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
          <me>
            <eval obj="theFunction"/>=0
          </me>
          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
          <me>
            h(x)=<eval obj="composition"/>
          </me>.
          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>