Geometry

Students explore measurements of geometric figures in two-and three-dimensions, finding area, surface area, and volume in mathematical and real-world problems.

Unit Summary

In Unit 7, sixth graders explore measurements in geometric space in both two-dimensional and three-dimensional figures. Throughout previous grade levels, students have been composing and decomposing geometric figures. In sixth grade, students apply those concepts of composition and decomposition to new and familiar shapes to formulate properties and formulas for finding area (MP.7). By understanding the area of rectangular arrays and using regularity in repeated reasoning, students are able to determine the area of parallelograms, triangles, and other polygons that are formed from these shapes (MP.8). Students also re-engage in major work of the grade in a few ways. They use their knowledge of the coordinate plane and absolute value to represent and measure polygons in a four-quadrant plane, they write equations to represent volume of rectangular prisms with fractional side lengths, and they write and evaluate numerical expressions to represent the surface area of prisms and pyramids.

In fifth grade, students explored volume as a measurement of a three-dimensional solid with whole-number side lengths. In this unit, students will reinvestigate how to find volume when packing solids now with fractional unit cubes. They will rely on their skills of working with fractions from fifth grade and earlier in their sixth-grade year. 

Throughout the geometry standards in sixth grade through eighth grade, students will encounter increasingly complex and multi-part geometric measurement problems, culminating in eighth grade with standard 8.G.9. Learning how to make sense of these complex problems, determine solution pathways, and organize information will be important skills for students to have as the demands and rigor levels increase (MP.1).

Pacing: 19 instructional days (17 lessons, 1 flex day, 1 assessment day)

For guidance on adjusting the pacing for the 2021-2022 school year, see our 6th Grade Scope and Sequence Recommended Adjustments.

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Assessment

This assessment accompanies Unit 7 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 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.
  • Do all Target Tasks. Annotate the Target Tasks for: 
    • 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.

Unit-Specific Intellectual Prep

Essential Understandings

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  • The area of any triangle is half the area of a rectangle with the same base and height: $${A={1\over2}bh}$$.
  • The area of polygons can be found by decomposing the polygon into familiar shapes or by building around the polygon and subtracting the extra area.
  • The volume of a prism with fractional side lengths is equal to the number of fractional unit cubes it takes to fill the prism multiplied by the volume of one fractional unit cube. The volume of any rectangular prism can be found by multiplying the length, width, and height.
  • A three-dimensional figure can be represented by a two-dimensional net; nets are valuable tools in understanding and determining the surface area of three-dimensional figures.

Materials

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  • Optional: Graph Paper (2-3 sheets per student)
  • Optional: Index cards (1 per student)
  • Optional: Geoboards (1 per student)
  • Template: Nets (1 per student)
  • Three-dimensional solids (Teacher set) — We suggest purchasing a teacher set of folding nets for this lesson in order to model to students.

To see more information about the materials in this unit, view the Unit Materials Overview.

Vocabulary

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surface area

polyhedron

height

face

right triangle

parallelogram

polygon

composition

decomposition

trapezoid

base

acute triangle

obtuse triangle

prism

unit cube

edge

vertex

pyramid

net

volume

area

To see all the vocabulary for this course, view our 6th Grade Vocabulary Glossary.

Lesson Map

Topic A: Area of Triangles, Quadrilaterals, and Polygons

1

    6.G.A.1

Find the area of parallelograms.

2

    6.G.A.1

 Find the area of right triangles.

3

    6.G.A.1

Find the area of acute triangles using height and base.

4

    6.G.A.1

Find the area of any triangle using height and base.

5

    6.G.A.1

Find the area of polygons using composition and decomposition.

6

    6.G.A.1

Solve real-world and mathematical problems involving area of polygons.

Topic B: Polygons in the Coordinate Plane

7

    6.G.A.3

Draw polygons in the coordinate plane and find area and perimeter (Part 1).

8

    6.G.A.3

 Draw polygons in the coordinate plane and find area and perimeter (Part 2).

Topic C: Volume of Rectangular Prisms

Topic D: Nets and Surface Area

14

    6.G.A.4

Describe features of and identify nets that match prisms and pyramids.

15

    6.G.A.4

Create nets and use them to find surface area of three-dimensional figures.

16

    6.G.A.4

Find the surface area of three-dimensional figures with and without nets.

17

    6.G.A.2

    6.G.A.4

Find surface area and volume in real-world problems.

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Geometry
  • 6.G.A.1 — Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

  • 6.G.A.2 — Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

  • 6.G.A.3 — Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

  • 6.G.A.4 — Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Foundational Standards

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Expressions and Equations
  • 6.EE.A.1

  • 6.EE.B.7

Measurement and Data
  • 3.MD.C.7

  • 3.MD.C.7.D

  • 4.MD.A.3

  • 5.MD.C.5

  • 5.MD.C.5.B

  • 5.MD.C.5.C

Number and Operations—Fractions
  • 5.NF.B.4

  • 5.NF.B.6

The Number System
  • 6.NS.A.1

  • 6.NS.C.6

Future Standards

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Geometry
  • 7.G.A.1

  • 7.G.A.2

  • 7.G.A.3

  • 7.G.B.4

  • 7.G.B.5

  • 7.G.B.6

  • 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.