Cover Up

Task 172 ... Years 4 - 8

Summary

Set in a game context, students soon find themselves looking for ways to get the highest score on any move so they can make the target of 45. They can't actually control the score they get, because it depends on the roll of the dice, but they soon begin to realise the most unfortunate outcomes of rolling the dice. The more number tiles you are able to cover, the lower your score, so the rolls that would give a score of 45 in one turn actually give a zero score. As one child said:
It's a bit twisted, but it makes you think.
The challenge is to be able to explain to someone else what they have learnt about the game, just as a mathematician would have to explain their investigation to peers. 		 

Materials

  • Nine tiles numbered 1 - 9
  • Nine unnumbered tiles in a different colour

Content

  • addition
  • arithmetic, addition / subtraction
  • arithmetic, multiplication / division
  • mental arithmetic
  • numbers, odd & even
  • numbers, properties & laws
  • patterns, number
  • probability calculations
  • probability experiences
  • probability, sample space
  • recording mathematics
  • sequences & series
Cover Up

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 		   

It soon becomes clear that to get the highest score on any turn you must cover the least number of tiles. But you have to cover the total of the dice roll and every pair that would make that total, so there will always be at least two tiles covered. A dice roll of 1 and 1 means exactly two tiles are covered. Tile 2 because it is the total and tile 1 because the only way to make a 2 is with 1 and 1. So the largest score on any turn is:
3 + 4 + 5 + 6 + 7 + 8 + 9

There will be several occasions in the task when students have to evaluate like this and it is always worth asking:

  • How did you calculate that?
  • Can you check it another way?
It is very important for a mathematician to be able to check their work another way - and it is their responsibility to do so; they can't ask their teacher, or anyone else, because they are always working on problems and problems, by definition, have not previously been solved.

For the series 3 + 4 + 5 + 6 + 7 + 8 + 9 the student could:

  • sequentially add beginning with 3.
  • sequentially add beginning with 9.
  • use a calculator.
  • make three 'rainbow pairs' of 12 - (3, 9) ... (4, 8) ... (5, 7) - then add or multiply to total them and add the 6.
  • use the middle number, 6, as a marker and make seven 6s by shifting the 'extras' from 7, 8 & 9 to 3, 2 & 1 so they also each become 6.
  • use the first number, 3, as a marker, count seven 3s, then sum the left over sequence:
    0 + 1 + 2 + 3 + 4 + 5 + 6 and add on this total.
  • use the last number, 9, as a marker, count seven 9s, then sum the left over sequence:
    6 + 5 + 4 + 3 + 2 + 1 + 0 and subtract this total.
  • work in multiples of 7 using (3, 4) ... (5, 9) ... (6, 8) and 7.
  • ...

If, on the other hand, 9 is rolled, it will be covered because it is the total and so will all pairs that make 9, which are (1, 8) ... (2, 7) ... (3, 6) ... (4, 5) and the score is zero. If only the score was the ones that were covered. Then the score would be:
1 + 2 + 3 + 4 + 5 + 6+ 7+ 8 + 9.

  • How many ways can you convince me that the total of this series is 45?
Exploring number series in this way is a natural outgrowth of the task that would make an excellent Investigation Guide. When you write it and trial it with your students please send it to us so we can share it through this page. It would be great if it could be accompanied by student work samples and/or comments.

Getting back to the task, if the smaller dice roll totals are required to get a higher score, what chance is there of them occuring? Obviously the total of 2 can only occur as (1, 1). Just one way. But a total of 3 can occur in two ways (1, 2) and (2, 1). (It helps to realise this if the dice are different colours). So there are three chances out of ... to get the lowest dice rolls and consequent highest scores. A table is a good way to list all the possibilities and fill in the 'out of ...' gap.

1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

The table reveals that on any roll you have 4 out of 36 chances of covering all the digits and scoring zero and only 3 out of 36 chances of getting one of the three highest scores. But is that a complete analysis?

What happens if you roll a total of 12? ... 11? ... 10?
Depending on age and experience students can be expected to include some or all of these points in their journal report as requested on the card.

Extensions

  1. What happens if the game is played with the rule that the person who reaches 45 first loses?
  2. What rules would you make for a similar game using tiles 1 - 12 and a standard cube dice?
  3. What rules would you make for a similar game using tiles 1 - 5 and a tetrahedral dice numbered 1 - 4?
  4. What rules would you make for a similar game using an icosohedral dice numbered 0 - 9 twice (or 1 - 10 twice)?
Also, the knowledge that the sum of digits 1 to 9 is 45 is used in the extensions of the tasks Steps and Number Tiles.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 		   

Some classrooms will have sets of 1 - 9 tiles already made up and plenty of other objects that can be used to cover them. If not, then the students simply write the digits 1 - 9 on each of their turns, roll the dice and cross off. Their total for that turn is the sum of the remaining numbers. Play the game a few times in pairs, then facilitate a class discussion to develop a class appreciation of the aspects above. Use these points to model an outline of how to write a maths report and then challenge the students to prepare their own report of the game to explain it to someone else.

This report could be written, but it could also be a poster, slide show, comic strip, photo sequence or could make use of software applications such as Explain Everything. See the Recording & Publishing link for samples of student work with other investigations.

At this stage, Cover Up does not have a matching lesson on Maths300 but the mathematics of sequences and series such as the example above summing 3 - 9 is the focus of Maths300 Lesson 12, Gauss Beats The Teacher.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner. 		   

The Cover Up task is an integral part of:

  • MWA Number & Computation Years 5 & 6

Green Line
Follow this link to Task Centre Home page.