GT Unit 11: Circles

Circle Geometry: Arcs, Angles, Circumference &Area

CIRCLES OVERVIEW

Go ahead and jump down to Part 1 and start taking some notes – I’ll get the rest of the info plugged in right now.

PART 1: CIRCLE ANGLES

(50 pts)  About 3 days

The first part of this unit looks at the different kinds of angles that can be constructed in and around circles.  It will include quite a bit of vocabulary, and then build into various types of calculation problems as we look at how those angles interact with one another.

PART 2: ARCS & SECTORS

(60 pts)  About 3 days

The next part of the unit is all about arc lengths and sector areas within circles.  These relationships, calculations, and properties will make it possible for us to understand how circles fit together.  This information is also super helpful when we try to build things with circles.

PART 3: CIRCLE EQUATIONS

(140 pts)  About 3 days

The final part of our unit looks at the more straight-forward calculations around circles.  We’ll focus on area and circumference, and think about how those equations relate to spheres as well!

PART 1: Circle Angles

Identifying and calculating angles within and around circles

Dilations & Similarity

The first part of our unit is all about similarity between figures and what happens when a shape undergoes the dilation transformation.  We’ll start by reviwing similarity and dilations.  Last semester we looked carefully at different translations (rotations, reflections, etc); now we get to look at mathematical operations that scale figures to different sizes.  You’ll start by taking a full page of notes on the four videos provided below.  These videos cover the essential components of similarity, and will give us a good foundation to begin our serious work in trigonometry.  In addition, you’ll work through three homework assignments on these topics before completing the part 1 Quiz 17: Similarity!

GRADING & PROCESS

 Use your engineering notebook to take a full page of notes on similarity between transformed shapes, dilations, and proving similarity

 Complete HW37: Dilations and Proof

 Add to your notes about similarity as you watch the Corresponding Parts and AA Similarity presentations

 Complete HW38: Corresponding Parts

 Complete HW39: AA Similarity

 Take Quiz 17: Similarity!

Central Angles

Finding Central Angles

Arcs & Central Angles

Inscribed Angle Theorem & Proof

Inscribed Angles 1

Inscribed Angles 2

Inscribed Angles 3

Circle Tangents

Measuring Circumscribed Angles

Circumscribed Angles 2

What’s Due In Similarity Part 1: Similarity

  • DIlations/Similarity Notes 1
  • HW37
  • Corresponding parts/AA Notes
  • HW38
  • HW39
  • Quiz 17: Similarity

Here’s what’s due in Part 1: Similarity

  1.  Take notes on dilations, transformations, and shape similarity
  2.  Complete HW37: Dilations & Proof
  3.  Next, add to your notes on similarity with information about corresponding parts and AA Similarity
  4.  Complete HW38: Corresponding Parts 
  5.  Complete HW39: AA Similarity
  6.  Take Quiz 17: Similarity!

PART 2: Arc Length & Sector Area

Calculating lengths along the circumference, and areas of wedge sectors

The second part of our unit is all about what happens when we divide triangles into smaller parts by subdividing opposing sides.  In short: if we draw a line through a triangle that is parallel to one of the sides, it creates a smaller triangle that is similar to the original triangle.  We’ll develop this idea more formally throughout the next few topics and assignments so that we can eventually turn these ideas into the six trigonometric ratios!

GRADING & PROCESS

 Use your Engineering Notebook to take a full page of notes on the Triangle Proportionality and Subdividing Triangles presentations

 Complete HW40: Triangle Proportionality

 Take one more complete page of notes on the Proportions in Triangles and Similarity in Right Triangles presentations.  Add examples from your homework to your notes on these topics!

 Complete HW41: Subdividing & Proportions

 Complete HW42: Right Triangle Similarity

 Take Quiz 18: Similar Triangles!

Arc Lengths

Arc Lengths 2

Sector Areas

Equations of Circles

Equations of Circles 2

What’s Due In Similarity Part 2: Similar Triangles

  • Triangle Proportionality Notes
  • HW40
  • Proportions & Right Triangle Notes
  • HW41
  • HW42
  • Quiz 18: Similar Triangles

Here’s what’s due in Part 2: Similar Triangles

  1.  Take notes on triangle proportionality and subdividing triangles!
  2.  Complete HW40: Triangle Proportionality
  3.  Take one more page of notes on proportions in triangles and similarity in right triangles
  4.  Complete HW41: Subdividing & Proportions
  5.  Complete HW42: Right Triangle Similarity
  6.  Take Quiz 18: Similar Triangles

PART 3: Model It!

Create a laser-cut series of models

The last part of our unit gives us the chance to build a series of laser-cut similar triangles and investigate their properties to confirm both the functions of Autodesk Inventor as well as the mathematical relationships between similar shapes!  First you’ll need to plan out your 4-triangle series in your engineering notebook, and then build the proper triangles of various sizes within Autodesk Inventor.  You’ll use the iProperties tab to record perimeter, area, and volume of each before exporting a .DXF file for use with the laser.  After you’ve laser-cut your 4 triangles, you’ll measure their perimeters, areas, and weights to see how these numbers compare to the values calculated by Autodesk.  You’ll also get to compare these values to see what they tell use mathematically about similar figures.

Of course our unit will also end with a unit review, a chance to get questions answered, and our final unit test to wrap things up!

GRADING & PROCESS

 Use your Engineering Notebook to make a plan for creating your custom-cut laser models.

 Use Autodesk Inventor to create your triangles each of the proper size.  Use the iProperties menu to record the perimeter, area, and volume of each of your four triangle models.

 Export .DXF files of your triangles and cut them on the laser out of acrylic.  Measure and record the perimeter, area, and weight of each of your four triangle models.

 Compare your measured values to those calculated by Autodesk, and to investigate the relationships between similar figures

 Complete the Unit 7 Review

 Schedule time to talk with Benshoof and get questions answered

 Take Unit Test 7: Similarity!

What’s Due In Similarity Part 3: Similar Laser Triangles!

  • Triangle Plan
  • Autodesk & iProperties
  • Laser Cut
  • Compare Triangles
  • Unit 7 Review
  • Question Time
  • Unit 7 Test

Here’s what’s due in Part 3: Build It!

  1.  Make a plan for creating your custom-cut laser models.
  2.  Use Autodesk Inventor & iProperties to create your triangles each of the proper size
  3.  Laser cut them out of acrylic & measure properties
  4.  Compare your measured values to Autodesk, and investigate the relationships
  5.  Complete the Unit 7 Review
  6.  Schedule time to talk with Benshoof and get questions answered
  7.  Take Unit Test 7: Similarity!