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PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}]} }, AutoDelete->False, FrameStyle->Thickness[Large], GridBoxFrame->{"Columns" -> {{True}}, "Rows" -> {{True}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], "Grid"], TraditionalForm]], "Output"] }, {2}]], Cell[TextData[{ "Lemniscate (rectangular): ", StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]}], ")"}], "2"], "=", RowBox[{"2", " ", SuperscriptBox["a", "2"], " ", "x", " ", "y", " "}]}], TraditionalForm]]] }], "Text"], Cell[TextData[{ "Lemniscate (polar):", StyleBox[" ", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["r", "2"], "=", RowBox[{ SuperscriptBox["a", "2"], " ", RowBox[{"sin", "(", RowBox[{"2", " ", "\[Theta]"}], ")"}]}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["r", "2"], "=", RowBox[{ SuperscriptBox["a", "2"], " ", RowBox[{"cos", "(", RowBox[{"2", " ", "\[Theta]"}], ")"}]}]}], TraditionalForm]]] }], "Text"], Cell[TextData[{ "Notice that the lemniscate has an \"", Cell[BoxData[ FormBox["r", TraditionalForm]]], "-squared\" rather than just \"", Cell[BoxData[ FormBox["r", TraditionalForm]]], ".\" This feature sets it apart from the rose curves, even though the \ lemniscate resembles a rose curve of 2 petals." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Goals", "Subsection"], Cell[CellGroupData[{ Cell["\<\ Students will be able to recognize from the equation whether the polar \ function is a rose curve, lemniscate, lima\[CCedilla]on, or a circle.\ \>", "Item"], Cell["\<\ Students will also be able to determine information about the polar graph, \ such as orientation, radius, diameter, number of petals, etc. from the \ equation.\ \>", "Item"], Cell[TextData[{ "Students will learn how to plot these graphs using ", StyleBox["Mathematica.", FontSlant->"Italic"] }], "Item"] }, Open ]] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Concept Exploration", "Section"], Cell[CellGroupData[{ Cell["Rose curves", "Subsection"], Cell["\<\ Look at the demonstration below. Try different functions by clicking on the \ buttons and then answer the questions below. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"PolarPlot", "[", RowBox[{ RowBox[{"3", " ", RowBox[{"trig", "[", RowBox[{"3", "\[Theta]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", RowBox[{"2", "\[Pi]"}]}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "5"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Thick", ",", "Orange"}], "}"}]}], ",", RowBox[{"PlotLabel", "\[Rule]", RowBox[{"Style", "[", RowBox[{ RowBox[{"\"\\"", "<>", RowBox[{"ToString", "[", "trig", "]"}], "<>", "\"\< 3\[Theta] \>\""}], ",", "Orange", ",", RowBox[{"FontSize", "\[Rule]", "20"}]}], "]"}]}]}], "]"}], ",", RowBox[{"{", RowBox[{"trig", ",", RowBox[{"{", RowBox[{"Sin", ",", "Cos"}], "}"}]}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`trig$$ = Sin, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`trig$$], {Sin, Cos}}}, Typeset`size$$ = { 360., {188., 192.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`trig$890$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`trig$$ = Sin}, "ControllerVariables" :> { Hold[$CellContext`trig$$, $CellContext`trig$890$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> PolarPlot[ 3 $CellContext`trig$$[3 $CellContext`\[Theta]], {$CellContext`\[Theta], 0, 2 Pi}, PlotRange -> 5, PlotStyle -> {Thick, Orange}, PlotLabel -> Style[ StringJoin["r = 3 ", ToString[$CellContext`trig$$], " 3\[Theta] "], Orange, FontSize -> 20]], "Specifications" :> {{$CellContext`trig$$, {Sin, Cos}}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{409., {236.625, 245.375}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output"] }, {2}]], Cell["\<\ What changes between the \"sine\" graph and the \"cosine\" graph? What stays \ the same?\ \>", "Example"], Cell["\<\ The graph seems rotated from one to the other. However, the size and shape \ of the graph remains unchanged.\ \>", "Answer"], Cell["What about the symmetry of the graphs?", "Example"], Cell[TextData[{ "The \"sine\" graph is symmetric about the y-axis, while the \"cosine\" \ graph is symmetric about the line ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", "x"}], TraditionalForm]]], "." }], "Answer"], Cell["\<\ Now look at a similar model and make some additional observations.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"PolarPlot", "[", RowBox[{ RowBox[{"3", " ", RowBox[{"Cos", "[", RowBox[{"a", " ", "\[Theta]"}], "]"}]}], ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", RowBox[{"2", "\[Pi]"}]}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "5"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{"Thick", ",", "Red"}], "}"}]}], ",", RowBox[{"PlotLabel", " ", "\[Rule]", RowBox[{"Style", "[", RowBox[{ RowBox[{"Row", "[", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Style", "[", RowBox[{"\"\\"", ",", "Italic"}], "]"}], ",", "\"\< = \>\"", ",", RowBox[{ RowBox[{"ToString", "[", "3", "]"}], "<>", "\"\< cos \>\"", "<>", RowBox[{"ToString", "[", "a", "]"}], "<>", "\"\<\[Theta]\>\""}]}], "\[IndentingNewLine]", "}"}], "]"}], ",", RowBox[{"{", RowBox[{"Red", ",", "20"}], "}"}]}], "]"}]}]}], "]"}], ",", RowBox[{"{", RowBox[{"a", ",", RowBox[{"{", RowBox[{"2", ",", "3", ",", "4", ",", "5", ",", "6"}], "}"}]}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ StyleBox[ DynamicModuleBox[{$CellContext`a$$ = 4, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`a$$], {2, 3, 4, 5, 6}}}, Typeset`size$$ = { 360., {188., 192.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`a$8783$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, TraditionalForm, "Variables" :> {$CellContext`a$$ = 2}, "ControllerVariables" :> { Hold[$CellContext`a$$, $CellContext`a$8783$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> PolarPlot[ 3 Cos[$CellContext`a$$ $CellContext`\[Theta]], \ {$CellContext`\[Theta], 0, 2 Pi}, PlotRange -> 5, PlotStyle -> {Thick, Red}, PlotLabel -> Style[ Row[{ Style["r", Italic], " = ", StringJoin[ ToString[3], " cos ", ToString[$CellContext`a$$], "\[Theta]"]}], {Red, 20}]], "Specifications" :> {{$CellContext`a$$, {2, 3, 4, 5, 6}}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{409., {236.40625, 246.59375}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], TraditionalForm], Manipulate`InterpretManipulate[1]], TraditionalForm]], "Output"] }, {2}]], Cell["\<\ What is the correlation between the number of petals and the coefficient in \ front of the \[Theta]?\ \>", "Example"], Cell["\<\ If the coefficient is even, there are twice as many petals as that number. If the coefficient is odd, there are exactly that many petals.\ \>", "Answer"], Cell["\<\ Now, open up the cell that contains the previous example and replace the \ \"cosine\" function with the \"sine\" function. Do you still get the same \ outcome?\ \>", "Text"], Cell["", "Spacer"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Examples: Rose curves 1", "Subsection"], Cell["\<\ Given the equations below, describe the petal length, number of petals and \ symmetry of the rose curves. Then check your answer by making a polar plot.\ \>", "Text"], Cell[BoxData[ FormBox[ RowBox[{"r", "=", RowBox[{"5", " ", RowBox[{"sin", "(", RowBox[{"4", " ", "\[Theta]"}], ")"}]}]}], TraditionalForm]], "Example"], Cell[TextData[StyleBox["Petal length: ", FontWeight->"Bold"]], "Text", Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[StyleBox["Number of petals: ", FontWeight->"Bold"]], "Text", Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[StyleBox["Symmetry: ", FontWeight->"Bold"]], "Text", Background->RGBColor[0.87, 0.94, 1]], Cell[BoxData[ RowBox[{"PolarPlot", "["}]], "Input"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{"r", " ", "=", " ", RowBox[{"2", " ", "sin", " ", RowBox[{"(", RowBox[{"5", "\[Theta]"}], ")"}]}]}], TraditionalForm]]]], "Example"], Cell[TextData[StyleBox["Petal length: ", FontWeight->"Bold"]], "Text", Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[StyleBox["Number of petals: ", FontWeight->"Bold"]], "Text", Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[StyleBox["Symmetry: ", FontWeight->"Bold"]], "Text", Background->RGBColor[0.87, 0.94, 1]], Cell[BoxData[ RowBox[{"PolarPlot", "["}]], "Input"], Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{"r", " ", "=", " ", RowBox[{"3.5", " ", "cos", " ", RowBox[{"(", RowBox[{"2", "\[Theta]"}], ")"}]}]}], TraditionalForm]]]], "Example"], Cell[TextData[StyleBox["Petal length: ", FontWeight->"Bold"]], "Text", Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[StyleBox["Number of petals: ", FontWeight->"Bold"]], "Text", Background->RGBColor[0.87, 0.94, 1]], Cell[TextData[StyleBox["Symmetry: ", FontWeight->"Bold"]], "Text", Background->RGBColor[0.87, 0.94, 1]], Cell[BoxData[ RowBox[{"PolarPlot", "["}]], "Input"], Cell["", "Spacer"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Examples: Rose curves 2", "Subsection"], Cell["\<\ Given the descriptions of the rose curves below, create a rose curve that \ matches those characteristics. Then check your answer by making a polar plot.\ \>", "Text"], Cell[TextData[{ "A rose curve with ", Cell[BoxData[ FormBox["y", TraditionalForm]]], "-axis symmetry that has 3 petals, each of length ", Cell[BoxData[ FormBox[ SqrtBox["7"], TraditionalForm]]], "." }], "Example"], Cell[TextData[StyleBox["", FontWeight->"Bold"]], "Answer", Background->None], Cell[BoxData[ RowBox[{"PolarPlot", "["}]], "Input"], Cell[TextData[{ "A rose curve with 8 petals, each of length 2. The rose curve has ", Cell[BoxData[ FormBox["y", TraditionalForm]]], "-axis symmetry.\n" }], "Example"], Cell[TextData[StyleBox["", FontWeight->"Bold"]], "Answer", Background->None], Cell[BoxData[ RowBox[{"PolarPlot", "["}]], "Input"], Cell[TextData[{ "A rose curve with petals of length 4. This 11-petaled curve must also have \ line ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", "x"}], TraditionalForm]]], " symmetry." }], "Example"], Cell[TextData[StyleBox["", FontWeight->"Bold"]], "Answer", Background->None], Cell[BoxData[ RowBox[{"PolarPlot", "["}]], "Input"], Cell["", "Spacer"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Concept Exploration 2", "Section"], Cell[CellGroupData[{ Cell["Lemniscates", "Subsection"], Cell["\<\ As we have seen already, lemniscates are a bit different in the structure of \ their equations. Let's look at the following demonstration:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"PolarPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ SqrtBox[ RowBox[{"9", RowBox[{"trig", "[", RowBox[{"2", "\[Theta]"}], "]"}]}]], ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"\[Theta]", ",", "0", ",", RowBox[{"2", "\[Pi]"}]}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", "5"}], ",", RowBox[{"PlotStyle", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"Thick", ",", "Blue"}], "}"}], ",", "Yellow"}], "}"}]}], ",", RowBox[{"PlotLabel", " ", "\[Rule]", RowBox[{"Style", "[", RowBox[{ RowBox[{"Row", "[", RowBox[{"{", "\[IndentingNewLine]", RowBox[{ RowBox[{"Style", "[", RowBox[{ "\"\<\!\(\*SuperscriptBox[\(r\), \(2\)]\)\>\"", ",", "Italic"}], "]"}], ",", "\"\< = \>\"", ",", RowBox[{ RowBox[{"ToString", "[", RowBox[{"9", " ", "trig"}], " ", "]"}], "<>", RowBox[{"ToString", "[", " ", "2", "]"}], "<>", "\"\<\[Theta]\>\""}]}], "\[IndentingNewLine]", "}"}], "]"}], ",", RowBox[{"{", RowBox[{"Blue", ",", "20"}], "}"}]}], "]"}]}]}], "]"}], ",", RowBox[{"{", RowBox[{"trig", ",", RowBox[{"{", RowBox[{"Sin", ",", "Cos"}], "}"}]}], "}"}]}], "]"}]], "Input"], Cell[BoxData[ FormBox[ TagBox[ FormBox[ StyleBox[ DynamicModuleBox[{$CellContext`trig$$ = Sin, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`trig$$], {Sin, Cos}}}, Typeset`size$$ = { 360., {190., 195.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`trig$16470$$ = False}, DynamicBox[Manipulate`ManipulateBoxes[ 1, TraditionalForm, "Variables" :> {$CellContext`trig$$ = Sin}, "ControllerVariables" :> { Hold[$CellContext`trig$$, $CellContext`trig$16470$$, False]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> PolarPlot[{(9 $CellContext`trig$$[2 $CellContext`\[Theta]])^ Rational[1, 2], 3}, {$CellContext`\[Theta], 0, 2 Pi}, PlotRange -> 5, PlotStyle -> {{Thick, Blue}, Yellow}, PlotLabel -> Style[ Row[{ Style["\!\(\*SuperscriptBox[\(r\), \(2\)]\)", Italic], " = ", StringJoin[ ToString[9 $CellContext`trig$$], ToString[2], "\[Theta]"]}], {Blue, 20}]], "Specifications" :> {{$CellContext`trig$$, {Sin, Cos}}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{409., {238.40625, 248.59375}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], TraditionalForm], Manipulate`InterpretManipulate[1]], TraditionalForm]], "Output"] }, {2}]], Cell["What do you notice about the symmetry of the graph?", "Example"], Cell[TextData[{ "The \"sine\" graph appears to be symmetric about either the line ", Cell[BoxData[ FormBox[ RowBox[{"y", "=", "x"}], TraditionalForm]]], " or the origin. The \"cosine\" graph appears to be symmetric about either \ the ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "-axis or ", Cell[BoxData[ FormBox["y", TraditionalForm]]], "-axis." }], "Answer"], Cell["\<\ What do you notice about the length of the petals in comparison with the \ equation?\ \>", "Example"], Cell["\<\ The coefficient in front of the trig function is the square of the \ petal-length.\ \>", "Answer"], Cell["What generalities can be made about lemniscates now?", "Text"], Cell[CellGroupData[{ Cell[TextData[{ "Lemniscates must \"start\" with an ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["r", "2"], "."}], TraditionalForm]]] }], "Item"], Cell["\<\ All lemniscates must have a 2 in front of the \[Theta].\ \>", "Item"], Cell["\<\ The coefficient in front of the \"sine\" or \"cosine\" is the square of the \ petal length.\ \>", "Item"], Cell[TextData[{ "We can generalize that the lemniscate behaves, in the way of symmetry, that \ the rectangular plot of sine and cosine do\[LongDash]sine is symmetric about \ the \norigin (the sine lemniscate is as well) and cosine is symmetric about \ the ", Cell[BoxData[ FormBox["y", TraditionalForm]]], "-axis (similarly with the cosine lemniscate)." }], "Item"] }, Open ]], Cell["", "Spacer"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Examples - Lemniscates 1", "Subsection"], Cell["\<\ Given the descriptions of the lemniscates below, create an equation that \ matches those characteristics. Then check your answer by making a polar plot.\ \ \>", "Text"], Cell[TextData[{ "A lemniscate with petals of length 4 and with ", Cell[BoxData[ FormBox["x", TraditionalForm]]], "-axis symmetry." }], "Example"], Cell[TextData[StyleBox["", FontWeight->"Bold"]], "Answer", Background->None], Cell[BoxData[ RowBox[{"PolarPlot", "["}]], "Input"], Cell["\<\ A lemniscate with origin symmetry and whose petals have length 2.5.\ \>", "Example"], Cell[TextData[StyleBox["", FontWeight->"Bold"]], "Answer", Background->None], Cell[BoxData[ RowBox[{"PolarPlot", "["}]], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Examples - Lemniscates 2", "Subsection"], Cell["\<\ Convert the following polar equations of lemniscates to rectangular \ equations. Make sure in the \"equation conversion\" cell below, that you hit \ Ctrl+Enter to move to the next line. This will keep the equal signs lined \ up. Then, type in the polar equation in the first entry box and the \ rectangular equation in the second box below. 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CellGroupData[{ Cell["Styles for Title and Section Cells", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Title"], CellMargins -> {{24, Inherited}, {0, 12}}, MenuPosition -> 1000, MenuCommandKey -> "1", FontFamily -> "Helvetica", FontSize -> 30, FontWeight -> "Bold", FontColor -> RGBColor[0.145098, 0.545098, 0.858824]], Cell[ StyleData["Title", "SlideShow"], CellMargins -> {{20, Inherited}, {0, 12}}, FontSize -> 42]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Subtitle"], CellMargins -> {{24, Inherited}, {12, 12}}, MenuPosition -> 1100, MenuCommandKey -> "2", FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Bold", FontColor -> RGBColor[0.301961, 0.631373, 0.890196]], Cell[ StyleData["Subtitle", "SlideShow"], CellMargins -> {{20, Inherited}, {12, 12}}, FontSize -> 28]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Section"], CellFrame -> {{0, 0}, {0, 0.5}}, CellMargins -> {{20, Inherited}, {6, 24}}, MenuPosition -> 1300, MenuCommandKey -> "4", FontFamily -> 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MenuCommandKey -> "6", FontFamily -> "Helvetica", FontSize -> 13, FontWeight -> "Bold", FontColor -> RGBColor[0.909804, 0.596078, 0.25098]], Cell[ StyleData["Subsubsection", "Presentation"], CellMargins -> {{80, Inherited}, {3, 15}}, FontSize -> 24], Cell[ StyleData["Subsubsection", "Printout"], CellMargins -> {{36, Inherited}, {Inherited, 12}}, FontSize -> 10], Cell[ StyleData["Subsubsection", "ColorPrintout"], CellMargins -> {{36, Inherited}, {Inherited, 12}}, FontSize -> 10]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Heading cells", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["TeacherName"], CellMargins -> {{24, Inherited}, {6, 24}}, CellFrameMargins -> 0, CellFrameLabels -> {{ Cell[ TextData["Teacher: "], "LabelStyleShort"], None}, {None, None}}, CellFrameLabelMargins -> 0, $CellContext`MenuPosition -> 1200, FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["TeacherName", "SlideShow"], FontSize -> 16]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["ClassName"], CellMargins -> {{24, Inherited}, {6, 0}}, CellFrameMargins -> 0, CellFrameLabels -> {{ Cell[ TextData["Class: "], "LabelStyleShort"], None}, { None, None}}, CellFrameLabelMargins -> 0, $CellContext`MenuPosition -> 1210, FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["ClassName", "SlideShow"], FontSize -> 16]}, Closed]], Cell[ StyleData["Course"], CellMargins -> {{24, Inherited}, {0, 7}}, CellFrameMargins -> 0, CellFrameLabels -> {{ Cell[ TextData["Course: "], "LabelStyle"], None}, {None, None}}, CellFrameLabelMargins -> 0, $CellContext`MenuPosition -> 1400, FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["ActivityLength"], CellMargins -> {{24, Inherited}, {0, 7}}, CellFrameMargins -> 0, CellFrameLabels -> {{ Cell[ TextData["Activity Length: "], "LabelStyle"], None}, {None, None}}, CellFrameLabelMargins -> 0, MenuPosition -> 1410, FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["NCTMStandard"], CellMargins -> {{24, Inherited}, {0, 7}}, CellFrameMargins -> 0, CellFrameLabels -> {{ Cell[ TextData["NCTM Standard: "], "LabelStyle"], None}, {None, None}}, CellFrameLabelMargins -> 0, MenuPosition -> 1420, FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["Prerequisites"], CellMargins -> {{24, Inherited}, {0, 7}}, CellFrameMargins -> 0, CellFrameLabels -> {{ Cell[ TextData["Prerequisites: "], "LabelStyle"], None}, {None, None}}, CellFrameLabelMargins -> 0, MenuPosition -> 1430, FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["Objectives"], CellMargins -> {{24, Inherited}, {0, 7}}, CellFrameMargins -> 0, CellFrameLabels -> {{ Cell[ TextData["Objectives: "], "LabelStyle"], None}, {None, None}}, CellFrameLabelMargins -> 0, MenuPosition -> 1440, FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["ActivityType"], CellMargins -> {{24, Inherited}, {0, 7}}, CellFrameMargins -> 0, CellFrameLabels -> {{ Cell[ TextData["Type of Activity: "], "LabelStyle"], None}, { None, None}}, CellFrameLabelMargins -> 0, MenuPosition -> 1450, FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["ReferenceMaterial"], CellMargins -> {{24, Inherited}, {0, 7}}, CellFrameMargins -> 0, CellFrameLabels -> {{ Cell[ TextData["Reference Material: "], "LabelStyle"], None}, { None, None}}, CellFrameLabelMargins -> 0, MenuPosition -> 1450, FontFamily -> "Helvetica", FontSize -> 14, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ CellGroupData[{ Cell[ StyleData["LabelStyle"], CellSize -> {136, Inherited}, CellBaseline -> Baseline, TextAlignment -> Right, FontFamily -> "Helvetica", FontWeight -> "Bold"], Cell[ StyleData["LabelStyle", 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StyleData["Example", "SlideShow"], FontSize -> 16]}, Closed]], Cell[ StyleData["Answer"], CellFrame -> 1., CellMargins -> {{18, Inherited}, {Inherited, 6}}, CellFrameMargins -> 10, CellFrameLabels -> {{ Cell[ TextData["Answer:"], "LabelStyleShort"], None}, {None, None}}, MenuPosition -> 1600, FontFamily -> "Helvetica", FontSize -> 16, FontColor -> RGBColor[0.196078, 0.196078, 0.196078], Background -> GrayLevel[1]]}, Closed]], Cell[ CellGroupData[{ Cell["Lists", "Subsection"], Cell[ CellGroupData[{ Cell["Bulleted", "Subsubsubsection"], Cell[ CellGroupData[{ Cell[ StyleData["Item"], CellMargins -> {{78, 10}, {7, 7}}, MenuPosition -> 1610, FontFamily -> "Helvetica", FontColor -> RGBColor[0.145098, 0.545098, 0.858824]], Cell[ StyleData["Item", "SlideShow"], CellMargins -> {{66, 10}, {8, 8}}, FontSize -> 16]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Subitem"], CellMargins -> {{102, 12}, {1, 2}}, MenuPosition -> 1620, FontFamily -> "Helvetica", FontSize -> 10, FontColor -> 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