Unit 6: Geometry
Students apply algebraic and proportional reasoning skills to investigate angle relationships, circle measurements, uniqueness of triangles, and solid figure application problems.
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)
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The following assessments accompany Unit 6.
Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.
Pre-Unit Student Self-Assessment
Have students complete the Mid-Unit Assessment after lesson 11.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Post-Unit Assessment Answer Key
Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.
Suggestions for how to prepare to teach this unit
Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning.
The central mathematical concepts that students will come to understand in this unit
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 7th Grade Course Material Overview.
Terms and notation that students learn or use in the unit
triangle inequality theorem
To see all the vocabulary for Unit 6, view our 7th Grade Vocabulary Glossary.
Topic A: Angle Relationships
Identify and determine values of angles in complementary and supplementary relationships.
Use vertical, complementary, and supplementary angle relationships to find missing angles.
Use equations to solve for unknown angles. (Part 1)
Use equations to solve for unknown angles. (Part 2)
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Topic B: Circles
Define circle and identify the measurements radius, diameter, and circumference.
Determine the relationship between the circumference and diameter of a circle and use it to solve problems.
Solve real-world and mathematical problems using the relationship between the circumference of a circle and its diameter.
Determine the relationship between the area and radius of a circle and use it to solve problems.
Solve real-world and mathematical problems using the relationship between the area of a circle and its radius.
Solve problems involving area and circumference of two-dimensional figures (Part 1).
Solve problems involving area and circumference of two-dimensional figures (Part 2).
Topic C: Building Polygons and Triangles
Draw two-dimensional geometric shapes using rulers, protractors, and compasses.
Determine if three side lengths will create a unique triangle or no triangle.
Identify unique and identical triangles.
Determine if conditions describe a unique triangle, no triangle, or more than one triangle.
Topic D: Solid Figures
Identify and describe two-dimensional figures that result from slicing three-dimensional figures.
Find the surface area of right prisms.
Find the surface area of right pyramids.
Find the volume of right prisms and pyramids.
Solve real-world and mathematical problems involving volume.
Distinguish between and solve real-world problems involving volume and surface area.
The content standards covered in this unit
— 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.
— Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
— 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.
— 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.
— 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.
Standards covered in previous units or grades that are important background for the current unit
— Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
— 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.
— 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.
— 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.
— 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.
— 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.
— Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
— Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
— Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
— Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
— Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
— Recognize and represent proportional relationships between quantities.
Standards in future grades or units that connect to the content in this unit
— Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
— Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
— Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
— Explain a proof of the Pythagorean Theorem and its converse.
— Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
— Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
— Make sense of problems and persevere in solving them.
— Reason abstractly and quantitatively.
— Construct viable arguments and critique the reasoning of others.
— Model with mathematics.
— Use appropriate tools strategically.
— Attend to precision.
— Look for and make use of structure.
— Look for and express regularity in repeated reasoning.
Percent and Scaling