Practice Conceptual Dependency in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
The relationship between concepts where understanding one requires prior understanding of another — the prerequisite structure of mathematical knowledge.
You cannot truly understand limits without understanding functions; you cannot understand derivatives without limits. Concepts form a dependency graph.
Showing a random 20 of 50 problems.
Example 1
easyTo compute , a student must understand common denominators. Is fraction-arithmetic a prerequisite for ? Give for yes.
Example 2
mediumNegative numbers are a prerequisite for subtraction yielding . What is , requiring that prerequisite?
Example 3
mediumTo understand 'eigenvalues of a matrix,' list the prerequisite concepts in dependency order.
Example 4
mediumThe quadratic formula depends on completing the square, which depends on squaring binomials. Expand .
Example 5
challengeExplain why a dependency graph with a cycle of length () admits zero valid learning orders.
Example 6
mediumConcept D depends on B and C; B and C both depend on A. Counting D, B, C, A, how many concepts total must be learned to reach D?
Example 7
easyCounting is a prerequisite for addition. True dependency direction: counting comes before addition. Give if that direction is correct.
Example 8
challengeA cycle A -> B -> A in a dependency graph is invalid because it has no starting point. How many valid learning orders exist for a true cycle of concepts? Give the count.
Example 9
mediumThe chain rule for derivatives depends on understanding composition of functions. Given , find .
Example 10
easyIf a DAG has concepts all needed before reaching the final concept , including how many concepts total are in the learning path?
Example 11
easyTo understand 'the derivative of is ', list the concepts you must already understand, and arrange them in dependency order.
Example 12
mediumIn a DAG of concepts where concept depends on all others (a star centered at ), how many distinct learning orders end at ?
Example 13
mediumLimits depend on functions; series depend on limits; both feed into convergence. If a student masters functions and limits, how many of {functions, limits} remain before tackling series? Give the count remaining.
Example 14
hardA student passes a calculus exam but cannot do basic factoring. This suggests their calculus knowledge is unstable. Which concept type is the gap: a prerequisite or a successor?
Example 15
mediumArrange these concepts in dependency order, explaining each link: set, element, subset, power set.
Example 16
easyTrigonometry depends on the Pythagorean theorem. Which concept must be learned first?
Example 17
mediumUnderstanding percentages depends on decimals. Convert to a percent.
Example 18
easyAddition is a prerequisite for multiplication (repeated addition). If you cannot add, can you reliably multiply? Give for yes.
Example 19
easyIn a chain , if a student masters , by inference how many earlier concepts are presumed mastered?
Example 20
mediumSolving depends on knowing division. Apply that prerequisite to give .