Sam and Holly's problem

A thinking mathematically context for practise opportunity focussed on using reasoning and additive thinking to work out hidden quantities


Syllabus outcomes and content descriptors from Mathematics K-10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2021


  • MAO-WM-01
  • MA1-CSQ-01

Collect resources

You will need:

  • something to write on
  • something to write with
  • 2 cups (optional)
  • 10 blocks or counters (optional)

Read about Sam and Holly playing a game

Sam and Holly were playing with combinations of numbers that make 10. They started by getting out 10 blocks.

Then, Sam covered some of the blocks and Holly had to work out how many were missing.

Holly decided she to organise what she could see to help her work out what was missing.

She imaged a ten-frame in her head and organised the blocks. This helped her see she needed 3 more to make 10.

Holly and Sam kept taking turns like this until Holly asked Sam if they could make it more challenging for them.


Watch Sam and Holly's problem video (1:06).

Make a game involving hidden numbers more challenging.

[On a grey carpet, 10 small green cubes are arranged in 2 columns of 5 each. On the right, 2 white coffee mugs are turned down on the carpet.]


Hi, Holly.




Do you want to play a game?




Sure. So we're gonna start off. Can you just tell me how many are here?




Alright. So I'm going to take some, put it under a cup, OK?


Is there a way you can make it a bit more harder for my brain?


Sure. Alright, so I'll do this. Alright. So close your eyes, OK? So don't peek. Close your eyes. Alright.

[A hand reaches over and takes 7 of the green cubes to put under the 2 mugs.]


So... So how many do you think is left? How many do you think are in here?

[Can you help solve Holly’s problem?’ In the bottom right, a white NSW Government ‘waratah’ logo.]

[Black text on white reads ‘How many cubes are hidden?’ and ‘How many cubes might be under each cup?’ Below, a colour image of the 2 white coffee mugs turned down and 3 green cubes to the left.]

[White text on blue reads ‘Over to you!’]

[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]

[End of transcript]

A problem for Holly

Sam made this for Holly to think about.


  • Can you help Holly work out how many blocks are hidden?

  • Can you help her work out how many of the hidden blocks might be underneath each cup?

  • What are all the possibilities? Record your thinking.


  • What possibilities did you find?

  • Could you work out exactly what Sam put under each cup for Holly?

    • Why?

    • Why not?

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