Fraction Pairs: Definitions of Strategies

This resource 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
• Residual Thinking
• Residual Thinking with equivalence
• Residual Thinking with some other proof
• Gap Thinking
• “Higher” or “larger” numbers
• Other (satisfactory explanation with a correct solution)
• Other (unsatisfactory explanation with either a correct or incorrect explanation)

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?”