Geometry

Unit Summary

In Unit 6, Three-Dimensional Measurement & Applications, students derive, describe, and use formulas for area and circumference of a circle, and volume and surface area of three-dimensional figures. In addition, students identify two-dimensional shapes that when spun around an axis will form a particular three-dimensional figure, identify cross-sections of three-dimensional figures, and analyze modeling situations.

In this unit, students build on their previous understanding of circles, volume, and surface area they developed throughout elementary and middle school to extend their reasoning into modeling situations, formula analysis, and deeper conceptual understandings. The primary foundational content students will need to have prior to beginning this unit are knowing the formulas for area and circumference of a circle from seventh grade; knowing the formulas for volume of a cone, cylinder, and sphere from eighth grade; and using Pythagorean Theorem from eighth grade and Unit 4.

The Unit begins with Topic A, Area and Circumference of Circles, where students refresh their understanding of area and circumference to solve problems. If students are proficient at these skills, the first three lessons may be skipped or combined. In Topic B, Three-Dimensional Concepts and General Volume, students build on their understanding of two-dimensional figures to develop an understanding of three-dimensional measurement and dimension through revolving two-dimensional figures around an axis, slices, and volume. Topic C, Cavalieri’s Principle, Spheres, and Composite Volume, explores Cavalieri’s principle with the purpose of comparing volumes of oblique and right figures and developing the underpinnings for the formula for the volume of a sphere, used in eighth grade Geometry. Topic C also challenges students to find the volume of nonstandard three-dimensional figures by adding or subtracting volumes of known figures. The unit concludes with Topic D, Surface Area, Scaling, and Modeling with Geometry, which focuses on modeling situations requiring the use of volume, surface area, density, rates, and unit conversions. Students are required to make general plans for the solution of problems, determine the measurements and formulas necessary to carry out these plans, evaluate the plans, and determine a final solution. The use of appropriate measurement is necessary for the estimates in this topic of the unit.

The material from this unit is foundational to applications in Algebra II and solids of revolutions through integration in Calculus.

Pacing: 20 instructional days (18 lessons, 1 flex day, 1 assessment day)

Assessment

This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

Internalization of Standards via the Unit Assessment

Take unit assessment. Annotate for:
- Standards that each question aligns to
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) that assessment points to

Internalization of Trajectory of Unit

Read and annotate the Unit Summary.
Notice the progression of concepts through the unit using the Lesson Map.
Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

Describe and use the formulas for area and circumference of a circle.
Define a three-dimensional figure in terms of its vertices, edges, and faces to find the volume, lateral surface area, and surface area.
Describe the relationship between two-dimensional slices, cross sections, and shapes to be revolved around an axis to form three-dimensional figures.
Use and develop formulas for finding volume of right and oblique shapes as well as comparing volumes.
Identify appropriate measurement precision and levels of calculation necessary to solve a modeling problem.
Develop and execute plans for multistep modeling problems with geometric concepts, including volume, surface area, density, rate, and unit conversions.

Vocabulary

Terms and notation that students learn or use in the unit

area of a circle	circumference of a circle
radius	diameter
level of precision	point (vertex)
line (edge)	plane (face)
polyhedron	right and oblique prism
right and oblique pyramid	apex
right and oblique cylinder	right and oblique cone
revolution around an axis	volume formulas
Pythagorean Theorem	cross-section
slice	sphere
Cavalieri's principle	truncated cone
truncated pyramid	lateral surface area
surface area	displacement
dimenional analysis	density

Materials

The materials, representations, and tools teachers and students will need for this unit

Massachusetts Comprehensive Assessment System Grade 10 Mathematics Reference Sheet

Lesson Map

Topic A: Area and Circumference of Circles

1

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

2

Calculate and justify composite area and circumference of circles.

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

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

4

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

G.CO.A.1

5

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

6

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

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

7

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

8

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

9

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

10

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

11

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

12

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

13

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

14

Use lateral surface area formulas to solve problems.

N.Q.A.2

15

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

16

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

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

17

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

18

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

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

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.

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.

Geometric Measurement and Dimension

G.GMD.A.1 — Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

Geometric Measurement and Dimension

G.GMD.A.1 — Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
G.GMD.A.2 — Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Geometric Measurement and Dimension

G.GMD.A.2 — Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
G.GMD.A.3 — Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Geometric Measurement and Dimension

G.GMD.A.3 — Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
G.GMD.B.4 — Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Geometric Measurement and Dimension

G.GMD.B.4 — Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

High School — Number and Quantity

N.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

High School — Number and Quantity

N.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.

High School — Number and Quantity

N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.
N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

High School — Number and Quantity

N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Modeling with Geometry

G.MG.A.1 — Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Modeling with Geometry

G.MG.A.1 — Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
G.MG.A.2 — Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Modeling with Geometry

G.MG.A.2 — Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
G.MG.A.3 — Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Modeling with Geometry

G.MG.A.3 — Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Seeing Structure in Expressions

A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Seeing Structure in Expressions

A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Similarity, Right Triangles, and Trigonometry

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Similarity, Right Triangles, and Trigonometry

G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

Standards covered in previous units or grades that are important background for the current unit

Congruence

G.CO.C.10

Congruence

G.CO.C.10 — Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Creating Equations

A.CED.A.2

Creating Equations

A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A.CED.A.4

Creating Equations

A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

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.
7.G.B.4

Geometry

7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.6

Geometry

7.G.B.6 — Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
8.G.B.7

Geometry

8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.C.9

Geometry

8.G.C.9 — Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Future Standards

Standards in future grades or units that connect to the content in this unit

Geometric Measurement and Dimension

G.GMD.A.2

Geometric Measurement and Dimension

G.GMD.A.2 — Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Modeling with Geometry

G.MG.A.1

Modeling with Geometry

G.MG.A.1 — Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
G.MG.A.2

Modeling with Geometry

G.MG.A.2 — Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
G.MG.A.3

Modeling with Geometry

G.MG.A.3 — Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.