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Students get to see the contexts for very large numbers and very small numbers.
i.e. State of California population (from a large number) 39,250,017 pop to 3.9 X 10¹

On the other hand, very small numbers such as; DNA size, Viruses, Bacteria, and atoms.
i.e. DNA size is 10-7 m or 0.000001 m.

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Positive exponents - students have been learning how to solve exponent with repeated multiplication; 10¹= 10 or 10/1, 102 = 100 or 100/1, etc.

But when it comes to negative exponents, students get confused.
10⁻¹= 1/10, 2-2 = 4/100

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1. Third-grade get to do basic order of operation with additions, subtractions, and multiplications.

2. In Six-grade, exponents are added to additions and subtractions, where students have to follow steps to order of operations.

3. One strategy for order of operation is the order of pyramid. There are three levels to order of pyramid; 1. Exponents, 2. Multiplication and Division (from left to right), 3. Addition and Subtraction(from left to right).

4. PEMDAS (parentheses, exponents, multiplication, division, addition, and subtraction)

5. As for other countries, BODMAS, BEDMAS, BIBMAS.

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1. Repeated multiplication of a number times itself; 3³ = 3X3X3

2. Exponents in Expression, where students get to work within parentheses; 2+5³ = 2 + (5X5X5).

3. Slowly, students get introduced to order of operation by solving exponents in expressions.

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Rational numbers: whole numbers, fractions, decimals, integers.

Fun Fact: Between any two numbers there are an infinite amount of rational numbers.

Irrational number: a number whose values is either cannot be written as a fraction or estimated.
8 th graders start exploring irrational numbers and their rational approximates. (real numbers)

High school:  consider square roots of negative numbers (imaginary numbers)
Rational Numbers:

Often referred to as fractions or decimals (terminating 3.415 and repeating 2.51414) & percentages.

Any rational number can be written as a faction and decimal.

6 th grade: recognize a rational number as a point on a number line.
7 th grades: able to think about a value across the representations and move between them flexibly.

4 3/5 is 4.6 because 3/5 = six-tenths of a whole so Four wholes and six-tenths so 4.6.

Leads to a/b is the same as a÷6

Square Roots & Cube Roots :

Middle School they mainly encounter irrational numbers in Pythagorean Theorem.

Real numbers fill in holes on the number line even when the holes are infinitesimally small.

7 th grade: encounter irrational numbers when learning about π

8 th grade: exploration of irrational numbers.

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Operations with positive and negative numbers!

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Operations with positive and negative numbers!

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It is often hard for students to connect representations when working with positive and negative numbers.
One way to help students connect equations and visual illustrations is to practice with a basic number line to their equation.

With addition and subtraction, it is good to introduce negative number values with real life situations, such as, golf scores and football field yardage.

In seventh grade, students are learning to solve multi-step real-life and mathematical problems that deal with positive and negative rational numbers in any form.

Although contexts are important places to begin, student really need to understand and be able to reason about the numbers themselves-- with that, it is important to focus on the meaning of opposites,

With any type of math problems, to understand the division portion, the student must have a good conceptual understanding of multiplication. REPEATED ADDITION!
Connect division of negative numbers with what the students already know about positive numbers.

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Typically: introduce negatives only as opposites of positive numbers
Result: weaker sense of what negative numbers really are

Real life connections: temperature, altitude, golf, money, timelines (B.C.), football yardage, etc.

TWO TYPES
Quantity contexts: matching opposites (4, -4)
Ex: money
Linear contexts: on a # line
Ex: temperature

Rule Based reasoning vs. True Meaning
Show definition of negative numbers in a number sentence.
Ex: the definition of -3 is the solution of 3 + ? = 0

Absolute Value: the #s distance from 0

What's the meaning of the negative sign?
Subtraction, Negative #s, & the opposite of

MODELS:
2 color sided counter chips - Number Lines

OPERATIONS w Positive & Negative #s
+ & -
-have students create their own stories to given number sentences

Multiplication
Positive factors = repeated addition - how many times added to 0
Negative factors = repeated subtraction - how many times subtracted from 0

Division
Treat like # sentences missing #s (fact families) and break it down to multiplication

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-exponents in expression and equations
-order of operations
-interfered exponents
-scientific notation

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1. Describe the strategies
2. Compare quantity and number line visuals
3. Illustrate and explain conceptual approaches
4. Construct visuals to illustrate relationships between different types of numbers