ACARA v9 CONTENT DESCRIPTION “create and use algorithms involving a sequence of steps and decisions and digital tools to experiment with factors, multiples and divisibility; identify, interpret and describe emerging patterns”
Builds on factors, multiples and number facts, and on the step-by-step methods used throughout Year 5. This unit makes those steps explicit as algorithms: clear sequences of instructions, sometimes including a decision, that can be followed exactly to get a result. Using algorithms and digital tools to experiment with factors, multiples and divisibility, children look for patterns that emerge, describe them, and see how a precise procedure and a noticed pattern support each other.
Algorithms: steps and decisions
An algorithm is a precise sequence of steps that solves a problem or completes a task. A recipe is an everyday algorithm, and so is a method for testing a number. Many algorithms include a decision, a point where the next step depends on a yes-or-no question. A divisibility test is a good example: choose a number, add its digits, then decide whether that sum is a multiple of three; if it is, the number is divisible by three, and if not, it is not. Writing a method out as ordered steps with clear decisions means anyone, or a computer, can follow it and reach the same answer.
Algorithms: steps and decisions
An ordered procedure, with a yes or no decision.
Choose a number N: the input the algorithm starts from.
Testing for divisibility
Divisibility tests are short algorithms for deciding whether one number divides another exactly, with no remainder. A number is divisible by two if it is even, by five if it ends in zero or five, and by ten if it ends in zero. It is divisible by three if the sum of its digits is a multiple of three: seventy-two passes because seven plus two is nine. These tests save dividing in full and turn a calculation into a quick decision. Combining tests answers questions like whether ninety is divisible by both two and three, which it is.
Testing for divisibility
A short rule decides if one number divides another.
Even for 2; ends in 0 or 5 for 5; digit sum a multiple of 3 for 3.
Factors and multiples
Factors and multiples are two sides of divisibility. The factors of a number are the whole numbers that divide it exactly: the factors of twelve are one, two, three, four, six and twelve. The multiples of a number are what is reached by counting in steps of it: the multiples of four are four, eight, twelve, sixteen and on. If one number divides another exactly, the first is a factor and the second a multiple. Listing factors or multiples is itself an algorithm, a systematic check of each candidate or a repeated step of counting on.
Factors and multiples
Factors divide exactly; multiples are reached by counting on.
Find the factors of 12.
Patterns that emerge
Experimenting with factors, multiples and divisibility makes patterns emerge. The multiples of five always end in zero or five; the even numbers are exactly the multiples of two; the multiples of three have digit sums that are themselves multiples of three. Spotting such a pattern, describing it clearly, and checking it on more examples is how mathematics is explored. A pattern noticed in the results of an algorithm can often be explained, and may even become a quicker rule, like a new divisibility test.
Patterns that emerge
Experimenting reveals rules you can describe and check.
List examples, notice the rule, then check it on more.
Running an algorithm
Running an algorithm means following its steps in order, often repeating a step to build a sequence. Starting at zero and adding six each time gives zero, six, twelve, eighteen and twenty-four, the multiples of six, a pattern produced by a simple repeated step. Starting at one and doubling gives one, two, four, eight and sixteen. Tracing the steps and recording each result shows the pattern emerging and makes it easy to predict the next term. Digital tools can run such steps quickly, letting larger patterns be explored than by hand.
Running an algorithm
Repeat a step to build a sequence and see the pattern.
Run the algorithm: start 0, add 6 each step.
Algorithms and patterns with confidence
Working with algorithms and patterns confidently means writing a clear sequence of steps, including any decision, and following it exactly to get a result. Divisibility tests, and lists of factors and multiples, are short algorithms that experiment with how numbers divide, and the patterns that emerge can be described and checked. Tracing an algorithm step by step reveals a sequence and its next term. With these habits a child can create and follow precise procedures, use digital tools to experiment, and describe the patterns that appear, ready for the algorithmic thinking of later years.
Quick self-check
1. An algorithm is...
2. A number is divisible by 3 when...
3. The factors of 12 are...
4. Multiples of 5 always end in...
5. Starting at 0 and adding 4 each step gives 0, 4, 8, 12, 16. This sequence is...