    5 C’s of Mathematical Engagement:
1. Curiosity
2. Connection making
3. Challenge
4. Creativity
5. Collaboration
• expose people to openness and flexibility of mathematics
• generate excitement
• create more opportunities for understanding
• provide support for high challenge tasks
Characteristics of Engaging Lessons with Rich Mathematical Tasks
• Task is challenging (high ceiling), but accessible (low floor).
• Students view task as a puzzle
• Visual thinking builds understanding
• Classroom culture values mistakes
• Students respect each other’s thinking
• Students use own ideas (not blindly follow given procedures)
• Students collaborate
• Students have different things to offer to team effort to solve problem
• Students don’t think they are finding a standard answer – think they are finding own solutions to a problem
• Good Timing (Just in Time Teaching)
• Let students explore applied problems first
• Observe students
• Introduce methods when they reach a stuck point that can be resolved by the method
• Helps students learn value of methods and develop intuitive sense for methods

• 18×5
• Solve 18×5 using mental math
• Display different solutions using simple thumbnail sketches
• Growing geometric patterns
• Explain visually how pattern is growing
• Share different growth pattern ideas
• Use growth pattern ideas to find 100th and nth iterations of patterns
• Build a fence out of 36 1-m planks
• Use 36 1-meter planks to build fence that encloses the most land
• Lemon volume
• Find the volume of a melon (precursor to teaching integration methods)
• Use Cuisenaire Rod to determine how many different trains can be made from a rod of prescribed length
• Find connection between Pascal’s triangle and numbers of trains in rods
• Find patterns in geometric patterns that go up in positive indexes and then down in negative indexes
• Figure out how to represent negative index results visually and numerically
• How Close to 100?
• Roll pair of dice – use pair of dice to draw an array on a 10×10 grid.  Write number sentence that related dice numbers to array area,
• Take turns rolling dice, drawing arrays and writing related number sentences
• Place until no more arrays can be fit unto 10×10 grid
• 1 Divided by 2/3
• Don’t apply the fraction rule of division
• Use visuals to make sense of answer
• Four 4’s
• Make all numbers from 1 to 20 using four 4’s and any operation
• For tips on how to stage this lesson, go here
• Paper Folding – Use paper folding of a square piece of paper to do the following
• Construct a square with exactly 1/4 the area of original square – convince partner that it is 1/4 the area
• Construct a triangle with exactly 1/4 the area of original square – convince partner that it is 1/4 the area
• Construct another triangle with exactly 1/4 the area of original square that is not congruent to the first triangle – convince partner that it is 1/4 the area

Developing Rich Mathematical Tasks: Questions to Consider:
• Open the task encourage multiple methods, pathways and representations.
• Ask students to make sense of solutions
• Make it an inquiry task.
• Ask students to come up with ideas, not follow a procedure
• examples
• write a book about y = mx + b
• create a coffee table book about similarity
• Ask the problem before teaching the method.
• Introduce methods after students have developed methods that relate or approximate new methods or after students reach a stuck point that can be unlocked with new method
• Use drawings, math manipulatives, etc to represent solutions and ideas
• Use color coding to feature common features in solutions such as “x” or to highlight relationships
• Extend the task to make it low floor and high ceiling.
• Low floor – ask students how they see the problem
• High ceiling – ask students to write a new question that is similar but more difficult
• Add the requirement to convince and reason
• Explain methods and why they make sense
• Reasoning is practicing mathematics
• 3 levels of convincing
1. Convince yourself
2. Convince a friend
3. Convince a skeptic Rich mathematical tasks can be used to engage all students in mathematical solving and sense making.  The principles behind good design for these tasks can be used by teachers to frame problems and lessons that invite students to apply their own ideas to problems and to relate these ideas to new concepts and methods. Preparation Steps
• Analyze standards and develop learning targets related to concepts and problem solving methods
• Use resources above to find rich and engaging mathematical tasks that relate to learning targets
• Build scaffolding and assessments around rich mathematical tasks that align to learning targets
Early Implementation Steps
• Use rich mathematical tasks to:
• create engagement
• introduce new concepts
• help students see need-to-knows in new content
• make sense of new content
• to get students to discuss their reasoning
• Use assessments to fine tune activities to improve student learning
• Ask students to reflect on how activities are improving their learning 