Unit 4 - circles & volume
Unit 4: Students will understand and apply theorems about circles, find arc lengths of circles, and find areas of sectors of circles. Students will develop and explain formulas related to circles and the volume of solid figures and use the formulas to solve problems. Building on standards from middle school, students will extend the study of identifying cross-sections of three-dimensional shapes to identifying three-dimensional objects generated by rotations of two-dimensional objects.
standards
Understand and apply theorems about circles -
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. Find arc lengths and areas of sectors of circles
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
Explain volume formulas and use them to solve problems -
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. Visualize relationships between two-dimensional and three-dimensional objects
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
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. Find arc lengths and areas of sectors of circles
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
Explain volume formulas and use them to solve problems -
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. Visualize relationships between two-dimensional and three-dimensional objects
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
Unit 4 vocabulary
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IXL circle vocabulary practice
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Reminder: for test bonus points...all vocabulary must be neatly defined with a picture. In IXL you must have a score of at least an 80 and have completed 20 questions. This is a +0 or +10 point opportunity. If you did anything less than defining all vocabulary with a picture and the IXL practice with at least a smartscore of 80 and the minimum of 20 question you will earn a 0 for this bonus point opportunity. DUE Friday at noon, no late work will be accepted. No partial work will be accepted. To complete IXL, you must be logged in with your Carroll County account! You will submit a screen capture of each part of the assignment in order to earn the bonus points.
Parts of a circle video
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Parts of a circle IXL practice
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IXL practice
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IXL practice
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Review of area and circumference of a circle
Try me!
Find 2 of these problems to copy, work out, and answer correctly
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Length of a sector and area of a sector
Try me too!
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Circle formulas, lines intersection inside & outside
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Circle formula (part 1 of 3)
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Circle formula (part 2 of 3)
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Circle formula (part 3 of 3)
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Try a minimum of 10 problems
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Circle and chord, segment, diameter, radius intersections
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Read the following about cross-sections of solids
Practice what cross-sections look like in figures
Geogebra 2-d and 3d rotation
How Cavalieri's principle can be used to find volume
Practice with Cavalieri's principle
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Cavalier's principle
Bonaventura Cavalieri - find a fact or two
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IXL – Identity the 3-dimensional shape
3 glass volume challenge
Try 10 composite volume problems.
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IXL – Calculate the volume, try 2 from each “skill”
3 volume problems
IXL - work with cross-sections
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Practice with segments and circles
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Try theses, there are 22!
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Volume practice
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Compound volume practice
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Do you recall your circles vocab?
Math options for next year:
Ideas to pick the correct math/s for your interests/career:
Think: what am I good at, what degree do I want to obtain, what career fields interest me, how much is my training/degree going to cost.....
Do NOT Think: what is my bff doing, what is my group of friends doing, money grows on trees and college is free....Have a seat and put your seatbelt on for a reality check (if you think like this)...your bff is not going to help you pay your bills, buy you a car, pay for your house/rent, and feed your kids.
Think: what am I good at, what degree do I want to obtain, what career fields interest me, how much is my training/degree going to cost.....
Do NOT Think: what is my bff doing, what is my group of friends doing, money grows on trees and college is free....Have a seat and put your seatbelt on for a reality check (if you think like this)...your bff is not going to help you pay your bills, buy you a car, pay for your house/rent, and feed your kids.
AP Calculus preview
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AP Statistics preview
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Honors PreCalculus preview
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Honors Algebra 2 preview
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