The Learning Task: finding maximum or minimum values for quadratic relations from real situations. The student will be able to recognize and translate a real situation involving maximum or minimum values (quadratic relationships) into a mathematical equation, then using previous skills to find the maximum or minimum value (at the vertex of the graph of that equation). Learned Capability Type: intellectual skill Associated Performance: given various scenarios stated verbally or in writing, and involving examples including maximizing floor area of a house and finding the maximum height reached by a projectile, translate the information into a quadratic function, find the vertex of that function, then state the maximum or minimum value. The performance will be in written form.

Type of Learning: cognitive strategy with both internal and external conditions (recalling rules, demonstrating solutions); ”verbal” information with both internal and external conditions (recall of larger meaningful context and presenting new information in a larger context); attitude with internal conditions (recall of intellectual skills) Type of Learning Outcome: intellectual skill. Essential prerequisites are the simpler component intellectual skills (translating into algebraic expressions and equations, substitution of expressions, completing the square on a quadratic expression, and so on). Supporting prerequisites include attitudes (this is important), verbal information (statement of the problem), and cognitive strategies (how do I know what type of problem this is).

Instructional Event: ask students how fast and how high they think they can throw a small rock. Use equation of projectile motion from physics (already known) to describe this mathematically, use previous skills to find the vertex, then test the maximum value against the students’ predicted height. Make adjustments to the equation as necessary. Next, ask students how large a rectangle can be formed with 40 metres of fencing to optimize grazing area for sheep. Combine equations for perimeter and area to develop quadratic function, then find the vertex. Students are usually surprised that a square encloses the maximum rectangular area. Students are then given two or more problems stated in writing to solve, their performance is monitored and guided as they work through their solutions, and the solutions are checked as a group activity at the end. Formal (summative) evaluation occurs on assignments and tests. Explanation: in the context of a pre-calculus course, these examples capture the students’ attention quickly (the “try this at home” exhortation usually gets a laugh), model the idea that real situations can be described mathematically with great accuracy, and integrate several well-developed skills in a new and meaningful context. Most of Gagne and Briggs’ 9 Instructional Events are contained within this lesson. Shared with Laurel Marsh, colleague