Geometry

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

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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 2020-2021 school year due to school closures, 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

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

Read Progressions for the Common Core State Standards in Mathematics, Geometry, K-6 for standards relevant to this unit.

Essential Understandings

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

Graph paper
Three-dimensional solids and foldable nets
Calculators
Template: Nets
Geoboards
index cards

Vocabulary

face

right triangle

parallelogram

polygon

composition

decomposition

trapezoid

base

acute triangle

obtuse triangle

prism

unit cube

edge

vertex

pyramid

net

volume

area

height

polyhedron

surface 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).

9

6.G.A.1

6.G.A.3

Solve real-world problems involving distance, area, and perimeter of polygons on and off the coordinate plane.

Topic C: Volume of Rectangular Prisms

10

6.G.A.2

Find volume of rectangular prisms with whole number and fractional edge lengths using unit and fractional unit cubes.

11

6.G.A.2

Determine the formulas for finding volume of rectangular prisms and use the formulas to solve for volume.

12

6.G.A.2

Apply volume concepts to solve real-world and mathematical problems, including finding missing measurements.

13

6.G.A.2

Apply volume concepts to solve real-world and mathematical problems, including finding volume of figures with composite 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

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.

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.

Geometry

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.

Geometry

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.

Geometry

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

Expressions and Equations

6.EE.A.1

Expressions and Equations

6.EE.A.1 — Write and evaluate numerical expressions involving whole-number exponents.
6.EE.B.7

Expressions and Equations

6.EE.B.7 — Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Measurement and Data

3.MD.C.7

Measurement and Data

3.MD.C.7 — Relate area to the operations of multiplication and addition.
3.MD.C.7.D

Measurement and Data

3.MD.C.7.D — Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
4.MD.A.3

Measurement and Data

4.MD.A.3 — Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
5.MD.C.5

Measurement and Data

5.MD.C.5 — Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
5.MD.C.5.B

Measurement and Data

5.MD.C.5.B — Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.
5.MD.C.5.C

Measurement and Data

5.MD.C.5.C — Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Number and Operations—Fractions

5.NF.B.4

Number and Operations—Fractions

5.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.B.6

Number and Operations—Fractions

5.NF.B.6 — Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

The Number System

6.NS.A.1

The Number System

6.NS.A.1 — Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
6.NS.C.6

The Number System

6.NS.C.6 — Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

Future Standards

Geometry

7.G.A.1

Geometry

7.G.A.1 — Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
7.G.A.2

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

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.5

Geometry

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

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.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.

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.

Equations and Inequalities

Statistics

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Geometry

Unit Summary

Assessment

Unit Prep

Intellectual Prep

Internalization of Standards via the Post-Unit Assessment

Internalization of Trajectory of Unit

Unit-Specific Intellectual Prep

Essential Understandings

Materials

Vocabulary

Lesson Map

Topic A: Area of Triangles, Quadrilaterals, and Polygons

1

2

3

4

5

6

Topic B: Polygons in the Coordinate Plane

7

8

9

Topic C: Volume of Rectangular Prisms

10

11

12

13

Topic D: Nets and Surface Area

14

15

16

17

Common Core Standards

Core Standards

Geometry

Geometry

Geometry

Geometry

Geometry

Foundational Standards

Expressions and Equations

Expressions and Equations

Expressions and Equations

Measurement and Data

Measurement and Data

Measurement and Data

Measurement and Data

Measurement and Data

Measurement and Data

Measurement and Data

Number and Operations—Fractions

Number and Operations—Fractions

Number and Operations—Fractions

The Number System

The Number System

The Number System

Future Standards

Geometry

Geometry

Geometry

Geometry

Geometry

Geometry

Geometry

Geometry

Standards for Mathematical Practice

Unit 6

Equations and Inequalities

Unit 8

Statistics