Partnership for Assessment of Readiness for College and Careers (PARCC) Mathematics: High School Geometry, Statistics, and Modeling Syllabus

Mathematics: High School Geometry, Statistics, and Modeling syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics: High School Geometry, Statistics, and Modeling in Partnership for Assessment of Readiness for College and Careers (PARCC), not a summary of it.

1. 1

   Congruence and Similarity

   3 topics

   • Rigid Motions and Congruence
     • Transformations in the plane
     • Triangle congruence criteria and proofs
   • Similarity and Dilations
     • Properties of dilations
     • Similarity proofs and applications
   • Geometric Constructions
     • Compass and straightedge constructions
     • Constructing inscribed and circumscribed figures
2. 2

   Right Triangles and Trigonometry

   3 topics

   • Pythagorean Relationships
     • Proofs and converse applications
     • Distance in the coordinate plane
   • Trigonometric Ratios
     • Sine, cosine, and tangent in right triangles
     • Solving real-world right-triangle problems
   • Laws of Sines and Cosines
     • Non-right triangle applications
     • Area of a triangle using trigonometry
3. 3

   Circles and Coordinate Geometry

   3 topics

   • Circle Theorems
     • Arcs, chords, and inscribed angles
     • Tangent and secant relationships
   • Equations of Conics
     • Deriving the equation of a circle
     • Parabola from focus and directrix
   • Geometry on the Coordinate Plane
     • Slope criteria for parallel and perpendicular lines
     • Partitioning segments and polygon properties
4. 4

   Geometric Measurement and Dimension

   3 topics

   • Volume and Surface Area Formulas
     • Prisms, cylinders, cones, and spheres
     • Cavalieri's principle
   • Two- and Three-Dimensional Relationships
     • Cross-sections of solids
     • Solids of rotation
   • Modeling with Geometry
     • Density and design problems
     • Geometric shapes in real-world contexts
5. 5

   Statistics and Probability in High School

   3 topics

   • Interpreting Categorical and Quantitative Data
     • Distributions and outliers
     • Two-way frequency tables
     • Linear regression and correlation
   • Making Inferences and Justifying Conclusions
     • Sampling and experimental design
     • Drawing conclusions from data
   • Conditional Probability and Rules of Probability
     • Independence and conditional probability
     • Addition and multiplication rules

Mathematics: High School Geometry, Statistics, and Modeling flashcards for Partnership for Assessment of Readiness for College and Careers (PARCC)

24 of 51 cards from the Mathematics: High School Geometry, Statistics, and Modeling deck — real questions with worked answers.

1. What is a rigid motion (isometry), and which properties does it preserve?

   A transformation that preserves distance and angle measure. The three rigid motions are translations, rotations, and reflections; they preserve length, angle measure, parallelism, and betweenness, so the image is congruent to the original.

2. What does it mean for two figures to be congruent in terms of transformations?

   Two figures are congruent if and only if one can be mapped onto the other by a sequence of rigid motions (translations, rotations, and/or reflections).

3. State the SAS, ASA, SSS, and AAS triangle congruence criteria.

   SAS: two sides and the included angle. ASA: two angles and the included side. SSS: all three sides. AAS: two angles and a non-included side. Each guarantees triangle congruence.

4. What is a dilation, and how does it affect lengths and angles?

   A dilation is a transformation with a center and scale factor k that stretches or shrinks a figure. It multiplies all lengths by |k|, preserves angle measures, and maps a line not through the center to a parallel line.

5. What is the definition of similar figures, and what do AA, SAS, and SSS similarity require?

   Similar figures have equal corresponding angles and proportional corresponding sides (related by a similarity transformation). AA: two pairs of equal angles. SAS similarity: two proportional sides with equal included angle. SSS similarity: all three pairs of sides proportional.

6. For similar figures with linear scale factor k, how do perimeter, area, and volume scale?

   Perimeter (and any length) scales by k, area scales by k², and volume scales by k³.

7. How do you construct the perpendicular bisector of a segment with compass and straightedge?

   Place the compass at each endpoint and draw arcs (radius more than half the segment) above and below; the two arc intersections define a line that is the perpendicular bisector.

8. How do you bisect an angle with compass and straightedge?

   From the vertex draw an arc crossing both sides; from those two intersection points draw equal arcs that cross inside the angle; the ray from the vertex through that crossing bisects the angle.

9. State the Pythagorean Theorem and its converse.

   For a right triangle with legs a, b and hypotenuse c: a² + b² = c². Converse: if a² + b² = c² in a triangle, then the triangle is right-angled (with the right angle opposite c).

10. What are the side ratios in 45-45-90 and 30-60-90 special right triangles?

    45-45-90: legs equal, hypotenuse = leg·√2, ratio 1 : 1 : √2. 30-60-90: sides opposite 30°, 60°, 90° are in ratio 1 : √3 : 2.

11. Define the sine, cosine, and tangent ratios for an acute angle in a right triangle.

    sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent (SOH-CAH-TOA).

12. What is the relationship between the sine and cosine of complementary angles?

    The sine of an angle equals the cosine of its complement: sin(θ) = cos(90° − θ), and cos(θ) = sin(90° − θ).

13. State the Law of Sines.

    In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite its corresponding angle.

14. State the Law of Cosines.

    c² = a² + b² − 2ab·cos C (and cyclic forms). It generalizes the Pythagorean Theorem and is used when you know two sides and the included angle, or all three sides.

15. When do you use the Law of Sines versus the Law of Cosines?

    Use the Law of Sines for ASA, AAS, or SSA cases. Use the Law of Cosines for SAS (find third side) or SSS (find an angle).

16. What is the formula for the area of a triangle using two sides and the included angle?

    Area = ½·a·b·sin C, where a and b are two sides and C is the included angle.

17. What is the relationship between an inscribed angle and the central angle subtending the same arc?

    The inscribed angle is half the central angle that subtends the same arc (Inscribed Angle Theorem).

18. What is the measure of an inscribed angle that subtends a semicircle (diameter)?

    It is a right angle, 90° (Thales' Theorem). Any angle inscribed in a semicircle is a right angle.

19. What is the relationship between a tangent line to a circle and the radius at the point of tangency?

    A tangent line is perpendicular to the radius drawn to the point of tangency.

20. State the formulas for arc length and area of a sector with central angle θ (in radians) and radius r.

    Arc length = rθ; sector area = ½r²θ. In degrees: arc length = (θ/360)·2πr, sector area = (θ/360)·πr².

21. What is the standard equation of a circle with center (h, k) and radius r?

    (x − h)² + (y − k)² = r².

22. What is the standard equation of a parabola with vertex (h, k) opening vertically, and what is p?

    (x − h)² = 4p(y − k), where p is the directed distance from the vertex to the focus (and to the directrix). It opens up if p > 0, down if p < 0.

23. What is the defining focus-directrix property of a parabola?

    A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

24. What is the distance formula between points (x₁, y₁) and (x₂, y₂)?

    d = √[(x₂ − x₁)² + (y₂ − y₁)²].

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