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Course Standards
General Course Information and Notes
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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:
https://cpalmsmediaprod.blob.core.windows.net/uploads/docs/standards/eld/ma.pdf
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
Educator Certifications
Student Resources
Original Student Tutorials
This is Part Two of a two-part series. Learn to identify faulty reasoning in this interactive tutorial series. 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
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
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.
Type: Original Student Tutorial
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!
Type: Original Student Tutorial
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
This is Part Two of a two-part tutorial series. In this interactive tutorial, you'll practice identifying a speaker's purpose using a speech by aviation pioneer Amelia Earhart. You will examine her use of rhetorical appeals, including ethos, logos, pathos, and kairos. Finally, you'll evaluate the effectiveness of Earhart's use of rhetorical appeals.
Be sure to complete Part One first. Click here to launch PART ONE.
Type: Original Student Tutorial
This is Part One of a two-part tutorial series. In this interactive tutorial, you'll practice identifying a speaker's purpose using a speech by aviation pioneer Amelia Earhart. You will examine her use of rhetorical appeals, including ethos, logos, pathos, and kairos. Finally, you'll evaluate the effectiveness of Earhart's use of rhetorical appeals.
Click here to launch PART TWO.
Type: Original Student Tutorial
Practice writing different aspects of an expository essay about scientists using drones to research glaciers in Peru. This interactive tutorial is part four of a four-part series. In this final tutorial, you will learn about the elements of a body paragraph. You will also create a body paragraph with supporting evidence. Finally, you will learn about the elements of a conclusion and practice creating a “gift.”
This tutorial is part four of a four-part series. Click below to open the other tutorials in this series.
- Drones and Glaciers: Eyes in the Sky (Part 1)
- Drones and Glaciers: Eyes in the Sky (Part 2)
- Expository Writing: Eyes in the Sky (Part 3)
- Expository Writing: Eyes in the Sky (Part 4)
Type: Original Student Tutorial
Learn how to write an introduction for an expository essay in this interactive tutorial. This tutorial is the third part of a four-part series. In previous tutorials in this series, students analyzed an informational text and video about scientists using drones to explore glaciers in Peru. Students also determined the central idea and important details of the text and wrote an effective summary. In part three, you'll learn how to write an introduction for an expository essay about the scientists' research.
This tutorial is part three of a four-part series. Click below to open the other tutorials in this series.
- Drones and Glaciers: Eyes in the Sky (Part 1)
- Drones and Glaciers: Eyes in the Sky (Part 2)
- Expository Writing: Eyes in the Sky (Part 3)
- Expository Writing: Eyes in the Sky (Part 4)
Type: Original Student Tutorial
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
Learn how to determine the shape of a cross-section created by the intersection of a slicing plane with a pyramid or prism in this ninja-themed, interactive tutorial.
Type: Original Student Tutorial
Learn to calculate the volume of a cone as you solve real-world problems in this ice cream-themed, interactive tutorial.
Type: Original Student Tutorial
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
Educational Software / Tool
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
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?
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Expert
Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Expert
Perspectives Video: Professional/Enthusiasts
Don't be a shrinking violet. Learn how uniform scaling is important for candy production.
Type: Perspectives Video: Professional/Enthusiast
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
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
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
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
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
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
Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)
Type: Problem-Solving Task
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
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
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
This task asks students to make deductions about a line after it has been dilated by a factor of 2.
Type: Problem-Solving Task
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
In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.
Type: Problem-Solving Task
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
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
This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. Multiple solution processes are presented.
Type: Problem-Solving Task
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
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
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
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
The goal of this task is to use geometry to study the structure of beehives.
Type: Problem-Solving Task
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
This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.
Type: Problem-Solving Task
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
This problem solving task asks students to find the area of a triangle by using unit squares and line segments.
Type: Problem-Solving Task
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
This problem solving task ask students to show the reflection of one triangle maps to another triangle.
Type: Problem-Solving Task
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
This task provides a concrete geometric setting in which to study rigid transformations of the plane.
Type: Problem-Solving Task
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
This activity uses rigid transformations of the plane to explore symmetries of classes of triangles.
Type: Problem-Solving Task
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
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
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
This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.
Type: Problem-Solving Task
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
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
This problem solving task asks students to explain certain characteristics about a triangle.
Type: Problem-Solving Task
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
This particular problem solving task exhibits congruency between two triangles, demonstrating translation, reflection and rotation.
Type: Problem-Solving Task
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
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
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
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
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
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
In this tutorial, students will use the SSS, ASA, SAS, and AAS postulates to find congruent triangles
Type: Tutorial
In this tutorial, students will use a scale factor to dilate one line onto another.
Type: Tutorial
This tutorial discusses the difference between a theorem and axiom. It also shows how to use SSS in a proof.
Type: Tutorial
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
In this video, students will learn about congruent triangles and the "Side-Side-Side" postulate.
Type: Tutorial
Students will investigate symmetry by rotating polygons 180 degrees about their center.
Type: Tutorial
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
This tutorial uses the midpoint of two lines to find the line of reflection.
Type: Tutorial
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
In this tutorial, students are introduced to the concept that three non-collinear points are necessary to define a unique plane.
Type: Tutorial
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
This tutorial is great practice for help in identifying parallel and perpendicular lines.
Type: Tutorial
In this tutorial we will learn the basics of geometry, such as identifying a line, ray, point, and segment.
Type: Tutorial
Video/Audio/Animation
This video illustrates how to determine if the graphs of a given set of equations are parallel.
Type: Video/Audio/Animation
Virtual Manipulatives
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
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
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
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
Congratulations
