# Draw arrows to show multiplication pattern (distributive property)

Based on this question: How to draw arrows between parts of an equation to show the Math Distributive Property (Multiplication)? I got some nice tools for illustrating to my students how to find the product of a number or a variable and a parenthesis.

However, the solution fails when I should have two sources in one line, for instance when expanding (a+3)(b+4). I would also be able to place a small number next to or above the arrowhead to indicate the first calculation, the second, etc. The second set of arrows could just as well go below the expression, not above it, to simplify things.

I could of course define myself a new set of macros similar to the existing ones, for instance \sourceTwo and \targetTwo, but it feels like this could be done more elegant.

MWE:

\documentclass[]{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}

\newcommand\source[1]{%
\tikz[remember picture,baseline,inner sep=0pt] {%
\node [name=source,anchor=base]{$#1$};
}%
\setcounter{target}{0}
}
\newcounter{target}
\newcommand\target[1]{%
\tikz[remember picture,baseline,inner sep=0pt] {%
\node [name=target-\thetarget,anchor=base]{$#1$};
}%
\stepcounter{target}%
}
\newcommand\drawarrows{
\tikz[remember picture, overlay, bend left=45, -latex] {
\foreach \i [evaluate=\i as \n using int(\i-1)] in {1,...,\thetarget} {
\draw (source.north) to (target-\n.north);
}
}
}

\begin{document}

$$(\source{a}+4)(\target{b}+\target{3}) = \drawarrows$$

\end{document}

• Can you include a picture that shows the desired output. It is not clear to me from the description. – Peter Grill Apr 1 '14 at 19:55

The following code maybe a starting point for you:

\documentclass[]{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}

\newcounter{source}
\newcommand\source[1]{%
\tikz[remember picture,baseline,inner sep=0pt] {%
\node [name=source-\thesource,anchor=base]{$#1$};
}%
\setcounter{target}{0}
\stepcounter{source}
}

\newcounter{target}
\newcommand\target[1]{%
\tikz[remember picture,baseline,inner sep=0pt] {%
\node [name=target-\thetarget,anchor=base]{$#1$};
}%
\setcounter{source}{0}
\stepcounter{target}%
}
\newcommand\drawarrows{
\tikz[remember picture, overlay, bend left=45, -latex] {
\foreach \j [evaluate=\j as \m using int(\j)] in {1,...,\thesource}{
\foreach \i [evaluate=\i as \n using int(\i-1)] in {1,...,\thetarget} {
\draw [red](source-0.north) to (target-\n.north) coordinate (UP);
}
\node [red] at (UP) [above] {1};
}
}

\tikz[remember picture, overlay, bend left=-45, -latex] {
\foreach \j [evaluate=\j as \m using int(\j)] in {1,...,\thesource}{
\foreach \i [evaluate=\i as \n using int(\i-1)] in {1,...,\thetarget} {
\draw [blue](source-1.south) to (target-\n.south) coordinate (DOWN) ;
}
\node [blue] at (DOWN) [below] {2};
}
}
}

\begin{document}

$$(\source{a}+\source{4})(\target{b}+\target{3})=\mbox{\drawarrows}$$

\end{document}


Output:

Update: Labeling each arrow:

\documentclass[]{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz,pgfplots}

\newcounter{source}
\newcommand\source[1]{%
\tikz[remember picture,baseline,inner sep=0pt] {%
\node [name=source-\thesource,anchor=base]{$#1$};
}%
\setcounter{target}{0}
\stepcounter{source}
}

\newcounter{target}
\newcommand\target[1]{%
\tikz[remember picture,baseline,inner xsep=0pt] {%
\node [name=target-\thetarget,anchor=base]{$#1$};
}%
\setcounter{source}{0}
\stepcounter{target}%
}
\newcommand\drawarrows{
\tikz[remember picture, overlay, bend left=45, -latex] {
\foreach \j [evaluate=\j as \m using int(\j)] in {1,...,\thesource}{
\foreach \i [evaluate=\i as \n using int(\i-1)] in {1,...,\thetarget} {
\draw [red](source-0.north) to (target-\n.north) ;
\node [red] at ([xshift=-5mm]target-\n.north) [above=2mm] {\i};
}

}
}

\tikz[remember picture, overlay, bend left=-45, -latex] {
\foreach \j [evaluate=\j as \m using int(\j)] in {1,...,\thesource}{
\foreach \i [evaluate=\i as \n using int(\i-1)] in {1,...,\thetarget} {
\draw [blue](source-1.south) to (target-\n.south)  ;
\pgfmathsetmacro{\ii}{\i+2)};
\node [blue] at ([xshift=-2mm]target-\n.south) [below=2mm] {\pgfmathprintnumber \ii};
}
}
}}

\begin{document}

$$(\source{a}+\source{4})(\target{b}+\target{3})=\mbox{\drawarrows}$$

\end{document}


and the output:

• It might by better to set only inner x sep=0pt instead of inner sep=0pt, because right now the arrows "touch" the figures which doesn't look good in my opinion. – Henri Menke Apr 1 '14 at 21:59
• @HenriMenke, you are right. My code is based on Holene's MWE. I didn't focused on details. – ferahfeza Apr 1 '14 at 22:20
• @ferahfeza This is sweet! Thank you, this solves my problem perfectly! I'm not quite understanding the way the code works though. How may I label each arrow (if its room for it)? – Holene Apr 2 '14 at 18:07
• @Holene, yourwelcome. I will edit my answer for labeling. – ferahfeza Apr 2 '14 at 18:28
• @ArneTimperman, right my code only for binomial * binomial. For your example, a new TiKz drawing code must be added, like \tikz[remember picture, overlay, bend left=-45, -latex] { \foreach \j [evaluate=\j as \m using int(\j)] in {1,2,...,\thesource}{ \foreach \i [evaluate=\i as \n using int(\i-1)] in {1,2,...,\thetarget} { \draw [orange](source-2.south) to (target-\n.south) ; \pgfmathsetmacro{\iii}{\i+4)}; \node [orange] at ([xshift=-2mm]target-\n.south) [below=5mm] {\pgfmathprintnumber \iii}; } } } – ferahfeza Oct 28 at 20:52

Just a more beautiful solution with PSTricks.

\documentclass[preview,border=1cm,12pt,varwidth]{standalone}
\usepackage{pst-node,amsmath}
\psset{nodesep=2pt,linearc=2pt,arrows=->,linecolor=blue,arrowinset=0}
\def\lbl#1{\ncput*{\text{\tiny #1}}}
\begin{document}
\abovedisplayskip=0pt\relax% don't use this line in your production
$(\rnode{A}{a}+\rnode{B}{b})(\rnode{C}{c}+\rnode{D}{d})=\ldots \ncbar[angle=90,offsetA=-1pt]{A}{C}\lbl{1} \ncbar[angle=90,offsetA=1pt,arm=16pt]{A}{D}\lbl{2} \ncbar[angle=-90,offsetA=1pt]{B}{C}\lbl{3} \ncbar[angle=-90,offsetA=-1pt,arm=16pt]{B}{D}\lbl{4}$
\end{document}


• Oh, dude... That was an elegant solution. I'm not into pstricks though, but I'm going to try to adapt the style of the solution to TikZ – Holene Apr 3 '14 at 6:58
• I cannot compile this code. I use texmaker and it said "undefined control sequence \ncbar...". How to fix it? – Kanegusa199 Jul 4 '18 at 23:41

Another solution with pstricks, compilable with pdflatex, if you launch it with the --enable-write18 switch if you're under MiKTeX, or -shell-escape (TeX Live, MacTeX). Alternatively, you can compile with XeLaTeX:

\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{mathtools}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pst-node}
\usepackage{auto-pst-pdf}

\begin{document}

\Large
\begin{postscript}
\begin{align*}
(\rnode{X1}{2x}+\rnode{Y1}{\smash[b]{3y}})(\rnode{X2}{4x}+\rnode{Y2}{5y}) & = (2x)(4x)+(2x)(5y)+(3y)(4x)+(3y)(5y) \\
& = 8x²+10xy+12xy+15y² \\
& = 8x²+22xy+15y²
\end{align*}
\psset{angle=90,nodesep=2pt, arrows=<->, arrowinset=0.2}
\ncbar[arm=15pt]{Y1}{Y2}\ncbar[border=1.5pt]{X1}{X2}
\psset{angle=-90,nodesep=4pt}
\ncbar{X1}{Y2}\ncbar[arm=8pt]{Y1}{X2}
\end{postscript}

\end{document}


• not able to compile the code...... What is the problem? Help me.. – Maths4Sandy May 26 '16 at 7:24
• Did you add the switch I mention at the beginning of my answer? Also, what does the error message say? – Bernard May 26 '16 at 9:55