Steps

Task 176 ... Years 2 - 10

Summary

Nine digits to place in nine boxes which are arranged as a staircase. Each of the four staircase parts must add to the same number.
  • How many solutions are there?
  • How do you know when you have found them all?
This cameo has a From The Classroom section which includes a photo of journal recording from one Year 3 class.
 

Materials

  • 9 numbered tiles

Content

  • arithmetic, addition / subtraction
  • arithmetic, multiplication / division
  • combination theory / ordered arrangements
  • concept of proof
  • mental arithmetic
  • patterns, number
  • recording mathematics
Steps

Iceberg

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

There are many solutions to this problem, but finding the first can take 15 - 20 minutes. Students may need encouragement to keep going, but it is not necessary to 'solve the problem today', so if they want to leave it for another time while they try another one, that's all right.

However, it should be a routine that their journal records the task name, date and something about what they have tried so far ... so that when you come back to this one another day, you don't waste time going over old ground. For example students should note that the 3 corner squares are each counted in two line totals.

The solution shown here was discovered by a Year 3 student at Essendon and Penleigh Grammar.

To go beyond finding solutions by guess and check, the second question on the card needs to be addressed. If we added all the numbers used in the problem, the total would be 45. This would have to be shared equally between the four lines, because each of the line totals is the same. But 45 can't be shared equally between 4 and give a whole number result.

The first 'grand total' that could work would be 48, meaning each line total would be 12. Where would the extra 3 come from? The answer is the corner numbers. So, given we only we have the digits 1- 9 to use, we need three corner numbers which add to 48 - 45 = 3. Possiblities are:

  • (1, 1, 1) - No, there is only one 1 to use.
  • (1, 2, 0) - No, there is no 0 to use.
  • (3, 0, 0) - No, that doesn't work either.
So, 48 can't be the grand total and 12 can't be the line total.

The next possible grand total is 52, meaning each line total would be 13. Now we need three corner numbers which add to 52 - 45 = 7. Possibilities are:

  • 1, 2, 4 - which is the only possibility using 1 because the remaining ways of making 6 are (1, 5) and (3, 3) neither of which work.
In fact, checking all the other possibilities for triplets that total to 7, shows this is only set of three numbers that might work for a line total of 13. Now we place them in the corners and try to jiggle the remaining numbers to make a solution. Try this:

1 --- ---
---
4 --- 2
---
---
If this is going to work we need one digit to combine with 1 and 2 to make 13 and one that combines with 2 and 4 to make 13. 1 and 2 would need 10, which we don't have, so this can't be the way to arrange (1, 2, 4). Try this:
1 --- ---
---
2 --- 4
---
---
That could work:
1 --- ---
8
2 7 4
---
---
But can the remaining digits - 3, 5, 6, 9 - be arranged to make the other two line totals equal 13? Yes:
1 9 3
8
2 7 4
6
5
Clearly the 9 & 3 and 6 & 5 could be swapped around to produce a variation, but are there any other unique solutions for a line total of 13?

To discover the remaining line totals and their related solutions requires continuing the reasoning described above. But how did the author of the card know that there are only four line totals that work?

  • The next possible grand total is 56, line total 14, and we would need triplets that add to 56 - 45 = 11. That's possible, for example (1, 2, 8), or the arrangement in the solution above. Others??
  • The next possible grand total is 60, line total 15, and we would need triplets that add to 60 - 45 = 15. That's possible too.
  • The next possible grand total is 64, line total 16, and we would need triplets that add to 64 - 45 = 19. That's possible - just - for example (9, 8, 2). Others??
  • The next possible grand total is 68, line total 17, and we would need triplets that add to 68 - 45 = 23. The only possibility that might work is (9, 8, 6). Try it.
All possible grand totals beyond this need triplets that add to numbers great than it is possible to make with digits 1 - 9. The largest possible triplet is made from (9, 8, 7), which give a total of 24, whereas the next possible grand total needs triplets that add to 27.

So there are only 4 possible line totals (because when line total 15 is explored further the digits can't be arranged to make the puzzle work) and they produce 12 families of solutions (unique solutions) with 8 variations in each, giving a total of 96 variations altogether.

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.
   

To turn this task into a whole class investigation you could make a class set of digit tiles. These would have many other uses. However, you could also ask students to rip up scrap paper so that they made nine pieces between each pair.

   
Teachers involved in Steps during a workshop day in Örnsköldsvik, Sweden

Start the problem with a large set of nine digit cards you prepared earlier. Gather the students in a central place (tables or floor space) and hand out the cards. Ask the students with cards to arrange them to make a staircase.

Now if we have put these down the right way, the total here ... will be the same as the total here ... and the total here ... and the total here ... All four line totals will be the same.
Encourage participation with the group tiles for a while, to try to achieve this result then promise it can be done and send students to their tables in pairs to see who can be the first team to find a solution with their torn up tiles. Continue the lesson guided by the information above. Of course, you are doing this not because the mathematics content in the problem (as might be described by government curriculum documents) is of any great consequence, but rather because it is yet another powerful opportunity to model how a mathematician works. Therefore it is important to draw the lesson together with the students by asking them to tell you How we have worked like a mathematician today.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 29, Steps, which also includes companion software.

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 Steps task is an integral part of:

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

The Steps lesson is an integral part of:

  • MWA Number & Computation Years 5 & 6
  • MWA Number & Computation Years 9 & 10

From The Classroom

Penleigh & Essendon Grammar

Karen McCasey
Year 3
Sample of student recording.

Green Line
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