Lesson 2.1 & 2.6 Integers, Absolute Value and Number Properties Essential Question: How do you order integers?

Before we start… The Global Positioning System (GPS) can be used to determine elevations. The table shows the lowest elevations of several countries. Which country in the table has the lowest elevation? Lowest Elevations Country Elevation (m) United States -86 Canada China -154 Bolivia 90 Czech Republic 115

What are whole numbers? The counting numbers 0, 1, 2, 3, 4, 5, …

What are integers? The numbers …, −4, −3, −2, −1, 0, 1, 2, 3, 4, … Positive integers are integers that are greater than 0 Negative integers are integers that are less than 0 0 is neither positive or negative.

What are opposites? 3 and −3 Two numbers that are the same distance from 0 on a number line but are on opposite sides of 0. 3 and −3

Write the opposite of the integer. 15 −8 −35 100

Write the integer that represents the situation Write the integer that represents the situation. Then write the opposite of that integer. A golf score of 3 under par A deposit of $100 into your savings account Scoring 20 points on a quiz Missing 7 questions on a quiz

How do you compare and order integers? To order integers, you have to compare them. You can use a number line to do so if needed. The numbers decrease as you move to the left and increase as you move to the right on a number line. Negative integers are less than positive integers.

Compare −2 −5 −6 1

Compare −4 −6 3 −9

Give two integers that make the statement true. 0<𝑑 𝑐<23 2<𝑝<7 9>𝑡>−2

Weather The table shows the average temperatures, in degrees Celsius, for six months in the Gobi Desert of Mongolia. Which of these months has the lowest average temperature? Month Nov. Dec. Jan. Feb. Mar. Apr. Average Temperature −6°𝐶 −14°𝐶 −15°𝐶 −12°𝐶 −3°𝐶 6°𝐶

Temperature Which state in the table has the coldest record low temperature? California Illinois Nevada Pennsylvania Record (°F) −45°𝐹 −36°𝐹 −50°𝐹 −42°𝐹

Order the integers from least to greatest. 8, −4, −6, 4, −3, 1

Order the integers from least to greatest. −7,−12,−16,−10,−8

Golf The scores with respect to par for 8 golfers after 36 holes of play are shown in the table. Which golfer has the highest score? Which golfer has the lowest score? Which golfer is in the lead? Golfer Patrick Steven Joe Dan Charles Brad Vincent Randy Score −3 −6 −2 1 2 −4 −5 3

𝑎 What is absolute value? The distance between the number and 0 on a number line It’s always positive 𝑎

Find the absolute value of the number. 8 −7 −12 43

Simplify the expression. −32 − −17

Simplify the expression. 18 − 6

Simplify the expression. −8 − −2

Compare 3 −3 − 7 7

Compare −8 − 8 − 4 − −4

What about properties for addition and multiplication? The words commute and associate have meanings in mathematics similar to their English definitions. Commutative properties allow you to change position of numbers. Associative properties allow you to group numbers together.

Commutative and Associative Properties Commutative Property of Addition Words In a sum, you can add the numbers in any order. Numbers 4+ −7 =−7+4 Algebra 𝑎+𝑏=𝑏+𝑎 Commutative Property of Multiplication Words In a product, you can multiply the numbers in any order. Numbers 8 −5 =−5 8 Algebra 𝑎𝑏=𝑏𝑎 Associative Property of Addition Words Changing the grouping of numbers in a sum does not change the sum. Numbers 9+6 +2=9+ 6+2 Algebra 𝑎+𝑏 +𝑐=𝑎+ 𝑏+𝑐 Associative Property of Multiplication Words Changing the grouping of numbers in a product does not change the product. Numbers 3∙10 ∙4=3∙ 10∙4 Algebra 𝑎𝑏 𝑐=𝑎 𝑏𝑐

Identity Properties Identity Property of Addition Words The sum of a number and the additive identity, 0, is the number. Numbers −6+0=−6 Algebra 𝑎+0=𝑎 Identity Property of Multiplication Words The product of a number and the multiplicative identity, 1, is the number. Numbers 4∙1=4 Algebra 𝑎∙1=𝑎

Tour Biking You are going on a 400 miles bike trip Tour Biking You are going on a 400 miles bike trip. You plan to cycle at an average speed of 12 miles per hour for 7 hours a day. Can you complete the trip in 5 days?

Homework Mr. Newman collects homework papers in three classes from 28 students, 25 students, and 22 students. Use mental math to find how many papers he collected altogether.

Identify the property. 2+0=2 21 ∙1=21

Identify the property. −5 6 =6 −5 7+5 +3=7+ 5+3

Identify the property. 2+ −9 =−9+2 2∙6 ∙5=2∙ 6∙5

Identify the property. 15+0=15 −8 ∙1=−8

How do you order integers?

Ticket Out the Door Order the integers from least to greatest. −1,−12,−34,−6