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School City of Hobart
School City of Hobart Mathematics
Mathematics - Geometry

Geometry
Points, Lines, Angles, and Planes
The learner will be able to find lengths and midpoints of lines. They describe and use parallel and perpendicular lines. They find slopes and equations of lines.
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Lines Master IDOE
  
G.1.1
The learner will be able to find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems. Example: Find the length and midpoint of the line joining the points A (3, 8) and B (9, 0). .
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Lines Master IDOE
  
G.1.2
The learner will be able to construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass, explaining and justifying the process used. Example: Construct the perpendicular bisector of a given line segment, justifying each step of the process. .
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Constructions Master IDOE
  
G.1.3
The learner will be able to understand and use the relationships between special pairs of angles formed by parallel lines and transversals.
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Angles Master IDOE
  
G.1.4
The learner will be able to use coordinate geometry to find slopes, parallel lines, perpendicular lines, and equations of lines. Example: Find an equation of a line perpendicular to y = 4x - 2. .
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Lines Master IDOE
  
Polygons
The learner will be able to identify and describe polygons, and measure interior and exterior angles. They use congruence, similarity, symmetry, tessellations, and transformations. They find measures of sides, perimeters, and areas.
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Polygons Master IDOE
  
G.2.1
The learner will be able to identify and describe convex, concave, and regular polygons. Example: Draw a regular hexagon. Is it convex or concave? Explain your answer. .
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Polygons Master IDOE
  
G.2.2
The learner will be able to find measures of interior and exterior angles of polygons, justifying the method used. Example: Calculate the measure of one interior angle of a regular octagon. Explain your method. .
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Polygons Master IDOE
  
G.2.3
The learner will be able to use properties of congruent and similar polygons to solve problems. Example: Divide a regular hexagon into triangles by joining the center to each vertex. Show that these triangles are all the same size and shape, and so find the sizes of the interior angles of the hexagon. .
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Polygons Master IDOE
  
G.2.4
The learner will be able to apply transformations (slides, flips, turns, expansions, and contractions) to polygons in order to determine congruence, similarity, symmetry, and tessellations. Know that images formed by slides, flips and turns are congruent to the original shape. Example: Use a drawing program to create regular hexagons, regular octagons, and regular pentagons. Under the drawings, describe which of the polygons would be best for tiling a rectangular floor. Explain your reasoning. .
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Polygons Master IDOE
  
G.2.5
The learner will be able to find and use measures of sides, perimeters, and areas of polygons, and relate these measures to each other using formulas. Example: A rectangle of area 360 square yards is ten times as long as it is wide. Find its length and width. .
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Polygons Master IDOE
  
G.2.6
The learner will be able to use coordinate geometry to prove properties of polygons such as regularity, congruence, and similarity. Example: Do these four points form a square: (2, 1), (6, 2), (5, 6), (1, 5)? .
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Polygons Master IDOE
  
Quadrilaterals
The learner will be able to identify and describe simple quadrilaterals. They use congruence and similarity. They find measures of sides, perimeters, and areas.
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Quadrilaterals Master IDOE
  
G.3.1
The learner will be able to describe, classify, and understand relationships among the quadrilaterals square, rectangle, rhombus, parallelogram, trapezoid, and kite. Example: Use a draw program to create a square, rectangle, rhombus, parallelogram, trapezoid, and kite. Judge which of the quadrilaterals has perpendicular diagonals and draw those diagonals in the figures. Give a convincing argument that your judgment is correct. .
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Quadrilaterals Master IDOE
  
G.3.2
The learner will be able to use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas. Example: Of two similar rectangles, the second has sides three times the length of the first. How many times larger in area is the second rectangle? .
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Quadrilaterals Master IDOE
  
G.3.3
The learner will be able to find and use measures of sides, perimeters, and areas of quadrilaterals, and relate these measures to each other using formulas. Example: A section of roof is a trapezoid with length 4 m at the ridge and 6 m at the gutter. The shortest distance from ridge to gutter is 3 m. Construct a model using a computer draw program showing how to find the area of this section of roof. .
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Quadrilaterals Master IDOE
  
G.3.4
The learner will be able to use coordinate geometry to prove properties of quadrilaterals such as regularity, congruence, and similarity. Example: Do these four points form a kite: (0, 0), (6, 0), (4, 2), (5, -2)? .
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Quadrilaterals Master IDOE
  
Triangles
The learner will be able to identify and describe types of triangles. They identify and draw altitudes, medians, and angle bisectors. They use congruence and similarity. They find measures of sides, perimeters, and areas. They apply inequality theorems.
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Triangles Master IDOE
  
G.4.1
The learner will be able to identify and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular. Example: Use a draw program to create examples of a right, acute, obtuse, scalene, isosceles, equilateral, and equiangular triangle. Identify and describe the attributes of each triangle. .
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Triangles Master IDOE
  
G.4.2
The learner will be able to define, identify, and construct altitudes, medians, angle bisectors, and perpendicular bisectors. Example: Draw several triangles. Construct their angle bisectors. What do you notice? .
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Triangles Master IDOE
  
G.4.3
The learner will be able to construct triangles congruent to given triangles. Example: Construct a triangle given the lengths of two sides and the measure of the angle between the two sides. .
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Triangles Master IDOE
  
G.4.4
The learner will be able to use properties of congruent and similar triangles to solve problems involving lengths and areas. Example: Of two similar triangles, the second has sides half the length of the first. The area of the first triangle is 20 cm2. What is the area of the second? .
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Triangles Master IDOE
  
G.4.5
The learner will be able to prove and apply theorems involving segments divided proportionally.
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Triangles Master IDOE
  
G.4.6
The learner will be able to prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. Example: In the last example, prove that triangles ABC and APQ are similar. .
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Triangles Master IDOE
  
G.4.7
The learner will be able to find and use measures of sides, perimeters, and areas of triangles, and relate these measures to each other using formulas. Example: The gable end of a house is a triangle 20 feet long and 13 feet high. Find its area. .
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Triangles Master IDOE
  
G.4.8
The learner will be able to prove, understand, and apply the inequality theorems: triangle inequality, inequality in one triangle, and hinge theorem. Example: Can you draw a triangle with sides of length 7 cm, 4 cm, and 15 cm? .
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Triangles Master IDOE
  
G.4.9
The learner will be able to use coordinate geometry to prove properties of triangles such as regularity, congruence, and similarity. Example: Draw a triangle with vertices at (1, 3), (2, 5), and (6, 1). Draw another triangle with vertices at (-3, -1), (-2, 1), and (2, -3). Are these triangles the same shape and size? .
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Triangles Master IDOE
  
Right Triangles
The learner will be able to prove the Pythagorean Theorem and use it to solve problems. They define and apply the trigonometric relations sine, cosine, and tangent.
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Pythagorean Theorem Master IDOE
  
G.5.1
The learner will be able to prove and use the Pythagorean Theorem. Example: On each side of a right triangle, draw a square with that side of the triangle as one side of the square. Find the areas of the three squares. What relationship is there between the areas? .
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Pythagorean Theorem Master IDOE
  
G.5.2
The learner will be able to state and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.
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Right Triangles Master IDOE
  
G.5.3
The learner will be able to use special right triangles (30° - 60° and 45° - 45°) to solve problems. Example: An isosceles right triangle has one short side of 6 cm. Find the lengths of the other two sides. .
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Right Triangles Master IDOE
  
G.5.4
The learner will be able to define and use the trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) in terms of angles of right triangles. Example: In triangle ABC, tan A = 1/5. Find sin A and cot A. .
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Right Triangles Master IDOE
  
G.5.5
The learner will be able to know and use the relationship sin2x + cos2x = 1. Example: Show that, in a right triangle, sin2x + cos2x = 1 is an example of the Pythagorean Theorem. .
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Pythagorean Theorem Master IDOE
  
G.5.6
The learner will be able to solve word problems involving right triangles. Example: The force of gravity pulling an object down a hill is its weight multiplied by the sine of the angle of elevation of the hill. What is the force on a 3,000-pound car on a hill with a 1 in 5 grade? (A grade of 1 in 5 means that the hill rises one unit for every 5 horizontal units.) .
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Right Triangles Master IDOE
  
Circles
The learner will be able to define ideas related to circles: e.g., radius, tangent. They find measures of angles, lengths, and areas. They prove theorems about circles. They find equations of circles.
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Circles Master IDOE
  
G.6.1
The learner will be able to find the center of a given circle. Construct the circle that passes through three given points (not in a straight line). Example: Given a circle, find its center by drawing the perpendicular bisectors of two chords. .
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Circles Master IDOE
  
G.6.2
The learner will be able to define and identify relationships among: radius, diameter, arc, measure of an arc, chord, secant, and tangent. Example: What is the angle between a tangent to a circle and the radius at the point where the tangent meets the circle? .
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Circles Master IDOE
  
G.6.3
The learner will be able to prove theorems related to circles. Example: Prove that the angle subtended by a chord at the center of a circle is twice the angle at the circumference. .
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Circles Master IDOE
  
G.6.4
The learner will be able to construct tangents to circles, and circumscribe and inscribe circles. Example: Draw an acute triangle and construct the circumscribed circle. .
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Circles Master IDOE
  
G.6.5
The learner will be able to define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents). Example: Find the measure of angle ABC in the diagram below. .
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Circles Master IDOE
  
G.6.6
The learner will be able to define and identify congruent and concentric circles. Example: Are circles with the same center always the same shape? Are they always the same size? .
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Circles Master IDOE
  
G.6.7
The learner will be able to define, find, and use measures of circumference, arc length, and areas of circles and sectors. Use these measures to solve problems. Example: Which will give you more: three 6-inch pizzas or two 8-inch pizzas? Explain your answer. .
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Circles Master IDOE
  
G.6.8
The learner will be able to find the equation of a circle in the coordinate plane in terms of its center and radius. Example: Find the equation of the circle with radius 10 and center (6, -3). .
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Circles Master IDOE
  
Polyhedra and Other Solids
The learner will be able to describe and make polyhedra and other solids. They describe relationships and symmetries, and use congruence and similarity.
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Congruence/Similarity/Symmetry Master IDOE
  
G.7.1
The learner will be able to describe and make regular and non-regular polyhedra. Example: Is a cube a regular polyhedron? Explain why or why not. .
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Three-Dimensional Solids Master IDOE
  
G.7.2
The learner will be able to describe the polyhedron that can be made from a given net (or pattern). Describe the net for a given polyhedron. Example: Make a net for a tetrahedron out of poster board and fold it up to make the tetrahedron. .
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Three-Dimensional Solids Master IDOE
  
G.7.3
The learner will be able to describe relationships between the faces, edges, and vertices of polyhedra. Example: Count the sides, edges, and corners of a square-based pyramid. How are these numbers related? .
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Three-Dimensional Solids Master IDOE
  
G.7.4
The learner will be able to describe symmetries of geometric solids. Example: Describe the rotation and reflection symmetries of a square-based pyramid. .
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Three-Dimensional Solids Master IDOE
  
G.7.5
The learner will be able to describe sets of points on spheres: chords, tangents, and great circles. Example: On Earth, is the equator a great circle? .
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Three-Dimensional Solids Master IDOE
  
G.7.6
The learner will be able to identify and know properties of congruent and similar solids. Example: Explain how the surface area and volume of similar cylinders are related. .
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Three-Dimensional Solids Master IDOE
  
G.7.7
The learner will be able to find and use measures of sides, volumes of solids, and surface areas of solids, and relate these measures to each other using formulas. Example: An ice cube is dropped into a glass that is roughly a right cylinder with a 6 cm diameter. The water level rises 1 mm. What is the volume of the ice cube? .
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Three-Dimensional Solids Master IDOE
  
Mathematical Reasoning, Problem Solving
The learner will be able to use a variety of strategies to solve problems.
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Problem Solving Master IDOE
  
G.8.1
The learner will be able to use a variety of problem-solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards. Example: How far does the tip of the minute hand of a clock move in 20 minutes if the tip is 4 inches from the center of the clock? .
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Problem Solving Master IDOE
  
G.8.2
The learner will be able to decide whether a solution is reasonable in the context of the original situation. Example: John says the answer to the problem in the first example is "12 inches." Is his answer reasonable? Why or why not? .
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Problem Solving Master IDOE
  
G.8.3
The learner will be able to make conjectures about geometric ideas. Distinguish between information that supports a conjecture and proof of a conjecture. Example: Calculate the ratios of side lengths in several different-sized triangles with angles of 90°, 50° and 40°. What do you notice about the ratios? How might you prove that your observation is true (or show that it is false)? .
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Problem Solving Master IDOE
  
G.8.4
The learner will be able to write and interpret statements of the form "if - then" and "if and only if." Example: Decide whether this statement is true: "If today is Sunday, then we have school tomorrow." .
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Problem Solving Master IDOE
  
G.8.5
The learner will be able to state, use, and examine the validity of the converse, inverse, and contrapositive of "if - then" statements. Example: In the last example, write the converse of the statement. .
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Problem Solving Master IDOE
  
G.8.6
The learner will be able to identify and give examples of undefined terms, axioms, and theorems, and inductive and deductive proof. Example: Do you prove axioms from theorems or theorems from axioms? .
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Problem Solving Master IDOE
  
G.8.7
The learner will be able to construct logical arguments, judge their validity, and give counterexamples to disprove statements. Example: Find an example to show that triangles with two sides and one angle equal are not necessarily congruent. .
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Problem Solving Master IDOE
  
G.8.8
The learner will be able to write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow chart, paragraph, two-column, and indirect.
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Proofs Master IDOE
  
G.8.9
The learner will be able to perform basic constructions, describing and justifying the procedures used. Distinguish between constructing and drawing geometric figures. Example: Construct a line parallel to a given line through a given point not on the line, explaining and justifying each step. .
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Constructions Master IDOE
  
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