Making Fractions 2

Task 177 ... Years 2 - 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.

 

Materials

  • One set of 16 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
  • powers of 2
  • spatial perception, 2D or 3D
Making Fractions 2

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 2 is based on halves, quarters, eighths and sixteenths of the largest square.

To record the biggest to smallest order along the top line of the chart, students can use drawings, or a code such as LS 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:

The task supplies half of what is needed for each shape to cover the largest square. However this is plenty to be able to work out the order of the pieces (by area) and their fraction relationships. When correctly filled in the table is:

A B C D E
Row 1 1 1/2 1/4 1/8 1/16
Row 2 2 1 1/2 1/4 1/8
Row 3 4 2 1 1/2 1/4
Row 4 8 4 2 1 1/2
Row 5 16 8 4 2 1

Encourage students to look for patterns in the table. They will find many doubling and halving connections and the rows might even be used to introduce powers of 2 and patterns in the way they are written. Consider Row 5 for example. It could be rewritten as 24, 23, 22, 21. Then, to fit pattern, 1 must surely be written as 20. Applying this new knowledge to Row 4 next makes us wonder if we could write 1/2 in this power format. And what about 1/4, 1/8and 1/16?

If encouraged to continue examining the table, students might see that cells reflected in the leading diagonal are reciprocals of each other - that is, the product of the two cell values is 1.

Example
Row 2E (1/8) and Row 5B (8) could be thought of as:
  • 1/8 of 8
  • 8 lots of 1/8
  • 1/8 x 8
  • 8 x 1/8
but in whichever way the answer is 1.

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 handling part cents when per unit prices like 61/4 cents occur. Should these be rounded (for example to 6 in this case) or would the manufacturer of the pieces leave the figure in the exact 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!

in non-rounded form, and using decimals, the table becomes:

A B C D E
Row 1 $1 050 025 0125 00625
Row 2 2 $1 050 050 0125
Row 3 4 2 $1 050 025
Row 4 8 4 2 $1 050
Row 5 16 8 4 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 such as 61/4 or 625? Which form would you prefer to use to check 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: Four of these equal parts make the whole so each part is ...worth one fourth (...called one fourth).

Now, other fraction statements (stories? / equations?) are obvious from the pieces such as:

  • one quarter + one one quarter + one quarter + one quarter = 1
  • 1 - three fourths = one fourth
  • 4 x one quarter = 1
  • three quarters (how many) one quarter(s) = 3
and it is better to encourage writing what is said (one quarter), 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 with the largest square as the whole. The pieces show that A can be partitioned into halves, fourths,eighths and sixteenths and there is a world of fractions equations to discover and record within the blocks. Students will soon be making up equations which imagine more pieces than are available.

Examples
Check these with the blocks:

  • two eighths = one quarter
  • three sixteenths + one eighth = five sixteenths
  • one eighth one quarter = one half
This 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.

A Little Extra

  1. It is possible to use four pieces to make a small symmetric, non-regular hexagon. Make it.
    1. Suppose this is the whole, work out the fraction value of each of the other pieces.
    2. Four small triangles have the same area as one large triangle. Write an equation to show this using the fraction values of this hexagon.
  2. It is possible to use all the pieces to make a large, symmetric, non-regular hexagon. Make it.
    1. Suppose this is the whole, work out the fraction value of each of the other pieces.
    2. Four small triangles have the same area as one large triangle. Write an equation to show this using the fraction values of this hexagon.

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 2 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 2 task is an integral part of:

  • MWA Number & Computation Years 5 & 6
  • MWA Number & Computation Years 7 & 8

Green Line
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