Unit 5 — Problem Solving with Geometry
Description
This unit applies algebraic and proportional reasoning to geometric contexts. Students begin by solving problems involving area and circumference of circles, deriving the relationship between these quantities. They use supplementary, complementary, vertical, and adjacent angle relationships to write and solve multi-step equations for unknown angles. Students then solve problems involving area of two-dimensional figures and volume and surface area of three-dimensional figures, including those composed of multiple shapes. Students use rulers, protractors, and technology to construct geometric shapes with given conditions, focusing on triangles and recognizing when conditions determine a unique triangle, more than one triangle, or no triangle. They analyze three-dimensional figures by describing two-dimensional cross-sections created by slicing at various angles. The unit includes work on transformations: students verify that rotations, reflections, and translations preserve length and angle measure, and use transformations to establish congruence and similarity. Students describe the effects of transformations on coordinates in the plane and use similarity to explain why slope is constant along a line, deriving equations y = mx and y = mx + b. Finally, students apply volume formulas to solve real-world problems involving cones, cylinders, and spheres.
Essential Questions
- How do algebraic equations help us solve geometric problems involving measurements and angle relationships?
- How do transformations help us understand congruence and similarity?
- How can we use coordinates and slope to represent linear relationships geometrically?
Learning Objectives
- Solve problems involving area and circumference of circles and derive their relationship.
- Write and solve multi-step equations involving supplementary, complementary, vertical, and adjacent angles.
- Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional figures.
- Construct geometric shapes with given conditions using freehand, mechanical, and technological tools.
- Recognize conditions determining unique triangles, more than one triangle, or no triangle.
- Describe all two-dimensional figures resulting from slicing three-dimensional figures at various angles.
- Verify properties of rotations, reflections, and translations experimentally.
- Describe congruence as a sequence of transformations.
- Describe the effects of dilations, translations, rotations, and reflections on coordinates.
- Describe similarity as a sequence of transformations including dilations.
- Use informal arguments to establish facts about angle sums, exterior angles, and angle-angle similarity of triangles.
- Derive the equations y = mx and y = mx + b using similar triangles and explain constant slope.
- Apply volume formulas for cones, cylinders, and spheres to solve real-world problems.
Supplemental Resources
- Rulers, protractors, and measuring tapes for geometric construction and measurement activities
- Grid paper and coordinate plane templates for graphing transformations
- Pattern blocks or tangrams for exploring area and similarity concepts
- Printed nets of three-dimensional shapes for understanding surface area
- Colored pencils and markers for creating visual representations of transformations and cross-sections
Expressions and Equations
Geometry
Ratios and Proportional Relationships
Students write in science notebooks, construct viable arguments, and critique reasoning of others using mathematical evidence and precise language to communicate mathematical thinking.
Students apply mathematical reasoning to analyze scientific data, represent relationships using equations and graphs, and solve real-world problems involving physical phenomena such as temperature, distance, and population dynamics.
Formative Assessments
- Exit tickets on angle relationships and area formulas.
- CPM checkups on 7.G, 8.G, and related 7.EE and 7.RP standards.
- Quizzes on transformations and coordinate geometry.
- Observations of student constructions and geometric reasoning.
- Pair-and-share discussions on similarity, congruence, and transformations.
Summative Assessment
Unit 5 test on geometric problem solving, transformations, and volume; performance assessment applying multiple geometric and algebraic concepts to a multi-step real-world problem.
Benchmark Assessment
Benchmark assessment on 8.G standards measuring geometric reasoning and spatial visualization.
Alternative Assessment
Students may respond orally to explain angle relationships or geometric properties in place of written work, with use of concrete models, manipulatives, or visual diagrams to support understanding. Reduced problem sets focusing on single-step or two-step equations and basic area or volume calculations may be provided, with sentence frames or labeled diagrams to organize thinking.
IEP (Individualized Education Program)
Students may benefit from graphic organizers that categorize angle relationships, formulas, and transformation properties, reducing the cognitive load of holding multiple geometric rules in working memory at once. Providing a reference sheet with labeled diagrams of angle types, circle parts, and three-dimensional figures supports access to content without requiring memorization of vocabulary in isolation. For multi-step equation writing and geometric constructions, breaking tasks into clearly sequenced steps with visual checkpoints helps students monitor their own progress. Extended time and the option to demonstrate understanding orally or through annotated diagrams—rather than written explanations alone—should be offered on assessments involving transformation descriptions and slope derivations.
Section 504
Students should have access to extended time on multi-step tasks involving angle equations, area and volume calculations, and coordinate-based transformation problems, as the sequential reasoning required in this unit can be demanding under timed conditions. Preferential seating near instructional demonstrations is especially useful during construction activities and lessons on cross-sections, where visual access to the teacher's modeling is critical. A reduced-distraction environment during assessments supports focus when students must hold geometric relationships in mind across several problem-solving steps.
ELL / MLL
Visual supports are particularly valuable in this unit—labeled diagrams of circles, angle relationships, three-dimensional figures, and coordinate grids help students connect content vocabulary to concrete representations before engaging with symbolic or written tasks. Key geometry terms such as supplementary, circumference, dilation, and congruence should be introduced with picture support and, where possible, connected to cognates or home-language equivalents. Directions for construction tasks and transformation activities should be given in short, clear steps, and students should have the opportunity to retell or demonstrate understanding before beginning independent work.
At Risk (RTI)
Connecting new geometric concepts to prior knowledge of area, basic equations, and coordinate graphing helps students build confidence before tackling more complex applications like volume of composed figures or transformation sequences. Offering entry points with partially completed diagrams, labeled figures, or pre-set up equations allows students to focus on the reasoning steps rather than getting stuck at the setup phase. Reducing the number of problems on practice tasks to prioritize depth of understanding over quantity—particularly for angle relationships, circle formulas, and slope derivation—helps students consolidate skills before moving to the next concept.
Gifted & Talented
Students who demonstrate early mastery of standard geometric problem solving can explore the relationships between transformations and coordinate algebra more formally, such as writing general coordinate rules for rotations or dilations and investigating why similarity transformations preserve angle measure. Extending the slope derivation work to non-linear contexts—asking students to explore what changes when the relationship is no longer proportional—encourages abstract reasoning beyond the unit's core expectations. Students may also pursue open-ended design challenges, such as creating and analyzing composed three-dimensional figures, justifying surface area and volume calculations, and presenting their reasoning using precise geometric and algebraic language.