Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Chocolate

There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?

Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Almost One

Stage: 3 Challenge Level:

The suggestions in these notes are adapted from Colin Foster's article, Sum Fractions.

Why do this problem?

Adding and subtracting fractions is a procedure which students often find very difficult to master. It is important to address the area without it feeling like an exact repetition of what they have done many times before.

One way to avoid the tedium of lots of repetitive practice is to embed practice in a bigger problem which students are trying to solve. This idea is explored in Colin Foster's article, Mathematical Etudes, and this problem is an example of a mathematical etude.

Possible approach

"What can you say about these six fractions?" $$\frac{1}{6} \quad \frac{1}{25} \quad \frac{3}{5} \quad \frac{3}{20} \quad \frac{4}{15} \quad \frac{5}{8}$$

Students might note that they are all different, that they are all less than 1, that they are all positive, that they are all expressed in their simplest terms, that four are less than a half and two are greater than a half, that they are not in order of size, and so on.

Encourage students to say as many things as they can think of. Questions like this are a good way to encourage students to be mathematically observant.

"Which fraction do you think is the largest? Which is the smallest? Why?"
Since all of the fractions are expressed in their simplest terms, it is easy to see that none of them are equal. Students may compare fractions by making their denominators equal or converting them to decimals.

Encourage students to use 'informal' methods of comparing fractions, and only calculate when it becomes absolutely necessary. For example, $\frac{1}{20}$ is bigger than $\frac{1}{25}$, so $\frac{3}{20}$ will certainly be bigger than $\frac{1}{25}$.

"Write a fraction that is equal to $\frac35$. And another, and another..."
Students could write their fractions on mini-whiteboards.

They will probably list equivalent fractions such as $\frac{6}{10}, \frac{30}{50}$ etc.
You could encourage a wider range of answers by introducing some constraints, for example... "Write down one with an odd denominator" or "Write down one where the numerator is a five-digit number that does not end in 0".

Then ask them to do the same with $\frac58$.

"How would you add $\frac35$ and $\frac58$ without a calculator?"
"The answer to $\frac35 +\frac58$ is a little bit more than 1. Is there any way that you could have predicted that the answer was going to be more than 1 without working it out exactly?"

Both fractions are more than $\frac12$, so their total must be more than 1.

This sort of reasoning can be very useful for estimating the size of an answer so that mistakes can be spotted. Estimation will be important in the main activity that follows.

Return to the original set of six fractions:
$$\frac{1}{6} \quad \frac{1}{25} \quad \frac{3}{5} \quad \frac{3}{20} \quad \frac{4}{15} \quad \frac{5}{8}$$
"Choose some of the fractions and add them together. You can use as many fractions as you like, but you can only use each fraction once."
"Can you get an answer that is close to 1?"
"What is the closest to 1 that you can get?"

Make it clear that calculators are not to be used!

If some students are unsure how to start, encourage them to talk to their partner.

Give students some time to work on the problem. This will be a good opportunity to circulate and see how students are getting on.

If students obtain an answer like $\frac{11}{12}$ (from $\frac{1}{6} + \frac{3}{5} + \frac{3}{20}$), they may think that they are as close as possible, as their answer is “only 1 away”, but because the “one” is “one twelfth” they are not really that close ($\frac{1}{12}$ is more than 8%), so they should aim to get even closer!

Allow plenty of time at the end of the lesson for students to share their approaches and reasoning.

Possible extension

Students could be asked to find the set of fractions which add up to as near to $\frac{1}{2}$ as possible.

For other rich contexts that offer students an opportunity to practise manipulating fractions see Peaches Today, Peaches Tomorrow and Keep it Simple.

Possible support

If students are not secure with equivalent fractions they could do some work with Fractional Wall.