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art A

Eggs are generally sold in batches of 12 (or dozens). The image shows the 3 different ways that 12 eggs can be packaged into arrays. Note that the array 3 by 4 is taken to be equivalent to the array 4 by 3 and therefore is not included.

If it was decided that eggs should be sold in batches of 24, then these 24 eggs could be packaged into a total of
different arrays.

Part B
Chloe made a conjecture that given any two numbers, the greater number can always be arranged into more arrays.

State whether you agree or disagree. Then explain why you think so by giving an example of two numbers that prove or disprove the conjecture.

explain it like a kindergardener

1 answer

Part A:

To find out how many different ways we can package 24 eggs into arrays, we can look at the pairs of numbers that can multiply together to make 24. Here they are:

  1. 1 x 24
  2. 2 x 12
  3. 3 x 8
  4. 4 x 6

Since we don't count the same pairs if we flip them (like 4 x 6 is the same as 6 x 4), we have these 4 different ways to arrange the eggs. So, we can package 24 eggs in 4 different arrays.

Part B:

Now, let’s think about Chloe’s idea! She said that if we have two numbers (let's say 12 and 24), the bigger number (24) can always make more arrays than the smaller number (12).

But we need to look closely!

  • For 12, we have these arrays:

    1. 1 x 12
    2. 2 x 6
    3. 3 x 4
      So, there are 3 ways to make arrays for 12 eggs.

  • For 24, we have:

    1. 1 x 24
    2. 2 x 12
    3. 3 x 8
    4. 4 x 6
      And that gives us 4 ways to make arrays for 24 eggs.

So in this case, 24 indeed has more ways than 12!

But let’s think of another pair:

  • If we take 10 and 20:
    • For 10, the pairs are:
      1. 1 x 10
      2. 2 x 5 So, we have 2 ways.
    • For 20, the pairs are:
      1. 1 x 20
      2. 2 x 10
      3. 4 x 5 So there are 3 ways for the 20.

Now, in this case, 20 has more ways than 10, which matches Chloe’s idea.

But here’s the tricky part! Let's look at 6 and 9:

  • For 6, the pairs are:

    1. 1 x 6
    2. 2 x 3 So that’s 2 ways.

  • For 9, the pairs are:

    1. 1 x 9
    2. 3 x 3
      So that’s 2 ways too!

Oops! 6 and 9 both have 2 ways!

This means Chloe’s idea is not always true. So I would disagree with her because sometimes two numbers can have the same number of ways to be arranged!

In summary, just like how sometimes two friends can have the same number of toys to play with, bigger doesn't always mean more arrays!