PreTeXt Sample Book Abstract Algebra (SAMPLE ONLY)

<exercises xml:id="fill-in-exercises">

                    <title>Fill-In Exercises</title>

                    <introduction>

                      <p>
This section has <term>fill-in-the-blank</term> (<init>FITB</init>) exercises using newer syntax, and the subject of active development.
                      </p>

                    </introduction>

                    <exercise label="fillin-numbers-many-tests">

                      <title>Fill-In, New Markup Numbers</title>

                      <statement>

                        <p>
I love <m>\pi</m>. What number am I thinking of, accurate to two decimal places?
                        </p>

                        <p>

                          <fillin width="5" answer="3.14"/>

                        </p>

                      </statement>

                      <evaluation> <evaluate> <test correct="yes"> <numcmp use-answer="yes"/>

                      <feedback>

                        <p>
The decimal approximation of <m>\pi</m> is <m>3.1415926535\ldots</m>, but to two decimal places we write <m>3.14</m>.
                        </p>

                      </feedback>

                      </test> <!-- Example B --> <test> <numcmp use-answer="yes" tolerance="0.1"/>

                      <feedback>

                        <p>
Your answer is within 0.1 of the value I wanted.
                        </p>

                      </feedback>

                      </test> <!-- Example C --> <test> <numcmp value="42"/>

                      <feedback>

                        <p>
That is a reasonable guess, but no.
                        </p>

                      </feedback>

                      </test> <!-- Example D --> <test> <numcmp min="3" max="4"/>

                      <feedback>

                        <p>
You chose a value between 3 and 4.
                        </p>

                      </feedback>

                      </test> <!-- Example E --> <test> <numcmp value="3" tolerance="0.5"/>

                      <feedback>

                        <p>
You chose a value that rounds to 3.
                        </p>

                      </feedback>

                      </test> </evaluate> </evaluation>

                    </exercise>

                    <exercise label="fillin-strings-">

                      <title>Fill-In, New Markup Strings</title>

                      <statement>

                        <p>
The word I'm thinking about is hinted at by the image.
                        </p>

                        <p>

                          <image source="cartoon-magic.jpg" width="50%"/>

                          </p>

                          <p>
What word am I thinking about? <fillin width="5" answer="magic"/> (Interactive feedback explores a variety of options: Try what happens if you mix the case, or type in a number, or include more than the word, or try <q>pizzazz</q>.")
                          </p>

                        </statement>

                        <evaluation> <evaluate> <!-- Example A --> <test correct="yes"> <strcmp use-answer="yes"/> </test> <!-- Example B --> <test> <strcmp use-answer="yes" case="insensitive"/>

                        <feedback>

                          <p>
Some of the characters used the wrong case.
                          </p>

                        </feedback>

                        </test> <!-- Example C --> <test> <strcmp>[0-9]+</strcmp>

                        <feedback>

                          <p>
You typed a word made out of digits.
                          </p>

                        </feedback>

                        </test> <!-- Example D --> <test> <strcmp use-answer="yes" strip="no"/>

                        <feedback>

                          <p>
Your answer includes the correct word but has extra text.
                          </p>

                        </feedback>

                        </test> <!-- Example E --> <test> <strcmp strip="no" case="insensitive">z.+z</strcmp>

                        <feedback>

                          <p>
Your answer includes text surrounded by z's.
                          </p>

                        </feedback>

                        </test> </evaluate> </evaluation>

                        <hint>

                          <p>
Do you really need a hint? Carefully reread the question.
                          </p>

                        </hint>

                      </exercise>

                      <exercise label="ex-demo-jscmp-primes">

                        <title>Fill-In, Javascript test of numbers</title>

                        <statement>

                          <p>
What is an example of a prime number less than 20? <fillin width="5" answer="13"/>
                          </p>

                        </statement>

                        <evaluation> <evaluate> <test correct="yes"> <jscmp><!-- test if number is in a list --> [2, 3, 5, 7, 11, 13, 17, 19].includes(Number(ans)) </jscmp>

                        <feedback>

                          <p>
Any number from the list <m>\{2, 3, 5, 7, 11, 13, 17, 19\}</m> is a prime number less than 20.
                          </p>

                        </feedback>

                        </test> <test> <jscmp><!-- find why it fails to be in list --> function(){ const val=Number(ans); if (val &lt;= 0) { return "You need to give a positive integer." } if (val &gt;= 20) { return "The integer must be less than 20." } for (let i=2; i&lt;=3; i++) { if (val % i == 0) { return `Your answer is composite; for example, it is divisible by ${i}.` } } return false; }() </jscmp> </test> </evaluate> </evaluation>

                      </exercise>

                      <exercise label="fillin-jscmp-palindrome">

                        <title>Fill-In, Javascript test of strings</title>

                        <statement>

                          <p>
What is an example of a palindrome? <fillin width="5" answer="radar"/>
                          </p>

                        </statement>

                        <evaluation> <evaluate> <test correct="yes"> <jscmp><!-- test if string is palindrome --> function(){ const r1 = new RegExp("^\\w+$"); var result=r1.test(ans); if (result) { let revAns = ans.split("").reverse().join(""); result = (ans === revAns); } return result; }() </jscmp>

                        <feedback>

                          <p>
Any word that is the same forward and backward is a palindrome.
                          </p>

                        </feedback>

                        </test> <test> <strcmp strip="no">\w+\s\w+</strcmp>

                        <feedback>

                          <p>
Your response needs to be a single word.
                          </p>

                        </feedback>

                        </test> </evaluate> </evaluation>

                      </exercise>

                      <exercise label="fillin-random-simple">

                        <title>Fill-In, Simple Randomization with Numbers</title>

                        <statement>

                          <p>
What is the square of <m>x=<eval obj="myNum"/></m>? <m>x^2=</m><fillin width="5" mode="number" ansobj="mySquare"/>
                          </p>

                        </statement>

                        <setup seed="1234"> <setupScript> v.myNum=RNG.randDiscrete(2, 12, 1); v.mySquare=v.myNum**2; </setupScript> </setup> <evaluation> <evaluate> <test correct="yes"> <numcmp use-answer="yes"/> </test> <test> <numcmp object="myNum"/>

                        <feedback>

                          <p>
You responded with the original number. Now square it.
                          </p>

                        </feedback>

                        </test> </evaluate> </evaluation>

                      </exercise>

                      <exercise label="fillin-math-solve-equation">

                        <title>Fill-In, Dynamic Math with Simple Numerical Answer</title>

                        <statement>

                          <p>
Solve the equation 
                            <md>

                              <eval obj="theFunction"/>=0

                            </md>

 to get the value of <m>x</m>.
                          </p>

                          <p>

                            <m>x = </m> <fillin width="5" mode="math" ansobj="theAnswer"/>

                          </p>

                        </statement>

                        <solution>

                          <p>
We want to isolate the <m>x</m> in the equation <m><eval obj="theFunction"/>=0</m>. Because addition of <m><eval obj="b"/></m> is the last operation, we apply the inverse by adding <m><eval obj="negB"/></m> to both sides. The new, but equivalent equation is now <m><eval obj="m"/>x = <eval obj="negB"/></m>. Dividing both sides of the equation by <m><eval obj="m"/></m>, we obtain the solution <m>x=<eval obj="theAnswer"/></m>.
                          </p>

                        </solution>

                        <setup seed="12345"> <de-object name="m" context="number"> <de-random distribution="discrete" nonzero="yes" min="-4" max="5" by="1"/> </de-object> <de-object name="b" context="number"> <de-random distribution="discrete" min="-10" max="10"/> </de-object> <de-object name="negB" context="number"> <de-number reduce="yes">-b</de-number> </de-object> <de-object name="theFunction" context="formula"> <de-expression mode="formula" reduce="yes">m*x+b</de-expression> </de-object> <de-object name="theAnswer" context="formula"> <de-expression mode="formula" reduce="yes">-b/m</de-expression> </de-object> </setup> <evaluation> <evaluate> <test correct="yes"> <mathcmp use-answer="yes"/> </test> <test> <de-expression mode="formula">{{b}}/{{m}}</de-expression>

                        <feedback>
Check for a sign error while isolating <m>x</m>.
                        </feedback>

                        </test> </evaluate> </evaluation>

                      </exercise>

                      <exercise label="fillin-math-find-derivatives">

                        <title>Fill-In, Dynamic Math with Formulas as Answers</title>

                        <statement>

                          <p>
Consider the function <m>f(x)=<eval obj="formula"/></m>. Find <m>f'(x)</m> and <m>f''(x)</m>.
                          </p>

                          <p>

                            <m>f'(x) = </m> <fillin mode="math" width="15" ansobj="correctD1" name="firstD"/> and <m>f''(x)=</m> <fillin mode="math" width="15" ansobj="correctD2" name="secondD"/>

                          </p>

                        </statement>

                        <solution>

                          <p>
The derivative of a constant is zero, so <m>\frac{d}{dx}[<eval obj="b"/>]=0</m>. The term <m>x^{<eval obj="n"/>}</m> is a power, so the power rule gives us <m>\frac{d}{dx}[x^{<eval obj="n"/>}]=<eval obj="n"/>x^{<eval obj="nm1"/>}</m>. Putting this together, we find <m>f'(x)=<eval obj="correctD1"/></m>. Applying the power rule a second time, we find <m>f''(x)=<eval obj="correctD2"/></m>.
                          </p>

                        </solution>

                        <setup seed="1234"> <de-object name="a" context="number"> <de-random distribution="discrete" min="-4" max="5" by="1" nonzero="yes"/> </de-object> <de-object name="n" context="number"> <de-random distribution="discrete" min="2" max="5"/> </de-object> <de-object name="nm1" context="number"> <de-number reduce="yes">n-1</de-number> </de-object> <de-object name="nm2" context="number"> <de-number reduce="yes">n-2</de-number> </de-object> <de-object name="b" context="number"> <de-random distribution="discrete" min="-10" max="10" nonzero="yes"/> </de-object> <de-object name="formula" context="formula"> <de-expression mode="formula">a*x^n+b</de-expression> </de-object> <de-object name="correctD1" context="formula"> <de-expression mode="derivative" reduce="yes"> <formula><eval obj="formula"/></formula> <variable name="x"/> </de-expression> </de-object> <de-object name="correctD2" context="formula"> <de-expression mode="derivative" reduce="yes"> <formula><eval obj="correctD1"/></formula> <variable name="x"/> </de-expression> </de-object> </setup> <evaluation> <evaluate submit="firstD"> <test correct="yes"> <mathcmp use-answer="yes"/> </test> <test> <mathcmp obj="correctD2"/>

                        <feedback>

                          <p>
You confused which derivative is which.
                          </p>

                        </feedback>

                        </test> </evaluate> <evaluate submit="secondD"> <test correct="yes"> <mathcmp use-answer="yes"/> </test> <test> <eval obj="correctD1"/>

                        <feedback>

                          <p>
You confused which derivative is which.
                          </p>

                        </feedback>

                        </test> </evaluate> </evaluation>

                      </exercise>

                      <exercise label="fillin-math-decompose-function">

                        <title>Fill-In, Dynamic Math with Interdependent Formula Checking</title>

                        <statement>

                          <p>
Consider the function 
                            <md>
h(x)=<eval obj="composition"/>
                            </md>.

 Find two nontrivial functions <m>f(x)</m> and <m>g(x)</m> so that <m>h(x) = f(g(x))</m>.
                          </p>

                          <p>

                            <m>f(x) = </m> <fillin mode="math" width="15" ansobj="outerFormula" name="fGiven"/> and <m>g(x)=</m> <fillin mode="math" width="15" ansobj="innerFormula" name="gGiven"/>

                          </p>

                        </statement>

                        <solution>

                          <p>
Noticing that the expression <m><eval obj="innerFormula"/></m> appears inside parentheses with a power, it makes sense to think of that as the inner function, defining <m>g(x) = <eval obj="innerFormula"/></m>. The outer function describes what happens to that. If we imagined replacing the formula <m><eval obj="innerFormula"/></m> with a box and then call that box our variable <m>x</m>, we find the outer function is given by <m>f(x) = <eval obj="outerFormula"/></m>.
                          </p>

                          <p>
This is not the only non-trivial composition. Can you find others?
                          </p>

                        </solution>

                        <setup seed="4321"> <de-object name="a" context="number"> <de-random distribution="discrete" min="-4" max="5" by="1" nonzero="yes"/> </de-object> <de-object name="n" context="number"> <de-random distribution="discrete" min="2" max="5"/> </de-object> <de-object name="b" context="number"> <de-random distribution="discrete" min="-10" max="10" by="1" nonzero="yes"/> </de-object> <de-object name="c" context="number"> <de-random distribution="discrete" min="-4" max="5" by="1" nonzero="yes"/> </de-object> <de-object name="d" context="number"> <de-random distribution="discrete" min="-10" max="10" by="1" nonzero="yes"/> </de-object> <de-object name="outerFormula" context="formula"> <de-expression mode="formula" reduce="yes">a*x^n+b</de-expression> </de-object> <de-object name="innerFormula" context="formula"> <de-expression mode="formula" reduce="yes">c*x+d</de-expression> </de-object> <de-object name="identityFunction" context="formula"> <de-expression mode="formula">x</de-expression> </de-object> <de-object name="composition" context="formula" reduce="yes"> <de-expression mode="substitution"> <formula><eval obj="outerFormula"/></formula> <variable name="x"><eval obj="innerFormula"/></variable> </de-expression> </de-object> </setup> <evaluation answers-coupled="yes"> <evaluate name="fGiven"> <test> <eval obj="identityFunction"/>

                        <feedback>

                          <p>

                            <m>f(x)=x</m> is not allowed for nontrivial compositions.

                          </p>

                        </feedback>

                        </test> <test> <logic op="not"> <mathcmp> <eval obj="composition"/> <de-expression context="formula" mode="substitution"> <formula><eval obj="fGiven"/></formula> <variable name="x"><eval obj="gGiven"/></variable> </de-expression> </mathcmp> </logic> <mathcmp> <eval obj="composition"/> <de-expression context="formula" mode="substitution"> <formula><eval obj="gGiven"/></formula> <variable name="x"><eval obj="fGiven"/></variable> </de-expression> </mathcmp>

                        <feedback>

                          <p>
You have composed in the wrong order.
                          </p>

                        </feedback>

                        </test> </evaluate> <evaluate name="gGiven"> <test> <eval obj="identityFunction"/>

                        <feedback>

                          <m>g(x)=x</m> is not allowed for nontrivial compositions.

                        </feedback>

                        </test> </evaluate> <evaluate all="yes"> <test correct="yes"> <mathcmp> <eval obj="composition"/> <de-expression context="formula" mode="substitution"> <formula><eval obj="fGiven"/></formula> <variable name="x"><eval obj="gGiven"/></variable> </de-expression> </mathcmp> <logic op="not"> <mathcmp> <eval obj="fGiven"/> <eval obj="identityFunction"/> </mathcmp> </logic> <logic op="not"> <mathcmp> <eval obj="gGiven"/> <eval obj="identityFunction"/> </mathcmp> </logic> </test> </evaluate> </evaluation>

                      </exercise>

                      </exercises>