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ENRICHMENT: Cutting up a circle – – 12

12 What is the greatest number of sections into which you can divide a circle, using only a particular number of straight line cuts?

a Explore the problem above. (Hint: For a given number of lines, you may need to draw several circles until you are sure that you have found the maximum number of sections.) Note: The greatest number of sections is required and, hence, only one of the two diagrams below is correct for three straight line cuts.

54 Incorrect. Not the maximum number of sections for 3 lines.

Copy and complete this table of values.

b Can you discover a pattern for the maximum number of sections created? What is the maximum number of sections that could be created with 10 straight line cuts?

c The formula for determining the maximum number of cuts is quite complex: sections = 1 2 × cuts2 + 1 2 × cuts + 1

Verify that this formula works for the values you listed in the table above. Using the formula, how many sections could be created with 20 straight cuts?

2J Tables and rules

Learning intentions for this section:

• To understand that a rule connects two varying quantities

• To be able to complete an input-output table given a rule

• To be able to find a rule for an input-output table

Past, present and future learning:

• This section consolidates and extends Stage 3 concepts which are used in Stage 4

• It prepares students for future work in Linear Relationships in Stage 4 and beyond

• Expertise with these concepts may be required in non-calculator examinations such as NAPLAN and industry aptitude tests

A rule describes a relationship between two values, often called input and output.

For example, the rule output = input + 3 describes a relationship between input and output. If the input value is 5, then the output value will be 8.

This is sometimes shown as a function machine: output

From tables of human heights and bone lengths, rules are found, e.g. height = 2.3 × femurlength + 65.5cm . By measuring bones from the Petra tombs, archaeologists calculated the height of ancient humans. + 3

If a different input is put in the function machine above, the output will change. We can use a table to list a selection of input and output combinations, as below:

Another example of a rule is output = 4 × input which we could use to create a table of values.

This can be drawn as a function machine too:

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