Planning tool

Students:

• create and investigate patterns using concrete materials and digital tools such as calculators and spreadsheets, and identify any emerging patterns, including similarities and differences
• design and use an algorithm that involves a sequence of steps to generate a number sequence and describe and explain relationships observed
• use a function machine to produce a table of values
• describe patterns produced by a function machine.

Some students may: 

• go into ‘auto-pilot’, simply adding all the numbers and ignoring the given context.
• have difficulty with the notion of variable and the ideas of ‘input’ and ‘output’ for a rule.
• have difficulty moving from an everyday language description of a process to a sequence of instructions or a template for a function machine.
• assign literal meanings to letters (variables), for example, 3a interpreted as 3 apples, rather than understanding that variables can be thought of as objects or ‘placeholders’ in a rule.
• have limited understanding of the properties of numbers and operations (for example, multiplication only understood in terms of groups of; division not seen as the inverse of multiplication).

To address these, provide multiple opportunities to use concrete materials to explore patterns and write rules for growing and shrinking patterns. Use function machines to develop algorithmic thinking.

Refer to the resource Common misunderstandings: Level 6 – Generalising. This was developed from a large body of research that examined the various difficulties students experience with algebraic thinking.

The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.

Learning intention: using a function machine

• We are learning to use a function machine to represent a sequence of instructions and fill in a table of values.
• We are learning to recognise and describe any emerging pattern that results from the use of a function machine.

Why are we learning about this?

• It is important in many practical situations to have a clear set of instructions for carrying out a process, such as a recipe for preparing a meal, putting together a piece of furniture, or calculating using a formula. Function machines provide a way of describing an input-output process using a simple diagram.

What to do

The following diagram shows a function machine that calculates output values for given input values.

1. Describe in words what the function machine does to the input value.

2. Complete the diagram below to show what the output value is for an input value of 4.

3. Use the function machine and a calculator to complete the table of values below:

4. Describe any pattern you observe in the output values from the table.

5. Find the input value which gives an output value of 67 and explain how you obtained the answer.

6. Below is a blank diagram of a function machine.

a. Use the diagram to construct a function machine that halves a number then adds one.

b. Use the machine above and a calculator, to complete the following table of values and describe any pattern you observe in the output values from the table.

c. Find the input value which gives an output value of 11.5 and explain how you obtained the answer.

Success criteria

I can:

• use a function machine to create outputs from inputs and complete a table of values
• describe an emerging pattern from a table of values created using a function machine.