# Geometry

Students apply algebraic and proportional reasoning skills to investigate angle relationships, circle measurements, uniqueness of triangles, and solid figure application problems.

## Unit Summary

In Unit 6, seventh-grade students cover a range of topics from angle relationships to circles and polygons to solid figures. The seventh-grade Geometry standards are categorized as additional standards, however, there are several opportunities throughout the unit where students are engaged in the major work of the grade. In the beginning of the unit, students use and solve equations to represent relationships between angles and find missing angle measures. Investigating circles, students discover the proportional relationship between the circumference of a circle and its diameter, and understand π as the ratio of these two quantities. Students will also use their expressions skills to write numerical expressions that can be used to find surface area and volume of three-dimensional figures.

Throughout the unit, students encounter several vocabulary words, such as complementary angles, vertical angles, radius, and circumference. Many of these words enable students to be more precise in their communications with each other (MP.6). Students will also encounter complex diagrams of angles and 3-D figures where they will need to understand what information they can glean from the diagram and plan a solution pathway before jumping in (MP.1). Students should have access to several tools they may opt to use throughout the unit, including rulers, protractors, compasses, and reference sheets (MP.5).

The foundational skills for the standards in this unit stem from fourth through sixth grades. In fourth grade, students studied the concepts of angle measurement and understood angle measure to be additive. In fifth grade, students developed an understanding of three-dimensional volume, which they further built on in sixth grade. Sixth-grade students also began to distinguish between the three-dimensional space an object takes up and the surface area that covers it.

In eighth grade, students will zoom in on right triangles and apply the Pythagorean theorem to determine side lengths in right triangles. They will also continue solving real-life applications of surface area and volume, with the addition of cones, spheres, and cylinders.

Pacing: 23 instructional days (21 lessons, 1 flex day, 1 assessment day)

For guidance on adjusting the pacing for the 2021-2022 school year, see our 7th Grade Scope and Sequence Recommended Adjustments. • Unit Launch
• Expanded Assessment Package
• Problem Sets for Each Lesson
• Student Handout Editor
• Vocabulary Package

## Assessment

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

## Unit Prep

### Intellectual Prep

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#### Internalization of Standards via the Post-Unit Assessment

• Take the Post-Unit Assessment. Annotate for:
• Standards that each question aligns to
• 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.
• Essential Understandings
• Connection to Post-Unit Assessment questions
• Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

### Essential Understandings

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• When two lines intersect, a pair of congruent vertical angles are created. This angle relationship, along with complementary and supplementary angle relationships, can be used to determine missing angle measures in diagrams.
• A circle is a closed shape that is defined by the set of points that are the same distance from the center of the circle. The distance from the center to any point on the circle is called the radius, and the distance across the circle through the center is called the diameter. The measurement around a circle is called the circumference and is proportional to the diameter of the circle with a constant of proportionality equivalent to ${\pi}$. The area of a circle can be found using the formula $A={\pi} r^2$.
• In any triangle, the sum of any two side lengths must be longer than the measure of the third side. Given different conditions about the side and angle measures of a triangle, one unique triangle may be formed, more than one triangle may be formed, or no triangle may be formed.

### Vocabulary

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supplementary angles

surface area

vertical angles

diameter

circumference

complementary angles

triangle inequality theorem

cross-section

volume

To see all the vocabulary for this course, view our 7th Grade Vocabulary Glossary.

### Materials

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• 180° Protractor (1 per student)
• String (1 per student)
• Compass (1 per student)
• Graph Paper (2-3 sheets per student)
• Blank Paper (2-3 sheets per student)
• Ruler (1 per student)
• Circles handout (1 per student)
• AngLegs or prepared wooden skewers (1 per student or small group) — See the blog post Triangle Inequality Skewers by Elissa Miller for guidance on how to prepare wooden skewers.

# 1

7.G.B.5

Identify and determine values of angles in complementary and supplementary relationships.

# 2

7.G.B.5

Use vertical, complementary, and supplementary angle relationships to find missing angles.

# 3

7.G.B.5

Use equations to solve for unknown angles. (Part 1)

# 4

7.G.B.5

Use equations to solve for unknown angles. (Part 2)

# 5

7.G.B.4

Define circle and identify the measurements radius, diameter, and circumference.

# 6

7.G.B.4

Determine the relationship between the circumference and diameter of a circle and use it to solve problems.

# 7

7.G.B.4

Solve real-world and mathematical problems using the relationship between the circumference of a circle and its diameter.

# 8

7.G.B.4

Determine the relationship between the area and radius of a circle and use it to solve problems.

# 9

7.G.B.4

Solve real-world and mathematical problems using the relationship between the area of a circle and its radius.

# 10

7.G.B.4

7.G.B.6

Solve problems involving area and circumference of two-dimensional figures (Part 1).

# 11

7.G.B.4

7.G.B.6

Solve problems involving area and circumference of two-dimensional figures (Part 2).

# 12

7.G.A.2

7.G.B.5

Draw two-dimensional geometric shapes using rulers, protractors, and compasses.

# 13

7.G.A.2

Determine if three side lengths will create a unique triangle or no triangle.

# 14

7.G.A.2

Identify unique and identical triangles.

# 15

7.G.A.2

Determine if conditions describe a unique triangle, no triangle, or more than one triangle.

# 16

7.G.A.3

Identify and describe two-dimensional figures that result from slicing three-dimensional figures.

# 17

7.G.B.6

Find the surface area of right prisms.

# 18

7.G.B.6

Find the surface area of right pyramids.

# 19

7.G.B.6

Find the volume of right prisms and pyramids.

# 20

7.G.B.6

Solve real-world and mathematical problems involving volume.

# 21

7.G.B.6

Distinguish between and solve real-world problems involving volume and surface area.

## Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

### Core Standards

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##### Geometry
• 7.G.A.2 — Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

• 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 — 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.5 — Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

• 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.

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• 5.G.B.3

• 6.G.A.1

• 6.G.A.2

• 6.G.A.4

• 7.G.A.1

• 4.MD.A.3

• 4.MD.C.5

• 4.MD.C.6

• 4.MD.C.7

• 5.MD.C.3

• 5.MD.C.5

• 7.RP.A.2

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• G.CO.B.7

• G.CO.B.8

• 8.G.A.5

• 8.G.B.6

• 8.G.B.7

• 8.G.C.9

### 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.