engageNY/Eureka Math Parent Workshop, Session 3...Parent Workshop, Session 3 Saratoga USD March 16,...

engageNY/Eureka Math

Parent Workshop, Session 3 Saratoga USD March 16, 2016

Outcomes

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• Review and understand the Van Hiele levels

of Geometric Reasoning

• Explore the progression of skills and

conceptual understandings in the domain

of Geometry

• Connect learnings in the domain to middle

school mathematics

• Receive resources to support your child’s

math education at home

Agenda

• Geometry Standards Card Sort

• Van Heile Levels of Geometric

Reasoning

• Geometry Domain , Grades K-5

• Geometry Domain, Grades 6-8

• Resources

• Closure

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COHERENCE OVERVIEW

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Focus

Shift 1: Focus

Coherence

Shift 2: Coherence

Rigor

Shift 3: Fluency

Shift 4: Deep Understanding

Shift 5: Application

Shift 6: Dual Intensity

Instructional Shifts Combined

Coherence: Think Across Grades

K 1 2 3 4 5 6 7 8 HS

Counting &

Cardinality

Number and Operations in Base Ten Ratios and Proportional

Relationships Number &

Quantity Number and Operations –

Fractions The Number System

Operations and Algebraic Thinking

Expressions and Equations Algebra

Functions Functions

Geometry Geometry

Measurement and Data Statistics and Probability Statistics &

Probability

Findwell, Bradford & Foughty, Zachary. “”Preparing to Implement the Common Core State Standards for Mathematics. Indiana Department of Education and Ohio Department of Education. March 30, 2011

Shift #2: Coherence: Think Across Grades, and

Link to Major Topics Within Grades

• Carefully connect the learning within and across

grades so that students can build new

understanding on foundations built in previous

years.

• Begin to count on solid conceptual understanding of

core content and build on it. Each standard is not a

new event, but an extension of previous learning.

Activity: Card Sort

• In each envelope there are six standards from

the domain of Geometry

• Work with your fellow parents and organize

the strips into the Progression of Standards from

Kindergarten to Grade 5

• Be prepared to share the sequence of your

card sort and your justification for selecting

that order with the whole group.

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Activity: Card Sort

Answers

• Kinder-Compose simple shapes to form larger

shapes. For example, "Can you join these two

triangles with full sides touching to make a

rectangle?"

• Grade 1-Distinguish between defining

attributes (e.g., triangles are closed and three-

sided) versus non-defining attributes (e.g.,

color, orientation, overall size); build and draw

shapes to possess defining attributes.

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Activity: Card Sort

Answers

• Grade 2-Recognize and draw shapes having

specified attributes, such as a given number of

angles or a given number of equal faces.1 Identify

triangles, quadrilaterals, pentagons, hexagons, and cubes.

• Grade 3-Partition shapes into parts with equal

areas. Express the area of each part as a unit fraction of the whole. For example, partition a

shape into 4 parts with equal area, and describe

the area of each part as 1/4 of the area of the

shape.

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Activity: Card Sort

Answers

• Grade 4-Draw points, lines, line segments,

rays, angles (right, acute, obtuse), and

perpendicular and parallel lines. Identify

these in two-dimensional figures.

• Grade 5-Represent real world and

mathematical problems by graphing points

in the first quadrant of the coordinate plane,

and interpret coordinate values of points in

the context of the situation.

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EngageNY Mathematics Overview PK-5 Curriculum Map: A Story of Units

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engageNY/Eureka Math

Geometry Modules (K-5)

Grade Module Title

K Two-Dimensional and Three-Dimensional Shapes Analyzing, Comparing, and Composing Shapes

1 Identifying, Composing, and Partitioning Shapes

2 Time, Shapes, and Fractions as Equal Parts of Shapes

3 Geometry and Measurement Word Problems

4 Angle Measure and Plane Figures

5 Addition and Multiplication with Volume and Area Problem Solving with the Coordinate Plane

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Geometry Connections

• Geometric and spatial thinking …

oConnect math with the physical world

o Support the development of number

and arithmetic concepts and skills

o Have roles in the physical sciences,

engineering

o Have strong aesthetic connections

VAN HIELE LEVELS OF GEOMETRIC REASONING

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Van Hiele Levels of Geometric

Reasoning

• Level 0: Visualization • Level 1: Analysis (Description) • Level 2: Informal Deduction • Level 3: Formal Deduction • Level 4: Rigor

Handout

Summary of the van Hiele Levels

https://youtu.be/fGjicKyxFn4

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Reasoning

Level 0: Visualization

Students see shapes as total entities, but do not recognize properties

(“It’s a rectangle, because it looks like a door”.)


Reasoning

Level 1: Analysis Students identify properties of figures,

and see figures as a class of shapes (“It’s a rectangle, because it has one

long set of sides and one short set of sides, and opposite sides are parallel, and …”.)

Transition from Level 0 to Level 1

• Examine examples and non-examples

• Sort and classify shapes

• Find hidden figures

• Re-arrange shapes into other shapes

(tangrams)

• Provide opportunities to build, draw,

make, put together and take apart shapes

Activity- Kinder


Reasoning Level 2: Informal deduction Students recognize relationships between

properties of shapes, and between classes of shapes; they also develop informal explanations using these relationships.

(“It’s a rectangle, because it is a

quadrilateral with four right angles”.)

Transition from Level 1 to Level 2

• Analyze classes of figures to determine

new properties

• Identify relationships by folding,

measuring and looking for symmetry

• Folding paper and predicting a shape

• Using geoboards and geometry

software


Reasoning

Level 3: Formal Deduction

Students understand the significance of deduction as a way of establishing geometric theory within an axiom system; they also see the interrelationship and role of undefined terms, axioms, theorems, and formal proof.


Reasoning

Level 4: Rigor

Students see geometry in the abstract,

even without concrete examples; they

compare geometric results in different

axiom systems (non-Euclidean

geometries).

Van Hiele Levels

• What resonates with you?

• What do you connect to based on

your experience and your working with

your child/children?

GEOMETRY PROGRESSION

The Geometry Progression (K-5)

Three Categories/Themes

• Geometric shapes, their components,

their properties and their

categorization (Van Hiele levels)

• Composing and decomposing

geometric shapes

• Spatial relations and spatial structuring

Spatial Reasoning in Everyday Life

“Spatial thinking is integral to everyday life.

People, natural objects, human-made

objects, and human-made structures exist

somewhere in space, and the interactions of

people and things must be understood in

terms of locations, distances, directions,

shapes, and patterns.”

(National Research Council, 2006, p. 5)

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Imagine This: Problem Solving with

the “Mind’s Eye” • Imagine a large cube floating in front of you. The cube is

made up of 64 smaller cubes and thus is a 4 x 4 x 4 cube.

Now, imagine that you are staring directly at the front of

the cube so that all you can see is the front face of the

cube – a 4 x 4 square face. You are now going to drill a

hole through the four corner cubes that are facing you,

all the way through to the back face. Now, imagine

looking down on the cube from above – a bird’s-eye

view. Again, your view is such that all you see is a 4 x 4

square. You drill a hole through the four corner cubes all

the way through to the other side. How many of the 64

cubes do not have holes drilled through them? Were

you able to visualize the solution?

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Geometry in Kindergarten (understanding shapes and space)

• Match two-

dimensional shapes

even when shapes

have different

orientations

(variants)

• Distinguish shapes

from non-examples

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Geometry in Kindergarten

• Increase their knowledge of a variety

of shapes with unequal bases and

non-parallel sides

• Model shapes in the world by building

shapes from components

• Name and describe three-dimensional

shapes

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Composing and Decomposing

• Students gain the ability to combine shapes

into pictures first

• They are able to synthesize combinations of

shapes into new shapes

• They decompose shapes by covering a part

of a shape to make another

• The are able to use a composed shape as a

new unit in making other shapes.

• Builds experience with properties (equal

sides, equal angles, etc.)

Geometry in Kindergarten

Composition of geometric figures

• Compose shapes to build pictures and

designs

• Solve problems using geometric

motions (slides, flips, and turns)

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engageNY Sample K Task

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Geometry in Grade 1

• Build on describing/classifying shapes

using geometric attributes (closed,

three-sided)

• Composing 2D and 3D shapes to

make to make a new shape

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Activity- Grade 1

Geometry and Fractional Parts

Progression

• 1G.3- partition circles an rectangles

into two and four equal shares…

• 2G.3- partition circles and rectangles

into two, three, or four equal shares…

• 3G.3- partition shapes into parts with

equal areas…

Activity- Grade 1

engageNY Grade 1 Sample Task

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Geometry in Grade 2

• Develop foundations for area, fraction

and proportion by recognizing the

composition of shapes as equal areas.

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Activity- Grade 2 Let’s Play!

Geometry in Grade 2

• Spatial Structuring- understanding how

a rectangle can be tiled with squares

in rows and columns

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Geometry in Grade 3

• Students analyze,

compare and classify

2D shapes by

properties.

• They can classify

using larger

categories such as quadrilaterals.

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Geometry in Grade 3

• Students develop more competence

in composition and decomposition of

rectangles into arrays.

• They count by the number of columns

and rows or use multiplication to

determine the number of squares in an

array.

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Engage NY example

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Grades 2 through 5

Geometry and Measurement

• Grade 2: Students begin the formal

study of measure, learning to use units

of length as attributes of shapes.

• Grades 3-5: measurement of angles

and parallelism with a focus on area

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Geometry in Grade 4

• Students build on their abilities to

describe, compare and classify 2D

shapes by explicitly using specific

angle size (acute, right, obtuse).

• They classify figures by length size

(equilateral, isosceles, scalene) and

by the absence or presence of parallel

and perpendicular lines.

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Geometry in Grade 4

• Students recognize a line of symmetry

and draw lines of symmetry.

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Geometry in Grade 5

• By the end of Grade 5, students

competencies in shape

composition/decomposition and

spatial structuring should be highly

developed

• These competencies form the

foundation for understanding

multiplication, area, volume and the

coordinate plane.

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Geometry in Grade 5

• Students understand the area of a

shape in square units

• Students extend their

spatial structuring by

applying it to 3D figures

and the continuous nature of the 2D

space in the coordinate plane.

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Geometry in Grade 5

• They apply their knowledge of number

and length to the order and distance

relationships in a coordinate grid.

• They know how to plot ordered pairs

according to the x-axis and the y-axis.

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Geometry in Grade 5

• Students analyze

and classify 2D

shapes explicitly by

their properties and

in hierarchies.

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CONNECTIONS TO MIDDLE SCHOOL MATH

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Transition to

Middle School Geometry

Areas of Elementary Geometry Geometric skills and knowledge in MS

Area and Volume Context for developing and using equations, find the volume of right, rectangular prisms with fractional edge lengths.

Shape composition and decomposition and spatial structuring of rectangular arrays

Understanding multiplication, formulas for area (triangles, parallelograms), volume and coordinate plane. Analysis and composition of polyhedral solids and describe the shapes of their faces, sides and vertices.

Identifying line segments, parallel and perpendicular lines

Identify relationships of line segments and angles within a figure Describe cross sections of 3D figures

Shape composition and decomposition Making scale drawings 58

Middle School Geometry Tasks

• Grade 6

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• Grade 7

60


• Grade 8

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HELPING STUDENTS WITH MATH (SPECIFICALLY WITH GEOMETRIC THINKING)

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Increasing Geometric Thinking

• Have blocks of different shapes readily available for

making designs and building. Point out shapes in

everyday objects and try to re-create them with

blocks.

• Sort and classify all kinds of items. Emphasize that

people create the categories for sorting.

• Ask questions: Why? Why not? What if? These

questions prompt your child to think about and

describe features of mathematical objects, such as

shapes. They also encourage looking at things from

another's point of view.

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Ways to increase

Geometric Thinking

Take advantage of everyday

opportunities to practice spatial thinking:

Will the groceries fit in one bag?

Which way does the sheet fit on the

bed?

What shapes do we get if we cut a

bagel (lengthwise or crosswise)?

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Ways to Increase

Geometric Thinking

• Play with tangrams and jigsaw puzzles.

• Encourage kids to use, create, and

explain maps.

• Try photography. (helps with

understanding of angles and scale)

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5 Powerful Questions to Ask

#1. What do you think?

#2. Why do you think that?

#3. How do you know this?

#4. Can you tell me more?

#5. What questions do you still have?

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Questions to Clarify

Understanding of Explanation

• Do you understand your solution?

• Can you explain what you’re thinking?

S

ANALYZING SOLUTIONS

Questions to promote analysis and reflection of

solutions:

– What do you see that is the same about

these solutions?

– What do you see that is different about

these solutions?

– How does this relate to ___?

– Ask students to think about how these

strategies relate to the mathematical

concept being discussed

S

DEVELOPING NEW MATHEMATICAL INSIGHTS

(ABSTRACT MATHEMATICAL CONCEPTS)

Questions to Promote Mathematical Insights

•Ask students to summarize key idea.

•Ask questions: Will the rule work all the time? (Making generalizations)

•Ask students to solve a related problem that extends the insights they had gained from the discussion.

•Ask “What if” questions.

S

Tips

• Emphasize skill, understanding and

application of math

• Talk through the math

• Value process along with final results

• Making mistakes is a part of doing

authentic mathematics

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Recognize math in everyday life

• How many more days?

• Reinforce multiplication and division by

helping your student plan for a major

purchase

• Cooking has endless possibilities to work with

fractions

• Shopping: Which discount or deal is better?

• Discuss the financial side of planning for

college

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Thank you for your time and participation!

Bernadette A. Salgarino, Ed.D.

[email protected]

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mailto:[email protected]

engageNY/Eureka Math Parent Workshop, Session 3...Parent Workshop, Session 3 Saratoga USD March 16,...

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