Sample Task Alignment



Sample Task Alignment

Unit: Geometric Relationships

Content Strands

G.G.1 Know and apply that if a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.

G.G.1a

Jim is a carpenter and would like to install a flagpole in his front yard. Carpenters use a tool called a level, shown in the figure below, to determine if objects are level (horizontal) or plumb (vertical). Describe how Jim can use a level to ensure that the flagpole appears vertical from any direction.

[pic]

Jim, being a carpenter who knows how to use a level, will properly use said level against alleged vertical flag pole from multiple “sides” of the pole (north and east).

G.G.1b

Study the drawing below of a pyramid whose base is quadrilateral [pic]. John claims that line segment[pic] is the altitude of the pyramid. Explain what John must do to prove that he is correct.

[pic]

John must show that [pic]or [pic] .

G.G.2 Know and apply that through a given point there passes one and only one plane perpendicular to a given line

G.G.2a

Examine the diagram of a pencil below:

[pic] [pic]

Explain how the pencil illustrates the fact that if two lines are perpendicular to the same line, then they must be parallel.

Two consecutive lines going down the length of the pencil are perpendicular to the line formed by the bottom of the pencil. The two lines are parallel.

[pic]

Explain how the pencil illustrates the fact that if two lines are parallel and the first line is perpendicular to a third line, then the second line must be perpendicular to the third line.

Same lines as above. The consecutives are parallel to each other. One is perpendicular to bottom. Viola second is also perpendicular.

G.G.3 Know and apply that through a given point there passes one and only one line perpendicular to a given plane

G.G.3a

Examine the diagram of a right triangular prism.

[pic]

Describe how a plane and the prism could intersect so that the intersection is:

a line parallel to one of the triangular bases (The plane touches only an edge of a triangle)

a line perpendicular to the triangular bases (A vertical plane touching one of the vertical edges)

a triangle (Plane parallel to triangle base, cutting through prism)

a rectangle (Plane parallel to a rectangular side, cutting through the prism)

a trapezoid (A plane intersecting both triangle bases)

G.G.4 Know and apply that two lines perpendicular to the same plane are coplanar

G.G.4a

The figure below is a right hexagonal prism.

[pic]

On a copy of the figure sketch a symmetry plane. Then write a description of the symmetry plane that uses the word parallel.

On a copy of the figure sketch another symmetry plane. Then write a description that uses the word perpendicular.

[pic]

G.G.4b The figure below in three dimensional space, where[pic]is perpendicular to [pic] and [pic] is perpendicular to [pic], illustrates that two lines perpendicular to the same line are not necessarily parallel. Must two lines perpendicular to the same plane be parallel? Discuss this problem with a partner.

G.G.5 Know and apply that two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane

G.G.5a

Julie wishes to construct a wall in the basement of her apartment.  Describe what she must do to ensure that the wall is perpendicular to the floor of the basement. Explain why Julie’s procedure will work, using appropriate mathematical terminology.

Julie must measure 3 feet up the wall and 4 feet on the floor out from the wall. The endpoints must be 5 feet apart. By the Pythagorean theorem, the wall and floor should form a right angle and therefore be perpendicular.

G.G.6 Know and apply that if a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane

G.G.6a

Justify the fact that if one edge of a triangular prism is perpendicular to its base then the prism is a right triangular prism.

Since the lateral edge is perpendicular to the base, it is an altitude, so a right angle is formed, therefore the prism is a right prism.

G.G.7 Know and apply that if a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane

G.G.7a

Examine the four figures below:

[pic] [pic] [pic] [pic]

Each figure has how many symmetry planes?

Describe the location of all the symmetry planes for each figure.

| |Rectangular prism |Pentagonal prism |Square puramid |Cone |

|# |3 |6 |4 |[pic] |

|Descriptions |Vertical left/right, |5 Vertical- one from each |2 intersecting midpoints |Vertical planes that |

| |Vertical front/back, |vertex to midpoint of |of base edges, |intersect the center of |

| |horizontal |opposite side , |2 intersecting opposite |the base |

| | |horizontal |vertices | |

G.G.8 Know and apply that if a plane intersects two parallel planes, then the intersection is two parallel lines

G.G.8a

The figure below is a right hexagonal prism.

[pic]

On a copy of the figure sketch a symmetry plane. Then write a description that uses the word parallel.

On a copy of the figure sketch another symmetry plane. Then write a description that uses the word perpendicular.

See G.G.4a

G.G.9 Know and apply that if two planes are perpendicular to the same line, they are parallel

G.G.9a

The figure below shows a right hexagonal prism.

[pic]

A plane that intersects a three-dimensional figure such that one half is the reflected image of the other half is called a symmetry plane. On a copy of the figure sketch a symmetry plane. Then write a description of the symmetry plane that uses the word parallel.

On a copy of the figure sketch another symmetry plane. Then write a description that uses the word perpendicular.

See G.G.4a

G.G.9b

Draw a line on a piece of cardboard. Use additional pieces of cardboard to construct two planes that are perpendicular to the line that you drew. Make a conjecture regarding those two planes and justify your conjecture.

The planes are parallel. If two planes are perpendicular to the same line, then the planes are parallel.

G.PS.1a

Obtain several different size cylinders made of metal or cardboard. Using stiff paper, construct a cone with the same base and height as each cylinder. Fill the cone with rice, then pour the rice into the cylinder. Repeat until the cylinder is filled. Record your data.

What is the relationship between the volume of the cylinder and the volume of the corresponding cone?

Collect the class data for this experiment.

Use the data to write a formula for the volume of a cone with radius r and height h.

3 cones worth of rice fit evenly into one cylinder with the same base and height.

[pic]

Does not match any of the content strands but goes with the topic.

G.PS.2c, G.PS.3a, G.PS.7b, G.RP.3a, G.RP.8a, G.CM.7a

Consider the following conjecture: The intersection of two distinct planes can be a point. Find a “real world” example that supports the conjecture or provides a counterexample to the conjecture. Share your example with a partner and use your knowledge of geometry in three dimensional space to justify it.

Not true. A wall and a ceiling represent planes. Their intersection is a line.

G.R.7 Use mathematics to show and understand social phenomena (e.g., determine if conclusions from another person’s argument have a logical foundation)

G.R.7a

Melvin claims that two lines in space that do not intersect must always be parallel. To support his conjecture he refers to the line of intersection of a wall and the ceiling and the line of intersection of the same wall and the floor. Discuss the validity of his conjecture and his justification.

He is wrong. The road under an overpass is not parallel to the overpass and does not intersect it. They are skew.

G.G.35 Determine if two lines cut by a transversal are parallel, based on the measure of given pairs of angles formed by the transversal and the lines

G.G.35a

Investigate the two drawings using dynamic geometry software. Write as many conjectures as you can for each drawing.

[pic]

[pic]

G.G.35b

In the following figure, certain angle measures are given. Use the information provided to show that [pic] is parallel to [pic] and that [pic]is not parallel to [pic].

[pic]

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Plane 1 – parallel to edges of prism, perpendicular to vertical faces it intersects and bases

Visualize the same symmetry plane is drawn in white ink.

Yes they must be parallel because two lines perpendicular to the same plane are coplanar and two lines perpendicular to the same line in the same plane are perpendicular.

• When the same side interior angles are not supplementary, the lines are not parallel.

• Vertical angles are congruent.

• Adjacent angles that form a line are supplementary.

• When the same side interior angles are supplementary, the lines are parallel.

• Vertical angles are congruent.

• Adjacent angles that form a line are supplementary.

[pic]are supplementary, therefore, [pic].

[pic] are not supplementary, therefore, [pic]is not // to [pic]

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