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Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Cut it Out

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Reasoned Rounding

Age 7 to 14
Challenge Level

Why do this problem?

The game offers a competitive context which provides pupils with lots of practice in rounding numbers to the nearest "..." and helps develop their reasoning by encouraging them to talk about their strategies for winning points whilst playing.

Possible approach
Play as a class first by projecting the recording sheet onto the board and asking pupils to:
roll dice, suggest what numbers could be made and explain to the class where they would want to put them and why.

Key questions
What two-digit numbers can you make from the digits you rolled?
Where might be a good place to put your number?
What numbers are you hoping might come up on your next turn?
Can you explain why you chose that circle?

Possible extension
Three more advanced versions of the same game are available here:

Reasoned Rounding to the nearest hundred recording sheet (PDF version)
This time the players roll the die 3 times each per turn and can be encouraged to record the different 3-digit numbers possible before making strategic decisions as to where they could best place their chosen number.

Reasoned Rounding to the nearest whole number recording sheet (PDF version)
Players roll the die twice and then place one number on each side of a decimal point. They round their chosen number to the nearest whole number and then place it in the appropriate circle as before.

Reasoned Rounding to the nearest tenth recording sheet (PDF version)
Players still roll the die twice, but then decide in which order to place them after a decimal point.  Having created a number which has 2 decimal places, they must then round it to the nearest tenth and place it in the appropriate space on the recording sheet.

What other variations can pupils come up with for themselves?  
What if they were to use different dice? 
What recording sheets would be needed now?

Possible support
Ask pupils to play in groups of four, one pair against another, in order to support their discussion of options whilst carrying out the rounding.