K-5 Algebra

1. K-5 Algebra

2. What is Algebra? Partners for Mathematics Learning

3. What Is Algebraic Thinking? Understanding patterns, relations and functions Representing and analyzing mathematical situations and structures using algebraic symbols Using mathematical models to represent and understand quantitative relationships Analyzing change in various contexts Source: Principles & Standards for School Mathematics, NCTM 2000

4. Principles and Standards for School Mathematics PSSM Algebra Standard: Understand patterns, relations and functions • Describe, extend, and make generalizations about geometric and numeric patterns • Represent and analyze patterns and functions, using words, tables, graphs

5. Principles and Standards for School Mathematics PSSM 8 + 4 + 6 = 5 + a Algebra Standard: Represent and analyze mathematicalsituations and structures using algebraic symbols • Identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers • Represent the idea of a variable as an unknown quantity using a letter or a symbol • Express mathematical relationships using equations 3 x 5 = 5 x 3 20 = 4 x

6. Principles and Standards for School Mathematics PSSM Algebra Standard: Use mathematical models to represent and understand quantitative relationships • Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions

7. Principles and Standards for School Mathematics PSSM Algebra Standard: Analyze change in various contexts • Investigate how a change in one variable relates to a change in a second variable • Identify and describe situations with constant or varying rates of change and compare them

8. Big Ideas in Algebra • Mathematical situations can be analyzed through patterns, functions, and other relationships; those mathematical situations can be represented in multiple ways (words, pictures, numbers, symbols) • Equations and inequalities are used to express relationships between quantities

9. Big Ideas in Algebra • Unknown quantities can be represented using variables that are letters or symbols • Operation properties are generalized guidelines for operating with numbers

10. Exploring Algebraic Thinking

11. Patterns Use SCOS to review Algebra objectives and relate to the Big Idea of Patterns

12. Woozles • What are the characteristics or attributes of a Woozle? • Sort through the Woozle cards to observe the attributes of the Woozle creatures • How can you use the attributes to determine how many different Woozles can be created?

13. Woozles • Select one attribute to focus on but do not reveal the “secret” attribute to your group • Place the cards that match your attribute inside the Venn • Allow group members to guess your attribute • Repeat with two rules and a double Venn diagram

14. Sorting and Classifying • Recognizing attributes allows students to sort and classify • Attributes describe particular properties which some objects have in common • Attributes describe the relationships of size, shape, color, texture, and other properties • Attributes describe ways materials can be sorted

15. Key to mathematical thinking and understanding Essential for making generalizations, seeing relationships, and understanding the logic and order of mathematics Fundamental to number systems Patterns 3 6 9 12 15 18 21 24

16. “Looking for patterns trains the mind to search out and discover the similarities that bind seemingly unrelated information together in a whole. A child who expects things to ‘make sense’ looks for the sense in things and from this sense develops understanding. A child who does not see patterns often does not expect things to make sense and sees all events as discrete, separate, and unrelated.” --Mary Baratta-Lorton Patterns

17. Create a repeating pattern Patterns

18. Simplest types of pattern Concrete (hat, glove, hat, glove) Oral (Old MacDonald) Physical (clap, snap, clap, snap) Combination of oral and physical (Hokey Pokey) Pictorial or Symbolic (wall paper) The core of the pattern (pattern unit) is the shortest string of elements that repeats Repeating Patterns

19. A a a B a A A b a A a B clap, clap, snap, clap, clap, snap, clap, clap, snap 1, 2, 3, 3, 1, 2, 3, 3, 1, 2, 3, 3 robin, dog, shark, wren, cat, flounder, cardinal, horse, goldfish, bluejay, pig… Repeating Patterns

20. Vocabulary for Patterns • In your copy of the SCOS, circle the verbs related to patterns that students must know • Are there other words or phrases that are important?

21. Create a growing pattern Patterns

22. Involve a progression from step to step Students must recognize the changes from one term to the next to determine the next term and begin the process of generalization Growing Patterns

23. clap, snap, clap, clap, snap, clap, clap, clap, snap 2, 6, 12, 20, 30… Growing Patterns

24. Arithmetic Patterns (Arithmetic Sequences) Consecutive terms have a common difference 8,12,16,20,24,… 5,7,9,11,13,15,… 25,22,19,16,13,… Geometric Patterns (Geometric Sequences) Consecutive terms have a common ratio or a common multiplier 2,4,8,16,32,… 1,4,16,64,256,… 200,100,50,25,… Growing Patterns

25. 1 yr. old 2 yr. old 3 yr. old Patterns Describe the pattern

26. Relationships can be described mathematically with pairs of numbers Function - set of ordered pairs in which no two pairs have the same first number Ways to describe a function: • table • formula • graph 1 yr. old Functions 2 yr. old 3 yr. old

27. Functions

28. Build these figures What comes next? Growing Triangles Figure 1 Figure 2 Figure 3

29. Describe the pattern Extend the pattern Figure 7____ Figure 12____ Growing Triangles

30. Figure 1 Figure 2 Figure 3 Growing Rhombuses What is your rule?

31. Label the y-axis as directed and plot the ordered pairs from the T-chart What do you notice about the points? Could (9,21) be a point on this graph if you follow the rule? Growing Rhombuses

32. Recursive Pattern Recursive Pattern +4 +4 +4 +4 +4 +2 +2 +2 +2 +2 Function Tables Rule: 4F Rule: 2F + 1

33. Function Tables Describe the pattern 2. Make a T-chart 3. What are the next two figures? 4. What is your rule?

34. Function Tables • What patterns do you see in the T-chart? • Create a visual representation for the progressive differences

35. Visual Representation of Differences Function Tables Figure 1 Figure 2 Figure 3 Figure 4 +4 +6 +8

36. Visual Representation of Generalization Function Tables Figure 1 Figure 2 Figure 3 Figure 4 Rule: n + n 2 n x n + n n(n + 1)

37. Functions Use SCOS to review Algebra objectives and relate to functional relationships, again noting verbs in the objectives

38. Function Tables What is the rule?

39. Multiple Representations Table Algebraic Formula Graphs Verbal Description Concrete or Pictorial Mari Muri, 2006

40. Equality =

41. 8 + 4 = + 5 Equality • What number would you put in the box to make this a true number sentence? • How would the students in your class respond to the question above?

42. Student Responses:30 Typical Elementary Classrooms Carpenter et al, p.9

43. Video Discussion Points: • What misconceptions do students have of equality and the equal sign? • What are the implications of these misconceptions and how do these misconceptions impact instruction in the classroom?

44. Equality Use SCOS to review Algebra objectives and relate to the Big Idea of Equality

45. True/False Number Sentences “True/false and open number sentences have proven particularly productive as a context for discussing equality. These number sentences can be manipulated in a variety of ways to create situations that may challenge students’ conceptions and provide a context for discussion.” Thomas Carpenter, Megan Franke, & Linda Levi Thinking Mathematically, p.15

46. True/False Number Sentences 3 + 5 = 8 3 + 5 = 8 = 3 + 5 8 = 3 + 8 = 8 8 = 3 + 5 = 3 + 5 3 + 5 = + 5 3 + 5 = 5 + 3 3 + 5 = + 3 3 + 5 = 4 + 4 3 + 5 = + 4

47. Balancing Equations 8 + 4 = + 5 9 + 7 = + 8 1 + = 8 + 3 9 + = 15 12 = x 3 4 x 4 = 8 x

48. Balancing Act What do you know about the objects on the scales?

49. Balancing Act

50. Balancing Act