Arithmagons 1

Task 188 ... Years 4 - 8

Summary

Using only the digits 1 to 10, numbers are placed at the corners of a triangle and the lines joining them are assigned a value that is the sum of the connected corners. If you know the line totals can you work out the numbers at each corner?
  • How many solutions are there?
  • How do you know when you have found them all?
  • What happens if you don't know all three line totals?
Arithmagons 1 is a partner to Arithmagons 2 which involves similar reasoning using a square as the shape.

This cameo has a From The Classroom section in which Matt Skoss describes how his Year 7 students reasoned their way through this problem, He also suggests the task is an opportunity to introduce a spreadsheet challenge.

 

Materials

  • Counters numbered 1 - 10

Content

  • addition
  • algebra, generalisation in words & symbols
  • arithmetic, addition / subtraction
  • concept of proof
  • flow charts
  • mental arithmetic
  • reasoning
  • recording mathematics
Arithmagons 1

Iceberg

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

Tasks exist to support a Working Mathematically curriculum - a curriculum with the core focus of students learning to work like a mathematician. Therefore, although this task involves mental arithmetic practise and the opportunity to generalise, its main focus is on the questioning, reasoning, justifying and communication skills that are the work of a mathematician.

Students won't have too much trouble finding solutions to questions 1 (a) - (d) since guess, check and improve can be used. So the implied challenge in these first questions is to find a process that will always produce the answers and be able to explain that process to someone else. There are several ways to think through this process. One is explained here, but it should be compared to how 'real' Year 7 students reasoned as explained in From The Classroom below.

Solving Arithmagons

  • Start with the biggest line total.
  • Make all the pairs of counters that equal this total and arrange them in order.
  • Start with the largest number in the first pair. Place it in the corner of the chosen line so that it is connected to the second largest line total.
  • Put the other counter from the pair in the circle at the other end of the largest line total.
  • Ask: Can this pair work?
    If it does you will be able to find a counter for the third corner and all the line totals will be correct.
  • Record the solution if there is one.
  • Whether or not it works, remove the third counter and swap the pair of counters to opposite ends of the highest line total.
  • Ask: Can this pair work?
    If it does you will be able to find a counter for the third corner and all the line totals will be correct.
  • Record the solution if there is one.
  • Repeat this process with the next pair of numbers and continue until all pairs have been checked.
This system will work for the Challenge questions too. Try it for yourself with the puzzles on the card. You should find, for example, these solutions:

1(a) 6 1(b) 5 Challenge 1 1 3
7 2 2 10 6 3 4 1

In Challenge 1, three pairs of numbers can make 7, the largest line total - (6, 1), (5, 2), (4, 3). The first gives one solution with the missing line total being 3, the second can't provide a solution (why?) and the third gives a solution with a missing line total of 5.

In these, and all other cases, checking for all solutions, requires explaining why other possibilities don't work. This is just as important to a mathematician as knowing why things do work.

If your students do create their own 'rules' - and them doing so is certainly the hope of the task - you might still introduce this process and ask them to compare it to their own. Then for either or both you have the opportunity to introduce the concept of a flow chart to explain this process in diagram form. The option is also there, especially if a flow chart works, to write a computer program to solve Arithmagon puzzles.

Extensions

The general picture for a triangle arithmagon is:

Which means that:

  1. ... A + B = tab
  2. ... B + C = tbc
  3. ... C + A = tac
Subtracting Equation 2 from Equation 1, for example, gives:
(A + B) - (B + C) = tab - tbc
which implies that:
A - C = tab - tbc

So, the difference between the two corner numbers is the same as the difference between the two totals that lead from those corners to the next corner.

This understanding can provide another way to solve an arithmagon puzzle.
In 1(a), tab = 13 and tbc = 9.
The difference between these values is 4, which means the difference between A and C is also 4.

So, to find A and C we need two numbers which sum to 8 and have a difference of 4. That would be 2 and 6, and trying them one way in the corners gives the solution above, but the other way can't work because the number in the third circle, B, would have to be 11 which is larger than the line total of 9 ( tbc ).

The generalised equations above also suggest another approach. If the three equations are added we get:
(A + B) + (B + C) + (C + A) = Total of the line totals
which means:
2(A + B + C) = Total of the line totals
which makes sense because each of the corner numbers A, B, C is counted in two line totals, therefore adding the line totals must include each corner number twice. This suggests that we could find the corner numbers by adding the three line totals, dividing by two and then looking for three counters that add to this number.

Check with 1(a) and you see the three line totals add to 30, so A + B + C = 15. We could now begin a process of checking every case. For example, let A = 1 and then find every pair in the remaining counters that sums to 14 and test them. Then let A = 2 and so on. However, consider this diagram for 1(a) and compare it to the answer above and a simpler, quicker process might emerge.

Why does it work?

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.
   

If you already have sets of tiles or counters numbered from 1 to 10, then all you need to do is design a triangle frame to fit them and make copies. If not, then it is quite easy for the students to draw a triangle frame on a scrap piece of paper as large as possible and fold and tear a second piece of paper into 10 parts and number them. One between two is all you need.

The lesson could start by one person in each pair secretly placing three corner numbers, then calculating the line totals and giving just those totals to their partner with the challenge: Can you work out my corner numbers?. This will lead you into a statement like:

A mathematician might be interested in finding a way to always be able to work out the corner numbers if you know the line totals. Our challenge today is to try to find a way to do that. So, keep exploring Arithmagon puzzles and when you discover something, let me know so we can share it.

The notes above will guide the lesson further. For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 63, Arithmagons, which also includes an Investigation Guide with answers and discussion. You might also find the Calculating Changes threaded activity Number Shapes to be a useful companion to Arithmagons.

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 Arithmagons 1 task is an integral part of:

  • MWA Pattern & Algebra Years 5 & 6

The Arithmagons lesson is an integral part of:

  • MWA Pattern & Algebra Years 5 & 6
  • MWA Number & Computation Years 7 & 8

From The Classroom

Centralian Middle School
Alice Springs

Matt Skoss
Year 7
A surprising strategy that arose through playing with Arithmagon problems was the kids making a decision to 'share the difference' between the two bottom vertices of the triangle. What impressed me was their sense of the three equations having to be balanced. Even though they didn't quite get to the point of using those words, they implied it by "if we add on one to each of the bottom numbers, we have to take one from the top number." Using rich problems like Arithmagons reinforces the incidental 'informal' mathematical thinking that is being developed by students. Our challenge as teachers is to be sufficiently 'tuned in' to pick up the nuances.

When pressed further, Matt explained...

In the past week, I've really enjoyed my Year 7 students' reactions to working on Arithmagon problems. After a lesson of 'playing with the problem,' my students came up with the approach of picking any number for the top (say 5), and then finding the missing numbers for the bottom two vertices (13-5 and 8-5 in the example). They then turned their focus to what their total for the bottom edge was, and what the problem 'wanted the total to be'. They came up with the idea of 'sharing the difference' by halving between the two bottom vertices, which then gave them the solution immediately. Having a 'sense of power' over the problem, especially when many adults will struggle with the problem for a while, was a very powerful outcome for these students.

Ta-Daa!

The nature of this problem helped them 'take a risk' which they are normally reticent to do. We played with a few Arithmagon problems using 'Write on-Wipe off' sleeves, which are more flexible than mini-whiteboards, because they enable any template to be inserted.

An unplanned journey I went on with my students was when their initial guess for the first number required them to have a negative number in one or both of the bottom vertices. I was pleasantly surprised when they were able to deal with the directed numbers in their heads, without 'breaking stride.' The fact that the need to use directed numbers was embedded in the 'story-shell' of an Arithmagon problem meant it was a tool they just drew upon without really thinking about it. The drive to find a solution didn't get in the way of them not having met directed numbers too often before.

In thinking about the very common 'classroom chestnut' of how do we differentiate learning and cater for diversity, I thought of developing an example spreadsheet to show students. This can be used to challenge students to develop their own generalised Arithmagon spreadsheet that would solve any problem. Using speech or thought bubbles on the example can help scaffold teachers or students who have limited experience in using spreadsheets in this way.

Many students won't necessarily need to or want to pursue developing a spreadsheet solution, but it would be a significant challenge for many. An extension to this problem might be to write a computer program to achieve the same outcome using free Scratch software, available from http://scratch.mit.edu.

Please feel welcome to contact me via matt@skoss.org if you'd like to follow up on the spreadsheet idea or 'Write on-Wipe off' sleeves.

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