Circles |
| HSG.C: Understand And Apply Theorems About Circles |
| HSG.C.2 | Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. |
Congruence |
| HSG.CO: Experiment With Transformations In The Plane |
| HSG.CO.1 | Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. |
| Prove Geometric Theorems |
| HSG.CO.10 | Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. |
Expressing Geometric Properties With Equations |
| HSG.GPE: Translate Between The Geometric Description And The Equation For A Conic Section |
| HSG.GPE.1 | Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. |
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| HSG.GPE.2 | Derive the equation of a parabola given a focus and directrix. |
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| HSG.GPE.3 | (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. |
Expressions And Equations |
| 6.EE: Apply And Extend Previous Understandings Of Arithmetic To Algebraic Expressions. |
| 6.EE.1 | Write and evaluate numerical expressions involving whole-number exponents. |
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| 6.EE.2 | Write, read, and evaluate expressions in which letters stand for numbers. |
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| 6.EE.3 | Apply the properties of operations to generate equivalent expressions. |
| Reason About And Solve One-Variable Equations And Inequalities. |
| 6.EE.6 | Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. |
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| 6.EE.7 | Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. |
Geometric Measurement And Dimension |
| HSG.GMD: Explain Volume Formulas And Use Them To Solve Problems |
| HSG.GMD.1 | Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. |
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| HSG.GMD.2 | (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. |
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| HSG.GMD.3 | Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★ |
| Visualize Relationships Between Two-Dimensional And Three- Dimensional Objects |
| HSG.GMD.4 | Identify the shapes of two-dimensional cross-sections of three- dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. |
Geometry |
| 6.G: Solve Real-World And Mathematical Problems Involving Area, Surface Area, And Volume. |
| 6.G.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. |
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| 6.G.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. |
| 7.G: Draw, Construct, And Describe Geometrical Figures And Describe The Relationships Between Them. |
| 7.G.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. |
| Solve Real-Life And Mathematical Problems Involving Angle Measure, Area, Surface Area, And Volume. |
| 7.G.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. |
| 8.G: Understand And Apply The Pythagorean Theorem. |
| 8.G.6 | Explain a proof of the Pythagorean Theorem and its converse. |
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| 8.G.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. |
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| 8.G.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. |
| Solve Real-World And Mathematical Problems Involving Volume Of Cylinders, Cones, And Spheres. |
| 8.G.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. |
Similarity, Right Triangles, And Trigonometry |
| HSG.SRT: Prove Theorems Involving Similarity |
| HSG.SRT.4 | Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. |
| Define Trigonometric Ratios And Solve Problems Involving Right Triangles |
| HSG.SRT.7 | Explain and use the relationship between the sine and cosine of complementary angles. |
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| HSG.SRT.8 | Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ |
The Number System |
| 6.NS: Compute Fluently With Multi-Digit Numbers And Find Common Factors And Multiples. |
| 6.NS.2 | Fluently divide multi-digit numbers using the standard algorithm. |
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| 6.NS.3 | Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. |
| Apply And Extend Previous Understandings Of Multiplication And Division To Divide Fractions By Fractions. |
| 6.NS.1 | Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? |
| 7.NS: Apply And Extend Previous Understandings Of Operations With Fractions To Add, Subtract, Multiply, And Divide Rational Numbers. |
| 7.NS.1 | Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. |
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| 7.NS.2 | Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. |
| 8.NS: Know That There Are Numbers That Are Not Rational, And Approximate Them By Rational Numbers. |
| 8.NS.1 | Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. |
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Menu for Plane Geometry.
About page for Circle shows basic definition and equations.
Calculator page for Divide on Paper shows input 3345 and 35, Calculate button, layout, and explanation.
Table page for Length shows unit lengths and sizes as well as empirical units.
