a triangle with two equal sides.

Obtuse Triangle: An obtuse triangle is a triangle with one angle more than 90°.

Right Triangle: A right triangle is a triangle with one right (90°) angle.

Scalene Triangle: A scalene triangle is a triangle with all three sides unequal.

Side-Angle-Side (SAS) Congruence: Side-angle-side (SAS) congruence states that if any two sides of a triangle are equal in length to two sides of another triangle and the angles bewteen each pair of sides have the same measure, then the two triangles are congruent; that is, they have exactly the same shape and size.

Side-Side-Side (SSS) similarity: he side-side-side (SSS) similarity test says that if two triangles have all three pairs of sides in proportion, the triangles must be similar.

Similar Triangles: Similar triangles are triangles that have the same shape but possibly different size. In particular, corresponding angles are congruent, and corresponding sides are in proportion.

Triangle Inequality: The triangle inequality says that for three lengths to make a triangle, the sum of the lengths of any two sides must be greater than the third length.

Notes

Part A Notes: Different Triangles
Part B Notes: Linkage-Strip Constructions
Part C Notes: Building Towers

Materials Needed

linkage strips
scissors
box of toothpicks
bag of mini-marshmallows
meter stick or ruler

You can make your own linkage strips from stiff paper with evenly spaced holes and paper fasteners, or you can purchase them from ETA/Cuisenaire under the name polystrips.

For homework, you’ll also need a ruler marked in inches and a protractor. Collect this set of materials for each group or individual working alone.

Polystrips

ETA/Cuisenaire
500 Greenview Court
Vernon Hills, IL 60061
800-445-5985 847-816-5050
800-875-9643 847-816-5066 (fax)
(Product not available online)

Series Directory

Private: Learning Math: Geometry

Credits

Produced by WGBH Educational Foundation. 2003.

Closed Captioning
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 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

In This Session

Learning Objectives

Key Terms

Previously Introduced

New in This Session

Notes

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 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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