attribute A5, the examinee must be able to map the graph of a familiar function with its corresponding function. In an item that requires attribute A5 (e.g., item 4), attribute A4 is typically required because the examinee must find random points in the graph and substitute the points into the equation of the function to find a match between the graph and the function. Attribute A6, on the other hand, deals with the abstract properties of functions, such as recognizing the graphical representation of the relationship between independent and dependent variables. The graphs for less familiar functions, such as a function of higher-power polynomials, may be involved. Therefore, attribute A6 is considered to be more difficult than attribute A5 and placed below attribute A5 in the sub-hierarchy. The second branch in the sub-hierarchy considers the skills associated with advanced substitution. Attribute A7 requires the examinee to substitute numbers into algebraic expressions. The complexity of attribute A7 relative to attribute A4 lies in the concurrent management of multiple pairs of numbers and multiple equations. Attribute A8 also represents the skills of advanced substitution. However, what makes attribute A8 more difficult than attribute A7 is that algebraic expressions, rather than numbers, need to be substituted into another algebraic expression. The last branch in the sub-hierarchy contains only one additional attribute, A9, related to skills associated with rule understanding and application. It is the rule, rather than the numeric value or the algebraic expression that needs to be substituted in the item to reach a solution. Cognitive model representation The Ratio and Algebra attribute hierarchy can also be expressed in matrix form. To begin, the direct relationship among the attributes is specified by a binary adjacency matrix (A) of order (k,k), where k is the number of attributes, such that each element in the A matrix represents the absence (i.e., 0) or presence (i.e., 1) of a direct connection between two attributes. The A matrix for the Ratio and Algebra hierarchy presented is shown below. Each row and column the A matrix represents one attribute; the first row and column represents attribute A1 and the last row and column represents attribute A9. The presence of a 1 in a particular row denotes a direct connection between that attribute and the attribute corresponding to the column position. For example, attribute A1 is directly connected to attribute A2 because of the presence of a 1