Making Fractions 3

Task 199 ... Years 4 - 8

Summary

Students 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.

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

  • One set of 27 blocks in 5 sizes - each size has a fraction relationship to the others
  • Recording board, marking pen and cloth

Content

  • arithmetic, multiplication / division
  • concept of proof
  • decimals, calculations
  • fractions, calculations
  • fractions, equivalence
  • fractions, reciprocal
  • fractions, whole & parts
  • fractions, value relations
  • measurement, area
  • mental arithmetic
  • money
  • spatial perception, 2D or 3D
Making Fractions 3

Iceberg

A 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:

  • Two of Block E fit on top of one of Block D, so D is double E or E is half of D (the reciprocal relationship)
  • Three of Block B fit on top of two of Block A, so A is one and a half times B or B is two thirds of A (the reciprocal relationship)
It helps to look at the stacks of blocks from the side rather than the top. For example consider A stacked on B.

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:

A B C D E
Row 1 1 2/3 1/2 1/3 1/6
Row 2 11/2 1 3/4 1/2 1/4
Row 3 2 11/3 1 2/3 1/3
Row 4 3 2 11/2 1 1/2
Row 5 6 4 3 2 1

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 cells reflected in the leading diagonal are reciprocals of each other - that is, the product of the two cell values is 1.

Examples
Row 4E (1/2) and Row 5D (2) could be thought of as:
  • 1/2 of 2
  • 2 lots of 1/2
  • 1/2 x 2
  • 2 x 1/2
but in whichever way the answer is 1.
That's pretty easy. Now apply similar reasoning to Row 2C (3/4) and Row 3B (11/3):
  • 3/4 of 11/3
  • 11/3 lots of 3/4
  • 3/4 x 11/3
  • 11/3 x 3/4
And the answer is ...?

Challenge
The card also asks students to try the same problem again by choosing the value of $1 for each piece in turn. This problem relates the fraction question to decimals (or money). It also involves making a decision about rounding off when per unit, prices like 162/3 cents occur. Should these be rounded or would the manufacturer of the pieces leave the figure in the fraction form on the grounds that pieces would be sold in large quantities?
Note: There are times, such as in currency exchange, that fractions of a cent are used ... and these small bits matter a lot if you are exchanging large amounts of currency!

If rounding off and decimals are used, the table becomes:

A B C D E
Row 1 $1 067 050 033 017
Row 2 150 $1 075 050 025
Row 3 2 133 $1 067 033
Row 4 3 2 150 $1 050
Row 5 6 4 3 2 $1
  • How do the reciprocal relationships across the leading diagonal work out now?
  • What would the table look like in 'fractions of a cent' form? What would happen to the reciprocal relationships?

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:

  • one third + one third + one third = 1
  • 1 - two thirds = one third
  • 3 x one third = 1
  • two thirds (how many) one third(s) = 2
and it is better to encourage writing what is said (one third), rather than symbols until the students can tell you that they know enough to use the symbols and explain them.

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.

  • Start with the two A blocks side by side to make a square. This is the new whole. Call this shape Z, say.
  • Z is obviously made of two equal pieces and the 'fraction language dictionary' tell us we can use the word 'half' for each piece.
  • Now stack the next size block on top. What fraction is each piece of Z?
  • Now the next biggest...
  • And the next biggest...
Now Z has been partitioned into halves, thirds, fourths, sixths and twelfths and there is an almost endless world of fractions equations to discover and record within the blocks. The journey could be a free exploration, but it also seems to be crying out for one or more teacher created Investigation Guides. When you write one and have trialled and modified it, please submit it to be shared with colleagues.

Whole Class Investigation

Tasks 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. Multiple copies of the task, or additional panels of blocks can be purchased from Mathematics Centre, or something similar can be made from wood or card. However, these approaches might place an excess demand on finances or time. Instead, consider using a work station approach where this task and several others with similar mathematics as suggested above in the 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 hands-on learning kits available from Years 3-10 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:

  • MWA Number & Computation Years 7 & 8

From The Classroom

St. Mary Mackillop College
Canberra

Neelima Gupta
Year 7
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 Reviews

Various 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.
Secondary teacher

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.
Secondary teacher

Green Line
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