This problem solving activity has a number focus.
In the diagram there are 6 circles arranged in the shape of an equilateral triangle.
You are given the numbers 1 to 6.
Put a different number in each circle.
How many ways are there of doing this so that the sums of the numbers on each side of the triangle are the same?
- Think logically about the sums of single digit numbers.
- Devise and use problem solving strategies (guess, and check work systematically).
At it's simplest this problem involves the students adding single digit numbers. However, as the students work towards finding all possible solutions they are involved in reasoning with these numbers. They may also apply their understanding of symmetry (reflective and rotational) to show that some answers are in fact the same.
- Digit cards (1-6)
- Copymaster of the problem (English)
- Copymaster of the problem (Māori)
The Problem
In the diagram there are 6 circles arranged in the shape of an equilateral triangle. You are given the numbers 1 to 6. Put a different number in each circle. How many ways are there of doing this so that the sums of the numbers on each side of the triangle are the same?
Teaching Sequence
- Introduce the problem to the class. Pose the problem as a game or puzzle. Ensure the students understand what is meant by 'equilateral triangle'.
- Brainstorm ideas for approaching the problem and keeping track of answers. Have available the digit labels for those students who choose to move instead of write the digits.
- As the students work on the problem in pairs, ask the following questions to extend their thinking:
What strategies helped you find the answer?
How can you use you knowledge about numbers here?
Are any of your answers similar – in what way? Are they different answers?
How did you know when you had the solution and could stop looking?
Can you think of a related problem to explore? - Share answers – write them on the board as they are given. Discuss those answers that are reflections or rotations. Ask the students to explain their reasoning for why there are only 4 different answers.
Extension
What other six numbers can be used to make equal sums for each side?
Solution
The students will probably very quickly discover two things. One, that some answers can be obtained from others by reflecting or rotating the triangle. Two, that there are only four different ways to do this, subject to reflections and rotations.
The difficult part is to show that there are only four different answers. In order to do this you can first show that there are only four possible sums that lie between 9 and 12 (inclusive). The reasons for this are (i) that 1 has to be somewhere; (ii) that the biggest sum that can be made using 1 is 1 + 5 + 6 = 12; and (iii) that 6 has to be somewhere; (iv) the smallest sum that can be made using 6 is 1 + 2 + 6 = 9. This can also be discovered in a number of other ways.
So why is there only one arrangement with a sum of 9? How can you make up 9? There are only three ways: 1 + 2 + 6, 1 + 3 + 5, 2 + 3 + 4. These can each be fitted into the three different sides. A similar argument can be used for sums of 10, 11, 12.
It’s important to try this last part so the students see how you can reason mathematically. How do you get to the bottom of something that appears quite complicated? You may want to do this as a whole class discussion.
The approach taken above is not the only way to do the problem. Explore the different methods that your students suggest.
Solution to the extension
There are many possible solutions to this extension.
- Adding 1 to each individual number will lead to a solution, since the total of each side will increase by 3. In fact they can add any number to each individual number to find a new solution.
- Having all 6 numbers the same will lead to a solution, as will having the three corner numbers the same and the three numbers in the middle of sides the same.
Your students may find other ways to solve the problem.