Denominator Dilemma

This island focuses on adding and subtracting fractions with the same denominator on a number line. Through attempting the problem, the fact that there are fewer halves in a whole than quarters are emphasised. This is because to be successful, they have to take this into account due to the numerators used.

There are lots of different starting points and directions possible with this set of islands. I like the idea of starting with a game. For this, we use two number lines that show halves and quarters between zero and two.

At the start of the game, a counter is placed on the value of one on both number lines. When the counter moves to each value, it is crossed off to indicate that the counter cannot return to that value. As such, initially, the value of one is crossed off on both number lines as the counter starts there.

There are twelve fractions underneath that can be added to or subtracted from the current value. However, only the numerators are specified. With each move, the player chooses to use either a denominator of two or four. The counter on that number line is then moved by that amount.

In the move below, two halves were added. As a result, this fraction at the bottom is completed and the value of two is crossed off on the halves number line and the counter moves there.

Let’s say that the next player then chooses to subtract one quarter. The counter is moved and three quarters is crossed off.

The first player then chooses to subtract three halves. The game continues until a player cannot move. In that case, they lose the game.

The key aspect of the game is that the numerators of five and six are only usable with quarters. This gives them a strategic element to focus on and use reasoning in their explanation of what moves are possible. It emphasises that there are more quarters within two wholes than halves. The halves number line will likely be crossed off more quickly.

This base game is there to introduce the concept and will not take too long to play.

What if we worked together and had to use the numerators in order?

Instead of a game, this variation requires the children to try and work out how many moves are possible in total. This could be done by allowing the fractions to be used in any order. In this case though, I have the constraint that the numerators are used in order in two sets of six. As before, for each move, the denominator could be two or four. Here is an example of three moves. Above each value on the number line, I have written the number of the move that it was reached on.

Things become trickier when they have to use the higher numerators as they can only be used with quarters. This is why I have enforced the order of the numerators. I do not want them to avoid using them as they are harder to use. I want the children to have to think about how they can be used.

What if we changed the sets of numerator values that we are using?

Instead of using one to six twice as the numerators, what if we tried one to four three times or one to three four times? Or some different pattern?

Once they have explored using two sets of one to six, they will have encountered the challenges involved. Therefore, this presents an opportunity to analyse whether or not making these changes will allow more moves or not. It would be natural to assume that by reducing the value of the maximum numerator, things will be easier as it could then be used on both number lines. However, it is not necessarily the case.

What if we changed the starting positions?

This small change could allow for further investigation into whether the number of moves possible could be increased by altering the starting positions.

What if we changed the denominators?

With this change, each increase in the value of a denominator will result in two extra values on the number line. As such, it is better to stick with lower numbers. It is the same principle if we altered the range from being zero to two.

What if we could move on both number lines at the same time?

For instance, if we were adding or subtracting two quarters, we could also add or subtract one half as they are equivalent. It throws up more possibilities.

What if we could use any denominator?

Instead of restricting it so that only halves could be used with the halves number line, we could change it so that any denominator could be used. This makes it easier in some ways but requires the use of equivalence to be more successful. It depends what you want to focus on in the lesson.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

The order that they choose to add or subtract the fractions matters a lot here and so they need to carefully consider this rather than just randomly selecting values. This is particularly emphasised when working together rather than playing the game.

Organise

One possible way of focusing on this is to compare the number of moves possible with different starting moves. Different members of the class could focus on an initial move and then results analysed as a class.

Discover

Whilst conjectures about the total number of moves possible are the most likely use of their Discover skills, they can also conjecture about the order of moves.

As changes are made to the rules, they can conjecture on the impact that it will likely have on the number of moves possible.

Investigate
Investigate inquiry reasoning skill in maths

When they develop strategies, they will need to investigate them to establish whether or not they work. There is a lot of potential for that here in both the game and the investigation that follows.

Argue

The use of numerators has been chosen to encourage discussion as to which moves are possible and when. The first few moves need to be carefully chosen to allow for the use of five and six as numerators.

Explore

There is so much potential for changes to the rules and further investigation. Children shouldn’t have any trouble coming up with ideas for what they could investigate further.

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