{"id":969,"date":"2017-11-14T03:26:10","date_gmt":"2017-11-14T03:26:10","guid":{"rendered":"http:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/?post_type=chapter&p=969"},"modified":"2018-01-19T00:12:53","modified_gmt":"2018-01-19T00:12:53","slug":"borders-on-a-square","status":"publish","type":"chapter","link":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/chapter\/borders-on-a-square\/","title":{"raw":"Borders on a Square","rendered":"Borders on a Square"},"content":{"raw":"[latexpage]\r\n\r\nHere\u2019s another problem:\r\n
\r\n

Problem 2<\/h3>\r\n

Here is a large square made up of 100 smaller unit squares.\u00a0 The unit squares along the border of the large square are colored red.\u00a0 Without counting one-by-one, can you figure out how many red squares there are in the picture?<\/p>\r\n

\"\"<\/p>\r\n

Clearly describe how you figured out the number of red squares, and how you know your answer is correct.<\/p>\r\n\r\n<\/div>\r\n

Justin calculated the number of squares as $(10 \\times 4) - 4$.\u00a0 He justified his answer this way:<\/p>\r\n\r\n

Since the dimensions of the big square are\u00a0$10\\times 10$, there are 10 squares along each of the four sides. So that gives me\u00a040\u00a0red squares. But then each corner is part of two different sides. I've counted each of the corners twice. So I need to make up for that by subtracting\u00a04\u00a0at the end.<\/blockquote>\r\n

Justin showed this picture to justify his work:<\/p>\r\n

\"\"<\/p>\r\n\r\n

\r\n

Think \/ Pair \/ Share<\/h3>\r\n
    \r\n \t
  • What do you think about Justin\u2019s solution? Are you convinced?\u00a0 Could he have explained it more clearly?<\/li>\r\n \t
  • Was Justin\u2019s solution different from your solution or the same?<\/li>\r\n \t
  • Notice the color coding in Justin\u2019s picture. What do the colors represent? Why did he use the colors the way he did?<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n
    \r\n

    Problem 3<\/h3>\r\n

    There are lots of different ways to calculate the number of colored squares along the border of a\u00a0$10\\times 10$\u00a0square. Below are the calculations several other students did. For each calculation, write a justification and draw a picture to show why it calculates the number of squares correctly. Think about using color in your picture to make your work more clear.<\/p>\r\n\r\n

      \r\n \t
    1. Valerie calculated $10 + 10 + 8 + 8$.<\/li>\r\n \t
    2. Kayla calculated $4 \\times 9$.<\/li>\r\n \t
    3. Linda calculated $(10\\times 10 ) - ( 8 \\times 8 )$.<\/li>\r\n \t
    4. Mark calculated $(4 \\times 8) + 4$.<\/li>\r\n \t
    5. Allan calculated $10 + 9 + 9 + 8$.<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
      \r\n

      Problem 4<\/h3>\r\n

      Now suppose that\u00a0 you have a large\u00a0$6\\times 6$\u00a0square with the unit squares along the border colored red.\u00a0 Adapt two of the techniques above to calculate the number of red unit squares.<\/p>\r\n

      For each technique you used, write an explanation and include a picture. Think about how to use colors or other methods to make your picture and explanation more clear.<\/p>\r\n\r\n<\/div>\r\n \r\n

      \r\n

      Problem 5<\/h3>\r\n

      Now suppose that you have a large\u00a0$25 \\times 25$\u00a0square with the unit squares along the border\u00a0 colored red.\u00a0 Adapt two of the techniques above to calculate the number of red unit squares.<\/p>\r\n

      For each technique you used, write an explanation and include a picture.\u00a0 Think about how to use colors or other methods to make your picture and explanation more clear.<\/p>\r\n\r\n<\/div>\r\n \r\n

      \r\n

      Problem 6<\/h3>\r\n
        \r\n \t
      1. Suppose that you have 64 red squares.\u00a0 Can you use all of those squares to make the border of a larger square in a picture like the one above?\u00a0 If yes, what are the dimensions of the larger square?\u00a0 If no, why not?<\/li>\r\n \t
      2. What if you have 30 red squares?\u00a0 Same questions.<\/li>\r\n \t
      3. What if you have 256 red squares? Same questions.<\/li>\r\n<\/ol>\r\n<\/div>\r\n \r\n
        \r\n

        Think \/ Pair \/ Share<\/h3>\r\n

        Describe some general rules:<\/p>\r\n\r\n

          \r\n \t
        • If you have a large\u00a0$n\\times n$\u00a0square with the border squares colored red, how many red squares will there be?\u00a0 Justify your answer with words and a picture.<\/li>\r\n<\/ul>\r\n
            \r\n \t
          • If you have $k$ red squares, is there a quick test you can do to decide if you can use all of those squares to make the border of a large square? Can you tell how big the square will be?<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"

            Here\u2019s another problem:<\/p>\n

            \n

            Problem 2<\/h3>\n

            Here is a large square made up of 100 smaller unit squares.\u00a0 The unit squares along the border of the large square are colored red.\u00a0 Without counting one-by-one, can you figure out how many red squares there are in the picture?<\/p>\n

            \"\"<\/p>\n

            Clearly describe how you figured out the number of red squares, and how you know your answer is correct.<\/p>\n<\/div>\n

            Justin calculated the number of squares as \"(10 \times 4) - 4\".\u00a0 He justified his answer this way:<\/p>\n

            Since the dimensions of the big square are\u00a0\"10\times 10\", there are 10 squares along each of the four sides. So that gives me\u00a040\u00a0red squares. But then each corner is part of two different sides. I’ve counted each of the corners twice. So I need to make up for that by subtracting\u00a04\u00a0at the end.<\/p><\/blockquote>\n

            Justin showed this picture to justify his work:<\/p>\n

            \"\"<\/p>\n

            \n

            Think \/ Pair \/ Share<\/h3>\n
              \n
            • What do you think about Justin\u2019s solution? Are you convinced?\u00a0 Could he have explained it more clearly?<\/li>\n
            • Was Justin\u2019s solution different from your solution or the same?<\/li>\n
            • Notice the color coding in Justin\u2019s picture. What do the colors represent? Why did he use the colors the way he did?<\/li>\n<\/ul>\n<\/div>\n

               <\/p>\n

              \n

              Problem 3<\/h3>\n

              There are lots of different ways to calculate the number of colored squares along the border of a\u00a0\"10\times 10\"\u00a0square. Below are the calculations several other students did. For each calculation, write a justification and draw a picture to show why it calculates the number of squares correctly. Think about using color in your picture to make your work more clear.<\/p>\n

                \n
              1. Valerie calculated \"10 + 10 + 8 + 8\".<\/li>\n
              2. Kayla calculated \"4 \times 9\".<\/li>\n
              3. Linda calculated \"(10\times 10 ) - ( 8 \times 8 )\".<\/li>\n
              4. Mark calculated \"(4 \times 8) + 4\".<\/li>\n
              5. Allan calculated \"10 + 9 + 9 + 8\".<\/li>\n<\/ol>\n<\/div>\n

                 <\/p>\n

                \n

                Problem 4<\/h3>\n

                Now suppose that\u00a0 you have a large\u00a0\"6\times 6\"\u00a0square with the unit squares along the border colored red.\u00a0 Adapt two of the techniques above to calculate the number of red unit squares.<\/p>\n

                For each technique you used, write an explanation and include a picture. Think about how to use colors or other methods to make your picture and explanation more clear.<\/p>\n<\/div>\n

                 <\/p>\n

                \n

                Problem 5<\/h3>\n

                Now suppose that you have a large\u00a0\"25 \times 25\"\u00a0square with the unit squares along the border\u00a0 colored red.\u00a0 Adapt two of the techniques above to calculate the number of red unit squares.<\/p>\n

                For each technique you used, write an explanation and include a picture.\u00a0 Think about how to use colors or other methods to make your picture and explanation more clear.<\/p>\n<\/div>\n

                 <\/p>\n

                \n

                Problem 6<\/h3>\n
                  \n
                1. Suppose that you have 64 red squares.\u00a0 Can you use all of those squares to make the border of a larger square in a picture like the one above?\u00a0 If yes, what are the dimensions of the larger square?\u00a0 If no, why not?<\/li>\n
                2. What if you have 30 red squares?\u00a0 Same questions.<\/li>\n
                3. What if you have 256 red squares? Same questions.<\/li>\n<\/ol>\n<\/div>\n

                   <\/p>\n

                  \n

                  Think \/ Pair \/ Share<\/h3>\n

                  Describe some general rules:<\/p>\n

                    \n
                  • If you have a large\u00a0\"n\times n\"\u00a0square with the border squares colored red, how many red squares will there be?\u00a0 Justify your answer with words and a picture.<\/li>\n<\/ul>\n
                      \n
                    • If you have \"k\" red squares, is there a quick test you can do to decide if you can use all of those squares to make the border of a large square? Can you tell how big the square will be?<\/li>\n<\/ul>\n<\/div>\n","protected":false},"author":21,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-969","chapter","type-chapter","status-publish","hentry"],"part":910,"_links":{"self":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/969","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":12,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/969\/revisions"}],"predecessor-version":[{"id":2767,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/969\/revisions\/2767"}],"part":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/parts\/910"}],"metadata":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapters\/969\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/media?parent=969"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/pressbooks\/v2\/chapter-type?post=969"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/contributor?post=969"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks-dev.oer.hawaii.edu\/math111\/wp-json\/wp\/v2\/license?post=969"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}