Lesson 4 | Three-Dimensional Measurement and Application | 10th Grade Mathematics

Objective

Describe the terms point, line, and plane. Define and classify polyhedrons, specifically prisms and pyramids.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Congruence

G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Foundational Standards

The foundational standards covered in this lesson

4.G.A.1

Geometry

4.G.A.1 — Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
5.G.A.1

Geometry

5.G.A.1 — Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
7.G.A.3

Geometry

7.G.A.3 — Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Recall “undefined terms” and identify features, examples, and nonexamples.
1. A point is a position in space. A point has no length or width (no dimensions).
2. A line is an infinite number of points that are connected together without any curves and that continues on in both directions infinitely. A line has no width and has an infinite length (one dimension).
3. A plane is a two-dimensional flat surface. A plane has infinite length, infinite width, and zero thickness (two dimensions).
Define planes in terms of points and lines.
Define polyhedrons in terms of points (vertices), planes (faces), and lines (edges).
Describe prisms as polyhedrons containing two parallel, congruent polygonal bases and parallelogram faces.
Describe pyramids as polyhedrons containing one polygonal base and triangular faces that meet at one point called the apex.
Classify prisms and pyramids.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson covers key concepts in the unit by building students’ vocabulary of three-dimensional figures.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Below is plane a with lines $$\overleftrightarrow {FB}$$ and $$\overleftrightarrow{EC}$$ perpendicular to plane $$a$$ at points $$B$$ and $$C$$, respectively. What is the relationship between $$\overleftrightarrow {FB}$$ and $$\overleftrightarrow{EC}$$? Justify your answer.

Guiding Questions

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Problem 2

Below is a right rectangular prism.

Define everything you see in terms of the following vocabulary:

Plane
Line
Point
Parallel
Perpendicular

Guiding Questions

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Problem 3

Below is a right triangular pyramid.

Define everything you see in terms of the following vocabulary.

Plane
Line
Point
Parallel
Perpendicular

Guiding Questions

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Using the diagram below:

What can be concluded about the relationship between line $$l$$ and plane $$P$$? Why?
What can be concluded about the relationship between planes $$P$$ and $$Q$$? Why?
What can be concluded about the relationship between lines $$l$$ and $$m$$? Why?
What can be concluded about $$\overline{AB}$$ and $$\overline{CD}$$?
Line $$j$$ lies in plane $$P$$, and line $$i$$ lies in plane $$Q$$. What can be concluded about the relationship between lines $$i$$ and $$j$$?

References

EngageNY Mathematics Geometry > Module 3 > Topic B > Lesson 5 — Exit Ticket

Geometry > Module 3 > Topic B > Lesson 5 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Illustrative Mathematics Defining Perpendicular Lines
EngageNY Mathematics Geometry > Module 3 > Topic B > Lesson 5 — Problem Set (especially Question #1)
Euler's Polyhedral Formula
Illustrative Mathematics Defining Parallel Lines
Illustrative Mathematics What's the Point? — (You will need to add rigor to these questions)

Lesson 3

Lesson 5

Lesson Map

Topic A: Area and Circumference of Circles

Describe and use the formulas for area and circumference of circles to solve problems.

A.SSE.A.1 G.GMD.A.1 N.Q.A.3

Calculate and justify composite area and circumference of circles.

N.Q.A.1 N.Q.A.2 N.Q.A.3

Solve multistep area and circumference of circles problems involving cost and other rates.

N.Q.A.1 N.Q.A.2 N.Q.A.3

Topic B: Three-Dimensional Concepts and General Volume

Describe the terms point, line, and plane. Define and classify polyhedrons, specifically prisms and pyramids.

G.CO.A.1

Define a general cylinder and general cone. Identify two-dimensional shapes that when revolved will form a cylinder.

G.CO.A.1 G.GMD.B.4

Use volume concepts and formulas to analyze and solve multistep problems with cylinders and prisms.

G.GMD.A.1 G.GMD.A.3

Define and calculate the volume of pyramids and cones. Describe the relationship between general cylinders and general cones with the same base area.

G.GMD.A.1 G.GMD.A.3

Use the Pythagorean Theorem to find missing measurements and calculate volume of pyramids, prisms, and compound shapes comprised of pyramids and prisms.

G.GMD.A.3 G.GMD.B.4 G.SRT.C.8

Topic C: Cavalieri's Principle, Spheres, and Composite Volume

Describe the cross-sections of prisms and cylinders and make conjectures about volume from the cross-sections.

G.GMD.A.3 G.GMD.B.4

Describe Cavalieri’s principle relating equal area cross-sections and volume, and how this relates to the formulas for volume. Derive the volume of a sphere using Cavalieri’s principle.

G.GMD.A.1 G.GMD.A.2

Identify cross-sections of pyramids and use the relationships between the cross-sections to determine the volume of truncated cones and pyramids.

G.GMD.A.2 G.GMD.A.3

Calculate the volume of a sphere and use this in the solution of problems.

G.GMD.A.1 G.GMD.A.2 G.GMD.A.3 N.Q.A.3

Calculate the volume of compound objects and those with subtracted solids. Determine how the volume will be affected by scaling one or more dimensions.

G.GMD.A.3 G.GMD.B.4 N.Q.A.3

Topic D: Surface Area, Scaling, and Modeling with Geometry

Use lateral surface area formulas to solve problems.

N.Q.A.2

Use the surface area and volume to solve application problems.

G.GMD.A.3 G.GMD.B.4 G.MG.A.1 G.MG.A.3

Solve multistep volume and surface area problems with rates and unit conversions.

G.GMD.A.3 N.Q.A.2 N.Q.A.3

Apply density concepts to surface area and volume problems.

G.GMD.A.3 G.MG.A.2 N.Q.A.2 N.Q.A.3

Apply constraints on volume, surface area, or cost to solve design problems with three-dimensional figures.

G.GMD.A.3 G.MG.A.2 G.MG.A.3 N.Q.A.2 N.Q.A.3

Three-Dimensional Measurement and Application

Objective

Common Core Standards

Core Standards

Congruence

Foundational Standards

Geometry

Geometry

Geometry

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Target Task

References

Additional Practice

Lesson Map

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