Less Than FractionsTask 240 ... Years 4  10SummaryNumber tiles (1  9) allow students to experiment with fractions less than 1 in a nonthreatening, openended way. Early success is guaranteed because there are 36 possible answers and the obvious one is 1/2. But the greater challenge is to add two fractions (each tile can be used only once) and still get an answer less than one.

Materials
Content

IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
Exploring questions 1 and 2 soon leads to breaking the problem into smaller parts.
ChallengeIf students have completed the first part of the task by breaking the problem into parts, then they will probably begin the Challenge with the same strategy. For example, if the first fraction is ^{1}/_{2}:
A key is that each fraction chosen to be the first one has a complement  a 'partner' that makes the total exactly equal to 1. That partner may or may not be in the set of fractions that can be made with the remaining tiles once the first fraction is chosen. For example if ^{1}/_{2} is chosen as the first fraction, its complement is ^{1}/_{2}. The 1 and 2 can't be used again, but the equivalent of ^{1}/_{2} could be made with ^{3}/_{6} or ^{4}/_{8} (but not with ^{5}/_{10}  why?) However we don't want a sum that exactly equals 1. We want a sum that is less than one. So, once the first fraction is chosen we want a second fraction made from remaining tiles that is less than the complement of the first fraction. Therefore our work will be cut down a little by:
We look forward to posting solutions from your students here. Extension

Whole Class InvestigationTasks 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. 
It is very easy to get a class started with this investigation. A pair of students only has to fold and tear a piece of paper into 9 parts (there's a little discussion about thirds hovering in just that request if you want it) and number them one to nine. Then use another piece of paper to draw up a pair of boxes (in the middle of a landscape page) large enough to fit their number tiles. Draw your own boxes on the board as you explain. It's easy to state the initial problem too: Now put your tiles in the boxes to make fractions. When you make a fraction less than one, come and write it on our board.Highlight your interest in less than fractions by including the inequality in the first part of the task and invite students to include it on their paper too. You might comment that the shape of the 'less than' sign is easy to remember because it is like a capital L turned a little. As the data grows there will be opportunity for discussion about how we know a fraction is bigger than, equal to, or less than 1. A number line could be introduced to mark approximately where some of the less than fractions lie. The underlying intent of the conversation is to illustrate that everyone of us knows something about fractions, together we know more about fractions and by asking and investigating questions we can help each other learn even more about them. The first of those questions is: How many less than fractions are there ... and how do we know when we have found them all? Follow this through and when the 36 possibilities have been discovered and recorded introduce the second challenge. Now let's see if we can use out knowledge to try something a little more challenging. Can we find any fractions that make this true...?Turn the inequation on the board into the second one on the card by including fraction boxes to the left of the middle and the plus sign. Invite the students to do the same on their paper. The discussion above (and the students' suggestions) will guide the rest of the lesson. As suggested above, we would love to see your class's work investigating the number of solutions to this challenge. At this stage, Less Than Fractions does not have a matching lesson on Maths300. However Maths300 does contain several lessons which explore fraction concepts and develop skill in operating with fractions. Those lessons are:

Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
Less Than Fractions is not in any MWA kit. However it can be used to enrich the Number & Computation kit at Years 5/6 and the Number & Computation kit at Years 9/10. 