Building a Bridge Between

Arithmetic and Algebra

~ Algebra in MAG ~

Heather ZetterbergMath Specialist

What thoughts do the word “Algebra” conjure

up?

What thoughts do the word “Algebra” conjure

up?• Success• Meaning• Memorization• Formulae• Equations• Terror• Unknown

• Joy• Bewilderment• Variables• Struggle• Despair• Pleasure• Puzzle

What is Algebra?

• A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set

– American Heritage Dictionary

The “father” of AlgebraAbdallāh Muḥammad ibn Mūsā al-

Khwārizmī • Wrote a book

called “Kitab al-jabr wa al-muqabalah”

• Book became the first text book on the subject of Algebra in Europe

• “Al-jabr” became Algebra or “reunion of broken parts”

Elementary Algebra…

• Elementary algebra is not watered down Middle School and High School algebra

• Elementary algebra is about building a solid foundation of understanding and skills so students can be successful in their later, formal study of algebra

Raising the BarShifting Attitudes

• Need to shift from finding the answer to thinking about the numerical relationships underlying the calculations students perform

• Need to develop thinkers who know what they are doing and why

• Need to guide students to see relationships in their answers and make generalizations

Big Ideas of Early Algebra

• Use mathematical models to represent, understand, and analyze quantitative relationships

• Understand and describe patterns and functional relationships

• Use operations, properties, and algebraic symbols to determine equivalence and solve problems (including finding unknown values)

Use Models to Represent Quantitative

Relationships• Model situations (using objects,

pictures, and symbols) that involve operations of whole numbers– Addition, Subtraction, Multiplication

• Model real-life situations that involve operations with whole numbers

• Describe and analyze qualitative and quantitative changes

Understand Patterns and Functional Relationships

• Sort, classify, and order objects by a variety of attributes

• Recognize, describe, create, and extend patterns and sequences

• Identify missing objects or numbers• Analyze how repeating and growing

patterns are generated

Patterns and Sequences

• Encourage children to– Look for the order in situations– Make conjectures– Predict beyond the information at hand– Develop and test generalizations

• Numerical growth sequences contribute to the development of number sense

Working with Patterns, Sequences, and

Functions

See Sample Activity Handout pages 1-16

Properties and Algebraic Symbols

• Illustrate principles and properties of operations

• Demonstrate an understanding of balance or equivalence

• Use concrete, pictorial, verbal, and numerical representations to demonstrate an understanding of symbolic notations

Properties andThe Concept of

Equivalence• 2 + 7 can be thought of at as 7 + 2• (2 + 7) + 3 can be thought of as (7 + 3) + 2• The role of the equals sign (=)

– For young children, it’s a signal to write an answer

– “Seven plus two makes nine”• Trouble when given 7 + = 9

– Many children will record 16 in the square.

Equivalence (cont.)

• Children will agree that 2 + 7 = 9• Children will insist 9 = 2 + 7 is incorrect• Children have trouble with the following

2 + 7 = + 4• All of these difficulties stem from a

child’s lack of understanding that the equals sign indicates equivalence between two quantities

Inequalities

• Compare quantities• Identify quantities as equivalent or

non-equivalent• Using symbols

Less than <Greater than >Not equal to ≠

Confusion with Symbols…

Working with Symbols,Properties, and

Equivalence

See Sample Activity Handout pages 17-35

Before You Know it…

There May Come a Day…