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2014 Mathematics SOL Institutes Grade Band: Geometry

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Title: 2014 Mathematics SOL Institutes Grade Band: Geometry


1
2014 Mathematics SOL InstitutesGrade Band
Geometry
2
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.

3
Agenda
  1. Defining Representations and Connections
  2. Doing the Mathematical Task
  3. Looking at Student Work  
  4. Planning Mathematics Instruction
  5. Closing

4
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.

5
I. Defining Representation and Connection
  1. Describe in words what the fourth arrangement
    would look like.

6
Defining Representation
  1. Using the cubes, build the fourth arrangement.

7
Defining Representation
  1. How many cubes are in the fourth arrangement?
  2. How many cubes are in the fifth arrangement?

8
Defining Representation
  1. How many cubes are in the 50th arrangement?
  2. How many cubes are in the nth arrangement?

9
Defining Representation
  • What does it mean for students to use
    mathematical representations?
  • Think/Pair/Share

10
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
11
(No Transcript)
12
  • 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.

13
Defining Connection
  • What does it mean for students to make
    mathematical connections?
  • Think/Pair/Share

14
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
15
  • 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

16
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.

17
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.

18
Doing the Mathematical Task
  • Solve each part of the problem in two ways.
  • Be prepared to share your solutions.

 
19
Orchestrating Mathematics Discussion
  1. Anticipating
  2. Monitoring
  3. Selecting
  4. Sequencing
  5. Connecting

20
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?

21
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.

22
Student Sample I
  • Part a) Part
    b)

23
Questions to Consider
  • Revisit your list of questions to consider when
    planning mathematics instruction.
  • Is there anything we need to add?

24
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
25
IV. Planning Mathematics Instruction
  • Afternoon goal Answer this question
  • How can we best integrate constructions into our
    Geometry curriculum?

26
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.

27
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?

28
Creating a construction
  • Paper/Pencil with Compass and Straightedge
  • Paper Folding
  • Virtually with TestNav available on VDOE Website
  • Dynamic Geometry Software

29
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

30
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

31
Integrating Constructions
  • Create a poster for your assigned construction.
    Include
  • Actual construction
  • Mathematical connections
  • Justifications
  • When we should teach this construction

32
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.

33
  • "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
34
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.

35
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.

36
  • 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.

37
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?

38
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.

39
Closing
  • Share your current and ideal practice form at
    your table.
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