Algebraic Thinking

• Algebraic Thinking

Algebraic Thinking

Chapter 14

Define Algebra

Algebra

1. A study of patterns and relationships
2. A way of thinking
3. An art
4. A language that uses carefully defined terms and symbols
5. A tool

Algebraic Thinking in Elementary Education 

• Discussion and questioning
  
• Helping students represent problems, patterns. and relationships
• Encouraging students to generalize
• Expecting students to justify their thinking and statements

Problems

Routine Problems

• Exercises for practicing computations

Nonroutine Problems

• Abstract problems such as "look-for-a-pattern problems,"

"number puzzles," etc.

Before her birthday, Jane had 8 toy trucks. At her birthday party, she was given some more trucks. That night when she recounted, she found that she had 15 trucks. How many trucks were given at her party?

Type the sentence you used and explain your process.

Same number:
Pick a number. Add 4. Double the sum. Subtract 6. Divide by 2. Subtract 1.
What did you get? Why does this work?

Patterns

Helps students organize their world and understand math

Having students explain their thinking is crucial 

• Helps children consider alternative ways of thinking
• Helps students solidify their understanding
• Builds flexibility in student thinking

Encourage students to look at patterns in different ways that would lead to different ways of extending the pattern.

Do NOT expect students to see patterns the way you do

1, 2, 4....
What are the next two numbers? Why?

Relations

Two Types

Properties of numbers

• Rules about math processes
• Communicative, associative, distributive, and identity

Functions

• A way of expressing a relationship using numbers and symbols

Which statement is not true?

1. If you add a number to a given number and then subtract that number from the sum, then you get that given number
2. If you subtract 0 from a given number and add the given number to the difference, then the sum is twice the given number.
3. If you multiply any two numbers, the product is larger than each of the two numbers.
4. If you divide a positive whole number by a proper fraction, the quotient is larger than the positive whole number.

Language and Symbols of Algebra

Expression: Representing a phrase

• a + 3

Equation: Representing a complete sentence

• Open sentence: n + 3 = 7 or _ + 3 = 7
• Closed Sentence: 4 + 3 = 7 

Children have difficulty solving equations because they do not understand the structure of the equations with open sentences or an equation using a variable. 

Closed Sentence	Open Sentence	Generalizations (Math Properties
79+0=79	37+__=37	When you add 0 and any number, you get that number
0x54=0	0x__=0	When you multiply and number and 0, you get 0
33-0=33	83-__=83	When you subtract 0 from any number, you get that number
67-67=0	3456-__=0	When you subtract any number from itself, you get 0
21x1=21	1002x__=1002	When you multiply any number times 1, you get that number

 Language and Symbols of Algebra

Variables

• Placeholder
• Generalizations
• Formulas and Functions

Use of Variable	Representation	Characteristics
Placeholder	3+a=7	Specific value of a
Generalizations	a-a=0	All values of a make a sentence true
Function	H=2xB	Each value of B produces one and only one value of H

What does this symbol mean?
=

Equality

• Children understand sharing equally but become baffled by the equal sign.
• Equal = Get the answer or a way of keeping track of work

5+3 = 8-6 = 2x5 = 10+14 = 24

• Equal = Balance

Algebraic thinking in elementary
education classrooms

As Educators

• Be careful about using quesionts like "What operation would you use to solve a problem?"
• Ask children to write open sentences AND a problem that could correspond to that sentence.
• Make sure that your students have experience writing all four types of problems
• Have students explain their way of thinking
• Encourage reasoning and sense making as they justify their answers or method
• Discuss patterns
• Do NOT rush students thinking
• Allow mistakes to happen, then explain why their mistake is a mistake

Any Questions?

Overall Lesson:

Children need to learn that they should not do something a certain way just because someone else tells them to; rather, they need to understand why doing it that way makes sense (or doesn't make sense)