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Definition: An exponent indicates the number of times the base is used as a factor, so it indicates repeated multiplication and it precedes other multiplication and division, as well as addition and subtraction.

Exponents are first introduced in 5th grade in terms of powers of 10 and place value

In 6th grade students learn to write and evaluate numerical expressions involving whole number exponents

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The rules are what confuse students because they might not remember whether to add or multiply the exponents when they raise a number to a given power.

This shows lack of understanding of operations and notation therefore, students need to explore exponents with whole numbers before they use exponents with variables.

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Working with exponents extends to order of operations

Definition:  is the rule that tells us the sequence in which we should solve an expression with multiple operations

As early as 3rd grade students need to know the order of operations for addition, subtraction, multiplication and division

By 6th grade, exponents are added to the order of operations

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We are taught to work through an expression left to right, AKA PEMDAS, but this rule misleads students into thinking "multiplication precedes division and addition precedes subtraction".

Instead students should use:

P =parenthesis

E =exponents

MD =multiplication and division (whichever is first from left to right)

AS =addition and subtraction (whichever is first from left to right)

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Any number to the 0 power is always 1 for example 10^0= 1

So what about negative exponents such as 10 ^ (-2) ?

Well… you change the number to the reciprocal and the base does not change ! but the exponent becomes positive so it would become 1/10^2.

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Definition: "a number is changed to be the product of a number greater than or equal to 1 and less than 10 multiplied by a power of 10. For example, 3,414,000,000 can also be written as " 3.414 x 10^9

Different notations have different purposes and values. For example, the population of the world on 6/24/17 was estimated to be 7,513,768,345 (U.S. Census Bureau, n.d.) This number can be written in the following ways depending on the purpose: "7,513 million, 7.5×10 ^ 9 and about 7.5 billion.

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Examples of Extremes: Large and Small Numbers

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Examples of Extremes: Large and Small Numbers

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The human body which has about 100 billion cells

Exploring lottery combinations

The estimated size of the universe is 40 billion light-years etc..
The chance of winning the lottery

"The length of a DNA strand in a cell is about 10^7 m. This is also measured as 1000 angstroms."

"Human hair grows at the rate of 10^8 miles per hour."

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Students must… Understand that positive and negative numbers are used together to describe quantities having opposite directions or values, use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

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Tools for Teaching Positive and Negative Numbers

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Tools for Teaching Positive and Negative Numbers

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Using two different colored counters, one (yellow) for positive and one (red) for negative helps students illustrate they are opposite values.

Yellow= +1

Red= -1

Number Lines are a great tool to represent the operation conceptually

Shows the distance from 0 or absolute value from a number

Arrows can be used to represent positive and negative numbers by showing the distance and direction they are going

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“Whole numbers, fractions and decimals, and integers are all rational numbers because they can all be written as a fraction with an integer over a nonzero integer”

"Irrational numbers are numbers such as 2—numbers whose value cannot be written as a fraction and whose exact value can only be estimated"