OpenLLM-France/wikimedia huggingface dataset:
# Ideas in Geometry/Constructible Polygons and Impossibilities

Lesson Twenty-Nine: Constructible Polygons and Impossibilities
While a compass and a straightedge allow for almost any constructible polygons, there are several impossible constructions. There are three main problems that are known as the “impossibilities” which cannot be solved with a compass and a straightedge.
The first impossibility is doubling the cube.
The goal of doubling the constructed cube is to double its volume. To do this, we must construct the roots of the equation.
To demonstrate this, we can begin with a cube that has a volume of 1. In order to “double the cube” we must take v=1*2 (v=2). This means we must construct a cube with a volume of 2. For a cube with this volume, each side length must equal ${\sqrt[{3}]{2}}.$ We know that the cube root of any number is not constructible so therefore we cannot double the cube. This is true for all volumes of a cube because each will have $x^{3}$ in its equation.
The second impossibility is squaring the circle.
The goal of squaring a constructed circle is to construct a circle with a given radius and then construct a square with the same area as the original circle.
To demonstrate this, we can begin with a circle that has a radius of 1 unit. Area of a circle is A=πr^2 so we know that the area of this circle is A=π(1^2) or A=π. In order to square the circle, we must construct a square with an area of π. We know that π is not a constructible number so this square cannot be constructed. This is true for all constructed circles because each will have π in its radius.
The third impossibility is trisecting an angle.
The goal of this construction is to trisect an angle. Though bisecting an angle is simple enough, to trisect one is impossible. Trisecting an angle is defined as an impossibility because there is no possible way to do it with just a compass and a straightedge. Though the concept of trisecting an angle cannot be constructed, there are certain angles in which it is possible to trisect.
One of these angles is a 45° angle. Given an angle ABC that equals 45°, we can construct an equilateral triangle at point B. This gives us a new angle DBE that is equal to 60°. Now all we need to do is bisect the 60° angle, giving us one 15° angle (formed between the 45° angle and the 60° angle) and a 30° angle. The last step is to bisect the 30° angle giving us three 15° angles, thus successfully trisecting a 45° angle.
# Ideas in Geometry/Constructions II

4.2 Trickier Constructions
This lesson focuses on constructing triangles. please note that some of the videos use a protractor to measure angles, we are using a compass to copy the angles.
We focus on four main constructions for triangles. This section gives us different constructions to make tirangles when given different parts of the triangle These constructions are done when you are given three different pieces of information. They can be completed using a compass and straight edge.
Side Angle Side (SAS Triangle)
This construction should be used when you are given a side, an angle adjacent to that side, and a side adjacent to that angle. Its like an angle sandwich! 1. Transfer one side that that it starts at the vertex of the angle 2. Transfter the other side so that it starts at the vertex 3. Connect the end points of all moved line segments
SAS video
SIde Side Side - you are given all three sides of the triangle, and no angles.
1. choose a side and select one of its endpoints 2. Draw a circle of radius equal to the length of the seond side aorund the choses end point 3. draw a circle of radius equal o the length of the third side around the other end point 4. connect the end pointd of the first side and the interection of the circles. This is the desired triangle
here is an example of how to do this side side side video
Side Angle Angle
1. start with given side and place adjacent angle at one of its end points 2. move second angle so that it shares a leg with the leg of the first angle- not the leg with the side 3. extend the side past the first angle, forming a new angle with the leg of the second angle 4. move this new angle to the other end point of the slide, extending the legs of this angle and the first angle will produce the desired triangle you are given a side, the angle adjacent, and the angle
here is an example video SAA
Angle Side Side is a construction to be careful of. it can produce more than one triangle, therefore making it an unreliable construction.
# Ideas in Geometry/Euclid's Axioms

Topic:Ideas_in_Geometry
# Ideas in Geometry/Instructive examples

This page will serve as an index of all problems worked out as instructive examples of the concepts involved in the course Ideas in Geometry/.
List the examples here as links.

## Group Examples

Proving the Pythagorean Theorem By Picture/
Pythagorean Theorem Proof by Picture-Leonardo da Vinci Proof
Proving the Area of a Quadrilateral
Proving the Area of a Trapezoid
Proving Parallel Bisectors in Quadrilaterals
Proving Pythagorean Theorem Using Similar Angels

## Individual Examples

/Section 1.1/#3
/Section 3.2/#7
/Section 1.3/#19
/Section 1.3/#21
Finding a Midset (21 from 1.3)
/Section 3.2/#26
Section 3.2 Number 28
/3.2/#44
/Section 2.1/#6
/Section 3.1/#23
9 from 2.1, Proof by Picture using Venn Diagrams
/Problem/#22 from Section 1.3, Drawing a Parabola
Section 2.1 Basic Set Theory number 10
Section 2.3 #4
Section 2.4 problem 23
/Section 2.1/#7
/Section 1.2/#11
/Continued Fractions, Section 3.2/#30
/Proving Sets Equal, Section 2.1/#11
/If-Then Truth Tables, Section 2.2/#3
/Proof by Vienn Diagrams Section 2.1/#5
/Section 2.2/#2
Section 2-3/#3 Tessellating Nonregular Hexagons
/Section 2.3/#6
Monty Hall Lesson
/Section 2.4/#14
/Section 2.4:/#24
/Section 2.1/#12
Section 3.2 Problem 42 - Lattice Area, Pick's Theorem, and Probability
Section 3.2 Problem 7-Using Brahmagupta's Formula
/Section 1.2 problem/#17
Lesson 10 logic
Lesson 17- Properties of Triangles
Lesson 15: Tessellations
Lesson 6: Introduction to City Geometry
Lesson 4: Algebraic Geometry
Lesson twenty Eight: Fields
Lesson 1: Euclid's Axioms
Lesson 21: Wacky Dice
# Ideas in Geometry/Instructive examples/9 from 2.1, Proof by Picture using Venn Diagrams

Prove that: (X-Y) u (Y-X)=(X u Y)-(X n Y)
First, look at each side of the equation separately. The first part of the left side is (X-Y). This means X complement Y, which is the set of all the elements that are in X and are not in Y. X will be shaded, but not where it overlaps Y. X-Y is shown in image 1 below.
The second half of the left side is Y-X. Y complement X is the set of all the elements that are in Y and not in X. Like the first half of the left side, Y will be shaded, but not where it overlaps with X. This is shown in image 2 below.
In between these two sets is a u, or union. This refers to all the elements in X-Y and Y-X. This means that everything in both pictures that we have constructed will combine and everything will be shaded. This is shown in image 3 below.
After we have finished with the left side of the equation, we can move on to the right side. The right side begins with (X u Y). As I stated before, union means that the set of all the elements in X and all the elements in Y. For this picture, we will shade both X and Y entirely. This is shown in image 4 below.
The second half of the right side of the equation is (X n Y). This means that given two sets X and Y, X intersect Y is the set of all the elements that are simultaneously in X and in Y. Pictorially, the only part that will be shaded is where X and Y overlap. For this, refer to image 5 below.
In between (X u Y) and (X n Y) on the right side is a complement. This means that the set of all the elements that are in X u Y and are not in X n Y will be shaded in. Look at image 6 below.
After we have finished working on both sides. We need to compare our final product. Looking at the pictures from each side of the equation (pictures 3 and 6), we can conclude that they are equal due to the fact that they are both shaded in the same areas.
# Ideas in Geometry/Hyperbolic Geometry

What is Hyperbolic Geometry? - Hyperbolic geometry allows us to follow Euclid's first four axioms then replacing the last one: Parallel Postulate. What are the axioms that hold true in this geometry then? 1. A line can be drawn from a point to any other point. 2. A finite line can e extended indefinately. 3. A circle can be drawn, given a center and a radius. 4. All right angles are 90 degrees. 5. Given a line and a point not on that line there exists more than one parallel to the given line through the given point. What does this mean? In hyperbolic geometry the Poincaré Half-Plane Model we work in the upper half-plane (x,y) when x > 0. - here points are still points the difference is that lines are actually arcs inside the boundary where the intersection point with the boundary forms a right angle. This causes a small distortion around the boundary causing these distances to look much smaller and closer together than they truely are when near the bourndary. The closer an object is to a boundary the larger it is with more area it can hold inside of it. This also means that the shortest distance between two points on a line lies between the two points the farthest from the boundary line.
# Instructional design/Tablet Technology/Introduction to IST
## Introduction

Introduction to IST design principles design principles can be used with tablet technology in the instructional design process
Objective: Define 2-3 (yet to be defined) IST design principles.
* Why they are important and helpful?
* IST design principles
* Examples and non-examples of how IST design principles can be used with tablet technology in the instructional design process
* Example lesson plan
This page will cover all of these key points in sequential order.

## Why are Design Principles Important and Helpful?

Design Principles are important....
Design principles are helpful....

## IST Design Principle(s)

Design principle number one.... - reason it fits - sources
Design principle number two... - reason it fits - sources

## Examples

So, taking these concepts and plugging them into a scenario, we will now example how IST design principles can be used with tablet technology in the instructional design process.

## Example Lesson Plan

Click the Next button to continue.
| ID Homepage | Tablet Technology | < Back | Next > |
# Instructional design/Task analysis/Objectives Are Not Subjective

| ID Homepage | 1. ID task analysis | 2. What is HTA? | 3. Anatomy of an HTA | 4. How HTA Works | 5. Rule of Thumb | 6. Objectives Are Not Subjective |
* Back

## Attainable, Measurable, and Goal-oriented

You are nearing the end of this lesson, but first we must round out our understanding of HTA with writing learning/performance objectives. Think of this as the so what? after the task analysis is completed. Things to keep in mind when writing any objective is to make sure they are attainable, measurable, and goal-oriented. So, what does that mean?
* Attainable: The objective must be realistic enough for the learner to accomplish what is being asked of him/her.
* Measurable: The objective needs to have a way to check for understanding, e.g. 80% completion, 5 out of 7 correct, etc...
* Goal-oriented: The objective must state clearly what the expectation is.
As a side note, be careful of using the word and in an objective. The word and indicates you are looking for two separate outcomes. Therefore, these two outcomes must be split into two separate attainable, measurable, goal-oriented objectives.
* Visit the Wikiversity lesson on ID Learning Objectives for all you need to know on writing awesome objectives!
* Back
| ID Homepage | 1. ID task analysis | 2. What is HTA? | 3. Anatomy of an HTA | 4. How HTA Works | 5. Rule of Thumb | 6. Objectives Are Not Subjective |
# Instructional design/Tablet Technology/Introduction
## Introduction
### The use of tablet in K-12 classroom

Mobile technologies have significantly developed and improved in past decades. It has impacted society in many ways. It is said that mobile technologies provide us with new opportunities, patterns of work, lives, and entertainments in academic and professional fields as well as in our daily lives. Mobile technologies enable people to utilize mobile devices as communication and collaboration tools.

### Possibilities

Mobile learning may have created new possibilities to take advantage of its technological innovations to enhance learners’ creative/critical thinking, positive attitude, spatial ability, and executive function as well as learner engagement.
Mobile learning games can be motivating for learners, providing satisfaction and desire to recommend it to others and to play again. There are mobile collaborative learning applications to promote higher-order thinking skills, such as discussion boards, blogs, outdoor education scenarios, and games. Since mobile devices have the capacity to support a sense of individuality and community as well as learner motivation to learn through participation in collaborative learning.

### Tablets

With the development of mobile technologies, specifically, tablets have created new instructional ideas.
Researchers in the mobile learning field have investigated how tablet technologies can enhance and improve the way people learn as well as interact each other. In the mean time, students’ use of tablet devices in classroom has encouraged their teachers to redesign traditional instruction. Tablet technologies can create an opportunity for students and teachers in classroom settings by supporting engagement, participation, interactions, or individualized learning.

## Reflection

Let’s think about the following topics.
* How many of your students have their own tablets?
* Have you ever thought about the use of tablets in your classroom?
** If so, what was the instructional goal of using tablets? How was it?
** If not, why haven’t you used it?
# Instructional design/Task analysis/A Rule of Thumb

| ID Homepage | 1. ID task analysis | 2. What is HTA? | 3. Anatomy of an HTA | 4. How HTA Works | 5. A Rule of Thumb | 6. Objectives Are Not Subjective |
| Back | Next |

## Identifying the components

Now that you know the structure of a completed HTA as seen in the last two sections, you should be able to identify the various components that make up an analysis. Since a subtask can be broken into two levels, the rule of thumb goes something like this: Treat everything as if it were a subtask. Then, if you can break one component down into more than two levels (going beyond steps), you're looking at more than one subtask. If it can be broken down into one supporting component, it is an element. Furthermore, if you're looking at a component that cannot be broken down any further, it is a step.
As mentioned previously in this lesson, once your HTA is completed, go through each component using what you have listed as steps, elements, and subtasks as a guide. Only act on what you wrote. If you're unable to take action because it is not written, you will know you missed a component in your analysis.

## Example Task: Converting a Word Document to a PDF

Get your HTA worksheet ready, your assignment is to observe a worker converting a Word document to a PDF. First, let's set the stage. It's Tuesday morning on May 25, 2010, and you're having a team meeting with your manager.
Now you're on your way to a local business that specializes in logistics. You reach your destination and get briefed from Mary the manager about Jake the desk jockey. She tells you that Jake often needs to e-mail documents in a way that prevents people from easily altering information. In order develop a job aid for new hires, you are asked to observe Jake during his day. His first task is to convert a Word document to a PDF. Print a new task analysis worksheet, and click 'play' on the image below when you're ready to observe Jake. Remember to identify subtasks, elements, and steps if applicable.

## Final Check

Answer the following questions before assessing your HTA. This will help you determine the accuracy of your HTA.

## How did you do?

Compare your HTA with this completed HTA of Jake creating a PDF.
If you recorded preliminary information about the task, person you are observing, and broke down at least three subtasks, you get an A!
If you are missing the preliminary information or did not break down one subtask, you get a B.
If you are missing the preliminary information or did not break down two subtasks, you get a C.
If you are missing the preliminary information and did not break down any subtasks, you should revisit the previous pages in this lesson and try again.

## What about the last part--Objectives?

I'm glad you asked! You are now finished with the HTA lesson. Click next to begin your lesson on objectives.
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| ID Homepage | 1. ID task analysis | 2. What is HTA? | 3. Anatomy of an HTA | 4. How HTA Works | 5. A Rule of Thumb | 6. Objectives Are Not Subjective |
# Instructional design/Task analysis/Anatomy of an HTA

| ID Homepage | 1. ID task analysis | 2. What is HTA? | 3. Anatomy of an HTA | 4. How HTA Works | 5. Rule of Thumb | 6. Objectives Are Not Subjective |
| Back | Next |

## Parts of a whole

Below is an image of a completed HTA (minus the objectives). The most basic HTA has three components:
* The Task = the main goal being observed.
* Subtasks = actions that must be completed before accomplishing the main goal/task.
* Elements = actions that must be completed before accomplishing a subtask.
In rare occasions, an element may be broken down into steps, in which case those steps must be completed in order to complete the element.
This image first appeared in the HCI textbook
An HTA begins with the task to be analyzed. It's a novel idea, I know. In this case, the task is making a cup of tea and is labeled with a 0. The task is then broken down into subtasks labeled 1-6. There may be times that a task to be analyzed will be fairly simple and not require subtasks. Shown here, the first subtask has been broken down into elements, which are labeled 1.1-1.4. These elements must be completed before moving to the next subtask of emptying the pot, which will have a new set of elements.
Click next to continue.
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| ID Homepage | 1. ID task analysis | 2. What is HTA? | 3. Anatomy of an HTA | 4. How HTA Works | 5. Rule of Thumb | 6. Objectives Are Not Subjective |
# Instructional design/Tablet Technology/Introduction to one instructional design theory
## Introduction to One Instructional Design Theory

The objective for this lesson is that the learner will be able to: Identify an example lesson developed with use of IST design principles and an instructional design theory
Process:
* Introduce the learner to one instructional design theory in text and graphic model form. • Provide definition and explanation of instructional design theory (include graphic model) • Present example scenario applying the instructional design theory
| ID Homepage | Tablet Technology | < Back | Next > |
# Instructional design/Tablet Technology/Lesson Summary
## Lesson Summary

The purpose of this lesson was to learn about technology integration, specifically the use of tablet in classroom. In this lesson you have:
* Learned about the use of tablet technology in the classroom, and the possibilities that they offer
* You've reflected on how many of your students have their own tablets and any past instructional experience where you used Tablet technology
* Examined the benefits and challenges of using tablet technology in the classroom
* Read two example scenarios, of how teachers added the use of tablet technology to their classroom, and reviewed what can be learned from each example
* Practiced paraphrasing what you've learned, related the knowledge gained to experiential knowledge you already possess, and developed an educational objective
* Developed a lesson plan that includes using a tablet app
Tablets give teachers access to diverse educational tools and instructional content that help them meet the instructional goals and the needs of today's students. We hope that you've enjoyed this lesson!
| ID Homepage | Tablet Technology | < Back |
# Instructional design/Task analysis

| ID Homepage | 1. ID task analysis | 2. What is HTA? | 3. Anatomy of an HTA | 4. How HTA Works | 5. Rule of Thumb | 6. Objectives Are Not Subjective |
| Back | Next |

## Introduction

Let's start off slow and simple. Imagine if you will the task of making a peanut butter and jelly sandwich. If you can get someone in the house to make you a sandwich, please tell me how to do this!
Like I said, we're starting simple here. Making a sandwich is a task that is comprised of many little tasks. Those little tasks are called subtasks. Those subtasks are made up of even littler parts called elements. Think about it... find the bread, get a knife, and so on. This will be a sample task later in this lesson. If you can break down making a PB&J sandwich, you will be able to handle nearly any HTA.
With that being said, let's get on with the lesson!
As a side note, there are several forms a task analysis can take. For the purpose of this lesson, we will be working with the hierarchical task analysis (HTA). An HTA is also known as task-decomposition.
After completing this lesson, you will be able to:
* state the conditions for using HTA.
* identify the elements of an HTA.
* state the procedure for conducting an HTA.
* apply a rule that classifies content as a subtask, element, or step.
* write a learning (performance) objective associated with the HTA.
Click Next to continue. The entire lesson should only take you approximately 20 minutes.
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| ID Homepage | 1. ID task analysis | 2. What is HTA? | 3. Anatomy of an HTA | 4. How HTA Works | 5. Rule of Thumb | 6. Objectives Are Not Subjective |
Back to Topic: Instructional Design