Rigor and DepthRigor and Depth

S. PetreyLISD August Staff Development

What is Rigor?What is Rigor?

Rigor can mean complex algorithmsRigor can mean conceptual complexityRigorous problems are not always deep

3 2(2 3 )a ab b If the volume of a cube is increased by a factor of 4, what is the resulting increase in the length of an edge?

28 ( 8) 4(3)( 5)2(3)

Find the volume of a regular square pyramid if all edges are 5cm.

What is Depth?What is Depth?

A problem is deep when it requires a

thorough understanding of

the underlying concepts to solve.

Why is adding Rigor and Depth Why is adding Rigor and Depth so Challenging?so Challenging?

Algorithmic complexity used in place of conceptual complexityDoing the work for the studentTime on TaskUnique and not common

Part 1: Get students ‘into’ Get students ‘into’ the problemthe problem

Graph a semicircle in the 1st and 2nd

quadrants that has an equation of

x2 + y2 = 16

Part 2, Build in different topicsBuild in different topics

Find the y-coordinate of a point on the

semicircle with an x-coordinate of 3.

Draw a radius to this point.

Part 3, Build up the rigorBuild up the rigor

What is the slope of the radius?

Draw a tangent line to the endpoint of the

radius. What is the equation of the tangent line?

Part 4, Add More LayersAdd More Layers

Find another tangent to the semicircle at

x = -3.

What is the equation of the tangent line?

Where does it intersect the first tangent line?

Part 5, Take the problem a Take the problem a different directiondifferent direction

Rotate the semicircle about the y-axis.

What solid figure is generated?

What is the volume and surface area of this

solid?

An Algebra ExampleAn Algebra Example

Evaluate the function at x = 2Graph the function if the domain is [0,4] What are the roots on the graph?What is the maximum?

2( ) 4f x x x

Introduce Vocabulary/NotationIntroduce Vocabulary/Notation

Create a secant line connecting f(0) and f(2).What is the slope of this secant?

2( ) 4f x x x

Use a pattern…Use a pattern…

Graph a new secant line containng f(1) and f(2).What is the slope of this secant?

2( ) 4f x x x

Encourage Intuition: Encourage Intuition: What will happen next?What will happen next?

Graph a 3rd secant line containing f(1.5) and f(2).What is the slope of this secant?

2( ) 4f x x x

Introduce New ConceptsIntroduce New Concepts

What happens to the slope of the secant line as the first point approaches f(2)?What is the limit for the slope of this secant as the first point approaches f(2)?

2( ) 4f x x x

Build Rigor into your classBuild Rigor into your classProvide problems with multiple parts that get students ‘into’ the problemGive students time to independently work on rigorous ‘new’ problemsLet students find solutions intuitively when possible instead of ‘telling’Allow students to collaborateMake rigor and depth a routine part of your class

Important things to noteImportant things to noteStudents can engage in very complex critical thinking without performing complex algorithmsStudents should be given opportunity to use concepts from other courses without you first ‘reteaching’ those concepts.Grading policy should build student confidence and encourage future rigorous problem solving

How could you change a lesson this How could you change a lesson this year to incorporate more rigor and year to incorporate more rigor and

depth?depth?

Homework?Quiz?Test?Project?Exam?

How can you grade these How can you grade these ‘layered’ problems?‘layered’ problems?

Rubric Style Grading‘Read with’ your studentRescue the Struggling and Reward the Excellent‘Find’ Good Critical Thinking

Questions?Questions?