# Informal Geometry   (#1206300)

Version for Academic Year:

## General Course Information and Notes

### Version Description

The fundamental purpose of the course in Informal Geometry is to extend students' geometric experiences from the middle grades. Students explore more complex geometric situations and deepen their explanations of geometric relationships. Important differences exist between this Geometry course and the historical approach taken in Geometry classes. For example, transformations are emphasized early in this course. Close attention should be paid to the introductory content for the Geometry conceptual category found in the high school standards. The Standards for Mathematical Practice apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas, organized into five units are as follows.

Unit 1- Congruence, Proof, and Constructions: In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. Students informally prove theorems, using a variety of formats, and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work.

Unit 2- Similarity, Proof, and Trigonometry: Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles, with particular attention to special right triangles and the Pythagorean theorem.

Unit 3- Extending to Three Dimensions: Students' experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas.

Unit 4- Connecting Algebra and Geometry Through Coordinates: Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines, which relates back to work done in the first course.

Unit 5- Circles With and Without Coordinates: In this unit students study the Cartesian coordinate system and use the distance formula to write the equation of a circle when given the radius and the coordinates of its center. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations, which relates back to work done in the first course, to determine intersections between lines and circles or parabolas.

### General Notes

Important Note: This Informal Geometry course content does not align with the End-of-Course Assessment required for graduation.

English Language Development ELD Standards Special Notes Section:
Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade level words, expressions, sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers and teachers which maximizes an ELL's need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please click on the following link:
http://www.cpalms.org/uploads/docs/standards/eld/MA.pdf

For additional information on the development and implementation of the ELD standards, please contact the Bureau of Student Achievement through Language Acquisition at sala@fldoe.org.

Additional Instructional Resources:
A.V.E. for Success Collection is provided by the Florida Association of School Administrators: http://www.fasa.net/4DCGI/cms/review.html?Action=CMS_Document&DocID=139. Please be aware that these resources have not been reviewed by CPALMS and there may be a charge for the use of some of them in this collection.

### General Information

Course Number: 1206300
Course Path:
Abbreviated Title: INF GEO
Number of Credits: One credit (1)
Course Length: Year (Y)
Course Type: Core Academic Course
Course Level: 2
Course Status: Course Approved
Grade Level(s): 9,10,11,12

## Educator Certifications

One of these educator certification options is required to teach this course.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this course.

## Original Student Tutorials

The Year-Round School Debate: Identifying Faulty Reasoning â€” Part Two:

Practice identifying faulty reasoning in this two-part, interactive, English Language Arts tutorial. You'll learn what some experts say about year-round schools, what research has been conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument, specifically the use of hasty generalizations.

Make sure to complete Part One before Part Two! Click HERE to launch Part One.

Type: Original Student Tutorial

The Year-Round School Debate: Identifying Faulty Reasoning â€“ Part One:

Learn to identify faulty reasoning in this two-part interactive English Language Arts tutorial. You'll learn what some experts say about year-round schools, what research has been conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument, specifically the use of hasty generalizations.

Make sure to complete both parts of this series! Click HERE to open Part Two.

Type: Original Student Tutorial

Evaluating an Argument â€“ Part Four: JFKâ€™s Inaugural Address:

Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence.

In Part Four, you'll use what you've learned throughout this series to evaluate Kennedy's overall argument.

Make sure to complete the previous parts of this series before beginning Part 4.

• Click HERE to launch Part One.
• Click HERE to launch Part Two.
• Click HERE to launch Part Three.

Type: Original Student Tutorial

Evaluating an Argument â€“ Part Three: JFKâ€™s Inaugural Address:

Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence. By the end of this four-part series, you should be able to evaluate his overall argument.

In Part Three, you will read more of Kennedy's speech and identify a smaller claim in this section of his speech. You will also evaluate this smaller claim's relevancy to the main claim and evaluate Kennedy's reasons and evidence.

Make sure to complete all four parts of this series!

• Click HERE to launch Part One.
• Click HERE to launch Part Two.
• Click HERE to launch Part Four.

Type: Original Student Tutorial

High Tech Seesaw:

Learn how to find the point on a directed line segment that partitions it into a given ratio in this interactive tutorial.

Type: Original Student Tutorial

Ready for Takeoff! -- Part Two:

Want to learn about Amelia Earhart, one of the most famous female aviators of all time? If so, then this interactive tutorial is for YOU! This tutorial is Part Two of a two-part series. In this series, you will study a speech by Amelia Earhart. You will practice identifying the purpose of her speech and practice identifying her use of rhetorical appeals (ethos, logos, pathos, Kairos). You will also evaluate the effectiveness of Earhart's rhetorical choices based on the purpose of her speech.

Please complete Part One before beginning Part Two. Click HERE to view Part One.

Type: Original Student Tutorial

Ready for Takeoff! -- Part One:

Want to learn about Amelia Earhart, one of the most famous female aviators of all time? If so, then this interactive tutorial is for YOU! This tutorial is Part One of a two-part series. In this series, you will study a speech by Amelia Earhart. You will practice identifying the purpose of her speech and practice identifying her use of rhetorical appeals (ethos, logos, pathos, Kairos). You will also evaluate the effectiveness of Earhart's rhetorical choices based on the purpose of her speech.

Please complete Part Two after completing this tutorial. Click HERE to view Part Two.

Type: Original Student Tutorial

Untangling Food Webs:

Learn how living organisms can be organized into food webs and how energy is transferred through a food web from producers to consumers to decomposers. This interactive tutorial also includes interactive knowledge checks.

Type: Original Student Tutorial

Comparing Mitosis and Meiosis:

Compare and contrast mitosis and meiosis in this interactive tutorial. You'll also relate them to the processes of sexual and asexual reproduction and their consequences for genetic variation.

Type: Original Student Tutorial

Ninja Nancy Slices:

Learn how to determine the shape of a cross section created by the intersection of a slicing plane with a pyramid or prism. This task is vital to those that work to create three dimensional objects.  Whether it is the inventor of a new toy or the architect of your next house, they must be able to convey their design on paper.  The drawings they make represent various cross sections of the finished product. Can you visualize the relationships between two-dimensional and three-dimensional objects? Imagine that Ninja Nancy will slice through this pyramid with her sword.  What two-dimensional figures will she reveal?

Type: Original Student Tutorial

I Scream! You Scream! We All Scream for... Volume!:

Have you ever ordered a scoop of ice cream in a cone and wondered how much ice cream actually fits inside the cone? By the end of this tutorial, you should be able to answer this question and solve other real-world problems by using the formula for the volume of a cone.

Type: Original Student Tutorial

Cancer: Mutated Cells Gone Wild!:

Explore the relationship between mutations, the cell cycle, and uncontrolled cell growth which may result in cancer with this interactive tutorial.

Type: Original Student Tutorial

Climbing Around the Hominin Family Tree:

Learn to identify basic trends in the evolutionary history of humans, including walking upright, brain size, jaw size, and tool use in "Climbing Around the Hominin Family Tree" online tutorial.

Type: Original Student Tutorial

## Educational Games

Volume of a Cylinder:

Students can play a game to solve for the volume of a cylinder and work backwards using the volume to find measures of a cylinder. Students may select Teach Me to learn how to find the volume of a cylinder or measures of it prior to beginning play. Hints and feedback are provided to players.

Type: Educational Game

Rotation on Coordinate Axes:

Students will play a game to discover the meaning of rotation. Students will use angle of rotation, direction and location of the center of rotation to turn a shape on a coordinate grid. Students may select Teach Me to learn about rotations prior to beginning play. Hints and feedback are provided to players.

Type: Educational Game

Population Density:

Play a game about density and population. Students may select Teach Me to learn about the density formula prior to beginning play. Hints and feedback are provided to players.

Type: Educational Game

Circle Up!:

This interactive game helps you learn about angles and segments, lines and arcs in a circle and how they are related. You will compete against yourself and earn points as you answer questions about radius, diameter, chord, tangent line, central angles and inscribed angles and intercepted arcs.

Type: Educational Game

## Educational Software / Tool

Transformations Using Technology:

This virtual manipulative can be used to demonstrate and explore the effect of translation, rotation, and/or reflection on a variety of plane figures. A series of transformations can be explored to result in a specified final image.

Type: Educational Software / Tool

## Perspectives Video: Experts

Mathematically Exploring the Wakulla Caves:

The tide is high!  How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?

Type: Perspectives Video: Expert

MicroGravity Sensors & Statistics:

Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Making Candy: Uniform Scaling:

Don't be a shrinking violet. Learn how uniform scaling is important for candy production.

Type: Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.

Type: Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

Bank Shot:

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.

Type: Problem-Solving Task

Are They Similar?:

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Type: Problem-Solving Task

Exstensions, Bisections and Dissections in a Rectangle:

This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.

Type: Problem-Solving Task

Toilet Roll:

The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints.

Type: Problem-Solving Task

Coins in a circular pattern:

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin.

Type: Problem-Solving Task

The Lighthouse Problem:

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Type: Problem-Solving Task

Solar Eclipse:

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Type: Problem-Solving Task

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Type: Problem-Solving Task

Seven Circles III:

This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.

Type: Problem-Solving Task

Dilating a Line:

This task asks students to make deductions about a line after it has been dilated by a factor of 2.

Type: Problem-Solving Task

Running around a track II:

The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.

Type: Problem-Solving Task

Running around a track I:

In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.

Type: Problem-Solving Task

Paper Clip:

In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.

Type: Problem-Solving Task

Ice Cream Cone:

In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

Type: Problem-Solving Task

How thick is a soda can? (Variation II):

This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can.

Type: Problem-Solving Task

How thick is a soda can? (Variation I):

This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.

Type: Problem-Solving Task

How many leaves on a tree? (Version 2):

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

Type: Problem-Solving Task

How many leaves on a tree?:

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

Type: Problem-Solving Task

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Type: Problem-Solving Task

Hexagonal pattern of beehives:

The goal of this task is to use geometry to study the structure of beehives.

Type: Problem-Solving Task

Global Positioning System II:

Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

Type: Problem-Solving Task

Eratosthenes and the circumference of the earth:

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

Type: Problem-Solving Task

Archimedes and the King's Crown:

This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

Type: Problem-Solving Task

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Type: Problem-Solving Task

Doctor's Appointment:

The purpose of the task is to analyze a plausible real-life scenario using a geometric model. The task requires knowledge of volume formulas for cylinders and cones, some geometric reasoning involving similar triangles, and pays attention to reasonable approximations and maintaining reasonable levels of accuracy throughout.

Type: Problem-Solving Task

Why Does ASA Work?:

This problem solving task ask students to show the reflection of one triangle maps to another triangle.

Type: Problem-Solving Task

When Does SSA Work to Determine Triangle Congruence?:

In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.

Type: Problem-Solving Task

Seven Circles II:

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

Type: Problem-Solving Task

Why Does SAS Work?:

This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points.

Type: Problem-Solving Task

Reflections and Isosceles Triangles:

This activity uses rigid transformations of the plane to explore symmetries of classes of triangles.

Type: Problem-Solving Task

Reflections and Equilateral Triangles II:

This task gives students a chance to see the impact of reflections on an explicit object and to see that the reflections do not always commute.

Type: Problem-Solving Task

Reflections and Equilateral Triangles:

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles

Type: Problem-Solving Task

Centerpiece:

The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario. In particular, students are in several places (implicitly or explicitly) to reason as to when making approximations is reasonable and when to round, when to use equalities vs. inequalities, and the choice of units to work with (e.g., mm vs. cm).

Type: Problem-Solving Task

Use Cavalieriâ€™s Principle to Compare Aquarium Volumes:

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

Type: Problem-Solving Task

Tennis Balls in a Can:

This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder

Type: Problem-Solving Task

Two Wheels and a Belt:

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Type: Problem-Solving Task

Right triangles inscribed in circles II:

This problem solving task asks students to explain certain characteristics about a triangle.

Type: Problem-Solving Task

Right triangles inscribed in circles I:

This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle: the fact that these triangles are always right triangles is often referred to as Thales' theorem.

Type: Problem-Solving Task

Why does SSS work?:

This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation.

Type: Problem-Solving Task

Building a tile pattern by reflecting octagons:

This task applies reflections to a regular octagon to construct a pattern of four octagons enclosing a quadrilateral: the focus of the task is on using the properties of reflections to deduce that the quadrilateral is actually a square.

Type: Problem-Solving Task

Building a tile pattern by reflecting hexagons:

This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern.

Type: Problem-Solving Task

Are the Triangles Congruent?:

The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles.

Type: Problem-Solving Task

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Type: Problem-Solving Task

Neglecting the Curvature of the Earth:

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Type: Problem-Solving Task

Regular Tessellations of the Plane:

This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.

Type: Problem-Solving Task

## Tutorials

Finding congruent triangles:

In this tutorial, students will use the SSS, ASA, SAS, and AAS postulates to find congruent triangles

Type: Tutorial

Dilation and scale factor:

In this tutorial, students will use a scale factor to dilate one line onto another.

Type: Tutorial

Using SSS in a proof:

This tutorial discusses the difference between a theorem and axiom. It also shows how to use SSS in a proof.

Type: Tutorial

Triangle congruence postulates:

This tutorial discusses SSS, SAS, ASA and AAS postulates for congruent triangles. It also shows AAA is only good for similarity and SSA is good for neither.

Type: Tutorial

Congruent Triangles and SSS:

In this video, students will learn about congruent triangles and the "Side-Side-Side" postulate.

Type: Tutorial

Rotating polygons 180 degrees about their center:

Students will investigate symmetry by rotating polygons 180 degrees about their center.

Type: Tutorial

Line of reflection:

Students are shown, with an interactive tool, how to reflect a line segment. Students should have an understanding of slope and midpoint before viewing this video.

Type: Tutorial

Line of reflection:

This tutorial uses the midpoint of two lines to find the line of reflection.

Type: Tutorial

Points after rotation:

Students will see what happens when a figure is rotated about the origin -270 degrees. Having a foundation about right triangles is recommended before viewing this video.

Type: Tutorial

Specifying planes in three dimensions:

In this tutorial, students are introduced to the concept that three non-collinear points are necessary to define a unique plane.

Type: Tutorial

The language of geometry:

Before learning any new concept it's important students learn and use common language and label concepts consistently. This tutorial introduces students to th point, line and plane.

Type: Tutorial

Identifying parallel and perpendicular lines:

This tutorial is great practice for help in identifying parallel and perpendicular lines.

Type: Tutorial

Basic Geometry Language and Labels:

<p>In this tutorial we will learn the basics of geometry, such as identifying a line, ray, point, and segment.</p>

Type: Tutorial

Skillswise Speaking: Communication Guidelines:

In this tutorial from the BBC, you will learn how to organize and express your ideas. The tutorial includes a short video, multi-level tutorial options, worksheets and answer keys, a game, and interactive quizzes to help you share your opinions in formal and informal situations and participate in a debate responding to others' views. After watching the video, simply scroll over the tabs to the right of the video to select your next activity.

Type: Tutorial

Skillswise Speaking: Giving a Presentation:

In this tutorial from the BBC, you will learn how to organize and express your ideas through the delivery of formal presentations. The tutorial includes a short video, multi-level tutorial options, worksheets and answer keys, a game, and interactive quizzes to learn tips for presenting information, staying on topic, and determining what information is relevant to share with a group. After watching the video, simply scroll over the tabs to the right of the video to select your next activity.

Type: Tutorial

## Video/Audio/Animation

Parallel Lines:

This video illustrates how to determine if the graphs of a given set of equations are parallel.

Type: Video/Audio/Animation

## Virtual Manipulatives

Which Holds More? :

This interactive manipulative will let you compare and calculate volumes of different solids.

Type: Virtual Manipulative

3-D Conic Section Explorer:

Using this resource, students can manipulate the measurements of a 3-D hourglass figure (double-napped cone) and its intersecting plane to see how the graph of a conic section changes.  Students will see the impact of changing the height and slant of the cone and the m and b values of the plane on the shape of the graph. Students can also rotate and re-size the cone and graph to view from different angles.

Type: Virtual Manipulative

Combining Transformations:

In this manipulative activity, you can first get an idea of what each of the rigid transformations look like, and then get to experiment with combinations of transformations in order to map a pre-image to its image.

Type: Virtual Manipulative

Space Blocks:

This virtual manipulative allows students to manipulate blocks, add or remove blocks, and connect them together to form solids. They can also experiment with counting the number of exposed faces, seeing what happens to the surface area when blocks are added or removed, and "unfolding" a block to create a net .

Type: Virtual Manipulative

Cross Section Flyer - Shodor:

With this online Java applet, students use slider bars to move a cross section of a cone, cylinder, prism, or pyramid. This activity allows students to explore conic sections and the 3-dimensional shapes from which they are derived. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Congruent Triangles:

This manipulative is a virtual realization of the kind of physical experience that might be available to students given three pieces of straws and told to make them into a triangle. when working with pieces that determine unique triangles (SSS, SAS, ASA). Students construct triangles with the parts provided. After building a red and a blue triangle, students can experience congruence by actually moving one on the top of the other.

Type: Virtual Manipulative

Transformations - Translation:

The user can demonstrate or explore translation of shapes created with pattern blocks, using or not using a coordinate axes and lattice points background, by changing the translation vector.
(source: NLVM grade 6-8 "Transformations - Translation")

Type: Virtual Manipulative

Transformations - Reflections:

The user clicks and drags a shape they have constructed to view its reflection across a line. A background grid and axes may or may not be used. The reflection may by examined analytically using coordinates. Symmetry may be displayed.

Type: Virtual Manipulative

Transformations - Dilation:

Students use a slider to explore dilation and scale factor. Students can create and dilate their own figures. (source: NLVM grade 6-8 "Transformations - Dilation")

Type: Virtual Manipulative

Transformations - Rotation:

Rotate shapes and their images with or without a background grid and axes.

Type: Virtual Manipulative

A Plethora of Polyhedra:

This program allows users to explore spatial geometry in a dynamic and interactive way. The tool allows users to rotate, zoom out, zoom in, and translate a plethora of polyhedra. The program is able to compute topological and geometrical duals of each polyhedron. Geometrical operations include unfolding, plane sections, truncation, and stellation.

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this course.

## Perspectives Video: Professional/Enthusiasts

Making Candy: Uniform Scaling:

Don't be a shrinking violet. Learn how uniform scaling is important for candy production.

Type: Perspectives Video: Professional/Enthusiast

Using Geometry and Computers to make Art with CNC Machining:

See and see far into the future of arts and manufacturing as a technician explains computer numerically controlled (CNC) machining bit by bit.

Type: Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

Bank Shot:

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.

Type: Problem-Solving Task

Are They Similar?:

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Type: Problem-Solving Task

Exstensions, Bisections and Dissections in a Rectangle:

This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.

Type: Problem-Solving Task

Toilet Roll:

The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints.

Type: Problem-Solving Task

Coins in a circular pattern:

Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin.

Type: Problem-Solving Task

The Lighthouse Problem:

This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.

Type: Problem-Solving Task

Solar Eclipse:

This problem solving task encourages students to explore why solar eclipses are rare by examining the radius of the sun and the furthest distance between the moon and the earth.

Type: Problem-Solving Task

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Type: Problem-Solving Task

Seven Circles III:

This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.

Type: Problem-Solving Task

Dilating a Line:

This task asks students to make deductions about a line after it has been dilated by a factor of 2.

Type: Problem-Solving Task

Running around a track II:

The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.

Type: Problem-Solving Task

Running around a track I:

In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.

Type: Problem-Solving Task

Paper Clip:

In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.

Type: Problem-Solving Task

Ice Cream Cone:

In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

Type: Problem-Solving Task

How thick is a soda can? (Variation II):

This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can.

Type: Problem-Solving Task

How thick is a soda can? (Variation I):

This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.

Type: Problem-Solving Task

How many leaves on a tree? (Version 2):

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

Type: Problem-Solving Task

How many leaves on a tree?:

This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

Type: Problem-Solving Task

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Type: Problem-Solving Task

Hexagonal pattern of beehives:

The goal of this task is to use geometry to study the structure of beehives.

Type: Problem-Solving Task

Global Positioning System II:

Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

Type: Problem-Solving Task

Eratosthenes and the circumference of the earth:

This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.

Type: Problem-Solving Task

Archimedes and the King's Crown:

This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

Type: Problem-Solving Task

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Type: Problem-Solving Task

Doctor's Appointment:

The purpose of the task is to analyze a plausible real-life scenario using a geometric model. The task requires knowledge of volume formulas for cylinders and cones, some geometric reasoning involving similar triangles, and pays attention to reasonable approximations and maintaining reasonable levels of accuracy throughout.

Type: Problem-Solving Task

Why Does ASA Work?:

This problem solving task ask students to show the reflection of one triangle maps to another triangle.

Type: Problem-Solving Task

When Does SSA Work to Determine Triangle Congruence?:

In this problem, we considered SSA. The triangle congruence criteria, SSS, SAS, ASA, all require three pieces of information. It is interesting, however, that not all three pieces of information about sides and angles are sufficient to determine a triangle up to congruence.

Type: Problem-Solving Task

Seven Circles II:

This task provides a concrete geometric setting in which to study rigid transformations of the plane.

Type: Problem-Solving Task

Why Does SAS Work?:

This problem solving task challenges students to explain the reason why the given triangles are congruent, and to construct reflections of the points.

Type: Problem-Solving Task

Reflections and Isosceles Triangles:

This activity uses rigid transformations of the plane to explore symmetries of classes of triangles.

Type: Problem-Solving Task

Reflections and Equilateral Triangles II:

This task gives students a chance to see the impact of reflections on an explicit object and to see that the reflections do not always commute.

Type: Problem-Solving Task

Reflections and Equilateral Triangles:

This activity is one in a series of tasks using rigid transformations of the plane to explore symmetries of classes of triangles, with this task in particular focusing on the class of equilaterial triangles

Type: Problem-Solving Task

Centerpiece:

The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario. In particular, students are in several places (implicitly or explicitly) to reason as to when making approximations is reasonable and when to round, when to use equalities vs. inequalities, and the choice of units to work with (e.g., mm vs. cm).

Type: Problem-Solving Task

Use Cavalieriâ€™s Principle to Compare Aquarium Volumes:

This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

Type: Problem-Solving Task

Tennis Balls in a Can:

This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder

Type: Problem-Solving Task

Two Wheels and a Belt:

This task combines two skills: making use of the relationship between a tangent segment to a circle and the radius touching that tangent segment, and computing lengths of circular arcs given the radii and central angles.

Type: Problem-Solving Task

Right triangles inscribed in circles II:

This problem solving task asks students to explain certain characteristics about a triangle.

Type: Problem-Solving Task

Right triangles inscribed in circles I:

This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles inscribed in a circle: the fact that these triangles are always right triangles is often referred to as Thales' theorem.

Type: Problem-Solving Task

Why does SSS work?:

This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation.

Type: Problem-Solving Task

Building a tile pattern by reflecting octagons:

This task applies reflections to a regular octagon to construct a pattern of four octagons enclosing a quadrilateral: the focus of the task is on using the properties of reflections to deduce that the quadrilateral is actually a square.

Type: Problem-Solving Task

Building a tile pattern by reflecting hexagons:

This task applies reflections to a regular hexagon to construct a pattern of six hexagons enclosing a seventh: the focus of the task is on using the properties of reflections to deduce this seven hexagon pattern.

Type: Problem-Solving Task

Are the Triangles Congruent?:

The purpose of this task is primarily assessment-oriented, asking students to demonstrate knowledge of how to determine the congruency of triangles.

Type: Problem-Solving Task

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

Type: Problem-Solving Task

Neglecting the Curvature of the Earth:

This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.

Type: Problem-Solving Task

Regular Tessellations of the Plane:

This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.

Type: Problem-Solving Task

## Video/Audio/Animation

Story of Pi:

This video dynamically shows how Pi works, and how it is used.

Type: Video/Audio/Animation

## Virtual Manipulatives

Space Blocks:

This virtual manipulative allows students to manipulate blocks, add or remove blocks, and connect them together to form solids. They can also experiment with counting the number of exposed faces, seeing what happens to the surface area when blocks are added or removed, and "unfolding" a block to create a net .

Type: Virtual Manipulative

Congruent Triangles:

This manipulative is a virtual realization of the kind of physical experience that might be available to students given three pieces of straws and told to make them into a triangle. when working with pieces that determine unique triangles (SSS, SAS, ASA). Students construct triangles with the parts provided. After building a red and a blue triangle, students can experience congruence by actually moving one on the top of the other.

Type: Virtual Manipulative

Transformations - Translation:

The user can demonstrate or explore translation of shapes created with pattern blocks, using or not using a coordinate axes and lattice points background, by changing the translation vector.
(source: NLVM grade 6-8 "Transformations - Translation")

Type: Virtual Manipulative

Transformations - Dilation:

Students use a slider to explore dilation and scale factor. Students can create and dilate their own figures. (source: NLVM grade 6-8 "Transformations - Dilation")

Type: Virtual Manipulative

Transformations - Rotation:

Rotate shapes and their images with or without a background grid and axes.

Type: Virtual Manipulative