Geometry

Grades K-2

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Goals:

Build an understanding of the mathematical concepts within the Geometry Domain

Analyze how concepts of Geometry progress through the grades

Discuss the van Hiele levels of geometric thought Explore Geometry Stations and Resources to Take

Back to Your Classroom

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Progression of GeometryKindergarten

Major Clusters Supporting/Additional ClustersCounting and Cardinality Know number names and the count sequence. Count to tell the number of objects. Compare numbers.Operations and Algebraic Thinking Understand addition as putting together and

adding to, and understand subtraction as taking apart and taking from.

Number and Operations in Base Ten Work with numbers 11–19 to gain foundations

for place value.

Measurement and Data Describe and compare measurable attributes. Classify objects and count the number of

objects in categories.Geometry Identify and describe shapes. Analyze, compare, create, and compose

shapes.

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Progression of Geometry

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Progression of GeometrySecond Grade

Major Clusters Supporting/Additional ClustersOperations and Algebraic Thinking Represent and solve problems involving addition

and subtraction. Add and subtract within 20. Work with equal groups of objects to gain

foundations for multiplication.Number and Operations in Base Ten Understand place value. Use place value understanding and properties of

operations to add and subtract.Measurement and Data Measure and estimate lengths in standard units. Relate addition and subtraction to length.

Measurement and Data Work with time and money. Represent and interpret data.Geometry Reason with shapes and their attributes.

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KindergartenIdentify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,

cylinders, and spheres).1. Describe objects in the environment using names of shapes, and describe the relative

positions of these objects using terms such as above, below, beside, in front of, behind, and next to.

2. Correctly name shapes regardless of their orientations or overall size. 3. Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”). Analyze, compare, create, and compose shapes.4. Analyze and compare two- and three-dimensional shapes, in different sizes and

orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).

5. Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.

6. Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”

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First Grade Reason with shapes and their attributes.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.

2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.1

3. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

_________________1 Students do not need to learn formal names such as “right rectangular prism.”

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Second Grade Reason with shapes and their attributes.1. 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.

2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

_________________1 Sizes are compared directly or visually, not compared by measuring.

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Progression of Geometry

• Read and make note of the the main concepts of grades K, 1, and 2

• Identify how the concept changes and increases in rigor and understanding for the student

• What connections can you make to other domains/concepts?

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Van Hiele Levels

•Pierre van Hiele and Dina van Hiele-Geldof•Created a 5 level hierarchy of ways of understanding spatial ideas•Each level describes the THINKING PROCESS used in geometric contexts.•0-Visualization•1-Analysis•2-Informal Deduction•3-Deduction•4-Rigor

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Level 0: Visualization

• The objects of thought at Level 0 are shapes and what they “look like”.

• The appearance of the shape defines it• A square is a square “because it looks like a square” • Turn it…and it’s a diamond• Sorting will be based on…pointy, fat, dented in, or

“looks like a house”• The products of thought at level 0 are classes or

groupings of shapes that seem to be “alike.”

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Level 1: Analysis

• The objects of thought at level 1 are classes of shapes rather than individual shapes.

• Students are able to consider all shapes within a class rather than a single shape.

• They can think about what makes a rectangle a rectangle (four sides, opposite sides parallel, opposite sides of the same length, four right angles, congruent diagonals, etc.)

• The irrelevant feathers (size, color, orientation) fade into the background.

• The products of thought at level 1 are the properties of shapes.

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Level 2: Informal Deduction

• The objects of thought at level 2 are the properties of shapes.• Students are able to develop relationships between and

among properties of geometric objects• Able to reason…”If all four angles are right angles, the shape

must be a rectangle. If it is a square, all angles are right angles. If it is a square, it must be a rectangle.”

• Begin to focus on logical arguments about the properties• Engage in “if-then” reasoning• The products of thought at level 2 are relationships among properties of geometric objects.

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Deduction and Rigor

• Level 3: Deduction• The objects of thought at level 3 are relationships among

properties of geometric objects• The products are deductive axiomatic systems of

geometry• Level 4: Rigor• The objects of thought at level 4 are deductive axiomatic

systems for geometry• The products of thought at level 4 are comparisons and

contrasts among different axiomatic systems of geometry

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Mason Article

• As you read…• What stuck out or interested you?• What implications do the van Hiele levels have

on your planning and teaching?

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• As you watch consider: 1) What types of questions is Dr. Mason posing to the students?2) On what van Hiele level do you think student is functioning?

• 3) What evidence in the video suggests that level?

van Hiele Videos

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van Hiele Videos

• Video 1 http://coedpages.uncc.edu/abpolly/5301/coursedocs/van-hielle-vids/927.MPG

• Video 2 http://coedpages.uncc.edu/abpolly/5301/coursedocs/van-hielle-vids/928.MPG

• Video 3 http://coedpages.uncc.edu/abpolly/5301/coursedocs/van-hielle-vids/930.MPG

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K-5 Math Teaching Resources Website

• http://www.k-5mathteachingresources.com

Let’s Play the Barrier Game!• Play with your partner• What are the strengths of this strategy?• What experiences would they need before using

this type of game?• Brainstorm 5 more ways to use this strategy?

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Online Resources

• Great Video Library!• http://www.engageny.org• Games for concept practice• http://www.sheppardsoftware.com/• Lessons and activities• http://illuminations.nctm.org/

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Time To Explore!

• Find a partner.• Explore the Geometry tasks for K-2• Record “Things I Want to

Remember”• Ask questions

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Not quite…