Level 4: Partitioning
Materials:
 24 Counters
 A ball of plasticine (enough to make a ‘pizza’ about 6 cm in diameter and 0.5 cm thick)
 An icypole stick or plastic knife (not a ruler as this might suggest measuring which could mask partitioning behaviour)
 4 small kindergarten squares of the same colour
 A copy of the Fraction Making Worksheet (pdf  24.83kb) cut in half along the dotted line
 Rectangular ‘tile’ approximately 3 cm by 5 cm made of thick card or wood (this needs to be thick enough not to be folded easily)
 Paper, pencil and eraser
Instructions:
Place the 24 counters in front of the student and say, “Could you give half of those to me please?” … Note student’s strategy. Regroup counters to a single group and say, “Now, could you give me 1 third please?” Note student’s strategy. If done fairly easily, regroup counters and ask, “What if you had to give me 5 eighths, how would you do that?”
Place the ball of plasticine in front of the student and say, “Can you make that look like a pizza please?” Once this as been done, hand the student an icypole stick and ask, “Can you show me where you would cut that to make 8 equal pieces?” Note student’s cutting strategy.
If the ‘pizza’ has been ‘cut’ reasonably accurately into 8 equal pieces, say , “If 3 quarters of this pizza was eaten, can you show me how much was eaten?” Note student’s response.
If the student successfully completed the ‘pizza’ task, roll plasticine back into a ball and place in front of student, saying, “Could you use this to show 5 thirds please?” Note student’s response, in particular, what is regarded as the unit.
If student appears hesitant, remove plasticine, place kindergarten squares in front of student, and ask , “Could you use this paper to make 2 and 5 sixths please?” Note student’s response.
If either one of these tasks is completed reasonably well, place the top half of the worksheet in front of the student and say, “Can you divide this rectangle into 3 equal parts please? … Can you tell me why you did it that way?” Note student’s strategy, in particular, whether or not halving is used to estimate. If parts reasonably equal, say, “Do you know what these parts are called? … Why?” Note student’s response.
Place the bottom half of the worksheet in front of the student and say, “Can you divide this line into 5 equal parts please? … Can you tell me why you did it that way?” Note student’s strategy (in particular, if they appeared to count from left to right or estimated using a halving strategy). If parts reasonably equal, ask, “Do you know what these parts are called? … Why?” Note student’s response.
If rectangle and line partitioned more or less accurately, place paper, pencil and rectangular ‘tile’ in front of the student and say, “If this tile is 2 thirds of a whole, could you draw the whole please?” Note student’s strategy, in particular, whether or not halving is used to estimate.
Draw a line (approximately 12 cm long) on the student’s paper, label one end 0 and the other end 6 fifths (written symbolically with a horizontal bar) and say, “Please mark the line to show where 1 would be … Can you tell me why you did it that way?” Note student’s strategy (in particular, if they appeared to count from left to right or estimated using a halving strategy).
4.3 Advice Rubric
Although most Year 4 students are able to recognise and name simple fractional parts in relation to physical models (chocolate, pizzas, sandwiches etc), the introduction of more formal models (eg, fraction diagrams and number lines) appears to be associated with a marked decline in student performance in this area. One of the possible reasons for this is that students do not have the strategies to generate their own, more formal fraction models and representations. Also, in the past, most of the practical work with fractions has focussed on proper fractions at the expense of mixed or improper fractions.
This task examines the extent to which students are able to generate their own fraction representations. It assumes some prior experience with practical models and a knowledge of fraction names.
Observed response 
Interpretation/Suggested teaching response 
Can identify 1 half and possibly 1 third of the counters but not 5 eighths, may partition ‘pizza’ and show 3 quarters but unable to complete much more beyond this 
May not be familiar with nonunitary fractions or fractions greater than 1 or have access to partitioning strategies beyond halving
 Provide plenty of opportunities to develop nonunitary fraction language through ‘realworld’, 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)
 Consolidate the halving partitioning strategy using paper folding (to represent region diagrams) and a rope and pegs (to represent line segments, eg, see below) and use this to support students’ capacity to generate their own fraction models and diagrams (see Partitioning (pdf  199.51kb) paper).
 Consider introducing the thirding strategy (ie, the thinking that 1 third is less than 1 half, estimate 1 third, leaving room for two more parts of the same size …)

Identifies 1 half and1 third of counters but may not recognise 5 eighths, partitions pizza and shows 3 quarters, able to make at least one mixed fraction model but experiences difficulty partitioning rectangle or number line on worksheet, may identify the whole for the tile task 
Suggests some understanding of partitioning in relation to familiar objects and halving
 Consolidate ‘cutting’ and sharing strategies using ‘realworld’ situations (eg, cutting up sandwiches), make and name mixed fractions by identifying the unit
 Introduce/consolidate the thirding and fifthing partitioning strategies (see Partitioning (pdf  199.51kb) paper) via realworld models (see above) and use these to support students capacity to generate their own fraction diagrams and number lines

Able to complete most tasks reasonably well but may use makeall strategy to determine 5 eighths and additive strategies to partition region diagram and line 
Suggests more developed understanding of partitioning but may be thinking additively instead of multiplicatively
 Review partitioning strategies and the thinking associated with these (eg, 1 fifth is less than 1 quarter, estimate 1 quarter, make 1 fifth slightly smaller than this, then halve remaining part and halve again …)
 Explore what happens when partitioning strategies are combined (eg, halving then thirding produces sixths, halving and then fifthing produces tenths etc), link region diagram representations to region idea for multiplication and number line representations to ‘for each’ idea for multiplication
 Practice making and naming an extended range of fraction representations (eg, make sevenths by thinking 1 seventh is smaller than 1 sixth, I can estimate sixths by halving and thirding …)
 Consider introducing/consolidating decimal fraction representations to hundredths by successively ‘tenthing’ region diagrams or number lines. Use to compare, order, sequence and rename decimal fractions

Completes all tasks confidently using known facts for counters tasks and multiplicative partitioning strategies based on halving (ie, does not count from left to right, but divides area or line into equal parts). 
Suggests multiplicative basis for partitioning and a sound knowledge of how to construct fraction representations
 Continue to explore what happens when partitioning strategies are combined (see above), link region diagram representations to ‘region’ idea for multiplication and number line representations to ‘for each’ idea for multiplication
 Consolidate and extend language and skills associated with partitioning, eg, Fraction Estimation from Maths300 (Curriculum Corporation, 2001)
 Consider introducing/consolidating decimal fraction representations to thousandths by successively ‘tenthing’ region diagrams or number lines. Use to compare, order, sequence and rename decimal fractions
