\documentclass[12pt]{article}
\usepackage[Mickael]{ammaths}

\begin{document}

\module{Constructible numbers}{3}{06}{1 period}
\en\nocouleur

\prereq{General rules and methods of ruler and compass geometry}

\object{\begin{itemize}
        \item Discover the concept of contructible numbers and study a few examples.
        \end{itemize}}

\mater{\begin{itemize}
	\item Task sheet.
	\item Answer sheet.
	\end{itemize}}

\modpart{The basic operations}{15 mins}

Students are working in groups of 3 or 4. They have to find out the method to construct the following quantities,
from the unit $1$ and two constructed numbers $a$ and $b$ :
\begin{itemize}
\itemb $a+b$ ;
\itemb $a-b$ ;
\itemb $ab$ ;
\itemb $\frac{a}{b}$ ;
\itemb $\sqrt{a}$.
\end{itemize}
Methods must be shown on the same answer sheet as part 2.

\modpart{Some special numbers}{40 mins}

Still working in groups, students have to construct the following numbers. At the end of the hour, an answer sheet by
group must be given to the teacher to be marked.

\begin{center}
\begin{tabular}{cp{0.5cm}cp{0.5cm}cp{0.5cm}c}
$\cfrac{5}{3}$ && $\cfrac{2}{7}$ && $\sqrt{2}$ && $\sqrt{3}$ \\[10pt]
$\sqrt{2}+\sqrt{3}$ && $\sqrt{3}-\sqrt{2}$ && $\cfrac{5}{3}\sqrt{2}$ && $\cfrac{2}{7}\sqrt{3}$ \\[10pt]
$\sqrt{2}\times\sqrt{3}$ && $\cfrac{\sqrt{2}}{2}$ && $\cfrac{1}{\sqrt{2}}$ && $\cfrac{\sqrt{3}}{2}$ \\[10pt]
$1+\sqrt{5}$ && $\varphi=\cfrac{1+\sqrt{5}}{2}$ && $\cfrac{1}{\varphi}$ && $\cfrac{-1+\sqrt{5}}{2}$ \\
\end{tabular}
\end{center}



\pagebreak

\moddocdis{Constructible numbers}{3}{06}{Answer sheet}

\vspace*{-1cm}

\sectionblack{\Large Part 1 -- Methods}

\begin{multicols}{2}
\begin{center}
\begin{tabularx}{7cm}{|X|}
\rowcolor{black}\centering\bf \textsf{\white Sum of two numbers $a+b$}\rule{0pt}{12pt}
\end{tabularx}
\begin{tabularx}{7cm}{|X|}
\hline
\psset{unit=0.8cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-3.92,-1.04)(4.26,7)
\psline(-3.46,6.2)(-2.46,6.2)
\psline(-3.46,5.5)(-1.62,5.5)
\psline(-3.46,4.82)(-0.08,4.82)
\psline(-2.24,1.06)(3.76,5.6)
\psdots(-3.46,6.2)(-2.46,6.2)
\uput[d](-2.98,6.2){\small $1$}
\psdots(-3.46,5.5)
\psdots(-1.62,5.5)
\uput[d](-2.54,5.5){$a$}
\psdots(-3.46,4.82)
\psdots(-0.08,4.82)
\uput[d](-1.78,4.82){$b$}
\end{pspicture*} \\
\hline
\end{tabularx}
\end{center}

\begin{center}
\begin{tabularx}{7cm}{|X|}
\rowcolor{black}\centering\bf \textsf{\white Difference of two numbers $b-a$}\rule{0pt}{12pt}
\end{tabularx}
\begin{tabularx}{7cm}{|X|}
\hline
\psset{unit=0.8cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-3.92,-1.04)(4.26,7)
\psline(-3.46,6.2)(-2.46,6.2)
\psline(-3.46,5.5)(-1.62,5.5)
\psline(-3.46,4.82)(-0.08,4.82)
\psline(-2.24,1.06)(3.76,5.6)
\psdots(-3.46,6.2)(-2.46,6.2)
\uput[d](-2.98,6.2){\small $1$}
\psdots(-3.46,5.5)
\psdots(-1.62,5.5)
\uput[d](-2.54,5.5){$a$}
\psdots(-3.46,4.82)
\psdots(-0.08,4.82)
\uput[d](-1.78,4.82){$b$}
\end{pspicture*} \\
\hline
\end{tabularx}

\end{center}
\end{multicols}

\begin{multicols}{2}
\begin{center}
\begin{tabularx}{7cm}{|X|}
\rowcolor{black}\centering\bf \textsf{\white Product of two numbers $a\times b$}\rule{0pt}{12pt}
\end{tabularx}
\begin{tabularx}{7cm}{|X|}
\hline
\psset{unit=0.8cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-3.92,-1.04)(4.26,7)
\psline(-3.46,6.2)(-2.46,6.2)
\psline(-3.46,5.5)(-1.62,5.5)
\psline(-3.46,4.82)(-0.08,4.82)
\psline(-2.24,1.06)(3.76,5.6)
\psdots(-3.46,6.2)(-2.46,6.2)
\uput[d](-2.98,6.2){\small $1$}
\psdots(-3.46,5.5)
\psdots(-1.62,5.5)
\uput[d](-2.54,5.5){$a$}
\psdots(-3.46,4.82)
\psdots(-0.08,4.82)
\uput[d](-1.78,4.82){$b$}
\end{pspicture*} \\
\hline
\end{tabularx}
\end{center}

\begin{center}
\begin{tabularx}{7cm}{|X|}
\rowcolor{black}\centering\bf \textsf{\white Quotient of two numbers $b/a$}\rule{0pt}{12pt}
\end{tabularx}
\begin{tabularx}{7cm}{|X|}
\hline
\psset{unit=0.8cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-3.92,-1.04)(4.26,7)
\psline(-3.46,6.2)(-2.46,6.2)
\psline(-3.46,5.5)(-1.62,5.5)
\psline(-3.46,4.82)(-0.08,4.82)
\psline(-2.24,1.06)(3.76,5.6)
\psdots(-3.46,6.2)(-2.46,6.2)
\uput[d](-2.98,6.2){\small $1$}
\psdots(-3.46,5.5)
\psdots(-1.62,5.5)
\uput[d](-2.54,5.5){$a$}
\psdots(-3.46,4.82)
\psdots(-0.08,4.82)
\uput[d](-1.78,4.82){$b$}
\end{pspicture*} \\
\hline
\end{tabularx}

\end{center}
\end{multicols}

\begin{center}
\begin{tabularx}{15.35cm}{|X|}
\rowcolor{black}\centering\bf \textsf{\white Square root of a number $\sqrt{a}$}\rule{0pt}{12pt}
\end{tabularx}
\begin{tabularx}{15.35cm}{|X|}
\hline
\psset{unit=0.8cm,algebraic=true,dotstyle=*,dotsize=3pt 0,linewidth=0.8pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-3.92,0)(9.26,7)
\psline(-3.46,6.2)(-2.46,6.2)
\psline(-3.46,5.5)(3.46,5.5)
%\psline(-3.46,4.82)(-0.08,4.82)
%\psline(-2.24,1.06)(3.76,5.6)
\psdots(-3.46,6.2)(-2.46,6.2)
\uput[d](-2.98,6.2){\small $1$}
\psdots(-3.46,5.5)
\psdots(3.46,5.5)
\uput[d](0,5.5){$a$}
%\psdots(-3.46,4.82)
%\psdots(-0.08,4.82)
%\uput[d](-1.78,4.82){$b$}
%\psdots(-5.08,11)
\end{pspicture*} \\
\hline
\end{tabularx}
\end{center}

\newpage

\sectionblack{\Large Part 2 -- Construct some numbers}

\psset{unit=2cm}

\begin{center}
\begin{tabularx}{14cm}{|X|p{11cm}|}
\hline
\vspace{-1.5cm}
$$1$$ &
\begin{pspicture}(-0.5,-0.4)(4.5,0.5)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(1,0)\psdot(1,0)\end{pspicture} \\
\hline
\vspace{-1.5cm}
$$\cfrac{5}{3}$$ &
\begin{pspicture}(-0.5,-0.4)(4.5,0.5)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{2}{7}$$ &
\begin{pspicture}(-0.5,-0.4)(4.5,0.5)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\sqrt{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\sqrt{3}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\sqrt{2}+\sqrt{3}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\sqrt{3}-\sqrt{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{5}{3}\sqrt{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{2}{7}\sqrt{3}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\end{tabularx}\end{center}

\newpage

\begin{center}
\begin{tabularx}{14cm}{|X|p{11cm}|}
\hline
\vspace{-1.5cm}
$$\sqrt{2}\times\sqrt{3}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{\sqrt{2}}{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{1}{\sqrt{2}}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{\sqrt{3}}{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$1+\sqrt{5}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\varphi=\cfrac{1+\sqrt{5}}{2}$$ &
\begin{pspicture}(-0.5,-0.15)(4.5,0.75)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{1}{\varphi}$$ &
\begin{pspicture}(-0.5,-0.4)(4.5,0.5)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{-1+\sqrt{5}}{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)\end{pspicture}\\
\hline
\end{tabularx}\end{center}

\newpage

\moddoc{Answers - to be printed on a transparent sheet of paper}

\begin{center}
\begin{tabularx}{14cm}{|X|p{11cm}|}
\hline
\vspace{-1.5cm}
$$1$$ &
\begin{pspicture}(-0.5,-0.4)(4.5,0.5)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(1,0)\psdot(1,0)\end{pspicture} \\
\hline
\vspace{-1.5cm}
$$\cfrac{5}{3}$$ &
\begin{pspicture}(-0.5,-0.4)(4.5,0.5)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(1.666666667,0)\psdot(1.666666667,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{2}{7}$$ &
\begin{pspicture}(-0.5,-0.4)(4.5,0.5)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(0.285714286,0)\psdot(0.285714286,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\sqrt{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(1.414213562,0)\psdot(1.414213562,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\sqrt{3}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(1.732050808,0)\psdot(1.732050808,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\sqrt{2}+\sqrt{3}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(3.14626437,0)\psdot(3.14626437,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\sqrt{3}-\sqrt{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(0.317837245,0)\psdot(0.317837245,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{5}{3}\sqrt{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(2.357022604,0)\psdot(2.357022604,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{2}{7}\sqrt{3}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(0.494871659,0)\psdot(0.494871659,0)\end{pspicture}\\
\hline
\end{tabularx}\end{center}

\newpage

\begin{center}
\begin{tabularx}{14cm}{|X|p{11cm}|}
\hline
\vspace{-1.5cm}
$$\sqrt{2}\times\sqrt{3}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(2.449489743,0)\psdot(2.449489743,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{\sqrt{2}}{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(0.707106781,0)\psdot(0.707106781,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{1}{\sqrt{2}}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(0.707106781,0)\psdot(0.707106781,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{\sqrt{3}}{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(0.866025404,0)\psdot(0.866025404,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$1+\sqrt{5}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(3.236067977,0)\psdot(3.236067977,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\varphi=\cfrac{1+\sqrt{5}}{2}$$ &
\begin{pspicture}(-0.5,-0.15)(4.5,0.75)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(1.618033989,0)\psdot(1.618033989,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{1}{\varphi}$$ &
\begin{pspicture}(-0.5,-0.4)(4.5,0.5)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(0.618033989,0)\psdot(0.618033989,0)\end{pspicture}\\
\hline
\vspace{-1.5cm}
$$\cfrac{-1+\sqrt{5}}{2}$$ &
\begin{pspicture}(-0.5,-0.2)(4.5,0.7)\psline[linewidth=0.25pt](0,0)(4.5,0)\psdot(0,0)
\psline[linewidth=1.5pt](0,0)(0.618033989,0)\psdot(0.618033989,0)\end{pspicture}\\
\hline
\end{tabularx}\end{center}



\end{document}
