Geometry and Algebra: Powerful When Together

1. Mix It Up Summer 2014 Geometry and Algebra:Powerful When Together

2. Goals for ALL Students National Council of Teachers of Mathematics (NCTM) Standards: • Learn to value mathematics • Become confident in their ability to do mathematics • Become mathematical problem solvers • Learn to communicate mathematically • Learn to reason mathematically Mix It Up Summer 2014

3. Geometry • Spatial sense – an intuition about shapes and the relationships among shapes • A “feel” for the geometric aspects of surroundings and shapes formed by the objects in the environment • Ability to mentally visualize objects and spatial relationships (to turn things around in your mind) • Geometry comprised of… • Shapes and their properties • Transformations • Location • Visualization Mix It Up Summer 2014

4. 3-D and 2-D Shapes and Figures • Laurie’s lesson on 3-d, name and describe a few • 2-d shapes, name and describe a few • Quadrilaterals – name and describe a few • Are these quadrilaterals? Mix It Up Summer 2014

5. Relationships Square Rectangles Trapezoids Parallelograms Quadrilaterals Mix It Up Summer 2014

6. Relationships Mix It Up Summer 2014

7. Area and Perimeter • What would your students say if you asked them to describe • Perimeter • Area • Do students confuse the two concepts? • Should the concepts be taught together? Mix It Up Summer 2014

8. Activity-area and perimeter • You will need: • Exactly 36 color tiles • Paper • Grid paper • Chart paper • With partner, create all possible rectangles with the 36 tiles Rectangle Not a Rectangle • Are these two different rectangles? • For today let’s consider both orientations when completing problem Mix It Up Summer 2014

9. Activity-area and perimeter • On chart paper with four-quadrant fold, illustrate your pattern with • a drawing • a table • a graph • a verbal description of maximum perimeter Mix It Up Summer 2014

10. Activity-area and perimeter • You will need: • Color tiles • Paper • Grid paper • With partner, create all possible rectangles with the 36 tiles Rectangle Not a Rectangle • Are these two different rectangles? • For today let’s consider both orientations when completing problem Mix It Up Summer 2014

11. Activity-area and perimeter • On chart paper with four-quadrant fold, illustrate your pattern with • a drawing • a table • a graph • a verbal description of maximum area Mix It Up Summer 2014

12. Area and perimeter • What did you notice about the perimeter when the area is fixed? • What did you notice about the area when the perimeter is fixed? • How will this help students understand difference in perimeter and area? Mix It Up Summer 2014

13. Area and perimeter • What did you notice about the perimeter when the area is fixed? • What did you notice about the area when the perimeter is fixed? • How will this help students understand difference in perimeter and area? Mix It Up Summer 2014

14. Levels of Geometric Thought • Van Hiele’s • Husband & wife • Dutch educators • Studied many students thinking while solving geometric tasks • Five hierarchal levels of geometric thinking • Visual Level – level 0 • Descriptive Level – level 1 • Relational Level – level 2 • Deductive Level - level 3 (usually high school) • Rigor – level 4 (usually college math major level) Mix It Up Summer 2014

15. Visual Level Characteristics (Level 0) The student • identifies, compares and sorts shapes on the basis of their appearance as a whole • For example, students recognize rectangles by its form but, a rectangle seems different to them then a square • At this level rhombus is not recognized as a parallelogram • solves problems using general properties and techniques (e.g., overlaying, measuring) • uses informal language • does NOT analyze in terms of components Mix It Up Summer 2014

16. Analysis/Descriptive Level Characteristics Level 1) The student • recognizes and describes a shape (e.g., parallelogram) in terms of its properties. • discovers properties experimentally by observing, measuring, drawing and modeling. • uses formal language and symbols. • does NOT use sufficient definitions. Lists many properties. • does NOT see a need for proof of generalizations discovered empirically (inductively). Mix It Up Summer 2014

17. Informal Deduction (Relational) Level Characteristics (Level 2) The student • can define a figure using minimum (sufficient) sets of properties. • gives informal arguments, and discovers new properties by deduction. • follows and can supply parts of a deductive argument. • does NOT grasp the meaning of an axiomatic system, or see the interrelationships between networks of theorems. • A square is a rectangle because it has all the properties of a rectangle • Students can conclude the equality of angles from the parallelism of lines: In a quadrilateral, opposite sides being parallel necessitates opposite angles being equal Mix It Up Summer 2014

18. Informal Deduction (Relational) Level Example Show that the sum of the angles in a triangle is 180° Find area of a triangle based on what you know about the area of a parallelogram Mix It Up Summer 2014

19. Deductive Level Characteristics (Level 3) – High School Geometry • Students prove theorems deductively and establish interrelationships among networks of theorems in the Euclidean geometry • Thinking is concerned with the meaning of deduction, with the converse of a theorem, with axioms, and with necessary and sufficient conditions • Students seek to prove facts inductively • It would be possible to develop an axiomatic system of geometry, but the axiomatics themselves belong to the next (fourth) level Mix It Up Summer 2014

20. Rigor Characteristics (Level 4) – Mathematics Majors • Students establish theorems in different postulational systems and analyze/compare these systems • For example, students can compare Euclidean geometry (plane geometry) with spherical geometry • Figures are defined only by symbols bound by relations • A comparative study of the various deductive systems can be accomplished • Students have acquired a scientific insight into geometry Mix It Up Summer 2014

21. Major Characteristics of the Levels • Levels are sequential and hierarchal • What is implicit at one level becomes explicit at the next level • Material taught to students above their level is subject to reduction of level • Progress from one level to the next is more dependant on instructional experience than on age or maturation • One goes through various “phases” in proceeding from one level to the next Students do not automatically progress; they gain abstraction and sophistication in their thinking as a result of their experiences. Mix It Up Summer 2014

22. Characteristics of Levels of Thinking • Students may reason at multiple levels or at intermediate levels. • Children advance at different rates for different concepts. • Each level has its own language/vocabulary, set of symbols, and network of relations. • Terms that are considered correct at one level may be modified at another. • Instruction and language at a level higher than the student may inhibit learning. Mix It Up Summer 2014

23. Levels of Geometric Thinking • Looking back to the levels of thinking, what level was the fixed area/perimeter activity? • Discuss with your table partners. • Be ready to share with class and support your thinking. Mix It Up Summer 2014

24. The TEKS • Let’s look at the TEKS • For the perimeter/area activity, create a chart describing • The grade level and TEKS the activity included • The prerequisite knowledge and skills (TEKS) that prepare students for the concepts in the activity • The level of geometric thinking involved in the specific TEKS • How to extend the activity to a higher grade level Mix It Up Summer 2014

25. Representations Revisited • Encourage students to develop and use multiple representations • Help them build connections across these representations so that they can transition from concrete to abstract ways of thinking. • “Seeing similarities in the ways to represent different situations is an important step toward abstraction” (NCTM 2000, 138). • Remember to ALWAYS move from • Concrete manipulative, • Pictorial/graphical, • Abstract (verbal/numerical) Mix It Up Summer 2014

26. Functions/Rules • Functions-the main concept in algebra in Texas • Describes how two (or more) quantities are related • Many times represented by tables and graphs • Example: number of legs that 7 dogs have Mix It Up Summer 2014

27. Recursive rule – each entry is found by adding subtracting, multiplying, dividing, etc. to previous entry (looks up or down a table) Explicit rule – uses a rule or equation based in the input term to determine the output (looks across the table) Mix It Up Summer 2014

28. Mix It Up Summer 2014

29. Using Pattern Blocks • Take out a yellow hexagon; Make a hexagon using the red trapezoids. How many trapezoids did it take to make a hexagon? • Make a larger hexagon with trapezoids. How many trapezoids did it take to make a hexagon? • Continue making the next 3 larger hexagons • Create a chart to illustrate. • Find a rule for the pattern. • State how many trapezoids it will take to make the 8th hexagon. Mix It Up Summer 2014

30. Using Pattern Blocks • Take out a yellow hexagon; Make a hexagon using the blue rhombus(es). How many rhombii did it take to make a hexagon? • Make a larger hexagon with rhombii. How many rhombii did it take to make a hexagon? • Continue making the next 3 larger hexagons • Create a chart to illustrate. • Find a rule for the pattern. • State how many rhombii it will take to make the 8th hexagon. Mix It Up Summer 2014

31. Using Pattern Blocks • Take out a yellow hexagon; Make a hexagon using the green triangles. How many trapezoids did it take to make a hexagon? • Make a larger hexagon with triangles. How many triangles did it take to make a hexagon? • Continue making the next 3 larger hexagons • Create a chart to illustrate. • Find a rule for the pattern. • State how many triangles it will take to make the 8th hexagon. Mix It Up Summer 2014

32. For your assigned pattern • On chart paper, illustrate your pattern • With a drawing • With a table • A graph • With a function (rule) • With a verbal description Mix It Up Summer 2014

33. Habits of Mind Mathematical Habits of Mind are productive ways of thinking that support the learning and application of formal mathematics. The learning of mathematics is as much about developing these habits of mind as it is about understanding established results of mathematics. Patti talked about the Science Habits of Mind Mix It Up Summer 2014

34. Mathematical Habits of Mind Geometric Habits of Mind • Reasoning with Relationships • Investigating Invariants • Generalizing Geometric Ideas • Balancing Exploration and Reflection Algebraic Habits of Mind • Doing-undoing • Building rules to represent functions • Abstracting from computation Mix It Up Summer 2014

35. Habits of Mind • Looking back activities, • What Algebraic Habits of Mind did the activities involve? • What Geometric Habits of Mind did the activities involve? • Discuss with your table partners. • Be ready to share with class and support your thinking. Mix It Up Summer 2014

36. How is mathematics taught in schools? Mix It Up Summer 2014 • Textbooks • Written with only the abstract phase which often coincides with the objective of the learning • Teachers mistakenly • Begin their teaching with this phase, a.k.a. “direct teaching” • Often present a definition to students rather than letting them discover it • Do not realize that their information cannot be understood by their pupils • Start instruction with revealing what students should discover through an activity • Teachers • Modeling correct use of terminology is essential

37. Where Is This In Schools? • Geometry Activities • Must be tiered so students first have exposure to shapes, begin sorting visually, and then talk about properties • Need appropriate experiences to advance • Inappropriate experiences inhibit learning • As students progress, geometry becomes more abstract. • Students must reach Van Hiele’s Levels 0, 1, and 2 in Grades K-8 • Research indicates that students may enter high school at Level 0 or 1 • Thus, they have great difficulty with formal proofs • The traditional form of teaching is time-efficient but not effective. • Students will have a firm grasp of geometry if they are allowed to “play” with the ideas and arrive at own conclusions. • Whole group discussion is important in clearing up misconceptions and presents alternate observations and understandings. Mix It Up Summer 2014

38. STOP HERE IN OCTOBER Mix It Up Summer 2014

39. Geometric Habits of Mind • In groups of four • Read your assigned papers • On chart paper, create an explanation of your assigned GHoM • Include examples • If your paper had a mathematics problem, solve and illustrate the problem (time permitting) • Relate to the mathematical levels of thinking (Levels 0-4) Mix It Up Summer 2014

40. Reasoning with Relationships • Actively looking for relationships within and between figures, such as congruence, similarity • Names properties of shapes • Constructs and deconstructs shapes • Reasons with symmetry • Asks questions concerning how alike, how different Internal questions include: • “How are these figures alike?” • “In How many ways are they alike?” • “How are these figures different?” • “What would I have to do to this object to make it like that object?” Mix It Up Summer 2014

41. Investigating Invariants • Invariant – remains unchanged as other tings vary. Could be … • Orientation • Location • Angles • Analyzing properties affected by a transformation • Performs transformations without prompting • Notices not all properties change during a transformation • Asks questions like • How did that get from here to there? • What changed? Why? • What stayed the same? Why? Mix It Up Summer 2014

42. Generalizing Geometric Ideas • Wants to understand and describe the “always” and “every” related to geometric concepts • Conjectures about every, always, and when • Tests conjectures • Makes convincing arguments to support conclusion • Uses one solution to generate another • Notices a rule that’s true for entire set of figures • Wonders what happens when context changes Asks questions like • “Does this happen in every case?” • “Why would this happen in every case?” • “Can I think of examples when this is not true?” • “Would this apply in other dimensions?” Mix It Up Summer 2014

43. Balancing Exploration and Reflection • Tries various approaches (often as result of hypothesis) • Regularly considers what worked, what learned • Modifies hypothesis • Asks questions like… • What would happen if I…? • What did the action or result tell me? Mix It Up Summer 2014

44. GHoM and Levels of Geometric Thinking • So far we have done two activities • Toothpick perimeter • Diagonals of quadrilaterals • What GHoM has each activity involved? • Discuss with your table mates and be prepared to defend your decisions Mix It Up Summer 2014

45. Analyze Activities • Go back to activities posted • What level of van Hiele is each activity? • Which Geometry Habits of Mind did you employ when doing activity? • What questions can you add to determine level? • How can you enrich activity to place on higher level? Mix It Up Summer 2014

46. Generalization: Goal of Algebraic Thinking • Algebra described as generalized arithmetic • Arithmetic effective in describing static situations • Algebra is dynamic and deals with how things change • Children can appreciate the significance of change and the need to describe and predict variation Mix It Up Summer 2014

47. Generalization: Goal of Algebraic Thinking Geometric Habits of Mind • Reasoning with Relationships • Investigating Invariants • Generalizing Geometric Ideas • Balancing Exploration and Reflection • The central goal of algebraic thinking • To get children to think about, describe, and justify what is going on in general with regard to some mathematical situation. • Children should be able to develop a generalization, a statement that describes a general mathematical truth about some set of data. • Three instructional strategies —representations, questioning, and listening—areall critical components in helping children build their own generalizations. Mix It Up Summer 2014

48. Three levels of Justification • Franke, and Levi (2003) describe three levels of arguments or justifications that children make: (1) appealing to an authority figure (a conjecture is true because “the teacher said so”) (2) looking at particular examples or cases (3) building generalizable arguments. What level of van Hiele’s? What GHoM? Mix It Up Summer 2014

49. Algebraic Thinking • Students use language to describe informal generalizations and connect symbols to the language, a process which produces formal (algebraic) generalizations • the language component is critical - students are required to articulate their own movement from concrete to abstract • Abstractions – beginning with the concrete enables students to then access higher levels of abstraction such as writing a linear function rule for the toothpick problem or generalizing properties of quadrilaterals/diagonals • The TEKS for mathematics require building and making connections among concrete, verbal, numeric, graphic, and symbolic representations of relationships between quantities Mix It Up Summer 2014

50. Trapezoid Table - perimeter • A restaurant has trapezoidal tables which seat five people. • If 2 tables are placed together as shown to form one table, how many people can be seated? • Using counters and the trapezoidal pattern blocks, create a concrete representation of this problem for 5 tables Mix It Up Summer 2014