INSTRUMENTAL UNDERSTANDING

P=Perimeter =2x + 2y = 30A=Area =xy + x(y-x) =2xy - x2

Isolate y in Perimeter Formula30 = 2x + 2y2y = 30 – 2xy = 15 – x

Eliminate y by substituting y from Perimeter equation into Area

equation.A = - x2 + 2xy = - x2 + 2x(15 – x) = -3 x2 + 30x = -3( x2 - 10x)

Complete the SquareA = -3( x2 - 10x + 25) + 75 = -3 (x−5)2 + 75

Interpret the Equation:A parabola that opens down with vertex at (5,75) so the maximum area is 75 m2 and this occurs when x = 5.

Use the Perimeter formula to solve for yP = 2x + 2y30 = 2(5) + 2y20 = 2yy = 10

Conclude:The maximum area is 75 m2 and is obtained when x = 5 and y = 10

Differentiate Area to show that (5,75) is a maximum:A(x) = -3 (x−5)2 + 75 A'(x) = -3(2)(x-5)A'(x) = -6x + 30

Solve for the x intercept:A'(x) = -6x + 30 0 = -6x + 30 6x = 30 x = 5

Choose a value to the left of x:x = 4A'(x) = -6x + 30A'(x) = -6(4) + 30A'(x) = 6 (positive)The parabola is increasing to the left of the vertex.

Choose a value to the right of x:x = 6

A'(x) = -6x + 30A'(x) = -6(6) + 30A'(x) = -6 (negative)The parabola is decreasing to the right of the vertex.

Conclude:Because the parabola increases to the left of the vertex and decreases to the right of the vertex, the vertex is a maximum not a minimum so the maximum area occurs at the vertex.

RELATIONAL UNDERSTANDINGINTRODUCE MANIPULATIVES

LET STUDENTS ENTER THE PROBLEM BY LOOKING AT

SPECIFIC CASESX Area1 272 483 634 725 756 727 638 489 27

SET UP THE GENERALIZATION

LET STUDENTS WORK IN GROUPS FROM HERE

WHEN SHARING SOLUTIONS AS A CLASS:

Point out that technology can be used so that we do not have to complete the square, interpret the parabolic equation, solve for y or differentiate.

Graph A = -3 (x−5)2 + 75

FOR STUDENTS THAT PLOTTED THE DERIVATIVE:

x f'(x)1 242 183 124 6

Use Meta-Calculator to find the derivative and graph it

Have students make connections between the

original graph and its derivative

Learner Outcomes

Math 20-1 Solve problems that involve quadratic equations (Program of Studies, p.18)

Math 31 Deriving f'(x) for polynomial functions up to the third degree, using the definition of the derivative (Program of Studies, p.19)

Analyze and synthesize information to determine patterns and links among ideas (Program of Studies, ICT Outcome 4.2)

Nature of Mathematics Patterns may be represented in concrete, visual, auditory or symbolic form (Program of Studies, p.7)

Mathematical Processes Number Visualization & Technology (Program of Studies, p.7); Problem Solving (Program of Studies, p.6); Communication & Connections (Program of Studies, p.5)