Big Ideas Linked to the Interview Tasks
Outlined below is a list of knowledge, skills and behaviours which students who have connected understandings of fraction ideas may possess. These capabilities have been divided these into overall ideas and those that relate to particular constructs of rational number.
As fractions is a major focus with the Interview, there is some occasions where only fractions is referred, while in other cases, the broader term rational number is used (to include fractions, decimals, percentages, etc.).
Note: The numbers in the parentheses refers to the corresponding questions in the Fractions and Decimals Online Interview.
OVERALL IDEAS
- Articulates rational number thinking using appropriate language [1-20]
- Forms and manipulates a variety of physical and mental models (areas and regions, sets, number lines, ratio tables, etc), in continuous and discrete situations [1-20]
- Understands the subconstructs of rational number (part/whole [1-4, 6], division [6], measure [8-11], ratio, and operator [4]) as well as their interrelationships
- Understands that rational numbers are largely about relationships [1-3, 6, 7, 9, 12-15, 20]
- Thinking multiplicatively rather than additively when appropriate (relative Vs absolute thinking) [17]
PART-WHOLE
- Understands that fractions are equal shares that are not necessarily congruent and that the subdivision of the whole must be exhaustive [1]
- Recognises that a given fraction (continuous/discrete) of a may not be the same size as that fraction of b [19]
- Moves from the whole to a given part, from the part to the whole and from the part to the part flexibly [2, 3, 6, 7]
- Understands that if a is a certain fraction of b, we can determine what fraction b is of a, through the reciprocal relationship
CONNECTING CONCEPTS WITH SYMBOLS/ EQUIVALENCES
- Understands the meaning attached to each part of a fraction (e.g., the denominator shows what ‘denomination’ is being counted, the numerator ‘enumerates’ how many of these parts) [9, 12, 14, 18]
- Understands that fractions (including whole numbers, mixed numbers and improper fractions) are entities that can be counted (e.g. 4/5 represents four things called “fifths”) and can recognise and use counting patterns and equivalences [3, 5, 9, 12, 14]
- Uses appropriate symbols to represent rational numbers (e.g. fractions, decimals and percentages) and can flexibly move between these as appropriate
FRACTIONS AS A NUMBER
- Understands and operates with the ‘density’ of rational numbers (meaning that between any two rational numbers there is an infinite number of rational numbers), relating them appropriately to whole numbers [11]
- Can identify a rational number on a number line with consideration to the calibrations and the intervals specified [10]
FRACTIONS AS DIVISION
- Recognises a/b as a divided by b [6, 16]
- Can solve whole number division problems understanding the significance of the size of the quotient (e.g. 4 ÷ 5 will result in an answer less than one) and/or treating remainders appropriately [6, 17]
- Has appropriate strategies in sharing-type problems [6]
RELATIVE SIZE/BENCHMARKING
- Readily compares and orders rational numbers, using efficient and understood strategies [5, 9-11, 13, 15]
- Relates a given rational number to key benchmarks (e.g. 0, ½, 1), using place value as appropriate [5, 8-11, 13, 15]
- Understands the inverse relationship between the denominator and the size of the parts [7, 9]
OPERATORS AND OPERATIONS
- Combines and partitions rational numbers using appropriate physical or mental tools, renaming as appropriate [4, 6, 10-12, 14, 16, 18]
- Estimates the answer appropriately in a rational number calculation [8, 16, 17, 19, 20]
- Can nominate a problem situation to which a particular rational number operation might apply, and conversely can represent a relevant rational number operation given the problem situation [17, 19, 20]