Common Misunderstandings - Level 5.5 Interpreting Rational Number Tool

Level 5: Proportional Reasoning



Introduce task by saying, “I am going to show you some numbers, one at a time, which I want you to put into a sentence.” Place the 2/5 Card in front of the student and say, “Can you put this in a sentence please? … Note student’s response. If school context is used (eg, “The teacher asked the children to shade 2/5 of the diagram”), say, “Can you put that number in a sentence that has something to do with what you might find in a supermarket?” Note student’s response.

Place the 0.37 Card in front of the student and say, “Can you put this in a sentence please? … If not already suggested, ask, “Can you think of a sentence with this fraction in it as a percentage?” Note student’s response.

Present the remaining cards, one at a time in the following order: , 4.675, 2:3, 250%. In each case, ask the student to put the number in a sentence. Note student’s response and explore student’s thinking as needed.

5.5 Advice rubric

While most middle years’ students are able to work with simple proper fractions, their relatively limited experience with mixed fractions, percentages larger than 1, and decimal fractions involving thousandths, can dramatically impact their willingness to attempt problems involving rational number more generally.

This task examines the extent to which students are able to generate meaningful contexts for different rational number representations. While not a guarantee of success, if students are able to recognise where such numbers might occur, they are more likely to be able to recognise and record proportional relationships in a form that supports efficient comparison and/or computation.

Observed response Interpretation/Suggested teaching response
Little/no response to most rational number representations, may be able to contextualise ⅖ and/or 3¾ Suggests a very limited understanding of how fractions are made/represented, named and recorded
  • Introduce/consolidate halving, thirding and fifthing partitioning strategies (see Partitioning (PDF - 215Kb) (pdf - 199.51kb) paper) using a variety of materials to make/represent, name and record a wide variety of fractions including mixed numbers, decimals and percents
  • Review key fraction generalisations, in particular, the generalisation to support fraction renaming based on fraction diagrams (see Partitioning paper referred to above)
  • Consolidate rational numbers in all their forms by comparing, ordering, sequencing, and renaming
Able to embed at least three of the rational numbers in an appropriate sentence, may experience difficulty with ratio and 4.675 Indicates some understanding of rational numbers and how they are represented
  • Directly address the meaning of the symbols and forms of representations used, consider developing a maths dictionary, display different representations on posters and annotate to elaborate different meanings
  • Demonstrate the value of rational numbers in measurement (eg, rates), chance and data topics (eg, probability measures, and space topics (eg, scale factors in transformations and map reading)
  • Review key generalisations and partitioning strategies, use to make, name and record an extended range of rational numbers
Able to embed all numbers in a sentence in a meaningful way Suggests a solid understanding of rational numbers and how they are represented
  • Introduce/review more generalised techniques for comparing and renaming fractions, ensure language in place (eg, factors, divisors, multiples, …)
  • Establish meaningful procedures for adding, subtracting, multiplying and dividing rational numbers
  • Consider introducing proportion more formally, provide opportunities for students to work with an expanded range of problems that require proportional reasoning and involve quantities expressed in different forms