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PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

Section 3.18 Projects and Friends

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“PROJECT-LIKE” refers to the elements/blocks <project>, <activity>, <activity>, and <exploration>. They are very similar to <exercise>, and here we stress that they can be interactive in all the same ways an <exercise> can. Key differences are that they are blocks, and so are peers of other blocks like paragraphs and examples, and are children of divisions. By default they have their own numbering scheme, though we expect this to convert to an elective behavior.
Both <exercise> and PROJECT-LIKE can be structured with <task>. It would be natural to have a <project> with multiple <task>, each a possibly different interactive problem type. We expect that to be possible soon. For now, <exercise> and <project> (and friends) need to be structured with a <statement> (and their associated signal) in order to be interactive.

Activity 3.18.1. Multiple-Choice, Not Randomized, Multiple Answers.

    Which colors might be found in a rainbow? (Note that the radio buttons now allow multiple buttons to be selected.)
  • Red
  • Red is a definitely one of the colors.
  • Yellow
  • Yes, yellow is correct.
  • Black
  • Remember the acronym…ROY G BIV. “B” stands for blue.
  • Green
  • Yes, green is one of the colors.
Hint.
Do you know the acronym…ROY G BIV for the colors of a rainbow, and their order?
Now an <exercise> with two <task>. Since the latter is a short-answer question, it will only be interactive on a capable platform.

Checkpoint 3.18.1. True/False and Explain.

Structured with task, we can have an introduction, but this does not migrate with the interactive portions.

(a) True/False.

    Every vector space has finite dimension.
  • True.

  • The vector space of all polynomials with finite degree has a basis, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) which is infinte.
  • False.

  • The vector space of all polynomials with finite degree has a basis, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) which is infinte.
Hint.
\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by Theorem 1.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)

(b)

Explain your reasoning in the previous question.
Now an <exploration> nested two-deep with the same two questions, two times each, just for testing purposes.

Exploration 3.18.2. Exploring Two-Deep.

(a) First Iterations of Each.

(i) Multiple-Choice, Not Randomized, One Answer (First Copy).
    What color is a stop sign?
  • Green
  • Green means “go!”.
  • Red
  • Red is universally used for prohibited activities or serious warnings.
  • White
  • White might be hard to see.
Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?
(ii) Python ActiveCode (First Copy).
Run the following program and observe the information provided at each step.
Answer.
We’re still not really sure.

(b) Second Iterations of Each.

(i) Multiple-Choice, Not Randomized, One Answer (Second Copy).
    What color is a stop sign?
    We include a spurious Python <program> element in the <statement> which should never convert this from a multiple-choice question into a programming exercise (on any host), but should still get syntax highlighting as part of rendering the exercise.
    sum = 0
    sum += 6
    for i in range(sum):
    print("Hello, World!")
    
  • Green
  • Green means “go!”.
  • Red
  • Red is universally used for prohibited activities or serious warnings.
  • White
  • White might be hard to see.
Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?
(ii) Python ActiveCode (Second Copy).
Run the following program and observe the information provided at each step.
Answer.
We’re still not really sure.
This is an <exercise> which has been structured deeply with <task>. It is here to allow for testing a tabbed viewer presentation in online outputs.

Checkpoint 3.18.2. A very structured exercise.

This is an over-arching introduction to the whole exercise. We follow with some tasks. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

(a) A super-simple task.

This first task is very simple, just a paragraph. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

(b)

Now three paragraphs. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

(c) A title of a task that has a subtask with an <answer> for the Solutions.

This second task is further divided by more tasks. This is its introduction. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
(i) A task with a title and an <answer> for the Solutions.
A really simple subtask. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
A short paragraph, before an answer.
Answer.
With a proof.
Proof.
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
And a bit more to say.
(ii)
A subtask with an answer. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
Answer.
Right! In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
(iii)
Three simple sub-sub-tasks. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
(B) A second three-deep subsubtask!
Second subsubtask. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
(C)
Third subsubtask. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
In interdum suscipit ullamcorper
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
The conclusion of the structured subtask. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
(iv)
A simple task as the last subtask. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
This concludes our structured second task. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.

(d)

This third top-level task is intermediate in complexity, you are reading the statement, which is followed by more items. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
Hint.
One hint. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
Answer 1.
First answer. In interdum suscipit ullamcorper.
Answer 2.
Second answer. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
Solution.
At last, the solution. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.
This is a conclusion where you could summarize the exercise. In interdum suscipit ullamcorper. Morbi sit amet malesuada augue, id vestibulum magna. Nulla blandit dui metus, malesuada mollis sapien ullamcorper sit amet. Nulla at neque nisi. Integer vel porta felis.