// Numbas version: exam_results_page_options
{"name": "Square and cube numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"x": {"templateType": "anything", "definition": "sort([x1,x2,x3,x4,x5])", "description": "<p>Sorted list of integers from 1 to 12.</p>", "name": "x", "group": "Ungrouped variables"}, "x1": {"templateType": "anything", "definition": "random(1..3)", "description": "", "name": "x1", "group": "Ungrouped variables"}, "y2": {"templateType": "anything", "definition": "y^2", "description": "<p>Squared number in part c).</p>", "name": "y2", "group": "Ungrouped variables"}, "x5": {"templateType": "anything", "definition": "random(10..12 except x1 except x2 except x3 except x4)", "description": "", "name": "x5", "group": "Ungrouped variables"}, "x4": {"templateType": "anything", "definition": "random(7..10 except x1 except x2 except x3)", "description": "", "name": "x4", "group": "Ungrouped variables"}, "x3": {"templateType": "anything", "definition": "random(5..7 except x1 except x2)", "description": "", "name": "x3", "group": "Ungrouped variables"}, "ly": {"templateType": "anything", "definition": "y2 - 2y + random(2..4)", "description": "<p>Lower bound in part c).</p>", "name": "ly", "group": "Ungrouped variables"}, "x2": {"templateType": "anything", "definition": "random(3..5 except x1)", "description": "", "name": "x2", "group": "Ungrouped variables"}, "uy": {"templateType": "anything", "definition": "y2 + 2y - random(2..5)", "description": "<p>Upper bound in part c).</p>", "name": "uy", "group": "Ungrouped variables"}, "y": {"templateType": "anything", "definition": "random(3..12 except x)", "description": "<p>Answer to part c).</p>", "name": "y", "group": "Ungrouped variables"}}, "type": "question", "name": "Square and cube numbers", "parts": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[0]*x[0]", "maxValue": "x[0]*x[0]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[1]*x[1]", "maxValue": "x[1]*x[1]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[2]*x[2]", "maxValue": "x[2]*x[2]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[3]*x[3]", "maxValue": "x[3]*x[3]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[4]*x[4]", "maxValue": "x[4]*x[4]", "marks": 1, "variableReplacements": []}], "prompt": "<p>Find the following:</p>\n<p>$\\var{x[0]}^2 =$&nbsp; [[0]]</p>\n<p>$\\var{x[1]}^2 =$ [[1]]</p>\n<p>$\\var{x[2]}^2 =$ [[2]]</p>\n<p>$\\var{x[3]}^2 =$ [[3]]</p>\n<p>$\\var{x[4]}^2 =$ [[4]]</p>\n<p></p>"}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[0]*x[0]*x[0]", "maxValue": "x[0]*x[0]*x[0]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[1]*x[1]*x[1]", "maxValue": "x[1]*x[1]*x[1]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[2]*x[2]*x[2]", "maxValue": "x[2]*x[2]*x[2]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[3]*x[3]*x[3]", "maxValue": "x[3]*x[3]*x[3]", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "x[4]*x[4]*x[4]", "maxValue": "x[4]*x[4]*x[4]", "marks": 1, "variableReplacements": []}], "prompt": "<p>Find the following:</p>\n<p>$\\var{x[0]}^3 =$&nbsp; [[0]]</p>\n<p>$\\var{x[1]}^3 =$ [[1]]</p>\n<p>$\\var{x[2]}^3 =$ [[2]]</p>\n<p>$\\var{x[3]}^3 =$ [[3]]</p>\n<p>$\\var{x[4]}^3 =$ [[4]]</p>"}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "scripts": {}, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "y^2", "maxValue": "y^2", "marks": 1, "variableReplacements": []}, {"correctAnswerFraction": false, "mustBeReduced": false, "type": "numberentry", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "minValue": "y", "maxValue": "y", "marks": "1", "variableReplacements": []}], "prompt": "<p>Find a square number $y^2$ between $\\var{ly}$ and $\\var{uy}$ and its integer root $y$.</p>\n<p>$y^2 = $&nbsp; [[0]]</p>\n<p>$y = $&nbsp; [[1]]</p>"}], "advice": "<h4>a)</h4>\n<p>Squared integers are called square numbers. It may be useful to remember the first few square numbers to be able to use them without a calculator.</p>\n<p>Here:</p>\n<p>\\[ \\begin{align} \\var{x[0]}^2 &amp;= \\var{x[0]} \\times \\var{x[0]} \\\\&amp;= \\var{x[0]^2} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[1]}^2 &amp;= \\var{x[1]} \\times \\var{x[1]} \\\\&amp;= \\var{x[1]^2} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[2]}^2 &amp;= \\var{x[2]} \\times \\var{x[2]} \\\\&amp;= \\var{x[2]^2} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[3]}^2 &amp;= \\var{x[3]} \\times \\var{x[3]} \\\\&amp;= \\var{x[3]^2} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[4]}^2 &amp;= \\var{x[4]} \\times \\var{x[4]} \\\\&amp;= \\var{x[4]^2} \\text{.}\\end{align}\\]</p>\n<p></p>\n<h4>b)</h4>\n<p>Cubed integers are called cubed numbers. To obtain these, we would typically always&nbsp;use a calculator.</p>\n<p>We can either cube the number $x$:</p>\n<p>\\[ \\begin{align} \\var{x[0]}^3 &amp;= \\var{x[0]} \\times \\var{x[0]} \\times \\var{x[0]} \\\\&amp;= \\var{x[0]^3} \\text{,} \\end{align}\\]</p>\n<p>or we can multiply the square number&nbsp;$(x_n)^2$&nbsp;from part <strong>a) </strong>by the appropriate $x_n$:</p>\n<p>\\[ \\begin{align} \\var{x[0]}^3 &amp;= \\var{x[0]}^2 \\times \\var{x[0]} \\\\&amp;= \\var{x[0]^2} \\times \\var{x[0]}\\\\&amp;= \\var{x[0]^3} \\text{.} \\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[1]}^3 &amp;= \\var{x[1]} \\times \\var{x[1]} \\times \\var{x[1]} \\\\ &amp;= \\var{x[1]^3} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[2]}^3 &amp;= \\var{x[2]} \\times \\var{x[2]} \\times \\var{x[2]} \\\\ &amp;= \\var{x[2]^3} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[3]}^3 &amp;= \\var{x[3]} \\times \\var{x[3]} \\times \\var{x[3]} \\\\ &amp;= \\var{x[3]^3} \\text{.}\\end{align}\\]</p>\n<p>\\[ \\begin{align} \\var{x[3]}^4 &amp;= \\var{x[4]} \\times \\var{x[4]} \\times \\var{x[4]} \\\\ &amp;= \\var{x[4]^3} \\text{.}\\end{align}\\]</p>\n<p></p>\n<h4>c)</h4>\n<p>Here is&nbsp;a table of square numbers for integers from $1$ to $15$:</p>\n<table style=\"border-color: #000000; width: 933px; height: 100px; margin-right: auto; margin-left: auto;\" cellspacing=\"1\" cellpadding=\"1\" border=\"1\">\n<tbody>\n<tr style=\"height: 11px;\">\n<th style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle; background-color: #d6d6d6;\">$y$</th>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$1$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$2$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$3$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$4$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$5$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$6$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$7$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$8$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$9$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$10$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$11$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$12$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$13$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$14$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$15$</td>\n</tr>\n<tr style=\"height: 1px;\">\n<th style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle; background-color: #d6d6d6;\">$y^2$</th>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$1$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$4$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$9$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$16$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$25$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$36$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$49$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$64$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$81$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$100$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$121$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$144$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$169$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$196$</td>\n<td style=\"width: 1px; height: 1px; text-align: center; vertical-align: middle;\">$225$</td>\n</tr>\n</tbody>\n</table>\n<p>The only square number between $\\var{ly}$&nbsp;and $\\var{uy}$ is $\\var{y2}$.</p>\n<p>To calculate $y$ we must&nbsp;calculate the square root of $y^2$,</p>\n<p>\\[ \\sqrt{\\var{y2}} = \\var{y} \\text{.}\\]</p>\n<p>This is our integer $y$.</p>\n<p></p>\n<p></p>", "tags": ["cube", "indices", "multiplication", "powers", "roots", "square", "taxonomy"], "preamble": {"js": "", "css": ""}, "rulesets": {}, "extensions": [], "functions": {}, "ungrouped_variables": ["x1", "x2", "x3", "x4", "x5", "x", "y2", "y", "ly", "uy"], "statement": "<p>Try the following questions on square and cube numbers.</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the squares, and cubes, of some numbers.</p>\n<p>Finally, find a square number between two given limits.</p>"}, "variablesTest": {"condition": "", "maxRuns": "1000"}, "contributors": [{"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}]}]}], "contributors": [{"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}]}