2014 Mathematics SOL InstitutesGrade Band: Geometry

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Making Connections and Using Representations

• The purpose of the 2014 Mathematics SOL Institutes is to provide professional development focused on instruction that supports process goals for students in mathematics.

• Emphasis will be on fostering students’ ability to make mathematical connections and use effective and appropriate representations in mathematics.

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Agenda

I. Defining Representations and ConnectionsII. Doing the Mathematical TaskIII. Looking at Student Work IV. Planning Mathematics InstructionV. Closing

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Welcome!

• Draw a geometric shape that represents you. Be able to justify your choice.

• Group yourselves with others who have like attributes. Discuss your rationale for your shape and your group classification.

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I. Defining Representation and Connection

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1. Describe in words what the fourth arrangement would look like.

Defining Representation

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2. Using the cubes, build the fourth arrangement.

Defining Representation

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3. How many cubes are in the fourth arrangement?

4. How many cubes are in the fifth arrangement?

Defining Representation

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5. How many cubes are in the 50th arrangement?

6. How many cubes are in the nth arrangement?

• What does it mean for students to use mathematical representations?

• Think/Pair/Share

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Defining Representation

Mathematical Representations

Students will represent and describe mathematical ideas, generalizations, and relationships with a variety of methods. Students will understand that representations of mathematical ideas are an essential part of learning, doing, and communicating mathematics. Students should move easily among different representations ⎯ graphical, numerical, algebraic, verbal, and physical and recognize that ⎯representation is both a process and a product.

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Virginia Department of Education. (2009). Introduction Mathematics Standards of Learning for Virginia Public Schools

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“Representations are useful in all areas of mathematics because they help us

develop, share, and preserve our mathematical thoughts.

They help to portray, clarify, or extend a mathematical idea

by focusing on its essential features.”

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. (p. 206). Reston, VA.

• What does it mean for students to make mathematical connections?

• Think/Pair/Share

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Defining Connection

Mathematical Connections

Students will relate concepts and procedures from different topics in mathematics to one another and see mathematics as an integrated field of study. Through the application of content and process skills, students will make connections between different areas of mathematics and between mathematics and other disciplines, especially science. Science and mathematics teachers and curriculum writers are encouraged to develop mathematics and science curricula that reinforce each other.

14Virginia Department of Education. (2009). Introduction Mathematics Standards of Learning for Virginia Public Schools

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“Connections are useful because they help students see mathematics as a unified body of

knowledge rather than a set of complex and disjoint concepts, procedures and processes. Real world contexts provide opportunities for students to connect what they are learning to

their own environment. Their mathematics may also be connected to other disciplines which

provides opportunities to enrich their learning.”

National Council of Teachers of Mathematics. 2000, p. 200. Principles and Standards for School Mathematics. Reston, VA

Planning for the Use of Representations

• How do we plan for the purposeful use of representations and connections in the classroom?

• What questions do we need to consider?

• Brainstorm as many questions as you can, and share on your chart paper.

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II. Doing the Mathematical Task

• Find the coordinates of the bus stop that is equidistant tothree houses A, B, and C. Explain and justify.

• Find other houses that are equidistant to the bus stop. Explain and justify.

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A

B

C

x

y

Doing the Mathematical Task

• Solve each part of the problem in two ways.• Be prepared to share your solutions.

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Orchestrating Mathematics Discussion

1. Anticipating2. Monitoring3. Selecting4. Sequencing5. Connecting

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Doing the Mathematical Task

In your groups, discuss:

• What mathematical representations could be used to solve this task?

• What mathematical connections can be made with this task?

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III. Looking at Student Work

For each student sample, identify:– The representation(s) used;– The connections that are evident; and – Any misconceptions evident in the student work.

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Student Sample I

Part a) Part b)

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Questions to Consider

• Revisit your list of questions to consider when planning mathematics instruction.

• Is there anything we need to add?

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Representation should be an important element of lesson planning. Teachers must

ask themselves,

“What models or materials (representations) will help convey the

mathematical focus of today’s lesson?”- Skip Fennell

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Fennell, F (Skip). (2006). Representation—Show Me the Math! NCTM News Bulletin. September. Reston, VA: NCTM

IV. Planning Mathematics Instruction

Afternoon goal – Answer this question:

How can we best integrate constructions into our Geometry curriculum?

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Perpendicular Bisectors/Circumscribed Circles

• On the handout, complete the constructions for:– Perpendicular bisector of a segment– Circumscribed circle of a triangle

• While completing the constructions, think carefully about why these constructions work.

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Perpendicular Bisectors/Circumscribed Circles

At your table, discuss:

• How can we justify these constructions?• When should we teach these constructions?• When should we use the bus stop task?

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Creating a construction

• Paper/Pencil with Compass and Straightedge• Paper Folding• Virtually with TestNav available on

VDOE Website• Dynamic Geometry Software

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Using Dynamic Geometry Software

• In tables, number from 1 – 4.• Rearrange to be with like numbers.

Watch for: 1: Teacher moves for connections2: Student moves for connections3: Teacher moves for representations4: Student moves for representations

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Using Dynamic Geometry Software

Watch video

• In numbered groups, share your observations.• Return to original tables. Share the consensus

from the breakout groups.

The Teaching Channel. Using Dynamic Geometry Softwarehttps://www.teachingchannel.org/videos/teaching-with-geometry-software

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Integrating Constructions

• Create a poster for your assigned construction. Include:– Actual construction– Mathematical connections – Justifications– When we should teach this construction

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Gallery Walk

• Designate one person in your group as the expert to stay at your poster and explain.

• Remainder of the group participate in the gallery walk.

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"Students representational competence can be developed through instruction. Marshall, Superfine, and Canty (2010, p. 40) suggest three specific strategies:

1. Encourage purposeful selection of representations.2. Engage in dialogue about explicit connections among

representations.3. Alternate the direction of the connections made among

representations."

National Council for Teachers of Mathematics. (2014). Principles to Actions. (p. 26). Reston, VA

The Role of the Teacher

• Create a learning environment that encourages and supports the use of multiple representations

• Model the use of a variety of representations• Orchestrate discussions where students share

their representations and thinking• Support students in making connections among

multiple representations, to other math content and to real world contexts

Van de Walle, J.A., Karp, K.S., Lovin, L.H. & Bay-Williams, J.M. (2013). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5

(2nd ed.). (Vol. II). Pearson.34

Role of the Student

• Create and use representations to organize, record, and communicate mathematical ideas

• Select, apply, and translate among mathematical representations to solve problems

• Use representations to model and interpret physical, social, and mathematical phenomena

Van de Walle, J.A., Karp, K.S., Lovin, L.H. & Bay-Williams, J.M. (2013). Teaching Student-Centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5

(2nd ed.). (Vol. II). Pearson.

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Students must be actively engaged in developing, interpreting, and critiquing

a variety of representations. This type of work will lead to better

understanding and effective, appropriate use of representation as a mathematical tool.

National Council of Teachers of Mathematics. (2000) Principles and Standards for School Mathematics. (p. 206). Reston, VA.

V. Closing

• Revisit questions to consider for planning mathematical instruction. Which questions should we add?

• How does your list compare with the list on the handout?

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Closing

• What practical implications does our work with constructions have for your classroom, school, or division?

• How will it inform instruction and pacing?• Complete the current and ideal practice form.

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Closing

• Share your current and ideal practice form at your table.

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