Grade 5

Geometry

Standard 5-4: The student will demonstrate through the mathematical processes an understanding of congruency, spatial relationships, and relationships among the properties of quadrilaterals.

The indicators for this standard are grouped by the following major concepts:

• Dimensional
• Plane and Transformational

The indicators that support each of those major concepts and an explanation of the essential learning for each major concept follows.

Dimensional

Indicators

5-4.1 Apply the relationships of quadrilaterals to make logical arguments about their properties.

5-4.2 Compare the angles, side lengths, and perimeters of congruent shapes.

5-4.3 Classify shapes as congruent.

5-4.4 Translate between two-dimensional representations and three-dimensional objects.

In 5^th grade, students apply the relationships of quadrilaterals to make logical arguments about their properties. This will include making and testing conjectures and explaining conclusions about quadrilateral properties and relationships. For example, are all squares rectangles? Are all rectangles squares? Why or why not?

In 4^th grade students used transformations to prove congruency. In 5^th grade, students will compare the angles, side lengths and perimeters of congruent shapes. Given congruent shapes, students will discover that the corresponding angles are the same, the corresponding side lengths are the same, and the perimeters are the same. Therefore, students will conclude that congruent shapes have the same shape and same size. In 6^th grade, students will compare the angles, side lengths, and perimeters of similar shapes. It is essential that students understand congruent shapes in order to progress to the concept of similarity in shapes.

Students in 5^th grade must also classify shapes as congruent. The definition of congruent is factual understanding. For students to classify shapes as congruent, they will need to have a conceptual understanding.

It is in 5^th grade that students begin exploring methods for translating between two-dimensional representations and three-dimensional objects. In 5^th grade, students should sketch the front, top, and side views of a three-dimensional object built with cubes. Students should be able to draw a net for a given three-dimensional shape and construct and/or state the three-dimensional shape when given its two-dimensional representation (net). Teachers should incorporate isometric dot paper in guiding the students to draw three-dimensional objects.

Plane and Transformational

Indicators

5-4.5 Predict the results of multiple transformations on a geometric shape when combinations of translation, reflection, and rotation are used.

5-4.6 Analyze shapes to determine line symmetry and/or rotational symmetry.

In 4^th grade, students have predicted the results of multiple transformations of the same type, but now they use multiple transformations of different types. "Students should consider three important kinds of transformations: reflections (flips), translations (slides), and rotations (turns). Younger students generally convince themselves that two shapes are congruent by physically fitting one on top of the other, but fifth grade students can develop greater precision as they describe the motions needed to show congruence ("turn it 90 degrees or flip it vertically, then rotate it 180 degrees). They should be able to visualize what will happen when a shape is rotated or reflected and predict the result. Students should also explore shapes with more than one line of symmetry. Students often create figures with rotational symmetry, but often have difficulty describing the regularity they see. They should be using language about turns and angles to describe these figures." (Principles and Standards for School Mathematics, 167-168)

For the first time, 5^th grade students are introduced to the concept of rotational symmetry. A shape that rotates onto itself before turning 360^o has rotational symmetry.

Students’ prior experiences have been limited to identification of lines of symmetry. Students will now analyze shapes to determine line symmetry and/or rotational symmetry. It is important to note that all regular polygons have rotational symmetry.