{"id":195,"date":"2024-08-22T16:35:49","date_gmt":"2024-08-22T16:35:49","guid":{"rendered":"https:\/\/learnx.ca\/mt\/?page_id=195"},"modified":"2025-08-18T08:02:56","modified_gmt":"2025-08-18T08:02:56","slug":"activities-numbers","status":"publish","type":"page","link":"https:\/\/learnx.ca\/mt\/activities-numbers\/","title":{"rendered":"PLACEMAT TASKS"},"content":{"rendered":"\n<p>[<a href=\"#s\">Solutions<\/a>&nbsp;are shown after the activities]<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 1<\/h4>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"398\" height=\"334\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a1.png\" alt=\"\" class=\"wp-image-215\" srcset=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a1.png 398w, https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a1-300x252.png 300w\" sizes=\"auto, (max-width: 398px) 85vw, 398px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 2<\/h4>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"218\" height=\"268\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a3.png\" alt=\"\" class=\"wp-image-217\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 3<\/h4>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"170\" height=\"227\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a4.png\" alt=\"\" class=\"wp-image-221\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 4<\/h4>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"338\" height=\"222\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a5.png\" alt=\"\" class=\"wp-image-219\" srcset=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a5.png 338w, https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a5-300x197.png 300w\" sizes=\"auto, (max-width: 338px) 85vw, 338px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 5<\/h4>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"269\" height=\"128\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a6.png\" alt=\"\" class=\"wp-image-220\"\/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 6<\/h4>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"178\" height=\"302\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a11.png\" alt=\"\" class=\"wp-image-230\" srcset=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a11.png 178w, https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a11-177x300.png 177w\" sizes=\"auto, (max-width: 178px) 85vw, 178px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading has-dark-gray-color has-text-color has-link-color wp-elements-fd657f999b5c075dfd639e73e4ea691e\" id=\"s\">SOLUTIONS<\/h1>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 1<\/h4>\n\n\n\n<p><strong>What students may see:<\/strong><\/p>\n\n\n\n<p>Students may notice  patterns that deal with:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Visual and concrete patterns for consecutive odd, even, and natural numbers.<\/li>\n\n\n\n<li>The resulting numeric and algebraic expressions for their sums. For example, the sum of odd numbers 1 + 3 + 5 + 7 = 4&#215;4 and more generally 1 + 3 + 5 + &#8230; + N = NxN = N<sup>2<\/sup>.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"527\" height=\"203\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a2.png\" alt=\"\" class=\"wp-image-216\" srcset=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a2.png 527w, https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a2-300x116.png 300w\" sizes=\"auto, (max-width: 527px) 85vw, 527px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 2<\/h4>\n\n\n\n<p><strong>What students may make:<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"214\" height=\"212\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a10.png\" alt=\"\" class=\"wp-image-228\" srcset=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a10.png 214w, https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a10-150x150.png 150w\" sizes=\"auto, (max-width: 214px) 85vw, 214px\" \/><\/figure>\n<\/div>\n\n\n<p>Using link cubes as well as markers, dabbers, and grid paper, students may create their own colourful and appealing arrangements of odd, even, and natural numbers.<\/p>\n\n\n\n<p>The <em>math<\/em> art may be shared at home, along with practiced dialogues that explain math patterns and concepts:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Odd numbers hide in squares. For example, the first 4 odd numbers hide in a 4&#215;4 square.<\/li>\n\n\n\n<li>Even numbers hide in rectangles where one side is one unit larger than the other. For example, the first 4 even numbers hide in a 4&#215;5 rectangle.<\/li>\n\n\n\n<li>Two sets of natural numbers fit in an even number rectangle.<\/li>\n<\/ul>\n\n\n\n<p><strong>Sample scaffolding prompts and questions:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>What math does this art represent?<\/li>\n\n\n\n<li>How else might you represent it?<\/li>\n\n\n\n<li>How would you explain it to someone at home, who hasn\u2019t seen this before?<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 3<\/h4>\n\n\n\n<p>How students may respond:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"147\" height=\"147\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a9.png\" alt=\"\" class=\"wp-image-226\"\/><\/figure>\n<\/div>\n\n\n<p>We don\u2019t know how many odd numbers are in the blue square.<\/p>\n\n\n\n<p>We may answer in two parts:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First, what is the smallest number of odd numbers that may fit in the square?\n<ul class=\"wp-block-list\">\n<li><span style=\"font-size: revert;\">The answer would be 1.<\/span><\/li>\n\n\n\n<li><span style=\"font-size: revert;\">The whole square could represent 1 blue cube or the odd number 1.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Second, what is the largest number of odd numbers that may fit in the square?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><span style=\"font-size: revert;\">Let\u2019s imagine that the blue square has a 10 x 10 blue grid that is invisible to us.<\/span>\n<ul class=\"wp-block-list\">\n<li><span style=\"font-size: revert;\">In this grid, we could fit the first 10 odd numbers.<\/span><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><span style=\"font-size: revert;\">If we then imagine drawing horizontal and vertical lines through each grid square, we can create a 20 x 20 grid.<\/span>\n<ul class=\"wp-block-list\">\n<li><span style=\"font-size: revert;\">In this grid, we could fit the first 20 odd numbers<\/span>.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>We can repeat this and keep increasing the number of odd numbers hiding in the square to infinity.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 4<\/h4>\n\n\n\n<p><strong>How students may respond:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"338\" height=\"129\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a7.png\" alt=\"\" class=\"wp-image-224\" srcset=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a7.png 338w, https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a7-300x114.png 300w\" sizes=\"auto, (max-width: 338px) 85vw, 338px\" \/><\/figure>\n\n\n\n<p>From the pattern above:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Stage 1 has 1 x 1 = 1 block<\/li>\n\n\n\n<li>Stage 2 has 2 x 2 = 4 blocks<\/li>\n\n\n\n<li>Stage 3 has 3 x 3 = 9 blocks<\/li>\n\n\n\n<li>Stage 4 has 4 x 4 = 16 blocks<\/li>\n\n\n\n<li>Stage 5 has 5 x 5 = 25 blocks<\/li>\n\n\n\n<li>Stage 10 has 10 x 10 = 100 blocks<\/li>\n\n\n\n<li>Stage 100 has 100 x 100 = 10 000 blocks<\/li>\n\n\n\n<li>Stage N has N x N = N<sup>2<\/sup> blocks<\/li>\n<\/ul>\n\n\n\n<p>The number of blocks represent the sum of the odd numbers. For example, 1 + 3 + 5 = 9 = 3&#215;3 = 3<sup>2<\/sup>.<\/p>\n\n\n\n<p><strong>In younger grades<\/strong><\/p>\n\n\n\n<p>We have engaged students as young as grade 1 with such activities.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>They successfully complete the patterns up to stage 10.\n<ul class=\"wp-block-list\">\n<li>Instead of using multiplication, they typically skip count by 10.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Some extend these patterns to complete the last 2 questions by describing the result:<\/li>\n\n\n\n<li>For stage 100, there are 100 rows of 100 blocks.<\/li>\n\n\n\n<li>For stage N, there are N rows of N blocks.<\/li>\n<\/ul>\n\n\n\n<p><strong>NOTE<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>N is a variable.<\/li>\n\n\n\n<li>It may represent any stage number.<\/li>\n\n\n\n<li>N x N = N<sup>2<\/sup> is an expression for the general rule for finding the sum of the first N odd numbers.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 5<\/h4>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"253\" height=\"92\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a8.png\" alt=\"\" class=\"wp-image-225\"\/><\/figure>\n\n\n\n<p>The image above lists the first 10 odd numbers, from 1 to 19, and groups them into 5 pairs, with each pair having a sum of 20.<\/p>\n\n\n\n<p>So, the sum of the first 10 odd numbers 1-19 is 5 x 20 = 100.<\/p>\n\n\n\n<p>We may also use this method to find the sum of the first 10 natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>We may create 5 pairs whose sum is 11: 1+10, 2+9, 3+8, 4+7, 5+6.<\/li>\n\n\n\n<li>So, the sum of the first 10 natural numbers is: 5 x 11 = 55.<\/li>\n<\/ul>\n\n\n\n<p>Similarly, we may find the sum of the first 10 even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>We may create 5 pairs whose sum is 22: 2+20, 4+18, 6+16, 8+14, 10+12.<\/li>\n\n\n\n<li>So, the sum of the first 10 even numbers is: 5 x 22 = 110.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\">TASK 6<\/h4>\n\n\n\n<p><strong>How students may respond:<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"230\" height=\"515\" src=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a12.png\" alt=\"\" class=\"wp-image-231\" srcset=\"https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a12.png 230w, https:\/\/learnx.ca\/mt\/wp-content\/uploads\/2024\/08\/numbers-a12-134x300.png 134w\" sizes=\"auto, (max-width: 230px) 85vw, 230px\" \/><\/figure>\n<\/div>\n\n\n<p>They grow at a constant rate.<\/p>\n\n\n\n<p>If we join the tops of the bars with a red dotted line, it will be straight.<\/p>\n\n\n\n<p>The sum of odd numbers grows by odd numbers: 1, 3, 5, 7, and 9. This makes sense, as the bars represent sums of odd numbers.<\/p>\n\n\n\n<p>The sums grow at an increasing rate (as the odd numbers added increase in size).<\/p>\n\n\n\n<p>If we join the tops of the bars with a red dotted line, it will be curved.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Next<\/strong><\/h4>\n\n\n\n<p>Coding puzzles &#8230; <a href=\"https:\/\/learnx.ca\/mt\/coding-numbers\/\">https:\/\/learnx.ca\/mt\/coding-numbers<\/a><\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>[Solutions&nbsp;are shown after the activities] TASK 1 TASK 2 TASK 3 TASK 4 TASK 5 TASK 6 SOLUTIONS TASK 1 What students may see: Students may notice patterns that deal with: TASK 2 What students may make: Using link cubes as well as markers, dabbers, and grid paper, students may create their own colourful and &hellip; <a href=\"https:\/\/learnx.ca\/mt\/activities-numbers\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;PLACEMAT TASKS&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-195","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/learnx.ca\/mt\/wp-json\/wp\/v2\/pages\/195","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/learnx.ca\/mt\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/learnx.ca\/mt\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/learnx.ca\/mt\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/learnx.ca\/mt\/wp-json\/wp\/v2\/comments?post=195"}],"version-history":[{"count":19,"href":"https:\/\/learnx.ca\/mt\/wp-json\/wp\/v2\/pages\/195\/revisions"}],"predecessor-version":[{"id":834,"href":"https:\/\/learnx.ca\/mt\/wp-json\/wp\/v2\/pages\/195\/revisions\/834"}],"wp:attachment":[{"href":"https:\/\/learnx.ca\/mt\/wp-json\/wp\/v2\/media?parent=195"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}