Level 4: Partitioning
Materials:
- A square piece of paper (approximately 20 cm × 20 cm) folded into 6 equal parts
- Large sheet of Butcher’s paper
- 24 coloured counters
Instructions:
Place folded paper in front of the student and say, ‘Can you unfold that please and tell me what you notice?’ Note student’s response. If student does not comment on the number of parts or equality of parts, say, ‘Do you know what we call these equal parts?’ If no response or a non-fraction name given, say, ‘Another student I talked to said these were eighths, do you agree? ... Why/why not?’
Place the sheet of Butcher’s paper in front of the student and ask him/her to fold it in half and in half again then say:
Place folded paper in front of the student and say, ‘Can you unfold that please and tell me what you notice?’ Note student’s response. If student does not comment on the number of parts or equality of parts, say, ‘Do you know what we call these equal parts?’ If no response or a non-fraction name given, say, ‘Another student I talked to said these were eighths, do you agree? ... Why/why not?’
Without unfolding the sheet of paper, how many parts has the sheet of paper been folded into? ... Do you know what these parts are called?’ Note student’s response. Ask
Place folded paper in front of the student and say, ‘Can you unfold that please and tell me what you notice?’ Note student’s response. If student does not comment on the number of parts or equality of parts, say, ‘Do you know what we call these equal parts?’ If no response or a non-fraction name given, say, ‘Another student I talked to said these were eighths, do you agree? ... Why/why not?’
If you folded it in half again, can you predict how many parts there would be? ... What would they be called? ... How do you know that?’ Note student’s prediction then ask him/her to fold the paper in half again. If student predicted 6 parts or sixths, unfold, check then refold. Ask the student to continue halving. Stop student after he/she has folded three more times (64 parts) and ask:
‘Without unfolding the sheet of paper, about how many parts do you think there might be now? ... If there were that many parts, what would they be called?’ Note response and invite the student to unfold the paper. When completed, ask:
‘How many parts did you make?’ Note student’s strategy, in particular, whether they count by ones or use the number of rows/columns, then ask:
‘What would these parts be called?’ If correct (sixty-fourths), ask
‘Which is bigger, three of these parts (indicating the sixty-fourths)
or 3 eighths? ... Why?’ Note response.
Place coloured counters in front of student and say, ‘There are 24 counters here’ ... take 4 and say, ‘This is my share’ ... give student 6 and say, ‘This is your share, what fraction of the counters did I get? ... What fraction of the counters did you get? ... Do we have equal shares?’
4.2 Advice Rubric
Many students do not make the connection between the number of parts or shares and the name of the parts. This is not surprising given that the most commonly recognised fractions (halves and quarters) do not fit the pattern which uses ordinal number names to indicate parts (eg, three parts, thirds). The use of ordinal names can also be confusing as is indicated by the following example, where the ‘fifth’ are interpreted as being equivalent to ‘five’, presumably on the basis that identifying the fifth in line is associated with counting to five.
Students may also fail to recognise the important generalisation that the larger the number of parts, the smaller they are in relation to the whole. This misconception can arise as a consequence of an undue emphasis on discrete models (eg, 3 out of 20 smarties) where the size of the actual object (a single smarty in this case) does not change if the size of the collection is changed but it’s relationship to the whole does (eg, 3 out of 50 smarties). That is, discrete models can lead to a confusion between wholes and parts which may mask the relationship between them. This is not to say they should not be used, but that they should be used in conjunction with continuous models where this generalisation is more readily observed.
This task is designed to evaluate the extent to which students’ recognise how fractions are named and appreciate the impact of increasing the number of parts on the relative size of the part.
| Observed response |
Interpretation/Suggested teaching response |
|---|
Little/no response, may correctly respond to first task after prompt (eg, says not eighths as 6 parts or sixths), incorrect or irrelevant responses to some or all aspects of the paper folding and counters tasks, eg, uses remaining counters (14) as the denominator in naming shares |
May not understand task
- Use paper-folding to make and name parts in the halving family (eg, halves, quarters, eighths, sixteenths, thirty-seconds, sixty-fourths etc)
- Make a table that records the number of parts and the name of the parts, eg,
Number of Parts |
Name |
1 |
whole |
2 |
halves |
4 |
quarters |
8 |
eighths |
- Discuss use of ordinal names, inconsistencies and peculiarities (eg, thirty-seconds sounds like 30 seconds as in a measure of time) and the generalisation about the number of parts in relation to their size
|
Names parts as sixths for first task, may predict sixths and/or use an additive strategy to determine the number of parts in paper-folding task (64). May not recognise 3 eighths as bigger. Recognises counter shares as 4 twenty-fourths and 6 twenty-fourths respectively. |
Suggests a reasonable understanding of halving and the implications of repeatedly halving, may not see part of a collection as a countable unit (eg, group of 4 counters not seen as 1 four of 6 fours)
- Extend paper-folding to make and name parts in the halving, thirding and fifthing families (see above)
- Use discrete fraction models and sharing to ensure equal parts understood in this context (ie, 4 counters seen as 1 of 6 equal parts)
- Use partitioning strategies to construct fraction diagrams and line representations for a broader range of fractions (see Partitioning (pdf - 199.51kb) paper)
- Connect fraction diagrams to the region model of multiplication, eg, three parts (thirds) by 4 parts (quarters) give 12 parts (twelfths) and more efficient mental strategies based on doubling and/or known facts
|
Able to complete all tasks fairly confidently, determines number of parts in paper-folding task either by repeat halving or by using region representation (eg, 8 rows of 8). Recognises counter shares as 1 sixth and 1 quarter respectively. |
Suggests that key ideas are well understood (ie, number of parts/shares names the part and each part/share get smaller as total number of parts/shares increases)
- Use partitioning strategies to construct fraction diagrams and line representations for a broader range of fractions (see Partitioning (pdf - 199.51kb) paper)
- Draw attention to the key generalisation and the links to multiplication and division implicit in the process of paper folding and sharing.
- Consider introducing/consolidating decimal fractions to hundredths by successively ‘tenthing’. Use to compare, order, sequence and rename decimal fractions
|