Making Fractions 3Task 199 ... Years 4  8SummaryStudents first arrange a set of blocks in order and record the order on the board. The purpose of biggest to smallest ordering is to make more obvious a relationship that connects the pieces when the puzzle is solved. The puzzle is based on value relations. That is, if we give this piece a certain value, what is the value of each of the other related pieces. The assigned value is always 1, in other words, the assigned piece becomes the whole for that row of the puzzle. Other pieces will be related to the whole as fractions or mixed numbers and the first challenge is to find these relationships in each row. The challenge section of the card introduces decimal equivalents by giving the chosen piece the value of $1.Use the Task Cameo Content Finder to discover other tasks involving value relations. Two with a structure almost identical to this one are Making Fractions 1 and Making Fractions 2. This cameo has a From The Classroom section which includes a slide show about Maths300 Lesson 135, Chocolate Cake, and contains reviews of Making Fractions 3 task by a student and teachers in various schools. 
Materials
Content

IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
The principle of this task is to take a whole, divide into parts, then choose any of those parts in turn to have a value of 1. In effect, the task is based on the concept that any piece, not necessarily the largest one, can be the whole. Making Fractions 3 is based on halves, thirds, quarters, sixths and twelfths of a whole which isn't actually included but can be easily made from sets of any of the shapes. To record the biggest to smallest order along the top line of the chart, students can use drawings, or a code such as LR for largest rectangle. The pieces are not intended to fit in these cells. However, some students do stand them on their edge outside the cells. An additional spatial challenge is provided by trying to fit the pieces back into the frame at pack up time. This should be encouraged because it is also a check that all the pieces are there. From largest to smallest the blocks are:
Some of the relationships between the blocks are straightforward, for example:
If A is the whole, then 2 wholes have to be divided between (or shared between) 3 B pieces. So, imagining each whole sliced into three and shared means each B receives two thirds whole. Two thirds is written as ^{2}/_{3}. Do those numbers look familiar? If B is the whole, we stack the other way. Now three wholes have to be divided between two pieces, so divide each whole in half and share. That means each A is worth one and a half of the wholes. One and half is the same as three halves which is written as ^{3}/_{2}. Do those numbers look familiar? The thinking in this approach is the same as that which students often use in Task 19, Cookie Count when there are 'left overs', or in Maths300 Lesson 135, Chocolate Cake (see From The Classroom below). When correctly filled in the table is:
Encourage students to look for patterns in the table. They might see the double/half connection between Columns D and E and also between Rows 4 and 5 (why?), but it may take a bit more prodding to see that cell values reflected in the leading diagonal are reciprocals of each other  that is, their product is 1. Examples
Challenge If rounding off and decimals are used, the table becomes:
More Fraction Calculations Fractions are all about knowing what the whole is, dividing the whole into equal parts and then, as a consequence of the partitioning, being in a position to choose and use the appropriate fraction language. For example: Three of these equal parts make the whole so each part is ...worth one third (...called one third). Now, other fraction statements (stories? / equations?) are obvious from the pieces such as:
If that same whole can be divided into equal parts in another way, then a different fraction word comes into play and equivalent fractions are possible. Taken all together, rather than in separate rows, the blocks in this task make a Fraction Set.

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. 
To convert this task to a whole class investigation each pair (or perhaps group of 4) will need a set of blocks. Or something similar can be made from wood or card and the eTask Pack provides a master for this purpose. However, creating enough sets in this way might place an excess demand on finances or time. Instead, consider using a work station approach where this task and several others with mathematics similar to that listed in Content Finder, are one station. The other work stations might be text based work on value relations and software involving value relations, such as that for the Maths300 companion lesson of Task 75, What's It Worth?. With a system like this, where the text and software stations tend to 'look after themselves' the teacher can often find time to spend with the task group to listen to, question and assess their mathematical discussion. At this stage, Making Fractions 3 does not have a matching lesson on Maths300. 
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. 
The Making Fractions 3 task is an integral part of:

St. Mary Mackillop College
Neelima Gupta 
Neelima has provided a PowerPoint showing us how Maths300 Lesson 135, Chocolate Cake, worked out in her classroom. Graciously she has included a photo which shows that she made a recording mistake on the board for Table 2 and didn't realise until the students questioned her. Can you spot the error? 
Task ReviewsVarious Schools 
It was a bit confusing at the start but after reading it again it made sense. It was good having the blocks to help because you could use different blocks to work out the others. It took a while to understand but I enjoyed it. Declan, Year 6
Once students figured out what to do, it was fairly simple and straightforward.
I thought they would find this too hard but all groups managed reasonably well. One boy in particular used the relationships between different sizes very interchangeably, eg: "There's three of these (blue) make one of those (dark green) so it's one and half of this (yellow)." They did not recognise the patterns in the tables. 