Grade 9 | Mathematics

Action steps

1. Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle.

2. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

3. Find the equation of a line parallel or perpendicular to a given line that passes through a given point.

Action steps

1. Use a compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc. to make various geometric constructions.

2. Construct perpendicular lines.

3. Construct a line parallel to a given line through a point not on the line.

Action steps

1. Represent transformations in the plane using, e.g., transparencies and geometry software.

2. Describe transformations as functions that take points in the plane as inputs and give other points as outputs.

3. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Action steps

1. Prove that vertical angles are congruent.

2. Prove that when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent.

Action steps

1. Verify experimentally the properties of dilations given by a center and a scale factor.

2. Show that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

3. Show that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

4. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar.

5. Explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Action steps

1. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

2. For parallelogramsprove that opposite sides are congruent.

3. For parallelograms prove thatopposite angles are congruent.

4. For parallelograms prove thatthe diagonals of a parallelogram bisect each other.

5. Prove that rectangles are parallelograms with congruent diagonals.

6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure.

7. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software.

Action steps

1. Explain and use the relationship between the sine and cosine of complementary angles.

2. Use trigonometric ratios and the Pythagorean Theorem to solve right triangle in applied problems.

3. Prove the Laws of Sines and Cosines and use them to solve problems.

4. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

5. Derive the formula A = 1⁄2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Action steps

1. Prove that all circles are similar.

2. Identify and describe relationships among inscribed angles, radii, and chords.

3. Construct a tangent line from a point outside a given circle to the circle.

4. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

5. Give an informal argument for the formulas for the circumference of a circle, area of a circle.

6. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality.

7. Derive the formula for the area of a sector.

8. Derive the equation of a circle of given center and radius using the Pythagorean Theorem.

9. Prove or disprove that a point which lies on a circle of radius equal to twois centered at the origin and contains the point (0, 2).

Action steps

1. Construct visuals that show the relation between two-dimensional and three-dimensional objects.

2. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

3. Give an informal argument for the formulas for the volume of a cylinder, pyramid, and cone.

4. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost).

Action steps

1. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

2. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified.

3. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

4. Collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade.

5. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

6. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

7. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

8. Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

9. Use permutations and combinations to compute probabilities of compound events and solve problems.

10. Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

11. Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulled a hockey goalie at the end of a game).