Balancing Act

This island on the equals sign using consecutive numbers expands easily and is one where a generalised proof is accessible to children. There are also multiple solutionwas for some sets. It also addresses the misconception that the equals sign means ‘the answer’.

We can start with a simple problem with a target number. For example, asking for a calculation with an answer of ten. Then ask for another, and another.

Show these examples as being equal – without the ten:

6 + 4 = 3 + 7

6 + 4 = 3 + 7 = 1 + 9

What if we had a set of numbers and tried to make them balanced?

For example, if we had 1, 2, 3 and 4, could we make a balanced equation?

4 + 1 = 3 + 2

What if we used 2, 3, 4 and 5?

2 + 5 = 3 + 4

It is evident that this is possible with any 4 consecutive numbers. The sum of the first and last number will always equal the sum of the middle two numbers.

What if we used 5 consecutive numbers?

With 5 consecutive numbers, we naturally bring in order of operations into the mix. We are connecting it to 4 consecutive numbers and wondering whether there is also a generalised strategy. Clearly, we can’t use the same one as before.

Starting with 1, 2, 3, 4 and 5 they might find solutions like this:

1, 2, 3, 4, and 5: (1 + 2) x 3 = 4 + 5

2, 3, 4, 5 and 6: 2 x 4 + 3 = 5 + 6

3, 4, 5, 6 and 7: 5 x (6 – 4) = 3 + 7

4, 5, 6, 7 and 8: 4 + 5 = 8 + 7 – 6

These are four different approaches. One of them works for all cases though. This is the strategy for 3, 4, 5, 6 and 7 and we can explain why.

We can do this algebraically and use cubes to assist. Let’s focus on the middle value initially. The first value is two fewer than the middle value. The last value is two greater than the middle value. Therefore, the first and last values are exactly double the middle value. We can then simply subtract the second value from the fourth value to get two and multiply that by the middle value.

Algebraically:

We can express the 5 numbers as:

a -2, a-1, a, a + 1, a + 2
a((a+1) - (a-1)) = a-2 + a+2
2a = 2a

What if we used six or seven numbers?

Seven numbers, with a bit of creative thinking, is just an extension of five numbers. We have two new numbers with a difference of one. Use that difference to multiply either side by one. For example:

1 + 5 x (7 – 6) = 3 x (4 – 2)

2 + 6 x (8 – 7) = 4 x (5 – 3)

This then extends to all sets with an odd amount of numbers.

Six numbers is trickier. But sometimes letting them struggle is important. It can result in more creative approaches.

What if we used two equals signs?

This then makes 6 numbers very possible. This is a ‘what if…?’ question that is so much better if it comes from the children. With enough experience, they will see ideas such as these as a result of the culture that is created.

1 + 6 = 2 + 5 = 3 + 4

Once found, we might choose to shift the focus again. Can we come up with more creative solutions?

What if we assigned points to different operations?

For example, we might give zero points for addition and subtraction, 1 point for multiplication and two points for division. For example, this equation would score two points:

1 + 5 x (7 – 6) = 3 x (4 – 2)

What if we used inequalities instead?

Obviously, on its own, this rule doesn’t really promote much mathematical investigation. It should be fairly simple to create two imbalanced calculations. We try and introduce other ideas to ensure that there is genuine mathematical investigation. One possible idea would be that we are looking for an equation where the side with the lower total, used a larger set of numbers:

5 – 4 – 1 < 3 x 2

Take it further by not allowing subtraction:

5 x 4 > 3 x 2 x 1

We are seeking challenge by altering what we are doing. The children are thinking about the impact of the operations. Does this still work with the numbers from 1 to 6 or 1 to 7?

Children’s possible use of reasoning skills

Search
Search inquiry reasoning skill in maths

Through this island, the children will be doing a lot of trial and improvement by making totals and seeing if they can create an equal value with other numbers. This includes elements of all three Search skills. Explicitly highlighting how they might know that certain totals (for example, having 5 x 4 when using 1, 2, 3, 4 and 5) is unlikely to produce the desired result.

Organise

Comparing similarities across solutions will provide them with the necessary clues to make generalisations. It might be that you have to explicitly encourage this though.

Discover

Conjectures are likely to be formed mainly through initially Noticing the similarities between existing examples. It would be easy for them to simply create a successful example and move on. Ensure that the children are taking the time to look at the existing examples to build a generalised approach.

Investigate
Investigate inquiry reasoning skill in maths

The main use of the Investigate skills here will be to see if approaches used with a set of numbers that have similar characteristics (4 consecutive numbers) can also apply to a different set (5 consecutive numbers). Then they will have to adjust conjectures accordingly.

Argue

Exploring why solutions work using cubes can ensure that children use Argue skills. This is an island where it is useful to slow the children down for a bit and focus on what they already have before they go off in too many directions themselves first. There are accessible proofs here for them.

Explore

As shown above, there are natural ways to deepen the inquiry by changing the constraints or sets used. However, there is also a lot of scope for creativity with new approaches such as the use of more than one equals sign. Children often come up with the unexpected.

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