Level 4: Partitioning
Group the pieces of the ‘Birthday Cake’ into a ‘cake’ in front of the student, remove 1 piece and say, “If this piece of cake was eaten, what fraction of the cake would be left?” Note student’s response. If correct, ask, “How would you write that as a fraction?” If written symbolically, circle the numerator and as you are doing this say, “What has this got to do with what you have got there? … Note student’s response then circle the denominator and repeat the question, noting the student’s response.
Re-assemble the ‘cake’ (if necessary) and say, “If 2 thirds of the cake was eaten, can you show me how much was eaten?” Note student’s response. If correct, ask, “How would you write that as a fraction?” … If written symbolically, circle the numerator and say, “What has this got to do with what you have there?” … Note response, then circle the denominator and repeat the question, noting the student’s response. If student appears hesitant, record 2/3 as a fraction (vertically aligned with horizontal bar) and say, “Some people might record it like this, would they be right? … Why? If student agrees, circle each of the numbers in turn and repeat the earlier questions, otherwise proceed to the next question.
Place the Fraction Recording Cards in front of student, point to one of the squares and say, “This is 1 whole… Can you write what is shown here as a fraction please?” Note student’s response. If written symbolically, circle each of the numbers recorded in turn and as you do this ask, “What has this got to do with what you have got there? … Note student’s response to each number circled.
Place the 15 square tiles in front of the student and say, “Please write down what fraction of the tiles are blue.” If 6/15 recorded, circle the 6 and say, “What has this got to do with what you have there?” … Note response, then circle the 15 and repeat the question, noting the student’s response. Ask, “Is there another way we could write that as a fraction?” …. If 2/5 recorded, circle each of the numbers in turn and repeat the earlier questions. If student appears hesitant, record 2/5 as a fraction (vertically aligned with horizontal bar) and say, “Some people might record the fraction of blue tiles like this, would they be right? … Why? If student agrees, circle each of the numbers in turn and repeat the earlier questions.
4.4 Advice Rubric
This task explores students’ capacity to recognise and record common fractions and, in particular, their meanings for the numerator and denominator. This is important as some students in the middle years tend to treat these two numbers as discrete whole numbers (eg, ‘5 over 6’) that you ‘do things to’, rather than seeing these as mutually dependent and indicative of a particular proportion or as a number in it’s own right (the result of the dividing the numerator by the denominator).
||Interpretation/Suggested teaching response|
Little/no response or incorrect (eg, says 5 halves or simply records 5 to indicate the number of pieces)
May not understand task, suggests little/no appreciation of how fractions are made and named
- Use paper-folding to make and name parts in the halving family (eg, halves, quarters, eighths, sixteenths, thirty-seconds, sixty-fourths etc)
- Focus on key generalisations regarding equal parts, number of parts and size of parts as number of parts increases (see Tools 4.1 and Tools 4.2),
- Use models and examples to distinguish between ‘how many’ (numerator idea) and ‘how much’ (denominator idea)
- Make a table that records the number of parts and the name of the parts.
- Consider introducing the thirding strategy (see Partitioning (pdf - 199.51kb) paper) using paper-folding, lengths of rope etc to make, name and count fractional parts to 1 and beyond
Recognises 5 sixths but unsure about how to record or, if written, refers to 5 as 5 pieces and 6 as 6 pieces (ie, little/no indication of relationship), may not recognise 2 thirds or be able to record mixed fraction, may recognise 6/15 in terms of a count of tiles
Suggests some understanding of how fractions are represented but a limited understanding of how they are recorded
- Consolidate ‘cutting’ and sharing strategies using ‘real-world’ material and situations (eg, cutting up sandwiches), make and name mixed fractions by identifying the unit
- Introduce/consolidate the halving and thirding partitioning strategies (see Partitioning (pdf - 199.51kb) paper) via real-world models (see above) and use these to support students capacity to generate their own fraction diagrams and number lines
- Explore what happens when halving and thirding are combined, make, name and count fractional parts
- Provide plenty of opportunities to develop non-unitary fraction language through ‘real-world’, informal fraction activities, labelling recognised parts and relating them to the whole (eg, 3 quarters of the orange, 2 thirds of the netball court),
- Extend work with fraction models to mixed fractions (eg, 5 thirds, 2 and 3 quarters), count forwards and backwards in fractional parts starting anywhere
Explains symbol for 2 thirds (eg, identifies 1 third as 2 sixths, shows 2), but may not be able to do this for 2 fifths, may recognise but not be able to record 2 and 3 eighths, records and justifies 6 fifteenths
Suggests some understanding of how fractions are recorded but may not have access to strategies to support fraction renaming
- Use partitioning strategies, region diagrams and lines to explore and justify fraction renaming (see Partitioning (pdf - 199.51kb) paper)
- Introduce, model and discuss the generalisation: if the total number of parts is increased/decreased by a certain factor then the number of parts required is increased/decreased by the same factor
- Practice making, naming and recording a wide range of fraction representations
Confidently completes most tasks, implicitly distinguishes between ‘how many’ and ‘how much’, uses fraction names routinely
Suggests a reasonably thorough understanding of fraction recording and renaming
- Consolidate and develop using a wide range of models and representations including mixed and improper fractions
- Consider introducing written procedures for adding and subtracting, like and related fractions
- Consider introducing decimal notation, tenths as a new place-value part