# Common Misunderstandings - Level 4.5 Decimal Fraction Naming and Recording Tool

## Level 4: Partitioning

### Instructions:

Place the length of streamer and the measuring tape in front of the student and ask, ‘If I hold the streamer and the measuring tape at this end (make a point of lining up the end of the streamer and the zero on the tape measure), could you measure the length of the streamer as accurately as possible please? ... How would you write that?’ Note student’s response. If 145 cm recorded, ask, ‘Could you write that in metres please?’ If 1.45 m recorded, circle the 1 and say, ‘What has this got to do with what you have there? ... Can you show me on the tape measure?’ Note response, then circle the 4 and repeat the questions noting the student’s response. If reasonable response provided, circle the 5 and repeat the questions noting the student’s response. If student appears hesitant, say ‘Another student I talked to wrote this (record 1.45 on paper) what do you think that tells us about the length of the string? ... Why? ... Can you show me on the tape measure.’ Note student’s response.

Place the Decimal Card showing 6 tenths in front of the student and say, ‘Can you tell me what fraction of this rectangle is shaded? ... Could you write that down please?’ If written as a common fraction, ask ‘Is there another way we could write that?’ If decimal form not offered, record 0.6 on student’s paper and ask, ‘Would this be correct? ... Why do you think that?’ Note student’s response.

If answered correctly, repeat with the card showing 63 hundredths. In this case, if 63% not offered at some point, record on student’s paper and ask, ‘Would this be correct? ... Why do you think that?’ Note student’s response.

Place the Decimal Card showing 1 and 79 hundredths (1.79) in front of the student and say, ‘If this is 1 whole (point to fully shaded rectangle), can you tell me what number is shown by the shaded areas altogether? ... Could you write that down please?’ If written as a whole number and a common fraction, ask ‘Is there another way we could write that?’ If not offered, record 1.79 on student’s paper and ask, ‘Would this be correct? ... Why do you think that?’ Note student’s response. If correct, circle the 7 and ask, ‘What has this got to do with what is shown there?’ Indicate 1.79 fraction card. Note student’s response.

Place the Number Line Card in front of the student and say ‘At what point does the green line end?’ ... Note student’s response. If answered correctly, ask the same question for the purple line noting student’s response. If incorrect or very general (eg, ‘it’s five and a bit’), ask student to explain his/her thinking.

There is a considerable research evidence to suggest that decimal numeration is not well understood at this level. One of the reasons for this is that decimals are often introduced before students are ready using inappropriate models such as money and MAB base 10 materials. Before working with decimals students need a solid understanding of common fractions, in particular how parts are formed and named. Tenths can be introduced as a new place-value part once students have access to appropriate partitioning strategies (eg, halving and fifthing, see Partitioning (pdf - 199.51kb) paper) and are able to construct their own fraction diagrams and line representations.

This task examines the extent to which students can recognise and record decimal fractions. It assumes that students have a reasonable understanding of common fractions and have been introduced to decimal recording.

Observed response Interpretation/Suggested teaching response

Measures streamer in terms of centimetres, may not be able to record in metres or recognise what 4 means in 1.45, may recognise and record 6 tenths

Suggests a fairly limited understanding of what decimals represent and how they might be used and recorded

• Use a range of measurement activities to explore tenths in the real world, review relationship between different units (eg, 1 cm is 10 mm, 1 metre is 100 cm)
• Use partitioning strategies (see Partitioning (pdf - 199.51kb)paper) to make and name models of ones and tenths (eg, 3 ones and 7 tenths), record using common fractions and decimal notation
• Review tenths as a place-value part and the idea more generally that 1 tenth of these is one of those, consolidate by providing opportunities to compare, order, sequence, count forwards and backwards in place-value parts, and rename

Explains and justifies 4 in 1.45 metres, recognises and records tenths (for diagram and green line) but may experience some difficulty with hundredths and/or per cent

Suggests a reasonable understanding of how decimals are used to measure length but more general understanding of decimals may be limited to tenths

• Extend activities with measurement to explore hundredths and thousandths and establish relationship between different units (eg, 1000 grams is 1 kilogram)
• Use per cent ‘benchmarks’ to support hundredths (eg, 50 per cent means 50 out of every hundred, 50 hundredths)
• Use partitioning strategies (see Partitioning (pdf - 199.51kb) ) to make and name models of ones, tenths and hundredths (eg, 2 ones, 8 tenths and 7 hundredths), link to ‘for each’ idea (ie, each tenth is divided into 10 parts, so 10x10 parts altogether), record using common fractions and decimal notation
• Review hundredths as a place-value part and the idea more generally that 1 tenth of these is one of those, consolidate (see above)