Download Lesson 1: Comparing and Ordering Integers

Lesson 1: Comparing
and Ordering Integers
The Number System
Power Up: Mental Math
• a) Calculation: Danielle drives 60 miles per hour for 2 hours. How far does she drive
•
•
•
•
•
•
in total?
b) Number Sense: 100 × 100
c) Estimation: 4.034 × 8.1301
d) Measurement: How long is the key?
e) Number Sense: 98 + 37
f) Percent: 50% 𝑜𝑓 50
g) Geometry: This figure represents a ___________.
Power Up: Problem Solving
• Problem: A robot is programmed to take two steps forward, then one step
back. The robot will repeat this until it reaches its charger unit, which is ten
steps in front of the robot. How many steps back will the robot take before
it reaches the charger unit?
New Stuff: Increasing Knowledge
• The numbers we use can be represented by points on a number line.
-5
-4
-3
-2
-1
0
1
2
3
4
5
• Points to the right of 0 (the origin) represent positive numbers. Points to the
left of 0 represent negative numbers. Zero is not positive or negative.
• The tick marks on the number line indicate the locations of integers, which
include the whole numbers, and the opposites of the natural numbers (or
counting numbers).
Classify
• Give an example of an integer that is not a whole number.
• Counting numbers, integers, and whole numbers are examples of different
sets of numbers.
• A set is a collection of objects. In math when we talk about sets, we are
usually talking about collections of numbers.
• We use braces {} to indicate a set, and we name them with a capital letter.
Subsets of Real Numbers
Natural Numbers 𝑁
The numbers we use to count things (these are also called the counting numbers):
𝑁 = {1, 2, 3, 4, … }
Whole numbers 𝑊
All the natural numbers and zero: 𝑊 = {0, 1, 2, 3, … }
Integers 𝑍
The whole numbers and the opposites of the natural numbers:
𝑍 = {… , −3, −2, −1, 0, 1, 2, 3, … }
Rational numbers 𝑄
Numbers that can be written in the form 𝑏, where 𝑎 and 𝑏 are integers and 𝑏 ≠ 0.
In decimal form, rational numbers either terminate or repeat. Examples:
1
2
,
0.
3,
−
, 0.125
2
3
Irrational numbers 𝑅 − 𝑄
Numbers that can’t be written as the quotient of two integers. Irrational numbers
do NOT terminate or repeat in decimal form. Examples:
3
5, 2, − 7, 3 3, 𝜋, 3𝜋, 𝑒
Real numbers 𝑅
The set including all rational and irrational numbers.
𝑎
Analyze
• How are the sets of whole numbers and integers different?
Absolute Value
• The absolute value of a number is its distance from 0 (the origin).
• Absolute value is always a positive number because distance is positive.
• For example, the absolute value of -5, denoted | − 5| is 5 because -5 is five
units away from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Ordering and Comparing Numbers
• We can use the number line to help us order numbers and compare them to
each other.
• We use the greater than/less than symbols (< and >) to show comparisons.
• The small end of the symbol should point to the lesser number.
Example 1
• Order these integers from least to greatest.
• −3, 3, 0, 1
-5
-4
-3
-2
-1
0
1
2
3
4
5
Example 2
• Compare:
• a) −3
−1
• b) −3
−1
Graphing numbers on a number line
• We graph numbers on a number line by putting dot at the location.
• Let’s graph the numbers -4, -2, 0, 2, and 4.
-5
-4
-3
-2
-1
0
1
2
3
4
5
• These numbers are all part of the set of even numbers. We can express this
set as a sequence, which is an ordered list of numbers that follows a rule.
• The set of even numbers is {… , −4, −2, 0, 2, 4, … }
Example 3
• Graph the numbers in this sequence on a number line.
• … , −3, −1, 1, 3, …
-5
-4
-3
-2
-1
0
1
2
3
4
5
Example 4
• Graph two numbers that are four units from 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Practice set
• Work through the questions in the problem set on your own. Show your
work and answers in your math notebook. Put a star next to the questions
you are unsure of.
Homework
• Show any necessary work.
• If you don’t have room in the boxes for your work, show your work on a
separate sheet of paper and just put your answer in the box.
• Make sure you have a red pen tomorrow so we can correct in class.
• Your homework is pg. 9-10, #1-15.
• Do the harder problems first. If you need help, place your help card on the
corner of your desk.