Crosses

Task 35 ... Years 4 - 8

Summary

Digits 1 - 9 are placed on arms of a cross so that the partial sum of each arm is the same.
  • How many solutions are there?
  • How do you know when you have found them all?
  

Materials

  • 9 tiles

Content

  • basic arithmetic skills
  • properties of odd and even numbers
  • problem solving strategies
  • number patterns
  • probability opportunities
Crosses

Iceberg

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

The task card only asks for three solutions, so clearly the iceberg is to find them all and to know when you have. Some of the key steps to doing this are to realise that:

  • The total of the 9 digits is 45.
  • The two partial sums will count the central tile twice.
  • If the sum of the partial sums is above 45, the excess must come from the second count of the central tile.
  • The smallest possible grand total above 45 is 46, which implies 1 in the centre has been counted twice and the partial sums must be 23.
  • The largest possible grand total above 45 is 54, which implies 9 in the centre has been counted twice and the partial sums must be 27.
  • It is not possible to have an even number in the centre because the second count would add an even number to 45, which would create an odd grand total. This could not be equally shared between the partial sums.
From here students could apply the strategy of try every possible case. For example, with 1 in the middle, the sum of the remaining four digits in each partial sum must be
23 - 1 = 22.
  • Is it possible to arrange the digits 2 - 9 in two sets which sum to 22?
  • In how many ways?
Students will need to make some decisions about what constitutes a 'different' or 'unique' solution.

A further investigation introduces the concept of chance. Turn the tiles face down and mix them up. Keeping them face down rearrange them into a cross. Turn the tiles over. Do they make a correct solution.

  • What do you predict are the chances of a correct solution?
Design an experiment to check your prediction.

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 state and easy to start. Begin with 9 large pieces of paper on which you have written the digits. Hand them out at random and indicate that you want the holders to simply put them on the floor to make a times sign (or a plus sign) - a cross.

If my team has put them down correctly, the total this way ... will be the same as the total this way.
Now the students know what the problem is, they return to their seats, tear a piece of paper into 9 pieces between two students and see which pair is the first to 'do it right'.

The first successful students record their solution on the board with their initials and the race is on to find, and 'own', a different solution. The class data opens up the broader investigation. The whole class investigation is explore in Maths300 Lesson 112, Crosses. The investigation about the chances of making a solution at random is recorded in Lesson 159, Chances With Crosses. Both have software support.

Visit Crosses on Maths300:
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Visit Chances With Crosses on Maths300:
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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 Crosses task is an integral part of:

  • MWA Number & Computation Years 3 & 4
  • MWA Number & Computation Years 9 & 10

The Crosses lesson is an integral part of:

  • MWA Number & Computation Years 9 & 10
The Chances With Crosses lesson is not in any MWA kit. However it can be used to enrich the Chance & Measurement kit at either Years 5/6 or Years 9/10.

Green Line
Follow this link to Task Centre Home page.