Eureka

Task 163 ... Years 4 - 8

Summary

The story shell about three prospectors finding gold nuggets stirs the imagination and is enhanced by the yellow counters. The observations made about the collection of nuggets are easy to understand and students quickly find themselves absorbed in a delightful number puzzle with many solutions. The card begs the questions:
  • How many solutions are there?
  • How do you know when you have found them all?
 

Materials

  • 12 'nuggets' each marked with their masses which are 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17 & 40

Content

  • addition
  • arithmetic, addition / subtraction
  • arithmetic, multiplication / division
  • combination theory / ordered arrangements
  • concept of proof
  • mental arithmetic
  • numbers, odd & even
  • numbers, properties & laws
  • recording mathematics
  • tree diagrams
Eureka

Iceberg

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

In total there are many solutions for Mary's nuggets. One is:

  • Peter: (5, 6, 9) = 20
  • Paul: (7, 8, 10, 15) = 40
  • Mary: (11, 14, 16, 17, 40) = 98
and another is:
  • Peter: (5, 8, 9) = 22
  • Paul: (6, 10, 11, 17) = 44
  • Mary: (7, 14, 15, 16, 40) = 92
but: How do we know when we have found them all?

To advance this question students need to see restrictions in the problem such as:

  • Peter's find 'drives' the problem because his total determines Paul's possibilities and the remainder after both of their's have been assigned must be Mary's.
  • Paul's total must be even because it is twice Peter's number.
Therefore, one approach that must work if it is carried out correctly is to systematically test all combinations for Peter from the lowest, (5, 6, 7) = 18, to the highest, (16, 17, 40) = 73. However this highest can't be correct because Paul's total would then be 146 made with 4 nuggets. For this to be true each one would have to be around 37 on average, and there clearly aren't four values like that in the list of nuggets.

So a further limit on the investigation is the largest number Paul's could be and the first case to test is Paul finding the four highest nuggets, (15, 16, 17, 40) = 88.

A systematic list would look something like this:

Peter Paul  
(5, 6, 7) = 18 ( _, _, _, _ ) = 36 Lowest for Peter, but it doesn't work for Paul because the lowest possible total from the remaining numbers is 38.
(5, 6, 8) = 19 ( _, _, _, _ ) = 38  
... ...  
( _, _, _ ) = 44 (15, 16, 17, 40) = 88 Highest for Paul, but there are not three nuggets left that could make the 44 needed for Peter.

Some results of continuing this systematic search are:

  • Lowest:
    Peter (5, 6, 9) = 20, Paul (7, 8, 10, 15) = 40, Mary (11, 14, 16, 17, 40) = 98
    and there is another way too.
  • Next Lowest:
    Peter (5, 6, 10) = 21, Paul (7, 8, 11, 16) = 42 or (7, 9, 11, 15) = 42
  • Highest:
    Peter (10, 14, 17) = 41, Paul (11, 15, 16, 40) = 82
    and there is another way too.

    Extending this approach to find all the solutions is not for the mathematically faint-hearted, so perhaps we can't expect every student to take up that challenge. On the other hand, a class record of the hunt for solutions not previously found by other pairs. It could be kept some sort of secret envelope in the task bag with a label along the lines of:

    DANGER!
    Do not open until you have found at least
    three answers for Mary's nuggets.
    TALK WITH TEACHER FIRST!

    The three solutions (or more) would occur when the students reached the bottom of the card, at which point the teacher can explore their current thinking and set them on the path to finding other solutions. The class record of known solutions could guide the students in deciding 'where to prospect' for a new solution.

    Extensions

    1. A computer could be programmed to test all the combinations for Peter's 3 nuggets against all the possible remaining combinations for Paul's 4 nuggets. A tree diagram representing the Peter choices would show 12 first branches, then 11 more from the end of each of these and 10 more from the end of each of those. That's 12 x 11 x 10 possibilities to test against the 9 x 8 x 7 x 6 choices of 4 for Paul from the 9 remaining after the Peter choices. Students might like to explore this calculation further, make an assumption about how many tests the computer could do in a minute and therefore get an idea of how long it would take a computer to use the strategy of trying every possible case.
    2. Students could be asked to add conditions to the problem so that it produced one and only one solution. For example, Mary's nuggets don't have any restrictions yet, other than being the ones left over.
    3. Students might like to try constructing their own similar problem.

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.
   

This task is easy to set up as a whole class lesson. Simply arrange the students in pairs and ask each one to tear a sheet of paper into six pieces. That gives the pair twelve pieces. Tell the story and write the weights of the nuggets on the board. Students copy the weights to their pieces of paper. Write the rules on the board and challenge the pairs to be first to find the solution.

It is very likely that once one solution is found, another will follow in a short time and this opens the door to finding more, developing the limits of the problem and setting up a team investigation to find all the solutions. The solutions would look great collected onto poster paper as a class display. Even an incomplete set of solutions looks pretty impressive as an outcome to such an apparently simple problem.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 71, Eureka, which includes an Investigation Guide with answers & discussion. More than 30 answers are listed in the discussion.

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 Eureka task is an integral part of:

  • MWA Number & Computation Years 5 & 6

The Eureka lesson is an integral part of:

  • MWA Number & Computation Years 5 & 6
This task is also included in the Secondary Library Kit. Solutions for tasks in the latter kit can be found here.

Green Line
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