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|Mathematics Unit Plan – Learning Progression Guide |
|Course No. |27.09720 |Course Name |GSE Analytic Geometry |
|Grade |10 |Unit # |3 |Projected |4 weeks |
| | | | |Timeline | |
|Unit Name |Circles and Volume |
|Unit Overview |
|In this unit students will: |
|• Understand and Apply theorems about circles |
|• Find Arc Length and Area of Sectors of circles |
|• Explain Volume Formulas and Use them to solve problems |
|Unit Curriculum Map |
|Unit Standards |MGSE9-12.G.C.1 Understand that all circles are similar. |
| |MGSE9-12.G.C.2 Identify and describe relationships among inscribed angles, radii, chords, tangents, and secants. 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. |
| |MGSE9-12.G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a |
| |quadrilateral inscribed in a circle. |
| |MGSE9-12.G.C.4 Construct a tangent line from a point outside a given circle to the circle. |
| |MGSE9-12.G.C.5 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; derive the formula for the area of |
| |a sector |
| |MGSE9-12.G.GMD.1 Give informal arguments for geometric formulas. |
| |a. Give informal arguments for the formulas of the circumference of a circle and area |
| |of a circle using dissection arguments and informal limit arguments. |
| |b. Give informal arguments for the formula of the volume of a cylinder, pyramid, and |
| |cone using Cavalieri’s principle. |
| |MGSE9-12.G.GMD.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and |
| |other solid figures. |
| |MGSE9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. |
| |MGSE9-12.G.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. |
|Content Learning Progression # __1__ |
|Topic __1__ out of __2___ |Area and Volume Formulas for 2-D and 3-D shapes |
| |(2 weeks) |
|Standards in this learning progression: |MGSE9-12.G.GMD.1 |
| |MGSE9-12.G.GMD.2 |
| |MGSE9-12.G.GMD.3 |
| |MGSE9-12.G.GMD.4 |
|Connections to other standards |1. Make sense of problems and persevere in solving them. |
|(Standards for Mathematical Practice, |2. Reason abstractly and quantitatively. |
|Literacy, etc: |3. Construct viable arguments and critique the reasoning of others. |
| |4. Model with mathematics. |
| |5. Use appropriate tools strategically. |
| |6. Attend to precision. |
| |7. Look for and make use of structure. |
| |8. Look for and express regularity in repeated reasoning. |
| | |
| |Resource 1: |
| |Resource 2: |
|Terms students should learn and use with|apothem, Cavalieri’s Principle, center of a circle, center of a polygon, center of a sphere, central angle of a regular |
|precision in this unit and progression: |polygon, circle, circumference, cross-section, cylinder, diameter, great circle, hemisphere, pi, prism, radius, radius of a|
| |sphere, sphere, volume |
| | |
| |GaDOE Formula Sheet: |
| |Area: |
| |Rectangle/Parallelogram [pic] |
| |Triangle [pic] |
| |Circle [pic] |
| | |
| |Circumference: |
| |[pic] [pic] [pic] |
| | |
| |Volume: |
| |Rectangular Prism/Cylinder [pic] |
| |Pyramid/Cone [pic] |
| |Sphere [pic] |
|Materials and tools students should use |Calculator, measuring tape, string, ruler, circular objects (used to model the relationship between circumference and |
|with precision in this unit and |diameter, i.e. pi), play dough, fishing line, 3-dimensional models of prisms, cylinders, and spheres |
|progression: | |
|Know – Understand – Do |
|(KUD) |
|By the end of this learning progression, students will be able to… |
|UNDERSTAND |
|Big Ideas, Essential Understandings, or Generalizations |
|Perimeter, Area, and Volume of Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and|
|proof. |
|Geometric transformations of shape (composing, decomposing or slicing) correspond to algebraic changes in their equations. |
|KNOW |DO |
|Facts and Procedural Knowledge |Skills |
|Know strategies for dissection and partitioning that support the visualizations |Develop and apply the formulas for the area and circumference of a circle. |
|necessary to build informal arguments. |Develop and apply the formula for area of a regular polygon. |
| |Identify the cross section of prisms and cylinders |
| |Learn and apply the formula for the volume of a prism |
| |Learn and apply the formula for the volume of a cylinder |
| |Learn and apply the formula for volume of a sphere |
| |Learn and apply the formula for surface area of a sphere |
|Content Learning Progression # __2__ |
|Topic __2__ out of __2___ |Circles |
| |(2 weeks) |
|Standards in this learning progression: |MGSE9-12.G.C.1 |
| |MGSE9-12.G.C.2 |
| |MGSE9-12.G.C.3 |
| |MGSE9-12.G.C.4 |
| |MGSE9-12.G.C.5 |
| | |
|Connections to other standards |1. Make sense of problems and persevere in solving them. |
|(Standards for Mathematical Practice, |2. Reason abstractly and quantitatively. |
|Literacy, etc: |3. Construct viable arguments and critique the reasoning of others. |
| |4. Model with mathematics. |
| |5. Use appropriate tools strategically. |
| |6. Attend to precision. |
| |7. Look for and make use of structure. |
| |8. Look for and express regularity in repeated reasoning. |
| | |
| |Resource 1: |
| |Resource 2: |
|Terms students should learn and use with|adjacent arcs, arc, Arc Addition Postulate, arc length, arc measure, central angle, chord, common tangent, concentric |
|precision in this unit and progression: |circles, congruent arcs, congruent circles, exterior of a circle, inscribed, inscribed angle, inscribed chord, inscribed |
| |polygon, intercepted arc, interior of a circle, major arc, minor arc, point of tangency, radian, secant segment, sector, |
| |semicircle, subtend, tangent circles, tangent line |
|Materials and tools students should use |Protractor, straightedge, compass, calculator |
|with precision in this unit and | |
|progression: | |
|Know – Understand – Do |
|(KUD) |
|By the end of this learning progression, students will be able to… |
|UNDERSTAND |
|Big Ideas, Essential Understandings, or Generalizations |
|Properties of Circles can be described by theorems that integrate algebraic and geometric understanding, modeling, and proof. |
|Properties of Circles can be used to derive an understanding of the radian measure of an angle |
|KNOW |DO |
|Facts and Procedural Knowledge |Skills |
|Know that all circles are similar. |Identify tangents, secants, and chords |
|Know strategies for geometric constructions. |Use properties of tangents to solve problems |
|Know theorems about circles |Apply properties of chords |
|Know how the application of proportional reasoning is used to develop the concept|Apply properties of arcs |
|of radian measure. |Find the area of sectors |
| |Find arc length |
| |Find the area of a segment |
| |Find the measure of an inscribed angle |
| |Use inscribed angles and their properties to solve problems |
| |Use proportions to convert angle measures from degrees to radians |
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