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Writing Digits

Lee was writing all the counting numbers from 1 to 20. She stopped for a rest after writing seventeen digits. What was the last number she wrote?

Number Detective

Follow the clues to find the mystery number.

Six Is the Sum

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

Less Is More

Age 5 to 11
Challenge Level

Less Is More

Find a partner and a 0-9 dice. The interactivity here can be used to simulate throwing different dice.

Each of you draw some cells that look like the picture below. (You don't have to make the cells on the left a different colour, but we will refer to those cells later.) Alternatively, you could print off this sheet of cells.

Version 1

Take turns to throw the dice. After each throw of the dice, you each decide which of your cells to put that number in.

Throw the dice eight times until all the cells are full.

If both of your number sentences are correct, you score a total of the two numbers in the blue boxes. 

If only the first number sentence is correct, then you score points equal to the number in the top two blue boxes.  If only the second number sentence is correct, you score points equal to the number in the bottom two blue boxes.

The winner is the person with the greatest number of points. 

Play the game several times so you get a good feel for it.
What do you notice?
Do you have any good strategies for winning?
How could you give advice to someone else before they play if you don't know what digits they might roll?

Version 2

Now, work with your partner, rather than against them.
If the eight numbers thrown are 5, 2, 8, 4, 1, 5, 1 and 9, and you are allowed to put them in the cells once all eight are known, where would you place them in order to get the highest possible score? How do you know this will give the highest score?

We would love to hear about your 'noticings' and strategies for playing the competitive game, as well as your solutions for the final part. Please do share your thinking!

Why play this game?

This game is thought-provoking and very engaging. It encourages discussion of place value, alongside valuable strategic mathematical thinking and it helps learners become more familiar with the mathematical symbols for 'greater than' and 'less than'.

The game also offers the chance to focus on any of the five key ingredients that characterise successful mathematicians. The collaborative version lends itself particularly to fostering a positive attitude to mathematics as learners' resilience may be tested!

Possible approach

This problem featured in an NRICH Primary webinar in October 2020.

This game can be played with a 1-6 dice but ideally would be played with a decahedral 0-9 dice or a spinner (interactive versions of dice and spinners are available here).

Invite volunteers (perhaps working in teams of two) to play the game on the board and explain the rules to them and the rest of the class. 

When the game is over, explain the scoring system and confirm who has won. Invite questions and encourage everyone to respond, rather than you always giving your answers. Once you feel that learners have grasped the rules, set them off on playing the game, working in pairs. 

Encourage learners to justify their strategies to their partners, and draw their ideas together in a mini plenary. Some learners might question whether it is 'allowed' to put a zero in the tens column boxes and if it does not come up naturally, you may want to pose the question yourself.

Then introduce the collaborative version of the game in which you can wait until you have thrown the dice all eight times before you decide where to place each number. Either throw a dice eight times, or use the eight rolls given in the problem, and challenge learners to place the numbers so that they score as highly as possible.

After allowing time to work on this, the final plenary can be an opportunity to share thinking. Listen out for learners who are using their knowledge of place value to make decisions, and encourage everyone to construct their reasoning carefully, leaving no room for doubt.

Key questions

How are you deciding where to place the numbers?
How are you trying to make sure each number sentence is true?
How could you give advice to someone else before they play if you don't know what digits they might roll?

Possible extension

You could challenge learners to come up with a set of eight numbers that would be easy to place in order to make the highest score, and a set of numbers that require deeper thought. What makes a set of numbers easier/harder?

Possible support

Provide learners with number cards that they can move around the grid to consider different options. For the competitive version of the game, allow pairs to play against another pair, so that partners can support each other.