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2014 Mathematics SOL InstitutesGrade Band

Geometry

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.

Agenda

- Defining Representations and Connections
- Doing the Mathematical Task
- Looking at Student Work
- Planning Mathematics Instruction
- Closing

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.

I. Defining Representation and Connection

- Describe in words what the fourth arrangement

would look like.

Defining Representation

- Using the cubes, build the fourth arrangement.

Defining Representation

- How many cubes are in the fourth arrangement?
- How many cubes are in the fifth arrangement?

Defining Representation

- How many cubes are in the 50th arrangement?
- How many cubes are in the nth arrangement?

Defining Representation

- What does it mean for students to use

mathematical representations? - Think/Pair/Share

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.

Virginia Department of Education. (2009).

Introduction Mathematics Standards of Learning

for Virginia Public Schools

(No Transcript)

- 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.

Defining Connection

- What does it mean for students to make

mathematical connections? - Think/Pair/Share

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.

Virginia Department of Education. (2009).

Introduction Mathematics Standards of Learning

for Virginia Public Schools

- 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.

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.

Doing the Mathematical Task

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

Orchestrating Mathematics Discussion

- Anticipating
- Monitoring
- Selecting
- Sequencing
- Connecting

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?

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.

Student Sample I

- Part a) Part

b)

Questions to Consider

- Revisit your list of questions to consider when

planning mathematics instruction. - Is there anything we need to add?

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

todays lesson? - Skip Fennell

Fennell, F (Skip). (2006). RepresentationShow 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?

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.

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?

Creating a construction

- Paper/Pencil with Compass and Straightedge
- Paper Folding
- Virtually with TestNav available on VDOE Website
- Dynamic Geometry Software

Using Dynamic Geometry Software

- In tables, number from 1 4.
- Rearrange to be with like numbers.
- Watch for
- 1 Teacher moves for connections
- 2 Student moves for connections
- 3 Teacher moves for representations
- 4 Student moves for representations

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

Software - https//www.teachingchannel.org/videos/teaching-wi

th-geometry-software

Integrating Constructions

- Create a poster for your assigned construction.

Include - Actual construction
- Mathematical connections
- Justifications
- When we should teach this construction

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.

- "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.

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.

- 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?

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.

Closing

- Share your current and ideal practice form at

your table.