Mathematics Online Interview and Fractions and Decimals Online Interview

• Articulates rational number thinking using appropriate language [Questions 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 [Questions 1-20]
• Understands the subconstructs of rational number (part/whole [Questions 1-4, 6], division [Question 6], measure [Questions 8-11], ratio, and operator [Question 4]) as well as their interrelationships
• Understands that rational numbers are largely about relationships [Questions 1-3, 6, 7, 9, 12-15, 20]
• Thinking multiplicatively rather than additively when appropriate (relative Vs absolute thinking) [Question 17]
  

• Understands that fractions are equal shares that are not necessarily congruent and that the subdivision of the whole must be exhaustive [Question 1]
• Recognises that a given fraction (continuous/discrete) of a may not be the same size as that fraction of b [Question 19]
• Moves from the whole to a given part, from the part to the whole and from the part to the part flexibly [Questions 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

• 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) [Questions 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 [Questions 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
  

• 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 [Question 11]
• Can identify a rational number on a number line with consideration to the calibrations and the intervals specified [Question 10]

• Recognises a/b as a divided by b [Questions 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 [Questions 6, 17]
• Has appropriate strategies in sharing-type problems [Question6]

• Readily compares and orders rational numbers, using efficient and understood strategies [Questions 5, 9-11, 13, 15]
• Relates a given rational number to key benchmarks (e.g. 0, ½, 1), using place value as appropriate [Questions 5, 8-11, 13, 15]
• Understands the inverse relationship between the denominator and the size of the parts [Questions 7, 9]

• Combines and partitions rational numbers using appropriate physical or mental tools, renaming as appropriate [Questions 4, 6, 10-12, 14, 16, 18]
• Estimates the answer appropriately in a rational number calculation [Questions 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 [Questions 17, 19, 20]
  

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.                                               

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.                                                   

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.                                             

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)”.                                                  

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.                                                   

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.                                                  

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.                                                  

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?”