engageny/eureka Math Parent Workshop, Session 3 Saratoga USD March 16, 2016

Transcription

1 engageny/eureka Math Parent Workshop, Session 3 Saratoga USD March 16, 2016

2 Outcomes 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 2

3 Agenda Geometry Standards Card Sort Van Heile Levels of Geometric Reasoning Geometry Domain, Grades K-5 Geometry Domain, Grades 6-8 Resources Closure 3

4 COHERENCE OVERVIEW 4

5 Instructional Shifts Combined Focus Coherence Rigor Shift 3: Fluency Shift 1: Focus Shift 2: Coherence Shift 4: Deep Understanding Shift 5: Application Shift 6: Dual Intensity

6 Coherence: Think Across Grades K HS Counting & Cardinality Number and Operations in Base Ten Number and Operations Fractions Operations and Algebraic Thinking Ratios and Proportional Relationships The Number System Expressions and Equations Functions Number & Quantity Algebra Functions Measurement and Data Geometry Statistics and Probability Geometry 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

7 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.

8 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. 8

9 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 threesided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes. 9

10 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. 10

11 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. 11

12 EngageNY Mathematics Overview PK-5 Curriculum Map: A Story of Units 12

13 engageny/eureka Math Geometry Modules (K-5) Grade K Module Title 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 13

14 Geometry Connections Geometric and spatial thinking o Connect 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

15 VAN HIELE LEVELS OF GEOMETRIC REASONING 15

16 Van Hiele Levels of Geometric Reasoning Level 0: Visualization Handout Level 1: Analysis (Description) Level 2: Informal Deduction Level 3: Formal Deduction Level 4: Rigor

17 Summary of the van Hiele Levels 17

18 Van Hiele Levels of Geometric 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.)

19 Van Hiele Levels of Geometric 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.)

20 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

21 Activity- Kinder

22 Van Hiele Levels of Geometric 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.)

23 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

24 Van Hiele Levels of Geometric 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.

25 Van Hiele Levels of Geometric Level 4: Rigor Reasoning Students see geometry in the abstract, even without concrete examples; they compare geometric results in different axiom systems (non-euclidean geometries).

26 Van Hiele Levels What resonates with you? What do you connect to based on your experience and your working with your child/children?

27 GEOMETRY PROGRESSION

28 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

29 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) 29

30 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? 30

31 Geometry in Kindergarten (understanding shapes and space) Match twodimensional shapes even when shapes have different orientations (variants) Distinguish shapes from non-examples 31

32 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 32

33 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.)

34 Geometry in Kindergarten Composition of geometric figures Compose shapes to build pictures and designs Solve problems using geometric motions (slides, flips, and turns) 34

35 engageny Sample K Task 35

36 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 36

37 Activity- Grade 1

38 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

39 Activity- Grade 1

40 engageny Grade 1 Sample Task 40

41 Geometry in Grade 2 Develop foundations for area, fraction and proportion by recognizing the composition of shapes as equal areas. 41

42 Let s Play! Activity- Grade 2

43 Geometry in Grade 2 Spatial Structuring- understanding how a rectangle can be tiled with squares in rows and columns 43

44 engageny Grade 2 Sample Task 44

45 Geometry in Grade 3 Students analyze, compare and classify 2D shapes by properties. They can classify using larger categories such as quadrilaterals. 45

46 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. 46

47 Engage NY example 47

48 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 48

49 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. 49

50 Geometry in Grade 4 Students recognize a line of symmetry and draw lines of symmetry. 50

51 engageny Grade 4 Sample Task 51

52 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. 52

53 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. 53

54 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. 54

55 Geometry in Grade 5 Students analyze and classify 2D shapes explicitly by their properties and in hierarchies. 55

56 engageny Grade 5 Sample Task 56

57 CONNECTIONS TO MIDDLE SCHOOL MATH 57

58 Transition to Middle School Geometry Areas of Elementary Geometry Geometric skills and knowledge in MS Area and Volume Shape composition and decomposition and spatial structuring of rectangular arrays Identifying line segments, parallel and perpendicular lines Context for developing and using equations, find the volume of right, rectangular prisms with fractional edge lengths. 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. Identify relationships of line segments and angles within a figure Describe cross sections of 3D figures Shape composition and decomposition Making scale drawings 58

59 Middle School Geometry Tasks Grade 6 59

60 Middle School Geometry Tasks Grade 7 60

61 Middle School Geometry Tasks Grade 8 61

62 HELPING STUDENTS WITH MATH (SPECIFICALLY WITH GEOMETRIC THINKING) 62

63 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. 63

64 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)? 64

65 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) 65

66 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? 66

67 Questions to Clarify Understanding of Explanation Do you understand your solution? Can you explain what you re thinking? S

68 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

69 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

70 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 70

71 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 71

72 Thank you for your time and participation! Bernadette A. Salgarino, Ed.D. 72