# Common Misunderstandings - Level 4.7 Rational Number Sequencing Tool

## Level 4: Partitioning

### Instructions:

Place the Rational Number Cards and rope in front of the student. Peg the ‘0’ card at one end of the rope and the ‘2’ card at the other end and say, “If this end of the rope is 0 and this end is 2, can you peg the cards on to the rope to show where the numbers live? … Can you tell me why you did it that way?” Note student’s response, in particular, whether or not the fraction is interpreted as an operator or as a number (eg, is &frac34; interpreted as &frac34; of the rope of or as a number in its own right less than 1, similarly with 50%)

Using the Negative Number cards, place the string in front of the student. Peg the ‘-3’ card at one end of the string and the ‘3’ card at the other and say, “This end of the rope is negative (or minus) 3 and this end is 3.” Give the ‘0.5’ card to the student and say, “Can you peg that on the rope to show me where this lives? … Can you tell me why you did it that way?” Repeat with the following cards in order, -2.5, 2/3 and -1&frac34;. Note student’s response.

This task should only be used where students have demonstrated some facility with fractions and decimals and have some appreciation of where negative numbers live. Student responses to this task indicate the extent to which they understand rational number recording and can compare, order/sequence, and rename common fractions and decimals. As such it is quite a powerful summative assessment tool.

Observed responseInterpretation/Suggested teaching response
Experiences difficulty locating numbers, may attempt to locate recognised numbers by counting from the left or guessing Suggests rational number not well understood
• Check the extent to which students can make, name and record common fractions and decimals (see Tools 4.1 to 4.5)
• Consolidate a sense of negative numbers and where they ‘live’ using real-world models (eg, temperature scales)
• Model and practice sequencing common fractions and decimals using a rope and pegs, play appropriate games, eg, adapt Place-Value Game (pdf - 16.36kb) to sequence hundredths
Locates some cards (eg, 0.2 and 1.5) reasonably accurately, but may use fraction as operator idea to locate &frac34; and 50% (eg, &frac34; of the rope), and/or experience some difficulty with negative number cards Suggests some confusion between fraction as operator and fraction as number
• Explore fraction renaming strategies (see Partitioning (pdf - 199.51kb) paper)
• Introduce/review link between fractions and division, in particular, the notion that m/n means m divided by n (n>≠0) , use to rename fractions as decimals and per cents
• Provide opportunities to practice the use partitioning strategies to locate fractions, decimals, and negative numbers
Able to locate most cards reasonably accurately Suggests sound understanding of rational numbers and access to appropriate partitioning and renaming strategies
• Extend sequencing activity to include more complex rational numbers
• Consider introducing/consolidating operations on common fractions and decimals (eg, addition, subtraction and multiplication)
• Consider introducing irrational numbers