Properties of Arithmetic

​Middle School Math (MSM)

AoPS Pre-Algebra Chapter 1

Properties of Arithmetic

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​Lesson 1A

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​Introductions

In this class

• No need to take notes unless the slide says so in this class

• All concepts & examples are from the book - AoPS Pre-Algebra

• The graphics, videos are all from google images search

• Watch the recording and take screenshots if you can't find it

• Keep papers & pencil handy to work out problems​

• Interact & Participate in this class - no other way to learn

• Stop & Ask questions, if something is not clear

• You are not in elementary school - teacher will not go around and check with each of you .. you need to speak up, take responsibility & learn.

• Teacher will provide a class participation score for each student - will be shared with your parents every few weeks

NEW THIS YEAR: Two class meetings each week ("Part A & Part B")

• ​Different topics in Part A & Part B classes

• Each class - 55 minutes long but you must sign in 3 to 5 minutes early

• Set reminders on your electronic device or home assistant

• Try to attend the same classes each week

• But if unavoidable, have your parents request alternate schedule

• Always plan ahead unless it is an unexpected situation

• It's your responsibility to watch recording & catchup, if you miss a class

• Homework assigned in Showbie - one set each for Parts A & B

Chapter 1 : Arithmetic

- Numbers

- Operations

- Properties

- Reciprocals

Easy topic ... or is it?

Let's start with something familiar to all of us

When we say "numbers", what comes to your mind?

Natural numbers or Counting numbers

- Positive integers that you can count with your fingers

- It goes forever to infinity

- The most ancient counting system

Before we go to next slide:

What about zero then?

Whole Numbers

- Adds "0" to the set of Natural numbers

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Zero

- In use in Mesopotamia in 3 B.C.

- Mayans used it around same time

- Used in Math by Mathematicians in India around 5th Century A.D.

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Before we go to next slide:

Are there other kinds of numbers?

Integers

Rational Numbers

Includes all integers and & any decimal number in between that can be expressed it as a fraction or ratio of two integers

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Obvious question: What about the decimal numbers that cannot be expressed as a fraction or ratio of two integers?

Something to think about for later:

Do "Real numbers" cover all known numbers in the universe?

(Hint: What can be something that is not real?)

Review Quiz

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​Get Ready ... Stay Sharp ... Fast Fingers!!

Multiple Select

Real numbers consist of :

(choose all that apply)

• Rational Numbers

• Irrational Numbers

• Arbitrary Numbers

Multiple Select

The following are examples of Integers

• (1/2, -3, 4, .75)

• (.987, .786, -9.9)

• (1, -1, 2, -2, 3, -3)

Multiple Choice

What is a repeating decimal or decimal that terminates called ?

• Natural Number

• Irrational Number

• Rational Number

• Whole Number

Multiple Choice

Which of the following is not a part of the set of rational numbers ?

• -42

• 5/0

• 0

• 14.7

Multiple Choice

Numbers that can be used for counting. They do not include zero or negatives. What kind of numbers are they?

• Whole

• Natural

• Integers

Multiple Choice

Any of the numbers {0, 1, 2, 3} etc.

There is no fractional or decimal part. There are no negatives. These would be:

• Integers

• Whole Numbers

• Irrational Numbers

Properties of Addition

Now, get ready for a quiz question

Let's review Example 1.3 (b)

Compute the sum 472 + (219 + 28)

(use the appropriate arithmetic property)

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Use the power of properties again for another quiz question.

Multiple Choice

1.4: Compute

(2 + 12 + 22 + 32) + (8 + 18 + 28 + 38)

Hint: Look for numbers that make a "perfect pair"?

• 120

• 160

• 200

Let's review Example 1.4:

Use an arithmetic property to solve

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(2 + 12 + 22 + 32) + (8 + 18 + 28 + 38)

Example 1.5: Find the sum 1 + 2 + 3 + ... + 19 + 20.

Reminder: The ellipsis ... means that we should include all the numbers in the pattern. So we are adding the positive integers from 1 to 20.

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Properties of

Multiplication

Example 1.9: Compute 25 x 125 x 4 x 6 x 8

How would you group the numbers differently to make the multiplication easier?

What property of multiplication will you apply?

Sharpen your fingers & get ready .... a quick 5 second quiz question ...

Multiple Choice

Example 1.11 (restated):

(5 + 6) x 7 = 5 + (6 x 7)

• True

• False

And the most useful property of 'em all ...

Distributive property is the basis for a lot of mental math

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Combines Multiplication & Addition

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- Can we replace Addition with Subtraction?

- Can we replace Multiplication with Division?

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Now ... get ready for a quiz question.

Example 1.13: Compute 51 . 9 + 51 . 31

This is a slightly harder version of the quiz question.

What does the "dot" mean?

Multiple Choice

Example 1.14:

Value of 17 . 13 + 51 . 13 + 32 . 13,

what is?

• 650

• 1300

• 2600

Review Quiz

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​Get ready ..... stay sharp .... Fast Fingers!!

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Multiple Choice

Which property of multiplication is shown below:

17 × 113 = 113 × 17

• Commutative Property

• Associative Property

• Identity Property

• Zero Property

Multiple Choice

Which property of multiplication is shown below:

1 × 83 = 83

• Commutative Property

• Associative Property

• Identity Property

• Zero Property

Multiple Choice

The commutative property of multiplication states that

the order of the numbers in an equation:

• Should go from least to greatest

• Doesn't change the product

• Should go from greatest to least

• Is always important

Poll

Example 1.19:

(-2) . (-3) = 2 . 3

After the timer stops, explain why

• True

• False

Poll

$$Is\ \frac{1}{-x}the\ same\ as\ -\frac{1}{x}?$$   After the timer stops, explain why or why not.

• Yes

• No