Algebra in Grades 6-8: Mystery in a Bag!

1. Algebra in Grades 6-8 “It is essential for students to learn algebra as a style of thinking involving the formalization of patterns, functions, and generalizations, and as a set of competencies involving the representation of quantitative relationships.” (Math Matters, p. 190)

2. Mystery in a Bag! There are several pattern blocks in each mystery bag Use the clues to determine the shape and number of each type of block in the bag 2

3. Mystery Bag #1 There are 3 colors of blocks in the bag ? ? ? ? ? ? 3

4. Mystery Bag #1 There are 3 colors of blocks in the bag The area of all the blocks is the same as the area of 18 green blocks ? ? ? ? ? ? 4

5. Mystery Bag #1 There are 3 colors of blocks in the bag The area of all the blocks is the same as the area of 18 green blocks The area of the red blocks is equal to 1/3 the of all the blocks ? ? ? ? ? ? 5

6. Mystery Bag #1 There are 3 colors of blocks in the bag The area of all the blocks is the same as the area of 18 green blocks The area of the red blocks is equal to 1/3 the of all the blocks The numbers of blocks for the 3 colors used are consecutive whole numbers ? ? ? ? ? ? 6

7. Mystery Bag #2 There are 7 blocks in the bag ? ? ? ? ? ? 7

8. Mystery Bag #2 There are 7 blocks in the bag 50% of 1 more than the total number of blocks are green ? ? ? ? ? ? 8

9. Mystery Bag #2 There are 7 blocks in the bag 50% of 1 more than the total number of blocks are green The blocks can be arranged to form two congruent figures, with exactly one right angle ? ? ? ? ? ? 9

10. Mystery Bag #2 There are 7 blocks in the bag 50% of one more than the total number of blocks are green The blocks can be arranged to form two congruent figures, with exactly one right angle There are no orange blocks in the bag ? ? ? ? ? ? 10

11. Focus on the Algebra Algebra involves… Recognizing relationships, particularly patterns and functions Generalizing relationships Communicating relationships with symbols 11

12. The Language of Algebra Look at the clues for each mystery bag Underline or circle phrases that were helpful in identifying relationships Why were these particular phrases helpful? Can we rewrite any of these phrases using symbols? Why is this important? 12

13. The Language of Algebra Which mathematical phrases sometimes pose difficulty for students to translate into symbols? Why is this? Students often ask, “Why do I have to learn algebra, I’m never going to use it!” How can teachers help students understand that algebraic reasoning is used everyday? 13

14. Algebraic Thinking Generalizing – about mathematical relationships Formalizing – ideas with mathematical language and symbols Justifying – mathematical claims, i.e. being able to explain how you know something is true …and we can do these things in all mathematics lessons, not just algebra 14

15. Inequalities Card Game Have one person from your group come get 1 deck of cards Divide the inequality cards among the group Taking turns, read each problem aloud, decide what the variable represents, and match the problem with an inequality 15

16. Inequalities Card Game For what grade level do you think this task is appropriate? What experiences would be important for students to have had in order to complete the task? 16

17. Inequalities Card Game What role could this task play in your algebra instruction? What goals would you hope this achieved? 17

18. Practice Makes Perfect Practice that engages students can be meaningful! 18

19. Big Ideas in Algebra 19

20. Big Ideas in 6-8 Algebra Variables can have different meanings depending upon their use as quantities that vary and change, as a specific unknown value, or as quantities that vary in relation to one another 20

21. Big Ideas in 6-8 Algebra A variety of representations (including tables, charts, graphs, number lines, expressions, equations, and inequalities) can be used to illustrate mathematical relationships, to model mathematical situations, or to describe and generalize patterns 21

22. Big Ideas in 6-8 Algebra Functions are rules that associate each member of one set with exactly one member of another set (i.e. one variable is defined in terms of the other Different representations provide a variety of ways to view functions; the usefulness of a particular representation depends on its intended purpose 22

23. Big Ideas in 6-8 Algebra The understanding of proportional reasoning and rates of change promotes algebraic thinking and development 23

24. Variables Variables can have different meanings depending upon their use as quantities that vary and change, as a specific unknown value, or as quantities that vary in relation to one another 24

25. Consider This… There are 3 feet in a yard If F represents feet and Y represents yards, which of the following is true? 3F = Y Or F = 3Y 25

26. Questions Which relationship did you choose and why? When numbers are substituted (F=3 and Y=1), which statement is correct? How would you describe the use of variables in this instance? ??? 26

27. Common Uses of Variables As a specific unknown To represent a set of numbers To illustrate general properties Joint variation 27

28. Variable as an Unknown Sometimes referred to as a missing value Used in mathematical equations (students have been working with these since first grade: 2 + 3 = ) Can you think of a different example? 28

29. Representing a Set Here they can truly vary One example is x2 = 4 Can you think of a different example? 29

30. Illustrating Properties Here too, they can truly vary One example is the Commutative Property of Addition How do we show that one is true for all numbers? Can you think of a different example? 30

31. Building on Number Sense Operation properties can be developed through pattern recognition Generalizing these patterns helps us write mathematical relationships that are satisfied by an infinite amount of numbers Some think of algebra as generalized arithmetic 31

32. Joint Variation Variables that represent quantities that vary in relation to one another Here’s an example that is not a function: x2 + y2=100 Can you think of a different example? 32

33. The Concept of Equality What is an Equation? Can you give an example? Can you give a non-example? We want to build students’ conceptual understanding of equality to ensure their fluency with solving equations 33

34. The Balancing Act Young children may be offered balance scales as one representation of the balance that must exist between both sides of the “=“ sign Complete the Balancing Act I and II handout Discuss your strategy for solving the problem with others 34

35. In Your Small Groups Share strategies that were used to solve the problem Discuss the various strategies and talk about which ones seem to be helpful Are there strategies that are still confusing for you? Determine the most unique strategy used and be ready to share with the whole group 35

36. Conceptual Understanding What value is there in using problems such as the Balancing Act to reinforce the concept of equality? How can teachers use tasks such as this to connect conceptual understanding for solving equations and inequalities to “standard algorithms”? 36

37. The Balancing Act “If we do not provide a variety of experiences to build conceptual understanding of equality then…” 37

38. Equations So what is an equation again? One definition is that an equation is any mathematical statement that involves an “=“ sign 38

39. The Problem with “=“ In the equation 7 + 5 = + 4 most students choose: A ) 16 B ) 12 C ) 8 D )6 39

40. Discussion Questions Many students chose 12. Why do students choose to place 12 in the box? What basic understanding do students lack when they choose 12? Justify how students might arrive at each of the other choices (16, 8, 6) 7 + 5 = + 4 40

41. Patterns in Sequences What kinds of patterns do we show our youngest customers (in say, kindergarten)? What kinds of patterns do we show the “big” kids? Sequences can represent mathematical relationships; what are some particular types of relationships we might represent? 41

42. Some Terminology Repeating Pattern; repetition of a pattern unit Growing Pattern; pattern terms change according to a rule Arithmetic Sequence; a numerical pattern created by repeatedly adding a constant Geometric Sequence; a numerical pattern created by repeatedly multiplying by a constant 42

43. Is a Pattern Developing? Use the cards on your table Work at your table to organize the patterns into two groups: Repeating Patterns and Growing Patterns How did you decide which patterns belong in which group? What distinguishes repeating patterns from growing patterns? 43

44. Classifying Sequences Working with the same patterns re-sort these patterns into three groups: Arithmetic Sequences, Geometric Sequences, or Neither Why is it important for students to be able to differentiate these types of patterns? 44

45. From Patterns to Functions We can build upon students’ variety of experiences with patterns in elementary school Generalizing numerical patterns and sequences lays the foundation for understanding functions and the use of variables in joint variation 45

46. Assembling Cubes Let’s build sequences from geometric relationships! You are a quality control manager in a cube building company You are in charge of ensuring that your team has the proper number and type of individual cubes used to produce larger cubes 46

47. Assembling Cubes Your team makes large blue cubes that are made of small individual cubes The individual cubes come to your team with various faces painted blue Some will have only one face painted blue, while others will have two or three faces painted blue and some will have no faces painted blue 47

48. Assembling Cubes Your team puts together cubes that are… 2 units on each edge, 3 units on each edge, 4 units on each edge, and 5 units on each edge Make a chart that your team can use to show the correct number and type of small cube needed to make each large cube 48

49. Discussion Questions What patterns emerged from your chart? Can you see patterns as you look both vertically and horizontally? Which patterns are easier to recognize, those patterns that are vertical or those that are horizontal? Write a general rule that will allow you to find the number of cubes needed for each type for any given size large cube. 49

50. Before Formalization… Even students who are not ready to formalize relationships with variables can be successful with some or part of this activity Students will notice patterns and can generalize those patterns with words instead of symbols 50