CHAPTER 14Algebraic Thinking: Generalizations, Patterns, and Functions

ALGEBRA

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 Research suggests three strands of algebraic reasoning,

all infusing the central notions of generalization and symbolization.

1. Study of structures in the number system 2. Study of patterns, relations, and functions 3. Process of mathematical modeling, including the

meaningful use of symbols

CONNECTING NUMBERS AND ALGEBRA 1. Generalization with Operations – Looking for generalizations

begins early with decomposition of numbers. Example: 13 dogs are born. How many might be female and how many

might be male? Generalization can be analyzed when data is recorded in a table. Example: How many possibilities do we have for the 13 dogs problem? Example: What if we had n dogs? Example: What if we had 13 dogs and f of them are female? How many

males? Example: Consider 2 x 4 and 4 x 2. How are they different? How are they

the same? Will this always be true? Example: What about 35 x 52 versus 52 x 35? Example: What is true in general?

CONNECTING NUMBERS AND ALGEBRA 2. Generalization in the Hundreds Chart

What is 15

What is 63

Generalizing things like 3 + 1 + 1 – 1 – 1 to n + 1 + 1 – 1 – 1

Task Examples When skip counting, which numbers make diagonal patterns? Which make column patterns?

Can you describe a rule for explaining when a number will have a diagonal or column pattern? If you move down two or over one on the hundreds chart, what is the relationship between the

original number and the new number? Can you find two skip-count patterns with one color marker “on top of” the other (that is, all of

the shaded values for one pattern are part of the shaded values for the other)? How are these two skip-count numbers related? Is this true for any pair of numbers that have this relationship?

Find a value on the hundreds chart. Add it to the number to the left of it and the one to the right; then divide by 3. What did you get? Why?

CONNECTING NUMBERS AND ALGEBRA 3. Generalization Through Explore a Pattern

One of the most interesting and perhaps most valuable methods of searching for generalization is to find it in the growing pattern represented with visual or concrete materials.

The Border Problem. Make an 8 x 8 grid with unifix cubes in your group.

Now many a border around the perimeter with different color squares.

How many different ways can you count the number on the border WITHOUT counting them individually.

Ultimate Question: What is the generalization?