Algorithms and flowchartsIB MYP Maths Extended: Revision notes

An algorithm is a precise list of steps to solve a problem or complete a task. It has an input (the starting values), a process (the steps) and an output (the result). The steps must be clear, in the right order, and always finish. Everyday examples are a recipe or the method for long division. In maths, algorithms are used to calculate values, generate sequences and solve equations.

A flowchart shows an algorithm as a diagram, with arrows showing the order. Standard shapes: an oval for start and stop, a parallelogram for input and output, a rectangle for a process (a calculation), and a diamond for a decision (a yes/no question with two exits). Follow the arrows from the start. In a written description, each box is a numbered step. A question about a flowchart can be given in words, such as: input $x$, double it and subtract $3$ to get $y$; if $y>10$, output 'Large', otherwise output $y$.

To trace an algorithm, follow the steps with real numbers and record the values in a trace table, one row per pass. For $n=5$, $total=0$ (add $n$ to $total$, reduce $n$ by $1$, repeat while $n>0$): after each pass $total$ is $5$, $9$, $12$, $14$, $15$, and $n$ is $4$, $3$, $2$, $1$, $0$. When $n=0$ the test $n>0$ fails and the output is $15$. A loop repeats steps until a condition stops it. Count the passes carefully: here Step 2 runs $5$ times, for $n=5,4,3,2,1$.

To write an algorithm: (1) say what the inputs are, (2) list the steps in order, using clear operations, (3) include any decision or repeat, (4) say what is output. Test it with an example you know. For the equation $px+q=rx+s$: input $p$, $q$, $r$, $s$; work out $p-r$; work out $s-q$; divide $(s-q)$ by $(p-r)$; output $x$. With $5x+7=2x+22$ this gives $15\div3=5$. Check by substituting back: $32$ on both sides.

An algorithm that repeats a rule generates a sequence. If $u_1=3$ and each term is twice the previous term plus $1$, the terms are $3,7,15,31,63,127,255,511,\ldots$. You can spot a pattern: each term plus $1$ is a power of $2$ ($4,8,16,32,\ldots$). A term-to-term rule (what to do to the previous term) is an algorithm. To find the first term greater than $500$, keep applying the rule until it is passed: $255<500<511$, so it is the $8$th term. Other rules, such as 'start at $5$ and add $3$', give $5,8,11,\ldots$ and the $n$th term is $5+3(n-1)$.

Solving $ax+b=c$ is an algorithm: undo the operations in reverse order. Subtract $b$ from $c$, then divide by $a$. For $3x+5=29$: $29-5=24$, then $24\div3=8$. The algorithm fails if $a=0$, because dividing by zero is undefined and the equation has no $x$ term. Working backwards through a flowchart uses the same idea: if a flowchart multiplies by $4$ then subtracts $6$ to output $10$, undo the steps in reverse: $10+6=16$, then $16\div4=4$.