# Pick A Box

### Task 91 ... Years 4 - 10

#### Summary

A collection of decorated boxes is sorted into 3 groups according to several criteria, the mathematical ones of which relate to consecutive numbers and finding their sum. There are actually several solutions to the problem, but students will often reach what they think is a solution, only to find that on a second or third check there is a condition not satisfied. Once the one solution has been found, the challenge becomes:
• How many solutions are there?
• How do I know when I have found them all?

#### Materials

• Blocks (or other packages) numbered from 1 - 15 and decorated with 12 different designs

#### Content

• consecutive numbers
• methods for summing consecutive numbers
• basic arithmetic operations
• odd and even numbers
• problem solving strategies, especially breaking the problem into parts and trying every possible case.

#### Iceberg

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

Due to the multiple conditions that need to be satisfied, it may be necessary to read the 'frustration level' for some students and decide to intervene.

Students at East Devonport Primary.
Their teacher has organised Mathematician Teams.

The third clue is one entry point for providing assistance. Adding numbers in the three groups counts each of the fifteen numbers once. Therefore the total of these totals must be the same as the total of the numbers 1 through 15, which is 120. Then we need three consecutive numbers adding to 120 and they must be 39, 40 and 41.

Finding the total of numbers from 1 to 15 will be a challenge for some students and great opportunity for the teacher to ask Can you check this another way?.

To extend the problem ask the students to consider the possibility of another solution. To date we know of these:

 1, 3, 5, 7, 9, 14 4, 6, 8, 10, 12 2, 11, 13, 15 3, 6, 8, 10, 12 1, 4, 7, 13, 15 2, 5, 9, 11, 14 1, 4, 7, 12, 15 3, 6, 8, 10, 13 2, 5, 9, 11, 14 1, 4, 7, 12, 15 3, 5, 8, 10, 14 2, 6, 9, 11, 13 2, 5, 7, 11, 14 1, 3, 9, 12, 15 4, 6, 8, 10, 13 2, 5, 7, 11, 14 4, 6, 8, 10, 12 1, 3, 9, 13, 15 1, 3, 5, 7, 9, 14 2, 6, 8, 11, 13 4, 10, 12, 15 If your class finds other solutions we would be happy to include them here.

Given there are multiple solutions, a further challenge would be to add a clue so that the problem has a unique solution. And of course, students could be challenged to create their own similar problem with the equipment or a subset of the equipment such as just the boxes numbered from 1 to 8.

### More Solutions

Fintona Girls' School
Year 5, Vanessa Stockley
Vanessa's class found lots of new solutions. What a great problem! Every group can be successful. And with so many solutions available, the new challenge of adding just one clue so that the problem has a unique solution is a great extension.

 3, 5, 7, 9, 15 2, 6, 8, 11, 13 1, 4, 10, 12, 14
 2, 4, 7, 12, 14 3, 6, 8, 10, 13 1, 5, 9, 11, 15
 2, 5, 7, 11, 14 3, 6, 8, 10, 13 1, 4, 9, 12, 15
 3, 5, 7, 10, 14 2, 6, 8, 11, 13 1, 4, 9, 12, 15
 2, 4, 9, 11, 13 1, 5, 7, 12, 15 3, 6, 8 10, 14

This solution was found by Catharin Martinsson, Henrietta Stigengård & Maria Olsson during a teachers' workshop:

 1, 6, 9, 11, 13 2, 4, 8, 12, 15 3, 5, 7, 10, 14

Extensions

• Can you create one extra clue so there will be only one solution?
• Use some or all of the blocks and make a Pick A Box problem of your own. Investigate its solutions.

#### 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.

Each pair (or perhaps group of 4) will need a set of 'boxes' before the lesson can become a whole class investigation. Cut rectangles of white card, provide markers and ask the students to design their own using these rules:

• The cards/boxes are numbered 1 to 15 in the same way (eg: always in the centre and the same size).
• There are only 12 designs altogether.
• Boxes 1 & 8 have the same design.
• Boxes 2 & 10 have the same design.
• Boxes 5 & 13 have the same design.
• All the other boxes have different designs.
Alternatively, the Maths300 lesson that partners this task has a set of designed boxes that can be printed in colour.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 70, Pick A Box.

#### 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 Pick A Box task is an integral part of:

• MWA Number & Computation Years 5 & 6

The Pick A Box lesson is not part of any Maths With Attitude kit, but it could be used to enhance:

• MWA Number & Computation Years 5 & 6
• MWA Number & Computation Years 7 & 8