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Mathematical Discourse: 5 Practices, 3-D Solids, Packaging Dilemma

Join us for a professional development session on mathematical discourse, 3-D solids, and the packaging dilemma. Learn and share ideas to improve practice and support student success.

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Mathematical Discourse: 5 Practices, 3-D Solids, Packaging Dilemma

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  1. Welcome to …

  2. TODAY’S AGENDA • 5 Practices for Mathematical Discourse • 3-D Solids • 3-D Nets • Packaging Dilemma • Lessons and Reflections

  3. Bring your ideas… • As a group of professionals we have made a commitment to helping children attain success in life and a voice in the world. • Many times the best part of these kinds of professional development is simply the chance to share ideas, raise questions, and work with other practitioners to improve our own understandings and practice. • Please bring your stories of children’s learning with you.

  4. Our Socio-mathematical Norms • Listen intently when someone else is talking avoiding distractions • Persevere in problem solving; mathematical and pedagogical • Solve the problem in more than one way • Make your connections explicit - Presentation Ready • Contribute by being active and offering ideas and making sense • Limit cell phone and technology use to the breaks and lunch unless its part of the task. • Be mindful not to steal someone else’s “ice cream” • Respect others ideas and perspectives while offering nurturing challenges to ideas that do not make sense to you or create dissonance. • Limit non-mathematical and non-pedagogical discussions

  5. Presentation Norms • Presenters should find a way to show mathematical thinking, not just say it • Presenters should indicate the end of their explanation by stating something like “Are there any questions, discussion, or comments?” • Others should listen and make sense of presenters’ ideas. • Give feedback to presenters, extend their ideas, connect with other ideas, and ask questions to clarify understandings

  6. 5 Practices for Orchestrating Productive Mathematics Discussions • Determining the Direction of the Discussion: Selecting, Sequencing, and Connecting Students’ Responses

  7. Making Progress on a Task • Teachers must: • Decide what aspects of a task to highlight • How to organize and orchestrate the work of the students • What questions to ask to challenge those with varied levels of expertise • How to support students without taking over the process of thinking for them; thus eliminating the challenge of the task. ”Principles to Actions: Ensuring Mathematical Success for All”, NCTM, 2014.

  8. Sequencing • Consider the posters on the wall from the “Calling Plan Task” from Friday. • Think about how you would sequence these for presenting in a class. • Discuss with your group. • Write your sequence on a post-it note on place on the chart stand at the front.

  9. Pizza Comparison Task

  10. Solving the Task • Solve the task individually in more than one way. • Share your thinking with your group. • Think about how your students might approach this problem. • Whole group discussion.

  11. Read & Reflect • Read the pages assigned to your group from the reading on pages 43 – 59. • Reflect on the reading. • Decide if you would change your original sequence order of the posters. • Discuss with your group. • Discuss questions #4 in your small group. • Whole group discussion

  12. Important Connections Through Mathematical Representations “Because of the abstract nature of mathematics, people have access to mathematical ideas only through the representations of those ideas”. National Research Council (2011)

  13. Developing Student Representation Competence • Specific Strategies: • Encourage purposeful selection of representations • Engage in dialogue about explicit connections among representations. • Alternate the direction of the connections made among representations. Marshall, Superfine, and Canty (2010, p.40)

  14. Break Time

  15. Naming 3-D Shapes • Common 3-D Shapes • Cube; Rectangular Prism; Triangular Prism; Cylinder; Cone; Pyramid; Sphere • Polyhedron (plural is polyhedra) • Geometric solid with flat faces • Each face is a polygon • Platonic solids • Polyhedra • Each face is the same regular polygon • Same number of polygons meet at each vertex

  16. 11 Nets for Cubes

  17. Rectangular Prism • Working with a partner draw on grid paper a flat pattern for a rectangular prism that is not a cube. • Test the pattern by cutting out and folding Describe the faces of the rectangular prism. • What are the dimensions of each face? What is the total area (surface area) of the prism? • Compare your results with others.

  18. Surface Area and Volume • The total area of all the faces is the surface area of the prism. • How many cubes will fit in a single layer at the bottom of the rectangular prism? • How many cubes are necessary to fill the prism? • How does the number of cubes in a single layer relate to the total number needed to fill the prism?

  19. Packaging Dilemma The Scrooge Shipping Company is looking to save money on the cost of shipping materials. To accomplish this, they have decided to use less material for the packages. Working with a partner and using the 1-centimeter cubes at your table, design a container that will allow for the maximum(greatest) number of cubes and the minimum(least) amount of packaging material.

  20. Lunch

  21. MAKING PILLOWS It takes 2/3 of a yard of material to make a pillow. How many yards of material would it take to make 15 pillows?

  22. RELATIONAL THINKING “A student who can express a number in terms of other numbers and operations on those numbers holds a relational understanding of the number.” - Empson and Levi, Extending Children’s Mathematics: Fractions and Decimals, p. 74 “When children use Relational Thinking to solve problems, they are drawing upon a small set of fundamental properties that govern how operations and equations work.” - Empson and Levi, Extending Children’s Mathematics: Fractions and Decimals, p. 76

  23. Problem Solving I am making Sub Sandwiches for some friends. There will be 13 of us eating sub sandwiches. I want to serve each person ¼ of a sub sandwich. How many sub sandwiches do I need to make all together? Mr. Davis is planning an art project for his class. Each student will need ¾ of a package of clay to do this project. If Mr. Davis has 12 students in his class, how many packages of clay would he need?

  24. MAKING RELATIONAL THINKING EXPLICIT Reading from Extending Children’s Mathematics: Fractions and Decimals Ms. Perez’s Fifth-Grade Lesson Pages 95 - 109

  25. RELATIONAL THINKING ACROSS THE MATHEMATICS CURRICULUM How are these alike? 70 + 40 0.07 + 0.04 7/5 + 4/5 7x + 4x What fundamental properties of operations and equality are apparent in the computation of each?

  26. Break Time

  27. Math Content for our Classrooms • Each day we will spend time with grade level teams making lesson plans. • Each of us will make one plan that is part of a unit of plans the grade level team is working on. • Each plan must have the following: • Connected mathematics content focus from Ohio’s Mathematics Learning Standards • A focus SMP • Designed to Orchestrate Productive Mathematics Discussions (The 5 Practices)

  28. Math Content for our Classrooms Three checks must be made for the completion of lesson plans: Check 1) Consult with Sandy and/or Mary about the mathematics content of the lesson and explain to her its mathematical appropriateness. When the lesson is complete Sandy, our resident mathematician, will sign off on its content (SMC’s). Check 2) Consult with Sherry about the design of the lesson to promote mathematical discourse (5 Practices). Sherry must sign off on the lessons discourse elements. Check 3) Consult with Dr. Matney about the design of the lesson having a focus Standard for Mathematical Practice. Dr. Matney must sign off on the lessons mathematics proficiency elements (SMP’s) ?Questions about COMP Lesson Plans?

  29. Air of Appreciation We want to pass on to each generation a sense of learning how to appreciate life, others, and learning. Let’s spend some time sharing one thing or experience that we appreciate: Examples: I appreciated when Ray didn’t give up on solving that hard problem. It encouraged me to keep thinking for myself to make sense of it.

  30. Time of Reflection • On a sticky note tell us one thing you learned today. • Tell us one think you liked or one thing you are still struggling with.

  31. Stay Safe • Please help us put the room in proper order. • Please leave your name tents for next time.

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