Course 1, Unit 6 - Patterns in Shape
The intent of this unit is to review, deepen, and extend students' understanding
of two- and three-dimensional shapes, their representations, their properties,
and their uses. The fundamental idea of this unit is one of shape—what
gives shapes their form and how the shape of an object often influences
its function. The unit provides an introduction to mathematical reasoning
as a way to discover or establish new facts as consequences of known
or assumed facts. As such, the unit lays the groundwork for ideas of
mathematical argument or proof that will be developed formally in Courses
2, 3, and 4. The focus here is on careful visual reasoning and explanations,
not on formal proof.
Key Ideas from Course 1, Unit 6
The Triangle Inequality: This relationship among the lengths
of the sides of a triangle is developed on pages 363-364.
(The quadrilateral analog to this inequality is developed in Problem 3 on page 365.)
Conditions that are sufficient for testing congruence of triangles: side-side-side,
side-angle-side, and angle-side-angle are developed on pages 369-371.
Students use the CPMP-Tools geometry
custom app, "Triangle Congruence," to use rigid transformations
to verify SSS, SAS, and ASA conditions for congruence of triangles (page 371).
The Pythagorean Theorem: If the lengths of the sides of a
right triangle are a, b, and c, with the side
length c opposite the right angle, then a2 + b2 = c2.
(This relationship is taught in most middle school classes. It
was reviewed in Course 1 Unit 1 on page 50 and
used in various review tasks in this course.)
The converse of a statement: The converse of an if-then statement
reverses the order of the two parts of the statement. For example,
the converse of the Pythagorean Theorem is: If the sum of
the squares of the lengths of two sides of a triangle equals the
square of the length of the third side, then the triangle
is a right triangle. The converse of the Pythagorean Theorem has
a very practical use - that of ensuring that a right angle is formed. (See
page 380 Problem 3.)
Visualize and represent two- and three-dimensional shapes: Students
build models of polyhedra and consider properties of polygons and
polyhedra such as symmetry and rigidity.
Name, analyze, and apply properties of polygons and polyhedra: Polygons
are frequently classified by the number of sides they have (page 400).
For example, a 10-sided polygon is called a decagon. Characteristics
of pyramids, prisms, cylinders, and cones are developed in Lesson
3 (pages 424-431).