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

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.

Strands of Algebraic Thinking Maria 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

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 49 + 18 can be found with the use of the hundreds chart. Moves on the chart can be recorded with arrows for up and down.

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 12 = 1 x 12 = 3 x 12 = 3 x 12 2 4 8 4 8 Solve and focus on what they notice. How are the problems alike and different? How does the difference in the problem affect the answer?

Structure in Number System: Properties

Structure in Number System: Properties

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

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 = 0 7 + 3 – 10 = 0 Follow- up with record five different ways to get your number Discuss what was true about all of the problems they wrote.

Patterns Rationale “Algebraic thinking or algebraic reasoning involves forming generalizations from experiences with number and computation, formalizing these ideas with the use of a meaningful symbol system, and exploring the concepts of pattern and function” (Van de Walle, Karp, & Bay-Williams, 2013, p. 258). “Patterns serve as the cornerstone of algebraic thinking” (Taylor-Cox, 2003, p. 15)

Repeating Pattern Examples Uno, dos, uno, dos, . . . Do, do, re, do, do, re, . . . A, B, A, C, A, B, A, C, . . . . . .

Repeating Pattern Examples A, B, B, A, A, B, B, ____ , ____ , B, ____ , A (Look up Romans 8:15 in the Bible, if you’d like.) clap, clap, stomp, clap, clap, _______ , clap, _______ , stomp, _______ , . . .

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

Growth Pattern Examples 1996, 2000, 2004, 2008, 2012, . . . When is the next summer Olympics? What were the host cities for these Olympics? And is the year 2030 a summer Olympic year? What about 2060? What pattern is associated with Winter Olympic years? Start with 2002.

Functional Thinking Rationale Blanton and Kaput (2005) wrote, “elementary teachers must develop ‘algebra eyes and ears’ as a new way of both looking at the mathematics they are teaching and listening to students’ thinking about it” (p. 440). They added, “Generalized arithmetic and functional thinking offer rich (and accessible) entry points for teachers to study algebraic reasoning” (p. 440).

Study of Patterns and Functions cont. Explore patterns that involve a progression from step to step (sequences) Look for a generalization or algebraic relationship between the step or term number and the number of tiles, dots, or the fraction. Geometric and algebraic patterns go “hand in hand” and often lead to functions. Include fractions, decimals, and integers in patterns and functions.

Study of Patterns and Functions cont. Three types of patterns Recursive-how a pattern changes from step to step, as in adding 1 repeatedly on the step number side (x) and as in adding 3 on the number of tiles (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 (x) to generate the output (y): ___ · x + ___

Study of Patterns and Functions cont.

Functional Thinking Examples The following table involves a number of dogs (D) and a corresponding number of legs (L). Find a function rule relating the two variables. (a) Find the missing value. (b) What type of number pattern or sequence is this in the “L row”? _________________________ (c) Find an equation relating L and D. L = ___________________ D 1 2 3 4 L 8 12

Linear Functions cont. Proportional Represented in these three growing patterns Non-proportional One value is constant For example, the “T” pattern, the pen problem (length, area),

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 y = 3x + 1

Study of Patterns and Functions cont. Growing patterns represented by 1. Physical models 2. Tables 3. Words (Verbal Descriptions) 4. Symbolic equations 5. Graphs

Linear & Quadratic Functions

Growth Pattern Examples How many blue tiles and how many white tiles are in the nth pattern? (from Billstein, Section 4.1) Graph the relationship between term number and white tiles; term number and blue tiles. Complete the table. Then find the rule for the relationship between x and y. Graph this relationship. x 1 2 3 4 y 8

Meaningful Use of Symbols In algebra, symbols represent real situations and are useful tools in representing situations. Equivalence and equal and inequality signs — help us understand and symbolize relationships in our number system, showing how we mathematically represent quantitative relationships — It is very nice to conceptualize equations as a balance and inequalities as a tilt. — Reinforce that equal means “is the same as”.

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

Relational Thinking Operational view-The equal sign means “do something”: 6 + 7 = ____ ____ = 7×8 Relational-computational view-The equal sign symbolizes a relation between two calculations 7 + n = 6 + 9 Since 6 + 9 is 15, I have to figure out 7 plus what equals 15. Relational-structural view-The equal sign signifies a numeric relationship 7 + n = 6 + 9 Since 7 is one more than 6 on the other side, n should be one less than 9.

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:

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

Using Expressions and Variables

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

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

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