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?

This cameo has two From The Classroom sections. The first is a lesson used at Buloba Primary School Uganda to prepare Year 6 children to investigate Crosses. The second confirms that Crosses is indeed a worthy problem for a professional mathematician and generalises the problem to crosses of any size.

 

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.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 112, Crosses and Maths300 Lesson 159, Chances with Crosses.

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.

From The Classroom

Buloba Primary School

nr. Kampala, Uganda
Year 6
Buloba Primary School is adjacent to Buloba Teachers College. When Vice-Principal Miiro Musoke William had the opportunity to invite a visiting Australian to spend a little time working with the children he was also able to organise several teachers to be part of Discussion Lesson. His own enthusiastic assessment of the lesson was to declare that all children could be involved. There were 8- children in the class, as is the case with the other classes in this K - 6 school of 1,100 students.

Five Numbers

The lesson followed the steps below and left the children prepared so that the teachers could follow up with Crosses. Our thanks to the staff and children for sharing the story of this experience.

  • Children worked in pairs to tear a sheet of A4 into nine parts and number them 1 - 9.
  • The teacher revealed 9 x 20cm square also marked 1 - 9 and asked 5 volunteers to select one card each.
  • Volunteers were then spread out along the board and had an 'igloo' drawn around them as they displayed their card in front of their chest.
  • The other children selected these numbers from their set and turned the others face down.
  • A plus sign was drawn between each 'igloo' with an equals sign at the end. Then all children were asked to 'add them' but keep the answer a secret from the teacher for now.
  • Volunteers returned to their place to help their partner and left their cards on the table.
  • The focus of this part of the lesson is now on ways of knowing the answer using these questions:
    • How do you know the answer?
    • Can you check it another way?
    • Share with another group. Do you have different ways of knowing the answer?
  • Select volunteers to come to the board to teach the teacher their way.
  • The teacher celebrates every suggestion and uses each, if possible, to extend discussion.
  • Emphasise there are lots of ways.
  • Congratulate the children for doing really well at this part of the lesson before leading into the next 'puzzle'.
  • Volunteers return and the teacher makes a show of secretly swapping their cards with others in the 1-9 set. They stand in their 'igloo' but don't show their number. (The teacher has already erased all except the 'igloos', plus and equals signs.
  • The teacher writes a random answer for the sum of these secret numbers. (What's the lowest and highest this could be?)
  • Challenge: What are the secret numbers?
  • As each pair selects the five numbers they think will work, they turn their unused ones face down.
  • Using just the five cards you have chosen, can you check your total a different way?
  • Select one pair to write their numbers in the 'igloos'. With all drama ask if anyone else chose the same numbers (order doesn't matter).
  • Ask another one or two pairs to repeat this process with the aim of asking:
  • How many solutions are there? How do you know when you have found them all?
  • If the teacher wishes these questions can be tackled by starting a display corner for Five Numbers and keeping an on-going record of the sets of 5 as they are discovered in the next few days.
Children are now ready to be introduced to Crosses in a future lesson.

               

               

               

               

 

Ulla Öberg

Consultant
Sweden
When Ulla was preparing Crosses to use in a professional development program she took the problem to one of her former Year 4 students. Jonas Månsson was now a mathematician at Lund Technical University. He thought the problem was very interesting and first calculated the total number of solutions. That is, as a mathematician he asked How many solutions are there? and to know how he had found them all he used the tool of combination theory after first identifying restrictions in the problem such as the:
  • number in the middle must be odd.
  • totals of the arms are limited by the fact that 1 + 2 + 3 ... + 9 = 45.
Jonas's calculation showed there are 20,736 solutions, the same value arrived at by the Maths300 software for this lesson which crunches away, albeit artistically, using the strategy of testing every possible combination.

Clearly, Jonas was applying the mathematician's questions listed in the Working Mathematically process. He continued his interest in the problem by asking, What happens if ... the cross is a different size?.

Other Crosses

The number tiles would be 1 to 5, so students can explore this problem as an extension of the task. There are solutions, for example, those shown here. But, how many solutions are there altogether and how do you know when you have found them all?

Then of course one could ask about 13 numbered tiles, 17 numbered tiles, 21 numbered tiles and ...

If I tell you any number of numbered tiles can you tell me the number of solutions?
Notice though that 'any number' must be of the form (4N + 1), where N is the number of tiles in one arm. Can students see why this is so? Perhaps students would like to try the challenge of finding five solutions to the 13 tiles problem. The general problem was the next level Jonas tackled.


Extracted with permission from Medlemsbladet nr 2 år 2006, Sveriges MatematikLärarförening, page 7.
The article continues by exploring other problems using 'nine pieces of paper', including Task (Mattegömma) 30, Truth Tiles 1 and Task 43, Number Tiles.
As we can see, he succeeded in providing a formula for calculating all the solutions for any Crosses problem of any size.

Ulla Explains

I took the problem to Jonas because I wanted to illustrate in my workshop that a good problem has many levels of exploration. It just waits for the next question to be asked.

Jonas gave me more than I was looking for. Not only did he ask and answer that next question, but he confirmed that Crosses is a worthy problem for a professional mathematician just as it is for any child learning to work like a mathematician.

Editor's Note

Ulla can't remember whether she used Crosses with Jonas's class when he was in Year 4. But why should the truth be allowed to get in the way of a good story? It feels good to think that she did.

Green Line
Follow this link to Task Centre Home page.