Task 176 ... Years 2 - 10
SummaryNine digits to place in nine boxes which are arranged as a staircase. Each of the four staircase parts must add to the same number.
IcebergA 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.
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:
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:
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?
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.
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.
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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.
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:
The Steps lesson is an integral part of:
From The Classroom
Penleigh & Essendon GrammarKaren McCasey