Implementing the 7th Grade GPS: Constructions, Transformations, and Three-Dimensional Figures

Presented by Judy O’Neal (joneal@ngcsu.edu)

North Georgia College & State University

Mathematics Topics Addressed

Basic constructions Copying a segment Copying an angle Bisecting a segment Bisecting an angle Perpendicular lines including

perpendicular bisector of a line segment

Line parallel to a given line through a point not on the line

Midpoint – A point that divides a line segment into two congruent () segments.

Perpendicular bisector – A line that is perpendicular (⊥) to a line segment and also divides the segment into two congruent () segments.

Angle bisector – A ray that divides an angle into two congruent () angles.

Transformations Translations Reflections Rotations

Three-dimensional figures formed by translations and rotations of plane figures through space

Cross sections Cones Cylinders Pyramids Prisms Spheres

Duplicating a Line Segment with Compass and Straightedge

Begin with a line segment PQ. Draw a line with a straightedge. Place a starting point on the line and

label it R Place the compass point on point P. Stretch the compass so that the

pencil is exactly on point Q. Without adjusting the compass span,

place the compass point on point R and swing the pencil so that it intersects the line.

Label the intersection point S.

S

Duplicating an Angle with Compass and Straightedge Begin with CAB. Draw a reference line and place a point A’ near the left end. With the compass on point A, stretch its width to point B. Without adjusting the compass, place the compass point on A’ and

draw a wide arc across the line. This establishes a new point B’. Place the compass point on B and stretch its width to point C. Without changing the compass span, place the compass point on

point B’ and draw an arc across the previously drawn arc. The intersection of these two arcs becomes point C’. Using a straightedge, draw a line from point A’ to C’ where the arcs

intersect. ∠C’A’B’ is (equal in measure) to CAB.∠

Perpendicular Bisector with Compass and Straightedge

Begin with a line segment AB. Place the compass point on point A. Stretch the compass along segment AB to a length greater than

half the segment length. Construct a circle (or wide arc) with center at point A. Without adjusting the compass span, place the compass point on

point B. Construct a circle (or wide arc) with center at point B. Mark and label the intersection points of the two circles as points

C and D. Using a straightedge, draw a line through points C and D. Line CD intersects line AB at the midpoint, M.

C

D

M

Segment Bisector with Patty Paper

Draw a line segment on patty paper using a straightedge.

Fold the paper over so that one endpoint lies on top of the other and pinch the line segment.

Open the patty paper and place a point at the pinch.

Questions for Students: What is true about the distances from the point at

the pinch to each segment endpoint? [Equal] What is the point at the pinch called? [Midpoint]

Perpendicular Bisector with Patty Paper

Use the segment bisector patty paper and fold the paper to align the segment endpoints.

Crease the paper forming a fold line and then open the paper.

Use a corner of another patty paper to determine if the angles formed by the crease and the line segment are right angles.

Perpendicular Bisector Student Investigation

Draw a point on the folded perpendicular bisector. Place a second patty paper on top and measure the

distance between the point on the perpendicular bisector and one endpoint of the segment.

Compare this distance to the distance between the point on the perpendicular bisector and the other segment endpoint.

Question for Students: What is true about the distances from a point on the

perpendicular bisector to each endpoint of a segment? [Equal]

Perpendicular to a Line Through a Point on a Line with Compass and Straightedge

Begin by drawing a line. Draw and label point C. Construct a circle (wide arc) with center

at point C that intersects the line. Where the circle intersects the line, label

the intersection points as A and B. Place the compass point on point A and

stretch the compass to a distance greater than AC.

Draw an arc above the line. Without adjusting the compass span,

place the compass point on B and draw an arc above the line, generating intersecting arcs.

Label the point D. Construct the perpendicular bisector of

segment AB.

A BC

D

Perpendicular to a Line Through an External Point with Compass and Straightedge Begin with a line and a point C not on

the line. Construct a circle (or wide arc) with

center at point C and radius greater than the distance from C to the line.

Where the circle intersects the line, label the intersection points as A and B.

Place the compass point on A and draw an arc below the line.

Without adjusting the compass span, place the compass point on B and draw an arc, generating intersecting arcs.

Label the intersecting point D. Construct a perpendicular bisector of

segment AB.

C

A B

D

Angle Bisection with Compass and Straightedge

Begin with an angle. Draw a circle (or wide arc) at point O with an arbitrary radius,

making certain the circle intersects both angle sides. Label the points on the angle sides as A and B. Draw a circle (or wide arc) at point A such that its radius is

more than half the distance between A and B. Without adjusting the compass span, place the compass point

on B and draw a circle (or wide arc). Mark and label the intersection point of the two arcs as point C. Using a straightedge, draw a line through points O and C.

O

A

B

C

Angle Bisection with Patty Paper

Draw an angle on a piece of patty paper using a straightedge.

Fold one side of the angle on top of the other.

Unfold and draw a point on the angle bisector (fold line).

Angle Bisector Student Investigation

Place a second piece of patty paper on top of the folded angle bisector with a corner of the top piece sitting on one angle side and the adjacent perpendicular side passing through the marked point on the angle bisector.

Mark the distance between the point on the angle bisector and the side of the angle on the patty paper.

Repeat the process with the other angle side.

Question for Students: What is true about the distances from a point on an angle

bisector to each side of the angle? [Equal]

Parallel Line through a Given Point with Compass and Straightedge

Begin with line AB and a point C not on the line.

Connect points A and C. Construct a circle with center at point

A and passing through point C. Where the circle intersects line AB,

label the intersection point as F. Construct a circle with center C and

radius AC. Construct a circle with center F and

radius AF. Label the circle intersection point I. Using a straightedge, connect points

C and I with a line.

Parallel Line through a Given Point with Patty Paper

Draw a line on patty paper using a straightedge. Draw a point on the patty paper that is not on the

given line. Fold the given line onto itself (forming

perpendicular lines) so that the fold passes through the point not on the line.

Unfold the paper and fold the new line onto itself so that it passes through the given point to form another line perpendicular to the folded line.

Unfold and view the parallel lines.

Translations with Patty Paper

Draw a figure on patty paper and place a point in its interior. Draw a ray from the interior point to the edge of the patty paper

(translation direction). Draw a second point on the ray. Place a second piece of patty paper on top of the first piece and

trace the figure, the interior point, and the ray. Slide the top patty paper along the ray’s path until the point on

the top patty paper coincides with the second point on the bottom patty paper (translation distance).

Measure the distance between corresponding points of the pre-image and image figures using another piece of patty paper.

Question for Students: What can be said about the distance between a pre-image

(original) point and its corresponding image point? [Always equals the translation distance]

Reflections with Patty Paper

Draw a figure on patty paper. Draw a line on the patty paper that does not intersect the drawn figure (line of

reflection). Fold the patty paper along the reflection line. Trace the figure onto the folded portion of your patty paper and unfold.

Image is on the reverse side of the patty paper. Draw a segment connecting a point in the pre-image figure and the

corresponding point in the reflected image. Repeat for a second set of corresponding points. Measure the distance between corresponding points of the pre-image and

image figures using another piece of patty paper.

Tasks/Questions for Students: Investigate the distance between any pre-image (original) point and its

corresponding image point. [Not always equal] What type of angle is formed by each segment and the line of reflection? [Right] How does the line of reflection divide each segment connecting a point and its

image? [Bisects it]

Rotations with Patty Paper

Draw a figure on patty paper. Draw a point on the figure or in its interior. Draw a second point on the patty paper to create a center of rotation. Draw a ray from the second point (center of rotation) that passes through the

first point. Draw a second ray from the center or rotation to create an angle of rotation. Place a second piece of patty paper on top of the first piece and trace the figure,

the points, and the first ray. Place a pencil tip on the center of rotation and turn the top patty paper through

the angle of rotation.

Investigations/Questions for Students: Measure the distance between corresponding points of the pre-image and

image figures using another piece of patty paper. Is the distance between any pre-image (original) point and its corresponding

image point always the same? Use another piece of patty paper to trace the angle with the point of rotation as

its vertex, one ray passing through a point in the original figure, and the other ray passing through the corresponding point in the rotated image.

Compare the resulting angle to the original angle of rotation.

Rotation of a Triangle Through Space

Rotating ABC about line d2 (axis of symmetry) produces a cone whose base diameter is equal to the length of side AC.

Beginning with the cone and slicing it vertically provides a cross-section view, which is shown as ABC.

Rotation of a Circle Through Space

Rotating O about point O produces a sphere whose radius is equal to the radius of O.

Beginning with the sphere and slicing it through its center provides a cross-section view (green), which is shown as O.

Translating a Circle Through Space

Translating A by translation distance BC produces a (slinky-like) cylinder.

Slicing the cylinder perpendicular to the axis of symmetry AD generates many circles congruent to A.

* Cabri 3D rendering created by Stephen F. West, State University College, Geneseo, NY

A

B

C

D

Translating a Triangle Through Space

Translating ABC by translation vector CD produces a triangular prism. Recall that a prism is a geometric

solid whose bases (green) are congruent, parallel polygons and whose lateral faces (white) are parallelograms.

Slicing the triangular prism generates many triangles (gray) congruent to ABC.

* Cabri 3D renderings created by Stephen F. West, State University College, Geneseo, NY

D

C

A

B

Translating a Square Through Space Consider the square CEDB and the

translation segment HK. Translating CEDB in the direction

and length of HK produces the green rectangular prism.

Beginning with the rectangular prism and slicing it perpendicular to segment AD provides a cross-section view, which is a square congruent to CEDB.

Task for Students: Investigate and describe how a

rectangular cross-section view could be generated from this rectangular prism.

D

E

H

K

E

Translating an OctagonThrough Space

Consider the blue octagon and the translation segment HK.

Translating the blue octagon in the direction and length of HK produces an octagonal prism.

Task for Students: Describe how the octagonal

prism must be sliced to produce a cross-sectional view of many congruent octagons.

* Cabri 3D rendering created by Stephen F. West, State University College, Geneseo, NY

E

HH

K

Transforming a Hexagon in Space

Task for Students: Describe how hexagon

A’ can be transformed into the hexagonal prism at the right.

Cross Sections of a Pyramid

Beginning with a pyramid and slicing it horizontally provides a cross-section view of concentric squares.

* Graphics animation available from Demos with Positive Impact - Volumes by Section Demo Gallery

Applications of Transformations and Cross Sections

Hurricane creation - http://observe.arc.nasa.gov/nasa/earth/hurricane/creation.html

Interior of the earth - http://pubs.usgs.gov/gip/interior/

Earthquakes – Mt. St. Helens http://www.geophys.washington.edu/SEIS/PNSN/HELENS/helenscs_yr.html

GPS Addressed M7G1

Perform basic constructions using both compass and straightedge. Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a lines segment; and constructing a line parallel to a given line through a point not on the line.

Recognize that many constructions are based on the creation of congruent triangles.

M7G2 Demonstrate understanding of translations, rotations, and

reflections and relate symmetry to appropriate transformations. M7G4

Describe three-dimensional figures formed by translations and rotations of plane figures through space.

Sketch, model, and describe cross sections of cones, cylinders, pyramids, and prisms.

GPS Addressed

M7P3 Communicate their mathematical thinking coherently and clearly

to peers, teachers, and others. Use the language of mathematics to express mathematical ideas

precisely.

M7P4 Recognize and use connections among mathematical ideas. Understand how mathematical ideas interconnect and build on

one another to produce a coherent whole. Recognize and apply mathematics in contexts outside of

mathematics.

Websites for Additional Exploration

Math Open Reference – Constructions http://www.mathopenref.com/tocs/constructionstoc.html

National Library of Virtual Manipulatives – Geometry (Translations, Rotations, Reflections) http://nlvm.usu.edu/en/nav/topic_t_3.html

Demos with Positive Impact – Cross Sections http://mathdemos.gcsu.edu/mathdemos/sectionmethod/pyramidcross.gif

Estimating the Circumference of the Earth - http://www.k12science.org/~jkoen/rwlo/Eratosthenes/Content%20Material/PartA.shtml