Task 43 ... Years 4 - 10
SummaryAdd two three digit numbers to make a third. The challenge is to do it using only the digits 1 - 9.
IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
Finding one solution can take 15 - 20 minutes, and students often ask if they can carry. There is nothing in the problem which says you can't, and there is everything in the problem to indicate it is a normal addition. Carrying introduces an extra 'unstated' digit into the problem, and this is the key to finding a solution, for example:
Five more solutions are:
248 654 573 397 596 +319 +327 +246 +251 +142 567 981 819 648 738
However, a more rigorous look at this method of solution would notice that it hasn't actually been proved that the answer line digits will always sum to 18. It has only been observed from limited data. There are actually 9x8x7x6x5x4x3x2x1 ways to rearrange the 9 digits into the 9 spaces on the card and without checking them all we can't be sure that one of these doesn't also sum to 18. The software available from the Maths300 lesson below can be used to investigate this question. It also explores all 362,880 rearrangements and confirms that there are 168 solutions - all of which have an answer line that sums to 18.
But is there another way to approach the problem? The answer is yes and here is an outline. Further detail is available in the Maths300 lesson.
All the solutions fit this pattern:
abc +def ghi where a, b, ... i are from the set of numbers 1-9
This is a general representation of the problem because it is the structure of every case. So far there has been no solution without carrying. The following proof shows there isn't one.
First, assume that there is at least one solution which has no carrying. So, given that there is no carrying, adding the columns must give:
c + f = i b + e = h a + d = gThe key now is to realise the unstated fact that:
Substituting the first 3 equations into line [A] gives:
This is a contradiction, so, since there is nothing wrong with the reasoning, the original assumption must be at fault. Therefore, if any solutions exist, they must involve carrying.
Part B of this approach means that a similar exploration must be carried out for these cases:
c + f = i + 10 b + e + 1 = h a + d = gwhich can be rewritten as:
c + f = i + 10 b + e = h - 1 a + d = gand then substituting into [A] above gives:
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.
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.
Start the problem with a large set of nine digits you prepared earlier. Gather the students in a central place (tables or floor space) and hand out the cards. Ask three students to place their cards to make a 3-digit number. Then ask another three to make a second 3-digit number underneath this:
Now we rule this off here (use a blackboard ruler as the line) and do a normal add up. If we have put the first cards down correctly, the answer will be the tiles left in our hands.Encourage participation with the group tiles for a while, 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.
For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 28, Number Tiles, 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 Number Tiles task is an integral part of:
The Number Tiles lesson is an integral part of:
From The Classroom
Ugandan StudentSolomon Kyeyune
Semanda Solomon Kyeyune was visiting Australia with his father who is Executive Director of the Rubaga Youth Development Association, Uganda. At nineteen, Solomon had recently completed his Year 12 studies and was waiting university placement. While visiting he explored a couple of tasks, one of which was Number Tiles. A couple of days later he asked:
Would you like to see my trick? I thought of it after using the tile problem.He explained it this way:
|Write down any three digit number you like.
Now write another one underneath.
Now I will write one.
Your turn again.
And finally one more from me.
Now I will tell you the sum of these numbers