{"id":218,"date":"2017-10-28T22:25:17","date_gmt":"2017-10-28T22:25:17","guid":{"rendered":"http:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/chapter\/division-explorations\/"},"modified":"2018-05-25T01:29:21","modified_gmt":"2018-05-25T01:29:21","slug":"division-explorations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/chapter\/division-explorations\/","title":{"raw":"Division Explorations","rendered":"Division Explorations"},"content":{"raw":"[latexpage]\n\nAnu refuses to tell anyone if she is working in a 1\u219010 system, or a 1\u21905 system, or any other system.  She makes everyone call it a 1 \u2190 x<\/i> system but won\u2019t tell anyone what x<\/i> stands for.\n\nWe know that boxes in a 1\u219010 have values that are powers of ten: 1, 10, 100, 1000, 10000...\n\n

And boxes in a 1\u21905 system are powers of five: 1, 5, 25, 125, 625...\n\n

So Anu\u2019s system, whatever it is, must be powers of x<\/em>: $1, x, x^2, x^3, x^4 \\dots$\n\n

\"\"\n\n

When Anu writes $2556 _x$ she must mean:\n\n

\"\"\n\n\\[2x^3 + 5x^2 + 5x + 6.\\]\n\n

And when she writes $12_x$ she means:\n\n

\"\"\n\n

\\[x + 2.\\]\n\n

Anu decides to compute $2556_x \\div 12_x$.\n\n

\"\"\n\n

She obtains:\n\n

\\[ (2x^3+5x^2+5x+6) \\div (x+2) = 2x^2+x+3.\\]\n\n

Problem 28<\/h3>
  1. Check Anu\u2019s division by computing the product \\[(x+2)(2x^2+x+3).\\]  Did it work?<\/li>
  2. Use Anu\u2019s method to find $(3x^2+7x+2)\\div(x+2)$.<\/li>
  3. Use Anu\u2019s method to find $(2x^4+3x^3+5x^2+4x+1)\\div (2x+1)$.<\/li>
  4. Use Anu\u2019s method to find $(x^4+3x^3+6x^2+5x+3)\\div(x^2+x+1)$.<\/li><\/ol><\/div>

    Anu later tells use that she really was thinking of a 1\u219010 system so that x<\/i> does equal ten. Then her number 2556 really was two thousand, five hundred and fifty six and 12 really was twelve. Her statement: \\[(2x^3+5x^2+5x+6)\\div(x+2)=2x^2+x+3\\] is actually 2556 \u00f7 12 = 213.\n\n

    Problem 29<\/h3>
    1. Check that 2556 \u00f7 12 = 213 is correct in base 10.<\/li>
    2. Keeping with the 1\u219010 system, what division problems did you actually solve in parts (b), (c), and (d) of Problem 28? Check that your answers are correct.<\/li><\/ol><\/div>

      Uh Oh! Anu has changed her mind. She now says she was thinking of a 1\u219011 system.\n\n

      Now $2556_x$ means $2\\cdot 11^3+5\\cdot 11^2+5\\cdot 11 +6 = 3328_{\\text{ten}}$.\n\n

      Similarly, $12_x$ means $1\\cdot 11+2 =13_{\\text{ten}}$, and $213_x$ means $2\\cdot11^2+1\\cdot 11+3 = 256_{\\text{ten}}$.\n\n

      So Anu\u2019s computation $2556_x\\div 12_x = 213_x$ is actually the (base 10) statement:\n\n

      3328 \u00f7 13 = 256.\n\n

      Problem 30<\/h3>
      1. Check that 3328 \u00f7 13 = 256 is also correct in base ten.<\/li>
      2. Keeping with the 1\u219011 system, what division problems did you actually solve in parts (b), (c), and (d) of Problem 28? Check that they are correct.<\/li><\/ol><\/div> \n\n

        Problem 31<\/h3>
        1. Use Anu\u2019s method to show that \\[(x^4+4x^3+6x^2+4x+1)\\div(x+1)=(x^3+3x^2+3x+1).\\]<\/li>
        2. What is this saying for x<\/i> = 10?  Check that the division is correct.<\/li>
        3. What is this saying for x<\/i> = 2? Check that the division is correct.<\/li>
        4. What is this saying for x<\/i> equal to each of 3, 4, 5, 6, 7, 8, 9, and 11? Check that each division is correct.<\/li>
        5. What is this saying for x<\/em> = 0?<\/li><\/ol><\/div>","rendered":"

          Anu refuses to tell anyone if she is working in a 1\u219010 system, or a 1\u21905 system, or any other system.  She makes everyone call it a 1 \u2190 x<\/i> system but won\u2019t tell anyone what x<\/i> stands for.<\/p>\n

          We know that boxes in a 1\u219010 have values that are powers of ten: 1, 10, 100, 1000, 10000…<\/p>\n

          And boxes in a 1\u21905 system are powers of five: 1, 5, 25, 125, 625…<\/p>\n

          So Anu\u2019s system, whatever it is, must be powers of x<\/em>: \"1, x, x^2, x^3, x^4 \dots\"<\/p>\n

          \"\"<\/p>\n

          When Anu writes \"2556 _x\" she must mean:<\/p>\n

          \"\"<\/p>\n

            <\/span>   <\/span>\"\[2x^3 + 5x^2 + 5x + 6.\]\"<\/p>\n

          And when she writes \"12_x\" she means:<\/p>\n

          \"\"<\/p>\n

          \n

            <\/span>   <\/span>\"\[x + 2.\]\"<\/p>\n

          Anu decides to compute \"2556_x \div 12_x\".<\/p>\n

          \"\"<\/p>\n

          She obtains:<\/p>\n

          \n

            <\/span>   <\/span>\"\[ (2x^3+5x^2+5x+6) \div (x+2) = 2x^2+x+3.\]\"<\/p>\n

          \n

          Problem 28<\/h3>\n
            \n
          1. Check Anu\u2019s division by computing the product\n

              <\/span>   <\/span>\"\[(x+2)(2x^2+x+3).\]\"<\/p>\n

              Did it work?<\/li>\n

          2. Use Anu\u2019s method to find \"(3x^2+7x+2)\div(x+2)\".<\/li>\n
          3. Use Anu\u2019s method to find \"(2x^4+3x^3+5x^2+4x+1)\div (2x+1)\".<\/li>\n
          4. Use Anu\u2019s method to find \"(x^4+3x^3+6x^2+5x+3)\div(x^2+x+1)\".<\/li>\n<\/ol>\n<\/div>\n

            Anu later tells use that she really was thinking of a 1\u219010 system so that x<\/i> does equal ten. Then her number 2556 really was two thousand, five hundred and fifty six and 12 really was twelve. Her statement: <\/p>\n

              <\/span>   <\/span>\"\[(2x^3+5x^2+5x+6)\div(x+2)=2x^2+x+3\]\"<\/p>\n

             is actually 2556 \u00f7 12 = 213.<\/p>\n

            \n

            Problem 29<\/h3>\n
              \n
            1. Check that 2556 \u00f7 12 = 213 is correct in base 10.<\/li>\n
            2. Keeping with the 1\u219010 system, what division problems did you actually solve in parts (b), (c), and (d) of Problem 28? Check that your answers are correct.<\/li>\n<\/ol>\n<\/div>\n

              Uh Oh! Anu has changed her mind. She now says she was thinking of a 1\u219011 system.<\/p>\n

              Now \"2556_x\" means \"2\cdot 11^3+5\cdot 11^2+5\cdot 11 +6 = 3328_{\text{ten}}\".<\/p>\n

              Similarly, \"12_x\" means \"1\cdot 11+2 =13_{\text{ten}}\", and \"213_x\" means \"2\cdot11^2+1\cdot 11+3 = 256_{\text{ten}}\".<\/p>\n

              So Anu\u2019s computation \"2556_x\div 12_x = 213_x\" is actually the (base 10) statement:<\/p>\n

              3328 \u00f7 13 = 256.<\/p>\n

              \n

              Problem 30<\/h3>\n
                \n
              1. Check that 3328 \u00f7 13 = 256 is also correct in base ten.<\/li>\n
              2. Keeping with the 1\u219011 system, what division problems did you actually solve in parts (b), (c), and (d) of Problem 28? Check that they are correct.<\/li>\n<\/ol>\n<\/div>\n

                 <\/p>\n

                \n

                Problem 31<\/h3>\n
                  \n
                1. Use Anu\u2019s method to show that\n

                    <\/span>   <\/span>\"\[(x^4+4x^3+6x^2+4x+1)\div(x+1)=(x^3+3x^2+3x+1).\]\"<\/p>\n<\/li>\n

                2. What is this saying for x<\/i> = 10?  Check that the division is correct.<\/li>\n
                3. What is this saying for x<\/i> = 2? Check that the division is correct.<\/li>\n
                4. What is this saying for x<\/i> equal to each of 3, 4, 5, 6, 7, 8, 9, and 11? Check that each division is correct.<\/li>\n
                5. What is this saying for x<\/em> = 0?<\/li>\n<\/ol>\n<\/div>\n","protected":false},"author":3,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-218","chapter","type-chapter","status-publish","hentry"],"part":128,"_links":{"self":[{"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/pressbooks\/v2\/chapters\/218","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/pressbooks\/v2\/chapters\/218\/revisions"}],"predecessor-version":[{"id":219,"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/pressbooks\/v2\/chapters\/218\/revisions\/219"}],"part":[{"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/pressbooks\/v2\/parts\/128"}],"metadata":[{"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/pressbooks\/v2\/chapters\/218\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/wp\/v2\/media?parent=218"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/pressbooks\/v2\/chapter-type?post=218"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/wp\/v2\/contributor?post=218"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.oer.hawaii.edu\/mathforelementaryteachers\/wp-json\/wp\/v2\/license?post=218"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}