CrossesTask 35 ... Years 4  8SummaryDigits 1  9 are placed on arms of a cross so that the partial sum of each arm is the same.
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
Content

IcebergA 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:
23  1 = 22.
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.

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.
 Visit the Home Page for more Background.
 For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.
Whole Class InvestigationTasks 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 handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
The Crosses task is an integral part of:
The Crosses lesson is an integral part of:

Buloba Primary Schoolnr. Kampala, UgandaYear 6 
Buloba Primary School is adjacent to Buloba Teachers College. When VicePrincipal Miiro Musoke William had the opportunity to invite a visiting Australian to run a 50 minute Discussion Lesson with the children he was also able to organise several college staff to be included. His own enthusiastic assessment of the lesson was to declare that all children could be involved. There were 70+ children in the class, as is the case with the other classes in this K  6 school of 1,100 students. 


Ulla ÖbergConsultantSweden 
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:

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?.
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? 
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 ExplainsI 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. 
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.