Use this tool to assess the mathematical understandings and strategies of fractions, decimals, ratio and percentage.
Access the Fractions and Decimals Online Interview
Via the Insight Assessment Platform
It's intended for students in years 5 to 8 but is also valuable for assessing high achieving students in year 4, or at-risk students in year 10.
The interview provides collated data that can be compared within schools to understand student achievement and monitor progress.
User guides
A full user guide is being developed and will be available here when it's complete.
Mapping the Interview to the Victorian Curriculum
This document links each task from the interview to the overarching big idea to which it relates.
The task and the big idea are then related to the content description from the Victorian Curriculum F-10: Mathematics which provides the best match. Key aspects of the proficiencies to the interview task and related overarching big idea are also included.
Mapping the fractions and decimals online interview to the Victorian Curriculum F-10 (docx - 887.58kb)
Big ideas linked to the interview
This 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.).
The numbers in the parentheses refers to the corresponding questions in the 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 ofb [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 a 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]
Fraction pairs: Definitions of strategies
This information will assist teachers in developing their understanding in relation to the types of strategies that students use to demonstrate their fraction understanding, for example residual thinking.
Benchmarking
Correct benchmarking is evidence that a student understands the relative size of fractions. It is also useful for comparing decimals.
When benchmarking, a student will compare a fraction to another well known fraction, usually a half, or to a whole number such as zero or one.
For example, when comparing
^{5}/_{8} and
^{3}/_{7};
^{5}/_{8} is greater than a half, and
^{3}/_{7} is less than a half, therefore
^{5}/_{8} is bigger.
Residual thinking
The term residual refers to the amount which is required to build up to the whole. For example
^{5}/_{6} has a residual of
^{1}/_{6}.
This thinking is useful for comparing the size of fractions such as
^{5}/_{6} and
^{7}/_{8}.
^{5}/_{6} has a residual of
^{1}/_{6} and
^{7}/_{8} has a residual of
^{1}/_{8}. Therefore
^{7}/_{8} is a larger fraction because it has the smaller residual – the smaller amount to make the whole.
Sometimes, however, residual thinking alone is not an efficient strategy. When comparing
^{3}/_{7} &
^{5}/_{8}, measuring up the residuals of
^{4}/_{7} &
^{3}/_{8} is not a helpful strategy as you are left with two residuals that are no easier to compare than the original pair. In this case, the residuals then need to be benchmarked to
^{1}/_{2} and 1 to prove which is larger. If students use residual thinking alone with this pair, it should be classified as an unsatisfactory explanation.
Residual thinking with equivalence
In order to use residual thinking effectively, creating an equivalent residual sometimes makes the justification clearer. For example, when comparing
^{3}/_{4} and
^{7}/_{9} a student may state that
^{3}/_{4} has a residual of
^{1}/_{4} or
^{2}/_{8}. Therefore the residual for
^{7}/_{9} (^{2}/_{9}) is smaller than the residual for
^{3}/_{4} (^{2}/_{8}). The fraction with the smaller residual is the larger fraction.
Residual thinking with some other proof
Sometimes residual thinking alone is not the most appropriate strategy. For example if a student uses residual thinking alone to compare
^{3}/_{4} and
^{7}/_{9}, they must then convince the interviewer that they can justify which of the residuals is bigger (^{1}/_{4} or
^{2}/_{9}).
An example of residual with proof might be, “I know one quarter of nine is more than 2 because 2 is a quarter of eight, so
^{2}/_{9} must be less than
^{1}/_{4} therefore
^{7}/_{9} is the bigger fraction”.
Please note: an explanation of residual thinking
without proof, should be recorded as “other (unsatisfactory explanation for either a correct or incorrect solution)”.
Gap thinking
This strategy is a form of whole number thinking, where the student compares the whole number difference between the numerator and denominator.
For example,
^{5}/_{6} and
^{7}/_{8} both have a difference of “one” between the numerator and denominator. A student using “gap thinking” might claim therefore that these fractions are the same size. When comparing
^{3}/_{4} and
^{7}/_{9}, a student using gap thinking would choose
^{3}/_{4} as larger because it has a smaller “gap”, thereby choosing incorrectly.
There are some instances where “gap thinking” will lead students to a correct choice. For example, comparing
^{3}/_{8} &
^{7}/_{8}. This is an inappropriate strategy for comparing the size of fractions.
“Higher” or “larger” numbers
With this strategy, fractions are deemed to be bigger if they contain larger digits. For example, when comparing
^{4}/_{7} and
^{4}/_{5} students may incorrectly claim that
^{4}/_{7} is larger because it has a “larger number”. Also in comparing
^{2}/_{4} and
^{4}/_{8}, a student would choose
^{4}/_{8} as it has “higher numbers”.
Sometimes students will directly compare the numerators or denominators and conclude a larger digit at the top or bottom of a fraction means that it is a larger fraction.
This is an inappropriate strategy for comparing the size of fractions.
Other (satisfactory explanation with a correct solution)
There are very few correct solutions with appropriate strategies that do not already fall into the provided categories, but it is possible for this to occur. For example: a student may be able to mentally convert a fraction to a decimal and then compare or use some other mathematically correct strategy.
This option is only for correct solution
and appropriate explanation.
Other (unsatisfactory explanation with either a correct or incorrect explanation)
There are many explanations (too numerous to mention) that may fall into this category. It is a “catch-all” for any strategy that cannot be placed in the other categories. Typically it will include any explanation that is mathematically incorrect, partially correct or vague.
Sometimes students relate fractions to an image of an area model. Their justification might be “^{7}/_{9} is larger than
^{3}/_{4} because if I imagined a picture of them,
^{7}/_{9} would look more”. This reasoning is not evidence of understanding the size of the fractions.
In a situation where the student provides a partially correct or vague explanation, it is appropriate to ask for further information in a non-leading way. For example, “can you tell me more about how you know? So you think it seems larger, but how can you be sure?”