Geometry Targets Matrix Guide
Curriculum Blueprint Washington State Standards
Geometry Priority Standards & Target Matrix
Deconstructed Priority Standards, Student "I Can" Targets, and Evidence of Mastery
Course: High School Geometry
Domains: Proofs • Similarity • Circles • Modeling
Aligned to WA OSPI & Smarter Balanced (SBAC)
WA OSPI Priority Standard Selection Rationale (Reeves & Ainsworth)
1. Endurance
Concepts retaining value far beyond Geometry into Pre-Calculus, Calculus, STEM pathways, and deductive problem solving.
2. Leverage
Foundational skills that bridge disciplines: algebraic proofs, coordinate modeling, and trigonometric functions.
3. Readiness
Essential prerequisite knowledge required for subsequent college-ready math and WA state high school graduation assessments.
Cluster 1: Geometric Proofs & Deductive Systems
G-CO.C.9 • G-CO.C.10 • G-CO.B.8
WA Priority Standards (SBAC Claim 3 - Reasoning Focus):
HSG-CO.C.9: Prove theorems about lines and angles (vertical angles, perpendicular bisectors, alternate interior angles).
HSG-CO.C.10: Prove theorems about triangles (measures of interior angles sum to \(180^\circ\), base angles of isosceles triangles, midsegments).
HSG-CO.B.8: Explain how triangle congruence criteria (ASA, SAS, SSS, AAS, HL) follow from rigid motions.
Deconstructed "I Can" Learning Targets
| Level | Student "I Can" Target | Success Criteria & Evidence of Mastery |
|---|
| Knowledge | | |
| DOK 1 | I can identify given geometric conditions, postulates, and definitions. | Extract hypotheses from diagrams; state definitions (bisector, midpoint, supplementary) and axioms without reference prompts. |
| Application | | |
| DOK 2 | I can construct formal two-column, flow, and paragraph proofs. | Provide valid mathematical reasons for every deduction; correctly execute triangle congruence proofs (SAS, ASA, SSS, AAS, HL) and apply CPCTC. |
| Reasoning | | |
| DOK 3 | I can critique geometric proofs to identify logical fallacies and gaps. | Spot invalid visual assumptions (e.g., assuming perpendicularity without markers); reconstruct flawed arguments with rigorous deductive logic. |
Common Student Misconceptions
- Assuming diagrams are to scale without given notation.
- Confusing a theorem with its converse (e.g., parallel lines).
- Invoking CPCTC before proving triangles congruent.
Scaffolded Instruction Anchor
- Begin with Card Sorts (ordering pre-written statements/reasons).
- Transition from Fill-in-the-Blank proofs to autonomous proofs.
- Emphasize color-coded markings directly on diagrams.
Washington State K–12 Mathematics Learning Standards (WA OSPI) Domain 1: Geometric Proofs & Deductive Systems Page 1 of 4
Domain Priority Focus WA OSPI • SBAC Claim 1
Similarity & Right Triangle Trigonometry
Dilation Invariance, Similarity Proofs, Proportional Reasoning, and Trig Ratios
G-SRT.A.2 • G-SRT.B.5 • G-SRT.C.6
WA Priority Standards:
HSG-SRT.A.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar.
HSG-SRT.B.5: Use congruence and similarity criteria for triangles (\(AA\sim, SAS\sim, SSS\sim\)) to solve problems and prove geometric relationships.
HSG-SRT.C.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios: \(\sin, \cos, \tan\).
Deconstructed "I Can" Learning Targets
| Target | Student "I Can" Target | Success Criteria & Evidence of Mastery |
|---|
| T 2.1 | I can verify similarity using dilations and establish the scale factor \(k\). | Demonstrate that similarity preserves angle measures while scaling side lengths proportionally (\(k = \frac{\text{image}}{\text{pre-image}}\)); calculate coordinates under center of dilation. |
| T 2.2 | I can prove triangles similar using \(AA\sim\), \(SAS\sim\), and \(SSS\sim\). | Write formal two-column and flow proofs demonstrating triangle similarity; write proper similarity statements matching corresponding vertices. |
| T 2.3 | I can set up and solve proportional equations to find missing side lengths. | Apply the Triangle Proportionality Theorem and geometric mean theorems (altitude and leg rules) to solve linear and quadratic geometric equations. |
| T 2.4 | I can calculate unknown sides and angles using right triangle trigonometry. | Accurately set up \(\sin\theta = \frac{\text{opp}}{\text{hyp}}\), \(\cos\theta = \frac{\text{adj}}{\text{hyp}}\), and \(\tan\theta = \frac{\text{opp}}{\text{adj}}\); solve for missing sides and use inverse trig for angles. |
Trigonometric Ratio Core Concept
Trigonometric ratios are constant for any given acute angle \(\theta\), regardless of triangle size, directly resulting from the \(AA\sim\) similarity criterion.
\(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)
Applied Real-World Application
Indirect Measurement: Modeling real-world situations with angles of elevation and depression (e.g., forestry tree height, surveyor triangulation).
\(\text{Height} = \text{Distance} \cdot \tan(\theta) + \text{Eye-Level Height}\)
Washington State K–12 Mathematics Learning Standards (WA OSPI) Domain 2: Similarity & Right Triangle Trigonometry Page 2 of 4
Domain Priority Focus WA OSPI • SBAC Claim 1
Circles & Geometric Measurement
Chords, Secants, Tangents, Arc Lengths, Sector Areas, and Circle Equations
G-C.A.2 • G-C.B.5 • G-GPE.A.1
WA Priority Standards:
HSG-C.A.2: Identify and describe relationships among inscribed angles, radii, and chords (e.g., inscribed angles intercepting the same arc are congruent; radius to tangent is perpendicular).
HSG-C.B.5: Derive using similarity that arc length is proportional to the radius; derive the formula for the area of a sector.
HSG-GPE.A.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find center and radius.
Deconstructed "I Can" Learning Targets
| Target | Student "I Can" Target | Success Criteria & Evidence of Mastery |
|---|
| T 3.1 | I can calculate central, inscribed, interior, and exterior circle angles. | Apply \(\theta_{\text{inscribed}} = \frac{1}{2}\text{arc}\); establish that tangent-radius intersections form \(90^\circ\); calculate angles formed by intersecting secants and tangents. |
| T 3.2 | I can calculate arc lengths and sector areas using proportional reasoning. | Differentiate between degree measure and arc length (\(s = \frac{\theta}{360}(2\pi r)\)); calculate sector area (\(A = \frac{\theta}{360}(\pi r^2)\)) in exact terms with \(\pi\). |
| T 3.3 | I can write and graph circle equations \((x-h)^2 + (y-k)^2 = r^2\). | Identify center \((h,k)\) and radius \(r\); complete the square from standard polynomial form \(x^2 + y^2 + Dx + Ey + F = 0\) to center-radius form. |
Arc Length vs. Degree Measure
A major hurdle is distinguishing angular measure (degrees, \(\theta\)) from linear distance along the curve (arc length, \(s\)).
\(\text{Arc Length } s = \frac{\theta}{360^\circ} \cdot 2\pi r \quad \Big(s = r\theta \text{ in rad}\Big)\)
Tangent-Secant Power Theorems
Segment length relationships derived via similar triangles created by shared intercepted arcs.
\(a \cdot b = c \cdot d \quad \text{and} \quad \text{Tangent}^2 = \text{External} \cdot \text{Secant}\)
Washington State K–12 Mathematics Learning Standards (WA OSPI) Domain 3: Circles & Geometric Measurement Page 3 of 4
Domain Priority Focus WA OSPI • SBAC Claim 4
Geometric Modeling & Implementation
3D Spatial Solids, Density Applications, Course Pacing, and Common Assessments
G-MG.A.1 • G-MG.A.2 • G-MG.A.3 • G-GMD.A.3
WA Priority Standards:
HSG-MG.A.1 & HSG-MG.A.2: Use geometric shapes and measures to describe objects; apply concepts of density based on area and volume.
HSG-MG.A.3 & HSG-GMD.A.3: Apply geometric methods to solve design problems; use volume formulas for cylinders, pyramids, cones, and spheres.
| Target | Student "I Can" Target | Evidence of Mastery & Rigor |
|---|
| T 4.1 | I can decompose complex physical objects into standard 3D solids. | Model real structures (silos, tanks, domes) by summing/subtracting volumes of cylinders, cones, prisms, and spheres. |
| T 4.2 | I can calculate and apply area and volume densities. | Solve multi-step problems using \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\) or \(\frac{\text{Population}}{\text{Area}}\); convert compound units accurately. |
| T 4.3 | I can optimize dimensions under geometric and cost constraints. | Analyze design trade-offs (e.g., minimizing surface area for canned goods packaging to lower raw material cost). |
Course Implementation & Common Assessment Matrix
| Priority Cluster | Pacing | Formative Checkpoints | Summative Capstone Assessment |
|---|
| 1. Proofs & Congruence | 4–5 Wks | Proof error-repair tasks; two-step deductive exit tickets. | Axiomatic Truss Defense: Formally prove structural rigidity in engineering truss diagrams. |
| 2. Similarity & Trig | 4–5 Wks | Scale factor checks; right triangle ratio drills. | Indirect Surveying Project: Measure inaccessible structures using clinometers and tangent ratios. |
| 3. Circle Theorems | 3–4 Wks | Arc length vs sector area exits; circle completing square. | Geolocation Triangulation: Model intersecting broadcast signals using coordinate circle equations. |
| 4. Modeling & 3D | 3–4 Wks | Composite volume exits; population density tasks. | Packaging Optimization Challenge: Design a container holding volume \(V\) with minimal surface area. |
Standards for Mathematical Practice (SMP) Integration
SMP 1: Persevere in decomposing non-routine 3D composite shapes.
SMP 3: Construct valid deductive arguments and critique false proofs.
SMP 4: Model real-world situations with trig ratios and material densities.
SMP 7: Identify structural invariants across dilations and circles.
Washington State K–12 Mathematics Learning Standards (WA OSPI) Domain 4 & Implementation Roadmap Page 4 of 4
