Bad apples

Show students the first 'bad apples' picture.

19 red apples in a line with the 10th apple coloured black

Day 1

19 red apples in a line with the 9th, 10th and 11th apples coloured black

Day 2

Some apples are sitting in one row. Each day, any bad apple will turn any good apple that is touching it bad.

  • On the first day, there is one bad apple. 
  • How many bad apples are there on the second day? 
  • On the third day, how many apples will be bad? 
  • How many bad apples will there be on the fourth day?

Ask students to explain their answers. What is the pattern of bad apples that is emerging? Ask pairs of students to talk about what generalisation could be made about the number of bad apples on any given day. Write it as a short 'rule'.

If there is a very long row of apples, how many bad apples will there be on the fifteenth day? Explain and justify the use of the rule.

 

Here is an array of apples — not just one row, but a whole tray of apples.

8 rows of 19 red apples in lines with the 10th apple in the first row coloured black

Day 1

8 rows of 19 red apples in lines with the 9th, 10th and 11th apples in the first row and the 10th apple in the second row coloured black

Day 2

  • On the first day, there is one bad apple. 
  • On the second day, there are four bad apples. Why?

Ask pairs of students to talk about what generalisation could be made about the number of bad apples on any given day.
Ask the students to explain and justify why their rules work. 

How concisely can the general rule be written?
Use the rule to work out how many bad apples there will be on the fifteenth day.

You can download 'Bad Apples': Extension Ideas for Secondary Students for more ideas. 

Curriculum links

Year 10A: (Possible extension) Apply understanding of polynomials to sketch a range of curves and describe the features of these curves from their equation

Year 1: Investigate and describe number patterns formed by skip counting and patterns with objects

Year 6: Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence

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