1. CHAPTER 14 Algebraic Thinking: Generalizations, Patterns, and Functions
Elementary and Middle School Mathematics Teaching Developmentally
Ninth Edition
Van de Walle, Karp and Bay-Williams
Developed by E. Todd Brown /Professor Emeritus University of Louisville

2. Big Ideas
Algebra is a useful tool for generalizing arithmetic and representing patterns in our world.
Methods we use to compute and structures in our number system can and should be generalized.
Symbols, especially involving equality and variables, must be well understood conceptually for students to be successful in mathematics.
Understanding of functions is strengthened when they are explored across representations (e.g. equations, tables, and graphs).Each mathematical model provides a different view of the same relationship.

3. Strands of Algebraic Thinking
Blanton, 2008; James Kaput- 2008
Research suggests three strands of algebraic reasoning, all infusing the central notions of generalization and symbolization.
Study of structures in the number system
Study of patterns, relations, and functions
Process of mathematical modeling, including the meaningful use of symbols

4. Connecting Number and Algebra
Number Combinations- looking for generalizations begins early with decomposition of numbers. For example Generalization can be analyzed when data is recorded in a table. Place Value- Sum of can be found with the use of the hundreds chart. Moves on the chart can be recorded with arrows for up and down.

5. Connecting Number and Algebra cont.
Algorithms- Sets of problems are good ways for students to look for and describe patterns across problems For example 1 x 12 = 1 x 14 = 1 x 12 = 3 x 12 = 3 x 12 = Solve and focus on what they notice. How are the problems alike and different? How does the difference in the problem affect the answer?

6. Structure in Number System: Properties

7. Applying the Properties of Addition and Multiplication
At the heart of what it means to do mathematics is
Noticing generalizable properties 1+6 = 6+1
Distributive property central to basic facts 6 x 8 can be split up (5 x 8) + (1 x 8)
Justifying a conjecture
provides exploration
of the number system

8. Try this one Activity 14.2 Five Ways to Zero
Materials- deck of cards, counters and number lines Directions- Draw a card and record five different ways to get to 0 using number sentences. For example if you have a 7 7 – 5 – 2 = – 10 = 0 Follow- up with record five different ways to get your number Discuss what was true about all of the problems they wrote.

9. Study of Patterns and Functions
Repeating patterns
— identifying patterns that have a core that repeats
— looking for patterns in number (place-value)
_ repeating patterns in seasons, days of week, simple singing do, mi, mi, do, mi,mi

10. Study of Patterns and Functions cont.
Growing patterns
Explore patterns that involve a progression from step to step (sequences)
Extend or identify the core
Look for generalization or algebraic relationship
Geometric good exemplars to see the pattern and students can manipulate the objects.
Include fractions and decimals

11. Study of Patterns and Functions cont.
Three types of patterns Recursive- how a pattern changes from step to step, x to y Covariational thinking- how two quantities vary in relation, as step increases by 1, number of tiles in T goes up by 3 Correspondence relationship- look across the table to see how to use the input to generate the output 3x + 1

12. Study of Patterns and Functions cont.
Growing patterns represented by
Physical models
Tables
Words
Symbols
Graphs

13. Study of Patterns and Functions cont.

14. Try this one Activity 14.9 Sketch a Graph
Materials- Graph paper
Directions- sketch a graph to match the following situations.
The temperature of a frozen dinner 30 minutes before removed from the freezer until removed from the microwave and placed on the table.
The value of a 1970 Volkswagen Beetle from the time it was purchased to the present.
The level of water in the bathtub from the time you begin to fill it to the time it is completely empty after your bath.
Profit in terms of number of items sold
The height of a thrown baseball from when it is released to the time it hits the ground
Speed of the same baseball

15. Linear Functions
Linear functions a subset of function Rate of Change and Slope

16. Linear Functions cont.
Proportional Represented in these growing patterns Non-proportional One value is constant For example; pattern block perimeter problem no matter what step number you are on there are two units (one on each end) that must be added.

17. Meaningful Use of Symbols
In algebra symbols represent real situations and are useful tools in representing situations
Equivalence and equal and inequality signs
— understand and symbolize relationships in our number system
— how we mathematically represent quantitative relationships
_ conceptualize as a balance
_ Reinforce that equal means “is the same as”

18. Try this one Activity 14.15 True or False
Materials- Equations Directions- More challenging problems use fractions, decimals, and larger numbers.

19. Relational Thinking
Operational view- equal sign means “do something” = ____ Relational- computational view- equal sign symbolizes a relation between two calculations 7 + n = is 15 so I have to figure out 7 plus what equals 15 Relational-structural view- equal sign signifies a numeric relationship 7 + n = is one more that 6 on the other side, so n should be one less than 9

20. Meaning of variables Variables can be used to represent
A unique but unknown quantity
A quantity that varies
Example of a variable used as an unknown:

21. Meaning of variables cont.
TRY THIS ONE Examples of problems with multiple scales with quantities and variables that may vary.

22. Using Expressions and Variables

23. Meaning of Variables cont.
Number line used
to build understanding
Table used to guide thinking and recording variables

24. Mathematical Modeling
Creating equations to describe a situation is an important skill.