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Algebraic Thinking
Algebraic Reasoning
What is Algebraic Thinking?
What is Algebraic Thinking?
 Algebraic thinking can be broken into
two categories:
1. Algebraic ideas
2. Mathematical thinking tools
Algebraic Ideas
 Algebraic ideas are the building blocks
of algebraic thinking:
1. Patterns
2. Functions
3. Variables
Mathematical Thinking Tools
 Mathematical thinking tools are the
analytical habits of mind, such as:
1. Problem solving
2. Representation
3. Reasoning
When do students begin to
think algebraically?
Algebraic Thinking through
the Grades
 Kindergarten
 Represent addition and subtraction in many ways
 Solve addition and subtraction problems within 10
 Break apart numbers into pairs in more than one way
(e.g. 5 = 4+1 5= 2 + 3
5 = 5 + 0)
 Find the number that makes 10 when added to given
number: (e.g. If you have 6 what number is needed to
make 10
 Fluently add and subtract with 5.
Algebraic Thinking Through
the Grades
 First Grade
 Add and subtract within 20, using 2 or 3 whole
numbers, to solve word problems
 Use commutative and associate properties
 Understand the meaning of the equal sign and
determine if equations are true.
 Find the missing number in an addition or
subtraction problem.
Algebraic Thinking Through
the Grades
 Third Grade

Represent and solve problems involving
multiplication and division

Understand properties of multiplication and the
relationship between multiplication and division

Multiply and divide within 100

Solve problems involving the four operations

Identify and explain patterns in arithmetic
Algebraic Thinking Through
the Grades
 Second Grade
 Add and subtract within 100 to solve one- and twostep problems.
 Mentally add and subtract within 20
 Determine odd or even numbers and write and
equation to express an even number.
 Skip count
 Use addition to find the total number of objects in
rows and columns.
Algebraic Thinking Through
the Grades
 Fourth Grade
 Use the four operations with whole numbers to solve
problems
 Gain familiarity with factors and multiples
 Generate and analyze patterns
Algebraic Thinking Through
the Grades
 Fifth Grade
 Write and interpret numerical expressions
 Analyze patterns and relationships
Algebraic Thinking Through
the Grades
 Sixth Grade
 Apply and extend previous understandings of
arithmetic to algebraic expressions.
 Apply and extend previous understandings of
numbers to the system of rational numbers.
 Reason about and solve one-variable equations and
inequalities.
 Represent and analyze quantitative relationships
between dependent and independent variables.
 Understand ratio concepts and use ratio reasoning to
solve problems.
Patterns, Functions,
Variables
 Patterns are key factors in understanding mathematical
concepts.
 The relationship between two sets of numbers is called a
function.
 Looking for patterns in the relationships between two sets of
numbers is a key way to develop students’ understanding of
functions.
 Variables
1.
2.
3.
Place holders for a specific number (2 + 3 = n)
Unknown values (A=lw or n + n = 7)
Represent quantities that vary (y = 2x)
An important convention to remember is that the same symbol in
an equation stands for the same number every time it occurs
in the equation.
Expressions and Equations
 Equations include an equal sign and state the
equality of two expressions.
 Expressions are symbolic statements. They can
either be arithmetic expressions such as 3 + 4, or
algebraic expressions such as 2x – 3.
 Coefficients, constants, terms
Ratios
 Ratios compare values
 A ratio says how much of one thing
there is compared to another thing.
3
:
1
There are 3 red squares to 1 yellow square.
Ratios
 Ratios can be shown in different ways:
Using “ : “ to separate the values
3:1
Instead of using the “ : “ we can use the word “to”
3 to 1
Or write it like a fraction 3 ⁄ 1
Rates
 A rate is a little bit different than a ratio. It is a
special ratio.
 It is a comparison of measurements that have
two different units.
2 cases of soda for $10.00
Sara types 10 words in 5 seconds
4 lbs of hamburger costs $20.00
I drove 250 miles in 4 hours.
Unit Rates
 Unit rates describe how many units of the first
type of quantity corresponds to ONE unit of the
second type of quantity.
Miles per hour
Earnings per week
Photos per page
People per vehicle
Cost per pound
$.75 for 3 apples = .75 ÷3÷ = 25¢ per apple
Proportions
 Proportion says that two ratios (or fractions) are
equal.