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

\begin{document}

\module{Finding Pick's formula}{3}{01}{1 period}
\en\nocouleur

%\prereq{}

\object{\begin{itemize}
        \item Experiment on a lattice to find Pick's formula.
	\end{itemize}}

\mater{\begin{itemize}
	\item Fact sheet about Pick's formula.
	\item Empty lattice.
	\item Beamer about Pick's formula.
	\item Paper for posters.
	\end{itemize}}

\modpart{Trying to find the formula}{45 mins}

Students work in teams of 3 or 4. They are given an empty lattice each and some hints on how to experiment. Other hints
may be asked from the teacher if no progress is made.

\modpart{Formula and examples}{10 mins}

The formula is given by the teacher, with a few examples of its use.

\pagebreak

\moddocdis{Pick's formula}{3}{01}{Lesson}

\begin{multicols}{2}
\defi{}{A lattice is a square grid, a discrete surface with a regular gap between any two points. A simple
example is the set of the points with whole coordinates on a
coordinate graph.}

\exe{\begin{center}
\psset{unit=0.5cm}
\pspicture(0,0)(10,4)
\multido{\n=0+1}{10}{
\multido{\d=0+1}{5}{
\psdot[dotstyle=o](\n,\d)
}}
\endpspicture
\end{center}}
\end{multicols}

\txt{Consider a lattice. We can draw in this lattice some
polygons whose vertices will be some of the lattice's points.
\emph{Pick's theorem} provides a simple formula for the area of any
such polygon\footnote{Inspiration for this chapter comes from
{\sl Another Fine Math You've Got Me Into\ldots} by Ian Stewart, a highly recommended
book for anyone interested in recreational but serious and thought-provoking mathematics.}.}

\defi{}{For any polygon $P$ in a lattice, we note $B_P$ the number of points on its border and $I_P$
the number of points strictly inside the polygon. The area of the polygon will be noted
$\mathscr{A}_P$.}

\theo{}{For any convex polygon $P$ in a square two-dimensional lattice, 
$$\mathscr{A}_P=\frac12B_P+I_P-1.$$
}

\begin{multicols}{2}
\exe{\begin{center}
\psset{unit=0.5cm}
\pspicture(0,-2)(10,5)
\multido{\n=0+1}{10}{
\multido{\d=0+1}{5}{
\psdot[dotstyle=o](\n,\d)
}}
\pspolygon(1,2)(2,3)(7,4)(8,0)(5,1)
\endpspicture
\end{center}

\begin{eqnarray*}
B_P &=& 5\\
I_P &=& 13\\
\mathscr{A}_P &=& \frac12B_P+I_P-1\\
\mathscr{A}_P &=& \frac52+13-1\\
\mathscr{A}_P &=& \frac{29}2
\end{eqnarray*}}
\end{multicols}

\pagebreak

\moddocdis{Finding Pick's formula}{3}{01}{Lattice}

\begin{center}
\psset{unit=0.5cm}
\pspicture(0,-10)(30,30)
\pspolygon[fillstyle=solid,fillcolor=lightgray](0,29)(1,29)(1,25)(0,25)
\pspolygon[fillstyle=solid,fillcolor=lightgray](2,29)(6,29)(6,25)(2,25)
\pspolygon[fillstyle=solid,fillcolor=lightgray](8,29)(17,29)(17,25)(8,25)
\pspolygon[fillstyle=solid,fillcolor=lightgray](18,29)(22,29)(22,25)
\pspolygon[fillstyle=solid,fillcolor=lightgray](23,29)(29,29)(29,25)
\pspolygon[fillstyle=solid,fillcolor=lightgray](12,24)(16,22)(17,23)
\pspolygon[fillstyle=solid,fillcolor=lightgray](0,24)(5,24)(4,23)(7,22)(4,19)(2,21)
\pspolygon[fillstyle=solid,fillcolor=lightgray](9,24)(8,23)(9,22)(8,21)(9,20)(8,19)(17,20)(14,22)
\pspolygon[fillstyle=solid,fillcolor=lightgray](18,22)(22,24)(23,23)(24,24)(29,24)(25,23)(26,19)
\multido{\n=0+1}{30}{
\multido{\d=-10+1}{40}{
\psdot[dotstyle=o](\n,\d)
}}
\endpspicture
\end{center}


\end{document}
